Paper Title: Initial Data Identification for Conservation Laws with Spatially Discontinuous Flux.



Substrate stabilization of Jahn–Teller distortion in a single molecule
    Andrea Donarini
    August 5, 2024
======================================================================



§ ABSTRACT 
We consider a scalar conservation law  
with a spatially discontinuous flux at a single point x=0,
and we 
study the initial data identification problem for AB-entropy solutions 
associated to an interface connection (A,B).
This problem consists in identifying the set of initial data 
driven by
the corresponding 
AB-entropy solution 
to a given target profile ω^T, at a time horizon T>0.
We provide a full characterization of such a set in terms of suitable integral inequalities,
and we establish structural and geometrical properties of this set.




A distinctive feature of the initial set is that it is in general not convex, differently 
from the case of conservation laws with  convex flux
independent on the space variable.
 The results rely on the properties
 of the AB-backward-forward evolution operator
 introduced in <cit.>,
 and on a  proper concept of AB-genuine/interface characteristic
for AB-entropy solutions provided in this paper.







§ INTRODUCTION
 
We are concerned with the 
initial value problem for a 


scalar conservation law in one space dimension


    u_t+f(x,u)_x=0,
    
       x ∈ℝ,    t ≥ 0,
    
       u(x,0) = u_0(x),      x ∈ℝ,



where u = u(x,t) is the state variable, and the flux f is a space discontinuous function of the form

    f(x,u) =
    f_l(u),      x < 0, 
    
    f_r(u),      x > 0 ,

with f_l, f_r: →
 twice continuously differentiable, uniformly  convex maps that satisfy

    f_l”(u), f_r”(u) ≥ a >0 ,

and (up to a reparametrization)

    f_l(0) = f_r(0),    f_l(1) = f_r(1) .

We assume also that 
the unique critical points 
θ_l, θ_r of f_l, f_r, respectively,
satisfy

    θ_l ≥ 0,   θ_r ≤ 1 .

It is well known that,
because of the nonlinearity of the equation,
in order to achieve 
global in time existence and uniqueness results for problems of this type one has to consider 
weak distributional solutions
that satisfy 
the classical Kružkov entropy inequalities
away from the  flux-discontinuity interface
x=0,

augmented by an appropriate interface entropy condition
at x=0. 


Here,   we will consider entropy solutions of AB-type, associated to a so-called interface connection (A,B)  (cfr <cit.> and see <ref>). Entropy solutions of AB-type form an 𝐋^1-contractive semigroup 
on the domain of L^∞ initial data <cit.>.
Thus, we adopt the semigroup notation 
u(x,t)≐ u_0 (x), t≥ 0, x∈, for the unique AB-entropy solution of (<ref>)-(<ref>),
for every initial datum u_0∈ L^∞(ℝ).


In this paper, we study the initial data identification problem (or inverse design problem) for AB-entropy solutions of the equation (<ref>). 
This problem consists in identifying the set of initial data for which the corresponding 
AB-entropy solution coincides with a given target profile ω^T, at a time horizon T>0.
Observe that we cannot expect to reach 
any desired profile ω^T∈ L^∞(ℝ).
In fact, even in the case where f_l=f_r,
since the work of Oleĭnik <cit.> it is well known that, 
because of the uniform convexity of the flux, 
the Kružkov entropy conditions imply 

that every entropy weak solution u of (<ref>)
must satisfy (in the sense of distributions)
the one-sided Lipschitz estimate

    ∂_x u(t, ·) ≤1/at, 
        in  D^',
      ∀ t>0.

Essentially, the nonlinearity of the flux forces characteristic lines to intersect which, together with the entropy condition, produces a regularizing effect
L^∞ to BV encoded in the Oleĭnik inequality (<ref>).
In the case  of equation (<ref>) 
with discontinuous flux (<ref>) where f_l≠ f_r, we have shown in <cit.> that
the set of reachable profiles at a time T>0:

    A^[AB](T) ≐{ S_T^[AB]+ u_0   |    u_0 ∈𝐋^∞(ℝ) },

is characterized in terms of suitable Oleǐnik-type estimates and
unilateral pointwise constraints. 
Note that a 

“loss of information” takes place when characteristic lines intersect 
into a shock:  there are infinitely many ways to create the same shock discontinuity at a given time T. Therefore the initial data identification problem for this type of equations is highly ill-posed:
multiple initial data can be stirred by (<ref>) into the same
attainable profile ω^T∈ A^[AB](T)
at  time T.


Our goal is to characterize and study the 

properties of 
the set 
of  initial data leading to a 
given profile ω^T∈ A^[AB](T)
at time T:

    I^[AB]_T(ω^T) ≐{ u_0 ∈𝐋^∞(ℝ)   |   S_T^[AB]+ u_0 = ω^T}.



In the case of 
 conservation laws with flux independent on the space variable, 
the initial data identification problem
was firstly studied for the Burgers equation in <cit.>, and next 
 for general uniformly convex flux in <cit.>,
where it is fully characterized the initial set of data evolving to a given profile, and it is shown
that such a set is convex. 
Similar results were obtained
in <cit.>
for Hamilton-Jacobi equations with convex Hamiltonian,
and in <cit.>
for  smoothly, space dependent,
conservation laws or Hamilton-Jacobi equations.


When the flux is of the form (<ref>)
with f_l≠ f_r, 

the initial data reconstruction problem
is more challenging 




because 

one has to deal with the
richer and more complicated near-interface
wave structure of AB-entropy solutions.
This is
due to the presence in the solution of waves that are reflected or refracted  
through the discontinuity interface x=0, as well as of shock discontinuities emerging from the interface
at positive times
(see the analysis in <cit.>).
Nonetheless,
we are still able to provide


a full characterization of the initial set (<ref>) 
by suitable integral inequalities,
for every given  ω^T∈ A^[AB](T), and we  show
that (<ref>)

shares almost the same geometric and topological
properties of the  initial set for
 conservation laws with uniformly convex flux
 independent on the space variable.
To 



establish these results 
we will rely on:


  - a suitable definition of backward evolution operator S^[AB]-_T given in <cit.>, and on the structural properties of the range of S^[AB]-_T therein analized; 

  - a proper concept of AB-genuine/interface characteristic
for AB-entropy solutions  which can “travel” along the discontinuity interface x=0 (see Definition <ref>).


Given an AB-entropy solution u,
a time horizon T>0, and a point x∈,
we will let 𝒞_0(u,x) denote the set of the initial
positions ζ(0) of the AB-genuine characteristics
ζ for u that
reach the point x=ζ(T) at time T
(cfr. (<ref>)). 
We recall that any element ω^T∈ A^[AB](T)
admits one-sided limits ω^T(x-), ω^T(x+) at every point
x∈, and that has at most countably many discontinuities (see <cit.>).
Then, we summarize the main results of the paper 
in the following 



































Given ω^T∈ A^[AB](T), set

    u_0^* ≐ S^[AB]-_T ω^T ,

and

    u^*(·,t) ≐ S^[AB]+_t u^*_0     ∀ t ≥ 0 .

Then, letting I^[AB]_T(ω^T) be the set defined in (<ref>), the following properties hold:

    
  (i) u_0∈ I^[AB]_T(ω^T)
    if and only if, for every point x of continuity of
    ω^T, there exists y∈𝒞_0(u^*, x) such that
    there hold
    
    ∫_y^y̅ u_0(x)  x ≤∫_y^yu^*_0(x)  x,     ∀   y < min𝒞_0(u^*,x),

and 

    ∫_y^y u_0(x)  x ≥∫_y^y u_0^*(x)  x,     ∀   y > max𝒞_0(u^*,x).

     
  (ii)
     The set I^[AB]_T(ω^T) is an infinite dimensional cone  which has vertex u_0^* and is in general not convex.

    

We will establish further geometric and topological properties of the initial set (<ref>)
besides the ones stated in Theorem <ref>-(ii), which are collected in Theorem <ref> stated in  <ref>.















Initial data identification problems 
are often formulated 
as least square optimization problems
associated to observable states at a final time
(also known in the literature as data assimilation problems).
These type of problems
arise naturally in environmental sciences <cit.>, 


but also in life sciences (see <cit.> and references therein), to improve the forecast of a model
or to refine numerical simulations.


















Similar issues, also related to parameter identification problems, arise  in traffic flow modeling <cit.>, in batch sedimentation <cit.>, or in petroleum reservoir engineering <cit.>.

Conservation
laws with spatially discontinuous flux have many relevant applications in physics and engineering including: 

porous media models
with changing rock types (for oil reservoir simulation) <cit.>;
sedimentation in waste-water treatment plants <cit.>; traffic flow dynamics
 with roads of variable width or surface conditions <cit.>;
Saint Venant models of blood flow in endovascular treatments <cit.>;
radar shape-from-shading models <cit.>.




The paper is organized as follows.

    
  - In <ref> we introduce the general setting of the problem.
    
  - In <ref> we introduce the characteristics for the discontinuous flux (<ref>) and state the main Theorems <ref> (integral inequalities) and <ref> (geometrical properties).
    
  - In <ref> we prove Theorem <ref>.
    
  - In <ref> we prove Theorem <ref>.




§ BASIC DEFINITIONS AND GENERAL SETTING






 §.§ Connections and AB-entropy solutions





We recall here the definitions and properties of interface connection and of admissible solution satisfying an interface entropy condition introduced in <cit.>. 
Throughout the paper, for the one-sided limits of a function u(x) we will use the notation

    u(x ±)≐lim_y→ x ± u(y).


[Interface Connection]

Let f be a flux as in (<ref>)
satisfying the assumptions (<ref>)-(<ref>).
A pair of values (A,B)∈ℝ^2 is called a connection if


  * f_l(A) = f_r(B),

  * f'_l(A)≤ 0,  f'_r(B)≥ 0.

 
We will say that a connection (A,B) is  critical if 
f'_l(A)= 0, or f'_r(B)= 0,
i.e. if
A=θ_l or B=θ_r.




























































































Clearly, condition (1) of Definition <ref> is equivalent to
A≤θ_l, B ≥θ_r.
For sake of uniqueness,

it is employed in <cit.> the  adapted entropy 

    η_AB(x,u)=|u-c^AB(x)|,      
        c^AB(x) ·=
    A,     x ≤ 0, 
    
    B,     x ≥ 0,


to select the unique
solution of the Cauchy problem (<ref>)-(<ref>) that satisfies the Kružkov-type 

entropy inequality

    |u-c^AB|_t + [sgn(u-c^AB)(f(x,u)-f(x,c^AB))]_x ≤ 0,   in  𝒟^' ,

in the sense of distributions, which leads to the following definition.




[AB-entropy solution]
Let (A,B) be a connection
and let c^AB be the function defined in (<ref>). A function u ∈𝐋^∞(ℝ× [0,+∞[) is said to be an AB-entropy solution of the problem (<ref>),(<ref>) if the following holds:



  * u is a distributional solution of (<ref>) on × ]0,+∞[, that is, for all test functions ϕ∈𝒞^1_c

with compact support 
contained in ℝ× ]0,+∞[,
there holds

    ∫_-∞^∞∫_0^∞{u ϕ_t+f(x,u)ϕ_x } x  t = 0 .


  * u is a Kružkov entropy weak solution of (<ref>),(<ref>) on (ℝ∖{0}) × ]0,+∞[, that is, t ↦ u(·,t) is a continuous map from [0,+∞[ to 𝐋^1_(ℝ),
 the initial condition (<ref>) is satisfied almost everywhere,
and:


  (2.a)
for any non-negative test function ϕ∈𝒞^1_c with compact support 
contained in ]-∞,0[ × ]0,+∞[, there holds


    ∫_-∞^0∫_0^∞{|u-k| ϕ_t + sgn(u-k)(f_l(u) - f_l(k)) ϕ_x } x  t ≥ 0,   ∀ k ∈ℝ ;


  (2.b)
for any non-negative test function ϕ∈𝒞^1_c with compact support 
contained in ]0,+∞[ × ]0,+∞[, there holds


    ∫_0^∞∫_0^∞{
    |u-k| ϕ_t + sgn(u-k)(f_r(u) - f_r(k)) ϕ_x
    } x  t ≥ 0,   ∀ k ∈ℝ .




  * u satisfies the interface

entropy inequality relative to the connection (A,B), that is, 
for any non-negative test function ϕ∈𝒞^1_c with compact support 
contained in × ]0,+∞[, there holds

    ∫_-∞^∞∫_0^∞{|u-c^AB| ϕ_t + sgn(u-c^AB)(f(x,u) - f(x,c^AB))ϕ_x } x  t ≥ 0 .










Since
the fluxes f_l,f_r in (<ref>) are uniformly convex, 

by Property (2) of Definition <ref> it follows
that, if u is
an AB-entropy solution, 








then u(· ,t) is a function of locally bounded variation on ℝ∖{0},
 
for any t>0.
On the other hand, 
relying on  <cit.>,


and because of the 𝐋^1_-continuity of the map t ↦ u(·,t),
we deduce that
u admits left and right strong traces at x = 0 for all t>0,  i.e. that 
there exist the one-sided limits  
    u_l(t)≐ u(0-,t),
    
         u_r(t)≐ u(0+,t),
    
        ∀ t>0 .


Moreover, 


by properties (1), (3) of Definition <ref>,

and thanks to the analysis in <cit.>,
we deduce that
u must satisfy at any time t>0 the




interface conditions

    f_l(u_l(t)) = f_r(u_r(t)) ≥ f_l(A) = f_r(B), 
       (u_l(t) ≤θ_l,    u_r(t) ≥θ_r)   ⟹  u_l(t) = A,   u_r(t) = B .





















It was proved in <cit.> (see also <cit.>)

that  AB-entropy solutions of (<ref>),(<ref>)
with bounded initial data are unique,
and  𝐋^1-contractive 
with respect to their initial data.
Thus, one can define a semigroup map

    : [0,+∞[ ×  L^∞() → L^∞(),
              (t,u_0)↦ u_0 ,

where the function u(x,t) ≐ u_0(x) provides the unique  AB-entropy  solution of the Cauchy problem (<ref>), (<ref>).
Such a map is L^1-stable also with respect to the time t and the
values A,B of the connection, as 
shown in <cit.>.












































 §.§ Backward 
solution operator
 

We  

review here the concept of 
backward solution
operator associated to a connection (A,B)
introduced  in <cit.>,
referring to <cit.>
for further details and properties.

Given a flux  f  as in (<ref>)
satisfying the assumptions (<ref>)-(<ref>),
and a connection
 (A,B),
 
 

observe that, setting

    B≐ 
        (f_r_| ]-∞, θ_r])^-1∘ f_r(B),       A≐ (f_l_| [θ_l, +∞[ )^-1∘ f_l(A) ,

where f_| I
denotes the restriction of the function f to the interval I,
the pair (B, A )
provides a connection for the symmetric flux

    f(x,u) = 
        f_r(u) ,     x ≤ 0, 
    
        f_l(u),     x ≥ 0 ,

(see Figure <ref>).

Then, letting
u_0(x) denote
the unique B A-entropy solution
of

    u_t+f(x,u)_x = 0     x ∈ℝ,    t ≥ 0, 
    
            u(x,0) = u_0(x)      x ∈ℝ,

evaluated at (x,t),
we shall define the backward 
solution operator associated to the connection  (A,B) in terms of the operator  as follows.


[AB-Backward solution operator]
Given a connection (A,B), the backward solution operator
associated to (A,B) is the map
S_(·)^  [A B]- :

[0,+∞)× L^∞()→ L^∞(),  defined by

    ω (x) ≐(ω(- · ))
    
         (-x)     x∈,  t≥ 0 .









§ GENUINE/INTERFACE CHARACTERISTICS
AND 

MAIN RESULTS





In this section we introduce the fundamental concept 
of genuine/interface characteristic
for an AB-entropy solution, 
and we collect the statement of the main results of the paper.
The definition of genuine/interface characteristic
extends to the 
setting of AB-entropy solutions the classical definition of genuine characteristic for a conservation law u_t+f(u)_x=0 (see <cit.>).
Throughout the following we fix a time T>0,
we
consider a fixed connection (A,B), and we set 

    γ≐ f_l(A) = f_r(B).


[AB-genuine/interface characteristics]


Let u ∈𝐋^∞(ℝ× [0,+∞[  ;   ) be an

AB-entropy solution of  (<ref>).



We 
say that a Lipschitz continuous function ζ:[0,T] →ℝ
is an AB-genuine/interface characteristic
(AB-gic) for u if  the following conditions hold:




    
  (i) for a.e. t ∈ [0,T] with ζ(t) ≠ 0 it holds
    
    ζ̇(t) = f^'(u(ζ(t)-,t),ζ(t))= f^'(u(ζ(t)+,t),ζ(t));


  (ii)  for a.e. t ∈ [0,T] with ζ(t) = 0, it holds 
    f_l(u(ζ(t)-,t) = γ  = f_r(u(ζ(t)+,t)).












    
    Applying the classical theory of generalized characteristics <cit.>
    it follows that any 
    
    AB-gic ζ∈Lip([0,T]  ;  ℝ) is a piecewise affine function 
    for which there exist 0≤τ_1≤τ_2≤ T,
    such that:
     
     
     
     
  - ζ(t)=0 
     and f_l(u_l(t))=f_r(u_r(t))=γ, for all t∈ [τ_1,τ_2];
     
  -  the restriction of ζ
     to [0, τ_1[ and to ]τ_2, T]
     is either a classical genuine characteristic 
     for the conservation law u_t+f_l(u)_x=0 
     on {x<0}, or it  
     is a classical genuine characteristic 
     for the conservation law u_t+f_r(u)_x=0 
     on {x>0}.
     
    Note that, if τ_1=τ_2∈{0,T}, then
    a curve defined by a function ζ satisfying the above conditions
    can cross the interface x=0 only
    at its starting or terminal points. Thus, in this case
    ζ
    is a a classical genuine characteristic 
     for u_t+f_l(u)_x=0 
    on {x<0} or for u_t+f_r(u)_x=0 
     on {x>0}, in the whole interval ]0,T[.
    
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
    

    
    
    By Definition <ref>
    an AB-gic
    
    can “travel" along the discontinuity interface x = 0 in an interval [τ_1, τ_2] only if in such an interval the flux of the solution is the minimum possible, i.e. if f_l(u_l(t))=f_r(u_r(t)) = f_l(A)=f_r(B)
    for all t∈ [τ_1, τ_2],
    with u_l, u_r as in (<ref>).
    
    This definition can be motivated by the following observation. 
    Let f be a smooth
   
   convex flux. Then,  
    relying on the  inequality 

    f(u) -f(v) -f^'(u) (v-u) ≥ 0     ∀  u, v ∈ℝ,

   
   
   
    one can verify that
    a classical genuine characteristic ζ:[0,T] →ℝ for a solution u of 
    the conservation law u_t+f(u)_x=0
    
    
    satisfies at  a.e. t ∈ [0,T] the equality

    f(u(ζ(t), t)) - ζ̇(t) u(ζ(t), t) = min_v ∈ℝ{f(v) - ζ̇(t) v}.





Therefore if the characteristic is “vertical" (i.e. ζ̇ = 0) we simply obtain 

    f(u(ζ(t), t)) = min_v ∈ℝ f(v),
       for a.e.  t∈[0,T].



In 
view of the interface constraint (<ref>)

for AB-entropy solutions, it is then natural to require in this setting that a “characteristic" 
lying on the interface x=0


be called 
“genuine" only if it minimizes the admissible flux
at the interface, i.e. if it
satisfies condition (ii) of 
Definition <ref>.




Next,  given an AB-entropy solution u of  (<ref>),

we consider
the set of AB-gic

passing through a point x∈ at time t=T, and the set of the corresponding initial points at time t=0,
setting:




    𝒞(u,x)
           ≐{ζ∈Lip([0,T]  ;  ℝ)
         |  ζ(T)=x   and
    
    
          ζ is an  AB-gic 
        for u},







and 


    𝒞_0(u,x) ≐{ζ(0)  | ζ∈𝒞(u,x)}.





The set 𝒞_0(u,x) is a fundamental tool to  analyze the set of initial 
data leading to an attainable profile
ω^T.
To this end, throughout this section we consider the initial
datum u_0^* in (<ref>) 



    defined as the image of ω^T
    through the backward solution operator S^[AB]-_T,
    and we let u^*(x,t) denote the corresponding AB-entropy solution 
    with initial datum u_0^*, defined in (<ref>).
    Moreover, we let A^[AB](T) denote the set of reachable profiles at time T>0 defined in (<ref>).
    We recall that any element of
    A^[AB](T)


has at most countably many discontinuities (see <cit.>).
    
    
    The first main result of the paper
    provides a characterization of the
    set I_T^[AB](ω^T)
    in (<ref>). 
    By the analysis in <cit.>
     we know that if ω^T∈ A^[AB](T),
     then u^*(·, T)=ω^T, which means that
     u_0^*∈ I_T^[AB](ω^T).
    Our next Theorem gives a characterization of the possible elements u_0∈ I_T^[AB](ω^T)
    which are different from u_0^*.
     
























Given  ω^T ∈ A^[AB](T),
let 𝒞_0(u^*, x) denote the set defined in (<ref>)
for the AB-entropy solution u^*.
Then u_0 ∈ I_T^[AB](ω^T) if and only if for every point x of continuity  of ω^T, there exists y∈𝒞_0(u^*,x) such that
there hold

    ∫_y^y u_0(x)  x ≤∫_y^yu^*_0(x)  x,     ∀   y < min𝒞_0(u^*,x)

and 

    ∫_y^y u_0(x)  x ≥∫_y^yu_0^*(x)  x,     ∀   y > max𝒞_0(u^*,x)





Given  ω^T ∈ A^[AB](T), one can verify that the set of initial data ℐ_T^[AB](ω^T)
shares the same topological properties
enjoyed by the  set
of initial data leading at time T to an attainable profile for a conservation laws with uniformly convex flux independent on the space variable
(see <cit.>).
Namely, 
with respect to the 𝐋^1_loc topology, 
    we have: 
    
        
  (i) for every M>0, the set ℐ_T^[AB](ω^T) ∩{u_0 : ‖ u_0‖_𝐋^∞≤ M} is closed, and ℐ_T^[AB](ω^T) is an F_σ set; 

  (ii) the set ℐ_T^[AB](ω^T) 

has empty interior.
    
    The first property follows immediately by the L^1-contractivity  of the semigroup  of AB-entropy solutions.
    Concerning property (ii), consider two points 0<x_1<x_2
    of continuity for ω^T,  such that the classical genuine characteristics ϑ_x_1, ϑ_x_2 :[0,T]→
    for u_t+f_r(u)_x=0,
    passing at time T through x_1, x_2, respectively, never cross the interface x=0.
    Let u^* be the AB-entropy solution defined in (<ref>).
    Then, by Remark <ref>, ϑ_x_1, ϑ_x_2 are the unique AB-gic for u^*  that reach at time T the points
    x_1, x_2, respectively. By definition (<ref>) this means that  
     𝒞_0(u^*,x_i) = {ϑ_x_i(0)}, i=1,2.
    Note that, by the non crossing property of genuine characteristics, we have
  θ_x_1(0) < θ_x_2(0). Next,
  applying the inequality (<ref>) for x = x_2, y = ϑ_x_1(0), we find that any element u_0∈ℐ_T^[AB](ω^T) satisfies

    ∫_ϑ_x_1(0)^ϑ_x_2(0)u_0(x)  x ≤∫_ϑ_x_1(0)^ϑ_x_2(0)u_0^*(x)  x

On the other hand, 
applying the inequality (<ref>) for x = x_1, y = ϑ_x_2(0), we find that any element u_0∈ℐ_T^[AB](ω^T) satisfies

    ∫_ϑ_x_1(0)^ϑ_x_2(0)u_0(x)  x ≥∫_ϑ_x_1(0)^ϑ_x_2(0)u_0^*(x)  x .

The inequalities (<ref>), (<ref>) together imply that every element u_0 ∈ℐ_T^[AB](ω^T) satisfies

    ∫_ϑ_x_1(0)^ϑ_x_2(0)u_0(x)  x = ∫_ϑ_x_1(0)^ϑ_x_2(0)u_0^*(x)  x.

Then, letting G: 𝐋^∞()→
be the linear map defined by G(u_0)=∫_ϑ_x_1(0)^ϑ_x_2(0)u_0(x)  x,
we deduce from (<ref>)
that

    ℐ_T^[AB](ω^T)⊂{u_0∈𝐋^∞() |  G(u_0)=G(u_0^*),
       }

which shows that ℐ_T^[AB](ω^T) has an empty interior since
is is contained 
in an hyperplane of L^∞()).




The second main contribution of this paper establishes some structural and geometrical 

properties of the set
I_T^[AB](ω^T).



Given  ω^T ∈ A^[AB](T), 
with the same notations of Theorem <ref>
the following properties hold.



  (i) The set ℐ_T^[AB](ω^T) reduces to the singleton 
{u_0^*}

if and only if |𝒞_0(u^*,x)| = 1 

for every x ∈ℝ. In particular, ℐ_T^[AB](ω^T)={u_0^*} only if (but not “if") ω^T is continuous outside the origin. 


  (ii) The set ℐ_T^[AB](ω^T) is an affine cone having u_0^* as its vertex and unique extremal point (i.e. the set ℐ_T^[AB](ω^T)-u_0^* is a linear cone). 


  (iii) If, setting

    L≐ L[ω^T, f_l]    ≐sup{ L < 0   :   x-T · f_l^'(ω^T(x)) ≤ 0   ∀  x ≤ L},
        
        
     R≐ R[ω^T, f_r]    ≐inf{ R > 0   :   x-T · f_r^'(ω^T(x)) ≥ 0    ∀  x ≥ R},

and


    𝒳
    ≐{
    x∈ |  x is a point of continuity of  ω^T  and  |𝒞_0(u^*, x)| = 1},


for every point x∈ ] L,  R[ of continuity  of ω^T there holds 








    𝒞_0(u^*, x̅) ∩cl(⋃_ x ∈𝒳 C_0(u^*,x)) ≠∅,

then,  the set ℐ_T^[AB](ω^T) is convex.




















    As shown in Example <ref>, 
   if condition (<ref>) is not verified
    the set I_T^[AB](ω^T) can well be in general non convex. Note that
    the stronger condition 
    
    |𝒞_0(u^*, x)| = 1    for every point x∈ ] L,  R[ of continuity of ω^T,

    clearly implies (<ref>)
    and thus ensures the convexity of I_T^[AB](ω^T). Actually, we will first show that the set I_T^[AB](ω^T)
    is convex under condition (<ref>).
    Next, we will extend the result to the case where (<ref>)
    is verified at every point x∈ ] L,  R[ of continuity  of ω^T.



    In <cit.> it is shown that the attainable set A^[AB](T) can be partitioned in classes of attainable profiles ω^T
    which depend on the quantities 
    L,  R defined in (<ref>)










    and on the relative positions of f'_l(A)/T with respect to L,
    or of f'_r(B)/T with respect to R.
    
    These classes of profiles do not provide a finer partition 
    than the one given by the two sets
    
    {ω∈ A^[AB](T) | ℐ_T^[AB](ω)  is convex},    {ω∈ A^[AB](T) | ℐ_T^[AB](ω)  is not convex}.

    
    
    In fact, there are profiles ω_1, ω_2 ∈ A^[AB](T) that belong to a same class
    described in <cit.>, but such that
    ℐ_T^[AB](ω_1) is convex
    while ℐ_T^[AB](ω_2) is not convex.
    For example, given L_0<0,
    letting v≐ v[ L_0,
    
    A, f_l]
    be the quantity defined in <cit.>, 
   that satisfies
   
    A< v
           
           < A

   (with A  defined as in (<ref>)), we
    consider the profile
    
    ω_1 (x) =  v
            
                x < L_0, 
    
            A     x  ∈ ] L_0, 0[ , 
            
     B      x > 0,

    where B is defined as in (<ref>), and we assume that 
    
    f'_l(A)<L_0/T<f'_l( v) .

    
    Note that, since f'_r(B)≤ 0
    it follows that R=  R[ω_1, f_r]=0,
    while (<ref>) imply L=  L[ω_1, f_l]=L_0.
    One can readly verify that ω_1 fulfills the conditions (i)-(ii) of <cit.>, as does 
    the profile ω_2
    in (<ref>) considered in Example <ref>.
    We will show in Example <ref>
    that the set of initial data ℐ_T^[AB](ω_2) is not convex. On the other hand, we will see here
    that, setting 
    
    u_0^* ≐ S^[AB]-_T ω_1,        u^*(·,t) ≐ S^[AB]+_t u^*_0     ∀ t ≥ 0 ,

    at every point x∈ ]L_0, 0[   there holds (<ref>).
    Thus, the set ℐ_T^[AB](ω_1) is convex because of Theorem <ref>-(iii).
    
    
    
    
    

    In order to determine the sets C_0(u^*,x), x∈
    
    (and then check (<ref>)),
    we construct explicitly the AB-entropy solution u^* defined in (<ref>), following the procedure described in <cit.>.
    Namely, because of condition (<ref>) the solution u^* contains a 
    shock curve  starting at the interface x=0,
    and then lying in the semiplane {x<0},
    which reaches the point x=L_0 at the time T. In fact,
    according with the analysis in <cit.>,
    there exist a constant  σ≐σ[ L_0, A, f_l], and a map
    γ : [σ, T]→ ]-∞, 0],  
    with the properties that γ(σ)=0,
    γ(T)=L_0, and that t→ (γ(t), t) is a shock curve for the conservation law
    u_t+f_l(u)_x=0, which connects 
    
    the left states 
    (f^'_l)^-1((γ(t)-L_0+T · f^'_l( v
    
    ))/t), t∈ [σ, T],
    with the right state A.
    On the left of γ(t) there is a rarefaction wave, connecting the left state 
    v with the right state A,
    and centered at the point (L_0-T · f^'_l( v), 0). Moreover, there holds
    
    σ=T· f'_l( v)-L_0f'_l(  A).

    Then,
    setting
    
    η_-(t)   ≐ L-(T-t) · f^'_l( v),   t∈ [0,T],
            
    η_+(t)   ≐ L-T · f^'_l( v)+ t · f'_l(  A),
               t∈ [0,σ],

    we find that the function u^* in (<ref>) is given by (see Figure <ref>)

    
    
    
    u^*(x,t)=
             v     if   x < η_-(t),   t∈ [0,T],
                
    
                (f^'_l)^-1(x-L_0+T · f^'_l( v
                )t)
                     if  {   η_-(t)<x<γ(t),   t∈ [σ, T],
                
       η_-(t)<x<η_+(t),
                  t∈ [0,σ],
                .
                
    
                A      if   γ(t)<x<0,   t∈ [σ, T],
                
    A     if   η_+(t)<x<0,   t∈ [0,σ],
                
    B     if    x>0,   t∈ [0,T].

    Observe that every AB-gic for u^* that reaches a point x∈ ]L_0, 0[ at time T
    has either the expression
    
    η__τ_1(t)=
                
                x+ (T-t)· f'_l(A)
                     if   t∈ [τ_2,T],
                
    
                0      if   t∈ [τ_1,τ_2],
                
    
                (t-τ_1) · f'_r(  B)
                     if   t∈ [0,τ_1],

    with τ_2≐ T- x/f'_l(A), and 
    τ_1∈ [0,τ_2], or the expression
   
    η__ τ_1(t)=
                
                x+ (T-t)· f'_l(A)
                     if   t∈ [τ_2,T],
                
    
                0      if   t∈ [τ_1,τ_2],
                
    
                (t-τ_1) · f'_l(  A)
                     if   t∈ [0,τ_1],

    with τ_2 as above and
    τ_1∈[0,σ].
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    By definition (<ref>) this means that 
    
    C(u^*,x)=
            {η__τ_1 | τ_1∈[0,τ_2]} ∪ {η__τ_1 | τ_1∈[0, σ]}.

    Since we have 
    
    {η_τ_1(0) | τ_1∈[0,τ_2]}   =
        [0,  (x/f'_l(A)-T)· f'_r(  B)],
        
    {η_τ_1(0) | τ_1∈[0, σ]}   =
        [-σ· f'_l(  A),0],

    
    by definition (<ref>) and by virtue of (<ref>)  we 
    then find that 
    
    C_0(u^*,x)=[ L_0 - T· f'_l( v),  (x/f'_l(A)-T)· f'_r(  B)]
                ∀ x∈ ] L_0, 0[ .

    On the other hand, 
    since ω_1 is constant for  x<L_0, there exists a unique AB-gic for u^* that reaches a point  x<L_0 at time T,
    which is a classical genuine characteristic
    
    ϑ_x(t)=x-(T-t)· f'_l( v),
                 t∈ [0,T],

    because it never crosses the interface x=0.
    Hence, we have
    
    C_0(u^*,x)={x-T· f'_l( v)}    ∀ x<L_0 .

    Therefore, from (<ref>), (<ref>), we deduce
    
    𝒞_0(u^*,  x) ∩cl(⋃_ x< L_0 C_0(u^*,x))
            ={ L_0 - T· f'_l( v)}    ∀ x∈ ] L_0, 0[ ,

    which proves (<ref>), and thus concludes the proof of the convexity of ℐ_T^[AB](ω_1).












We end the section with some Remarks.

In the following of this paragraph we consider the connection (A,B) as fixed and we simply write S^+_T,  S^-_T for the forward and backward operators, and in general omit the dependence of the object from the connection. As already said, the map S_T^+:  A(-T) → A(T) is invertible and bi-Lipschitz, with inverse given by S^-_T. Exploiting Theorem <ref>, we can give a geometric meaning to the backward forward operator 

    S^+_T ∘ S_T^-:𝐋^∞(ℝ) → A(T)

For an element χ∈ A^(-T), we introduce the backward cone of initial data

    I^-_Tχ≐{χ_0 ∈𝐋^∞(ℝ)  |  S^-_Tχ_0 = χ}

Notice that it is indeed a cone by Theorem <ref> and the definition of S_T^-. Then for any ω∈ A^(T), with χ≐ S_T^-ω∈ A^(-T), it holds

    {ω̃ |  S^+_T ∘ S^-_Tω̃= ω} =  I_T^-χ

and in particular the left hand side set is a cone with ω at its vertex. In order to prove (<ref>) notice the following. If ω̃ satisfies S_T^+∘ S_T^-ω̃= ω, then 

    S_T^-ω̃=  S^-_T∘ S^+_T ∘ S^-_Tω̃=  S^-_Tω = χ

where the first equality holds because since S_T^-ω̃∈ A^(-T), then it is a fixed point of the operator S^-_T∘ S^+_T.
Therefore ω̃∈ I_T^- χ. If instead ω̃∈ I_T^- χ, then 

    S^+_T ∘ S^-_Tω̃=  S_T^+χ =  S_T^+ ∘ S_T^-ω = ω

that proves the equality (<ref>). Therefore we have a partition of 𝐋^∞(ℝ) as a union of cones

    𝐋^∞(ℝ) = ⋃_χ∈ A^(-T) I_T^- χ

and the map S^+_T ∘ S^-_T sends each element ω̃∈ I_T^- χ to the vertex of its cone I_T^-χ. Finally, notice that 

    {Vertex of I_T^-χ} =  A^(T) ∩ I_T^- χ










  §.§.§ Minimal Entropy Solutions

Consider the discontinuous flux problem (<ref>) with fixed connection (A,B). Let in the following 

    η^AB(u,x) = |u-A|,    if  x < 0, 
    |u-B|,    if  x > 0

be the adapted entropy with corresponding adapted entropy flux

    q^AB(u,x) = sgn(u-A)(f_l(u)-f_l(A)),    if  x < 0, 
    sgn(u-B)(f_r(u)-f_r(B)),    if  x > 0

With this notation (<ref>) rewrites 

    η^AB_t(u,x) + q^AB(u,x)_x = 0,     in  D^'

Therefore for every initial datum u_0 we find

    ∫_ℝη^AB( S_T^AB u_0,x )  x ≤∫_ℝη^AB(u_0,x)  x ,     ∀  T ≥ 0

(the right hand side is possibly +∞ in which case the inequality is always true).
We claim that the initial datum S^AB_-T satisfies 

    ∫_ℝη^AB( S^AB_-Tω,x) = min{∫_ℝη^AB(u_0,x)  x   :   S^AB_T u_0 = ω}

On the one hand (<ref>) holds, therefore every initial datum u_0 such that S_T^AB u_0 = ω satisfies 

    ∫_ℝη^AB(ω,x)  x ≤∫_ℝη^AB(u_0,x)  x ,     ∀  u_0   :   S^AB_T u_0 = ω

On the other hand, let u_0^* =  S_-T^ABω and as usual u^b(x,t) = S̃_t^B̅A̅ω(-·)(x). Then 

    ∫_ℝη^AB( S_-T^ABω,x)  x    
          
    
          =   ∫_ℝη^AB(S̃_T^B̅A̅ω(-·)(-x), x)  x    by definition of S_-T^AB
          
    
         =    ∫_ℝη^AB(S̃_T^B̅A̅ω(-·)(x), -x)  x    change of variable x ↦ -x
          
    ≤   ∫_ℝη^AB(ω(-x), -x)  x    u^b entropy admissible in {x > < 0}
          
    
         +    ∫_0^T  (sgn(u^b_r(t)-A)(f_l(u^b_r(t))-f_l(A))    
          
    
         -   sgn(u_l^b(t)-B)(f_r(u_l^b(t))-f_r(B)) )  t

    
    
Now, it is easy to see that since ω∈ A^AB(T), then the backward solution u^b has to satisfy 

    sgn(u^b_r(t)-A)(f_l(u^b_r(t))-f_l(A))-sgn(u_l^b(t)-B)(f_r(u_l^b(t))-f_r(B)) = 0,   for a.e.  t ∈ (0,T)

A quick way to see this is that if sgn(u^b_r(t)-A) = sgn(u_l^b(t)-B) then our claim follows. If sgn(u^b_r(t)-A) >0 and sgn(u_l^b(t)-B) < 0 then (u_l^b(t), u_r^b(t)) = (B̅,A̅) for almost every t by the (B̅, A̅)-entropy admissibility of u^b. If sgn(u^b_r(t)-A) <0 and sgn(u_l^b(t)-B) > 0, then tracing the backward genuine characteristics from (0,t) at the left and at the right of the interface for the backward solution u^b we find a contradiction with the fact that ω was attainable. 
Then we deduce

    ∫_ℝη^AB( S_-T^ABω)  x ≤∫_ℝη^AB(ω(x), x)  x

Then we deduce that (<ref>) holds true and 

    ∫_ℝη^AB( S_-T^ABω)  x =   ∫_ℝη^AB(ω(x), x)  x,


    t ↦∫_ℝη^AB( S_t^AB∘ S_-T^ABω)  x   is a constant η^AB-entropy trajectory

In particular the forward solution u^* =  S_t^AB∘ S_-T^ABω does not dissipate the η^AB-entropy along the interface and any shock  in the region {x  < 0} must have either the left or the right state equal to A, and any  shock in the region {x > 0} must have either the left or the right state equal to B. 










§ PROPERTIES OF GENUINE/INTERFACE CHARACTERISTICS






In this section we 
establish some basic properties 
enjoyed by the AB-genuine/interface cha­ra­cteristics
for an AB-entropy solution u, and by the sets 𝒞(u,x), 𝒞_0(u,x), introduced in  <ref>.




















Let u be an AB-entropy solution to (<ref>). Then the following properties hold.

 
  (i) C(u,x) ≠∅  
 for all x∈;
 
  (ii) the  map x ↦ C(u, x) has closed graph as a set-valued map from
     into the space Lip([0,T]  ; ℝ) with the topology of uniform convergence;
    








  (iii) C_0(u,x) is a closed set
for all x∈;

  (iv) the maps x ↦min C_0(u, x),
x ↦max C_0(u, x) are monotone nondecreasing.












 Throughout the proof we set ω^T(x)≐ u(x,T), x∈,
and we let u_l(t), u_r(t) denote the one-sided traces of u(t,·) at x=0.

 1. Proof of (i). 
 
 Given x>0, consider the minimal backward characteristic  
 for the conservation law u_t+f_r(u)_x=0,
 in the semiplane {x>0},
 starting from (x,T),
 defined by ϑ_x,-(t)=x-(T-t) · f_r^'(ω^T(x-)).
 
 If 
x-T· f_r^'(ω^T(x-)) ≥ 0, then 


ϑ_x,- is  a classical genuine characteristic for u 

in the whole interval [0,T], since it never 
crosses the interface x=0 but at most at t=0.
Therefore, according with Definition <ref>,
the map

    ζ(t)=x-(T-t) · f_r^'(ω^T(x-)),
         t∈ [0,T],

is an AB-genuine/interface characteristic,
and hence by (<ref>) it holds ζ∈ C(u,x).
Otherwise, we have x-T· f_r^'(ω^T(x-)) < 0, and thus ϑ_x,-
 impacts the interface at the time:

    τ_-(x)≐ T-x/f_r^'(ω^T(x-))>0.













Then, consider the set

    E ≐{ t ∈ [0, τ_-(x)]  | either  u_l(t)>θ_l  or  u_r(t)<θ_r},





























and 
let 

    τ≐sup E,

where we understand that τ=0 when E=∅.
Because of the  non-intersection 
property of classical genuine characteristics in the domains {x<0}, {x > 0}, 
and since uniform limit of classical genuine characteristics is a classical genuine characteristic as well (e.g. cfr. <cit.>), 
we deduce that 

    τ∈ E    if   
        E≠∅.


Thus, when E≠∅, if u_l(τ)>θ_l we can consider the minimal backward characteristic  
 for the conservation law u_t+f_l(u)_x=0,
 in the semiplane {x<0},
 starting from (0,τ),
 defined by ϑ_τ,-(t)=(t-τ) · f_l^'(u_l(τ)).
 Otherwise, if u_r(τ)<θ_r we can consider the maximal backward characteristic
 for the conservation law u_t+f_r(u)_x=0,
 in the semiplane {x>0},
 starting from (0,τ),
 defined by ϑ_τ,+(t)=(t-τ) · f_r^'(u_r(τ)).
 On the other hand, by definition of E, and recalling the interface condition (<ref>), we find that
 
    u_l(t)= A,    u_r(t) =B     
    ∀ t ∈ ] τ, τ_-(x)].

 Note in particular that
 
    τ<τ_-(x)  ⟹   u_r(τ_-(x))=ω^T(x-)=B.

Therefore,  the piecewise affine map

    ζ(t) = 
    x-(T-t) · f_r^'(ω^T(x-)),     t ∈ [τ_-(x), T], 
    
    0 ,     t ∈ ]τ, τ_-(x)[, 
    
    (t-τ) · f_l^'(u_l(τ)),     t ∈ [0, τ],  if u_l(τ)>θ_l,
    
    
    (t-τ) · f_r^'(u_r(τ)),     t ∈ [0, τ],  if u_r(τ)<θ_r,

satisfy the conditions of Definition <ref>,
and thus it is  an AB-gic

belonging to the set C(u,x).
Note that it may well happen that τ=τ_-(x),
in which case there will be in (<ref>) no nontrivial interval where the
characteristic is travelling along the interface x=0. Instead, in the case 
τ=0, the AB-gic
in (<ref>) lies on the interface x=0
in the whole interval [0,τ_-(x)].

Clearly, the same analysis can be carried out to show that C(u,x) ≠∅ also for  x<0. It remains to consider the case
x=0. If we assume that 

ω^T(0-)>θ_l,
then the minimal backward characteristic for
u_t+f_l(u)_x=0,
 in the semiplane {x<0},
 starting from (0,T), 
is  a classical genuine characteristic for u 

in the whole interval ]0,T], and hence it 
it is  an AB-gic

belonging to the set C(u,0).
Similarly, if ω^T(0+)<θ_r,
then the maximal backward characteristic for
u_t+f_r(u)_x=0,
 in the semiplane {x>0},
 starting from (0,T), 
is  a classical genuine characteristic for u 

in the whole interval [0,T], and hence 
it is  an AB-gic

belonging to the set C(u,0).
Finally, if ω^T(0-)≤θ_l
and ω^T(0+)≥θ_r, by
 the interface condition (<ref>), we
 deduce that ω^T(0-)=A, ω^T(0+)=B.
 Then, set
 
    τ=sup E,
           
    E ≐{ t ∈ [0, T]  | either  u_l(t)>θ_l  or  u_r(t)<θ_r}.

With the same type of analysis 
as above we find that τ∈ E
and that the map

    ζ(t) = 
    0 ,     t ∈ ]τ, T], 
    
    (t-τ) · f_l^'(u_l(τ)),     t ∈ [0, τ],  if u_l(τ)>θ_l,
    
    
    (t-τ) · f_r^'(u_r(τ)),     t ∈ [0, τ],  if u_r(τ)<θ_r,

is  an AB-gic

belonging to the set C(u,0), thus completing the proof of (i).











































 2. Proof of (ii). 
 The closed graph property 
 
 of the map x ↦ C(u, x) is equivalent to 
 
 
    (x_n → x,   ζ_n ∈ C(u,x_n),  ζ_n →ζ  uniformly)   ⟹  ζ∈ C(u,x).

Then, let {x_n}_n be a sequence 

converging to x≥ 0, and
consider a sequence of AB-gic

ζ_n∈ C(u,x_n), that 
converge uniformly to some ζ∈Lip([0,T]  ; ℝ). By Remark <ref>, for every n there will be
0≤τ_1,n≤τ_2,n≤ T, such that

    ζ_n(t)=0,     f_l(u_l(t))=f_r(u_r(t))=γ     ∀ t∈ [τ_1,n, τ_2,n],

and such that
the restriction of ζ_n
     to ]0, τ_1,n[ and to ]τ_2,n, T[
     is either a classical genuine characteristic 
     for  u_t+f_l(u)_x=0 
     on {x<0}, or it 
     is a classical genuine characteristic 
     for  u_t+f_r(u)_x=0 
     on {x>0}.
     This, in particular, implies that
     
    ζ̇_n(t)=
             x_nT-τ_2,n     ∀ t ∈ 
                  ]τ_2,n, T[ ,
                  
    
                  -ζ_n(0)τ_1,n     ∀ t ∈ 
                  ]0,τ_1,n[ .

     Possibly considering a subsequence we can assume that {τ_i,n}_n converge to some τ_i∈ [0,T], i=1,2, with τ_1≤τ_2.
     Suppose that τ_1>0, τ_2<T. The
     cases where τ_1=0, or/and τ_2=T
     can be treated with entirely similar
     and simpler arguments.
     Up to extracting a further subsequence
     we may also assume that x_n>0, and that
     
    ζ_n(t)<0    ∀ t∈ [0, τ_1,n[ ,        ζ_n(t)>0    ∀ t∈ ]τ_2,n, T],

Again, the cases where ζ_n(t)>0 for all t∈ [0, τ_1,n[, or/and x_n<0, ζ_n(t)<0 for all t∈ ]τ_2,n, T], can be analyzed in an entirely similar way.
By the uniform convergence of ζ_n to ζ
and since τ_i,n→τ_i, i=1,2, it follows from (<ref>) that

    ζ(t)=0,     f_l(u_l(t))=f_r(u_r(t))=γ     ∀ t∈ ]τ_1, τ_2[ .

and

    ζ(t)≤ 0   ∀ t∈ [0, τ_1],      ζ(t)≥ 0   ∀ t∈ [τ_2, T].

Moreover, we have 

    ζ(T)=x,

since x_n→ x and
x_n=ζ_n(T)→ζ(T).
Note also that, because of (<ref>), there holds

    ζ̇(t)=lim_n ζ̇_n(t)=
             xT-τ_2     ∀ t ∈ 
                  ]τ_2, T[ ,
                  
    
                  -ζ(0)τ_1     ∀ t ∈ 
                  ]0,τ_1[ .

     
Now, if we assume that x>0, 
it follows from (<ref>) that
that ζ(t)>0 for all t∈ ]τ_2, T].







On the other hand, since 
uniform limit of classical genuine characteristics is a classical genuine characteristic as well,
we deduce that the restriction of ζ to 
]τ_2, T] is a classical genuine characteristic for u_t+f_r(u)_x. 

Next, if we assume that x=0, then
the uniform convergence of ζ_n to ζ, together with (<ref>), (<ref>), imply that

    ζ(t)=ζ̇(t)=0    ∀ t ∈ 
                  ]τ_2, T[ ,

and

    f'_r(u_r(t))=
    lim_n f'_r(u(ζ_n(t),t))=lim_n ζ̇_n(t)=0
            ∀ t ∈ 
                  ]τ_2, T[ .

In turn, (<ref>) implies that u_r(t)=θ_r=B for all t ∈  ]τ_2, T[, and that (A,B) is a critical connection. On the other hand, because of 
the interface condition (<ref>), it follows that f_l(u_l(t))=f_r(u_r(t))=γ
for all t ∈  ]τ_2, T[,
which proves that the restriction of ζ
to the interval [τ_2, T] satisfies the condition (ii) of Definition <ref>.
With entirely similar arguments one can show that the restriction of ζ
to the interval [0,τ_1] satisfies the condition (i) or (ii) of Definition <ref>,
which, together with (<ref>), completes the proof that ζ is
an AB-gic

belonging to the set C(u,x). This completes the proof of (<ref>).




























3. Proof of (iii). Let {y_n}_n
be a sequence of elements of C_0(u,x)
converging to some point y∈.
Then, there will be a sequence of AB-gic 

ζ_n∈ C(u,x),
such that y_n=ζ_n(0) for all n.
Observe that by Definition <ref>
 it follows that 
 
    |ζ_n(t)|≤ |x|+LT,     |ζ̇_n(t)|≤
         L,    ∀ t∈[0,T],  ∀ n ,

 for some constant L>0 depending on u_ L^∞. Hence, applying Ascoli-Arzelà Theorem, we deduce that up to a subsequence {ζ_n}_n  converges uniformly to some ζ∈Lip([0,T]  ; ℝ).
 Thus, 
 in particular we have
 
    ζ(0)=lim_mζ_n(0)=lim_n y_n=y.

Then, in view of property (ii) established 
at previous point, we find that ζ∈ C(u,x), and (<ref>) implies
y∈ C_0(u,x), completing the proof of (iii). 


 4. Proof of (iv).
Given x_1<x_2, let y_1=max C_0(u, x_1), and consider  ζ_1 ∈ C(u, x_1) such that ζ_1(0) = y_1. Choose any ζ_2 ∈ C(u, x_2) 

and define 

    ζ(t) ≐max{ζ_1(t),  ζ_2(t)}     t ∈ [0,T].

Observe that, by definition the maximum of two AB-gic is still an AB-gic, and ζ(T) = x_2, so that one has ζ∈ (u,x_2). Moreover:

    max C_0(u, x_1) =y_1 = ζ_1(0) ≤ζ(0) ≤max C_0(u, x_2).





















































































This proves (iv), and 
thus concludes the proof of the proposition.






























































































































































































The following Proposition states that  
the AB-entropy solution u^* defined in (<ref>) has the maximal amount of interface characteristics.


Given ω^T ∈𝒜^[AB](T),
let u^* be the 
 AB-entropy solution defined by (<ref>)-(<ref>),
and let u be any other 
 AB-entropy solution




to (<ref>) 
with initial datum u_0∈ I^[AB]_T(ω^T).

Then, there holds 

    𝒞(u^*, x) ∩𝒞(u, x) ≠∅    ∀ x∈ .




To fix the ideas, we will assume that,
letting L,  R be the quantities
defined in (<ref>), there holds
L=0, R∈ ]0, T· f'_r(B)[ , and that
ω^T
fulfills the conditions (i)'-(ii)' of <cit.>
for a non critical connection
(A,B), which in particular require
    
    ω^T(x-)   ≥ω^T(x+)
                 ∀ x≠ 0,
             
    ω^T(x)   ≥ B        ∀ x∈ ]0, R[ .

The cases where ω^T belongs to other classes of reachable profiles described in <cit.> can be analyzed with entirely similar arguments.
Throughout the proof we  let u_l(t), u_r(t), and u^*_l(t), u^*_r(t), denote the one-sided traces of u(t,·)
and u^*(t,·), respectively, at x=0.


1. 

Relying on the fact that any sequence
{ζ_n}_n of AB-gic 
(for u^* and u) 
admits a subsequence uniformly convergent to some 
ζ∈Lip([0,T]  ; ℝ)
(see point 3. of the proof of Proposition <ref>),
and since the map
x→𝒞(u^*, x) ∩𝒞(u, x) has closed graph by
Proposition <ref>-(ii),
it will be sufficient to show that
𝒞(u^*, x) ∩𝒞(u, x) ≠∅ holds for all point x of continuity for ω^T.
Moreover, for every point x∈ ]-∞, 0[ ∪ ] R, +∞[  of continuity for ω^T, 
there exists a unique AB-gic for u^* and u that reaches the point x at time T,
    which is a classical genuine characteristic ϑ_x
    for u^* and u (since it never crosses the interface x=0,
    but at most at t=0,
    by definition (<ref>)).
    Thus we have 𝒞(u^*, x) ∩𝒞(u, x)={ϑ_x} for all point 
    x∈ ]-∞, 0[ ∪ ] R, +∞[  of continuity for ω^T. As a consequence, in order to establish (<ref>) it will
    be sufficient to show
    
    𝒞(u^*, x) ∩𝒞(u, x) ≠∅      for all x∈ ]0,  R[ of continuity for ω^T.

To this end, given any x∈ ]0,  R[ of continuity for ω^T, we consider the AB-gic
ζ∈𝒞(u, x)
 defined in (<ref>),
 
 with τ as in (<ref>)
 and 
 
    τ(x) ≐ T-x/f_r^'(ω^T(x)),

 in place of τ_-(x). We will show that ζ also belongs to 𝒞(u^*, x).
 Note that by definition of R at (<ref>) we have τ(x) >0. 
  2. 


We determine here  explicitly
the 
 AB-entropy solution u^* defined by (<ref>)-(<ref>)
 when ω^T
satisfies the conditions (i)'-(ii)' of <cit.>
for a non critical connection,
with
L=0, R∈ ]0, T· f'_r(B)[ . These conditions require in particular that

    ω^T(0-)   ≥π(ω^T(0+)),
    
    ω^T(x)   ≥ B,    ∀ x∈ ]0,  R[ ,
        
    ω^R( R-)   ≥ u ,


where 

    π(u)≐  (f_l_| [θ_l,+∞))^-1∘ f_r(u),     u∈,

and u≐ u[ R, B, f_r]
is the quantity defined in <cit.>
   that satisfies
   
    B> u
           
           > B,
                 
           f'_r( u)< R/T,

   (with B  defined as in (<ref>)).
Because of condition (<ref>), according with the analysis in <cit.> the solution u^* contains a shock curve
starting at the interface x=0
and reaching the point R at time T,
which is parametrized by a map
    γ : [τ, T]→ [0,∞[,  
    with the properties that γ(τ)=0,
    γ(T)= R, 
    where τ≐τ[ R, B, f_r]
    is a quantity defined as
    in <cit.>.
    The curve
     t→ (γ(t), t) is the location of a shock  for the conservation law
    u_t+f_r(u)_x=0, which connects 
    the left state B
    with the
    right states 
    (f'_r)^-1((γ(t)- R+T · f'_r( u
    ))/t), t∈ [τ, T].
    On the right of γ(t) there is a rarefaction wave, connecting the left state B 
    with the right state u,
    and centered at the point ( R-T · f'_r( u), 0). Moreover, there holds
    
    τ=T· f'_r( u)- Rf'_r(  B).

    Following the procedure described in <cit.>, in order to define u^* we introduce some notations 
    for the polygonal lines along which 
    u^* takes constant values
    in each region {x<0}, 
    {x>0}
    (that correspond to AB-gic for u^*).
    We define

    ϑ_ 0, -(t)    ≐
     (t-T)· f'_l(ω^T(0-)),
        
    ϑ_ 0, +(t)    ≐
        (t-T)· f'_l(π(ω^T(0+))),
        
    ϑ_ R, -(t)    ≐ R-(T-t) · f'_r(ω^T( R-),
                 if  τ_-( R)≤ t≤ T,
            
    (t-τ_-( R))
     · f'_l∘π(ω^T( R-)),
          if   0≤ t≤τ_-( R),
        
    ϑ_ R, +(t)    ≐ R-(T-t) · f'_r(ω^T( R+)),

and, for every y ∈ ]-∞, 0 [ ∪ ] R, +∞[ , we define
    
    ϑ_y, ±(t) ≐
    y-(T-t) · f_l^'(ω^T(y±)),     if  y < 0,    0≤ t ≤ T,
    
      y-(T-t) · f'_r(ω^T(y±)),     if  0< y <  R,   τ_±(y) 
      ≤ t ≤  T,
    (t-τ_±(y))
     · f'_l∘π(ω^T(y±)),     if  0< y <  R,    0 ≤ t < τ_±(y),
    
        y-(T-t) · f'_r(ω^T(y±)),     if  y>   R,    0≤ t ≤ T,

    where
    
    τ_±(y)   ≐
        T-y/f'_r(ω^T(y±)),     y>0.

    Moreover, letting {y_n}_n denote the (at most) countably many discontinuity points of ω^T
    in the intervals ]-∞, 0],
    ] R, +∞[, we set 
    
    I^n_0   = ]x_n^-,x_n^+[ ,    x_n^±=ϑ_y_n,±(0),
       y_n∈ ]-∞, 0] ,
    
    ℐ^n_ R   = ]x_n^-,x_n^+[ ,    x_n^±=ϑ_y_n,±(0),
       y_n∈ ] R,+∞[ ,

    (here we consider the possibility of a jump 
    of ω^T in x=0 when ω^T(0-)>π(ω^T(0+))).
    The intervals I^n_0, ℐ^n_ R, consist of the starting points of compression waves in u^* that generate a shock at (y_n, T).
    
    Next, we introduce the polygonal lines
    connecting two points (z,0), (y,T)
    (that correspond to compression fronts
    for u^* generating a shock at the point 
    (y,T)) defined by
    
    η_y,z≐
                y -(T-t)·(y-z)/T,     if   y∈ ]-∞, 0] ∪ ] R, +∞[ ,   0≤ t ≤ T,
                
    
                 y -(T-t)· f'_r(u_y,z)
                      if   0<y< R,
                    T-y/f'_r(u_yz)≤ t≤ T,
                 
    (t-T+y/f'_r(u_yz))· f'_l∘π(u_y,z)
                     if   0<y< R,
                    0≤ t<T-y/f'_r(u_yz),

    where u_y,z is the unique constant u≥ (f'_r)^-1(y/T) satisfying 
    
    (y/f'_r(u)-T)· f'_l∘π(u)=z

    (see <cit.>).
    
    
    
    
    
    Finally, we set
    
    r_-(t)   ≐ R-T · f'_r( u)+ t · f'_r(  B),
               t∈ [0,τ],
            
    
            r_+(t)   ≐ R-(T-t) · f'_r( u),   t∈ [0,T].

    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    Then, the function u^* defined by (<ref>)-(<ref>) is given by

    u^*(x,t)=
            ω^T(y±),    if  x=ϑ_y,±(t) for some  y ∈ ]-∞, 0[ ∪ ] R, +∞[,
    ω^T(y±),    if  x=ϑ_y,±(t)>0 for some  y ∈ ]0,  R[,
    π
    (ω^T(y±)),    if  x=ϑ_y,±(t)<0 for some  y ∈ ]0, R[,
    
    (f_r^')^-1(y_n-z/T),    if  x = η_y_n, z(t) for some  z ∈ I^n_ R,
    
    (f_l^')^-1(y_n-z/T),    if  x = η_y_n, z(t) for some  z ∈ I^n_0,
    
     B    if {   ϑ_ R, -(t)≤ x<γ(t),   t∈ [τ_-( R), T],
                
       0<x<γ(t),
                  t∈ [τ, τ_-( R)],
                .
                
      A     if  ϑ_ R, -(t)≤ x<0,   t∈ [0,τ_-( R)],
    
      B     if  0<x≤ 
    r_-(t),   t∈ [0, τ],
    
    
    (f'_r)^-1(x- R+T · f'_r( u
                )t)
                     if {   γ(t)<x<r_+(t),   t∈ [τ, T],
                
       r_-(t)<x<r_+(t),
                  t∈ [0,τ],
                .
    
    
    (f'_r)^-1( R-x/T-t)    if  
    r_+(t)≤ x≤ϑ_ R, +(t),   t∈ [0, T].
 


















































Observe that the left and right traces of u^* satisfy

    u^*_l(t)   ≥ A,     u^*_r(t)≥ B,
            ∀ t∈ ]τ_-( R),T],
        
    
        u^*_l(t)   =  A,     u^*_r(t)= B,
            ∀ t∈ ]τ,τ_-( R)],
        
        
        
    
        u^*_l(t)   =  A,     u^*_r(t)=  B,      ∀ t∈ ]0,τ].

Moreover, since x is a point of continuity for ω^T, 
it follows that 
the restriction of ζ 
to ]τ(x), T]
is a (classical) genuine characteristic both for u
and u^* 
with slope f'_r(ω^T(x))>0
so that, 


recalling (<ref>), there holds

    u_r(τ(x))=ω^T(x)=u^*_r(τ(x))
        
        >θ_r,
              
        f_l(u_l(τ(x)))=f_r(u^*_r(τ(x))).






Now, we will distinguish two cases
according with the position of τ with respect to the time 
τ_-( R) defined as in (<ref>).




Note that by definition of R at (<ref>) we have 
τ_-( R) ≥ 0.
 3. 
 Assume that τ≥τ_-( R), and suppose first that τ(x)=τ> τ_-( R). Note that, since τ>0
 is an element of the set E
 in (<ref>), and
 because of
 
  (<ref>), 
 (<ref>),
 we have u_l(τ(x))=u^*_l(τ(x))>θ_l. Therefore, also
 the restriction of ζ 
to [0,τ(x)[
is a (classical) genuine characteristic both for u
and u^*. Hence, when 
τ(x)=τ>τ_-( R),
the map ζ in (<ref>)
 is an AB-gic also for u^*,
 proving that ζ∈𝒞(u^*, x) ∩𝒞(u, x).
 
 Next, consider the subcase 
 τ(x)>τ≥τ_-( R), and 
observe that by (<ref>), (<ref>), 
(<ref>),
we have

    u_l(t)=A,  
    u_r(t)=B    ∀ t∈ ] τ, τ(x)],
           u^*_r(τ(x))=B .

Moreover, we claim that

    ω^T(z)   =B,    ∀ z∈
        [x,  x[ ,
        
           x ≐ 
        (T-τ)· f'_r(B),
        
    
        u_r^*(t)   =B,
            ∀ t∈ ] τ,τ(x)] .

Note that the first equality in (<ref>)  implies the second one by tracing the backward (genuine) characteristics for u^* 
at time T, from points z∈ [x,  x[.

In order to prove the first equality in (<ref>), we trace 
the minimal backward characteristic 
ϑ_z,- for the solution u, at time T, from points z ∈ [x,  x].

Observe that 
ϑ_z,-
impacts the interface x=0 at time 

τ_-(z)≐ T-z/f'_r(ω^T(z-)).



Moreover, since
ω^T( R-)≥ B because of (<ref>),
we deduce from τ≥τ_-( R) that

    x ≤ R.

Furthermore, 
we  have

    τ_-(z)≥ 
    T-z/f'_r(B)≥τ      ∀ z ∈ [x,  x],

since
ω^T(z-) ≥ B
by virtue of (<ref>). 
On the other hand, by (<ref>)


we know that
the (genuine) characteristic ϑ_x for u, starting at time T from the point x,
reaches the interface x=0 at time τ(x).







Since ϑ_x, ϑ_z,-
are (classical) genuine characteristics 
that cannot cross in the
domain {x>0}, it follows that

    τ(x)≥τ_-(z)
    
    
              ∀ z ∈ [x,  x] .

Combining together (<ref>), (<ref>),
we deduce that, 
for every z∈ [x,  x],

the minimal backward characteristics ϑ_z,-
reaches the interface x=0 
at time τ_-(z)∈ [ τ,τ(x)].
Hence, because of (<ref>),
we find that for all z∈ [x,  x]

there holds ω^T(z-) = u_r(τ_-(z))=B, and this 
yields the first equality in (<ref>), concluding the proof of claim (<ref>).

 Relying on (<ref>)
we will show now that 

ζ 

satisfies the condition of an AB-gic
also for u^*
on the interval [0, τ(x)].
To this end, observe that (<ref>) (<ref>) together imply

    u_l^*(t)= A,
            ∀ t∈
         ] τ,τ(x)] .
 
Hence, because of (<ref>),
(<ref>), ζ satisfies the condition (ii) of Definition <ref>
of an AB-gic for u^* on the interval
]τ, τ(x)].

Moreover, let ϑ_t_n^*
denote the (classical genuine) backward characteristic for u^*, on the
region {x<0}, starting at time t_n∈ ] τ,τ(x)]
from x=0,
for a sequence t_n↓τ.
Note that, because of (<ref>), all 
ϑ_t_n^* have slope f_l'(  A). Thus {ϑ_t_n^*}_n
converges uniformly to a function ϑ^*: [0,τ]→
that is as well a (classical) genuine characteristic 
for u^* with slope f_l'(  A) and such that ϑ^*(τ)=0.
This in turn implies that

    u^*_l(τ) = 
     A .



Next, we will prove that 

    u_l(τ) 
        =  A .


 
 To this end
 observe that (<ref>), 
 (<ref>), 
 (<ref>), (<ref>)
 together imply
ω^T( x-)= B=ω^T( x+).
This means that the 
   characteristic
 ϑ_ x starting
 at time T from x,
 and reaching x=0 at time τ,

is a (classical) genuine characteristic for u
(on the semiplane {x>0}),
and hence we deduce that 

    u_r(τ)=ω^T( x)=B.

Recalling (<ref>)
and the definition (<ref>)
of the set E, we derive from (<ref>)
and from 
condition (2) of Definition <ref>
that 

    u_l(τ)>θ_l.

In turn, condition (<ref>), together with (<ref>),
implies (<ref>) by a blow-up argument as in <cit.>. 
Namely, we can consider 
the blow ups 
of u at the point (0, τ ):

    u_n(x,t) ≐ u(x/n, τ +(t-τ)/n)
             x∈,  t≥ 0 ,  n∈ℕ,
 
and observe that,
because of (<ref>),
the left and right traces of u_n(·, t) at x=0
satisfy

    (u_n,l(t), u_n,r(t))=(A,B)      ∀ t∈]τ,  τ +n(τ(x)-τ)[ .

When n→∞,

up to a subsequence, 
the blow ups u_n(·, t)
converge in L^1_loc  to a limiting AB entropy solution v(·, t),
for all t>0,
and there holds 

    v(x, τ )= 
    
        u_l(τ),    if x < 0,
    
        u_r(τ),    if x > 0,


    v(0-,t)∈{A, A },         v(0+,t)∈{B, B },    ∀ t>τ .

Then, by a direct inspection
we find that, if 
an AB entropy solution
of a Riemann problem for (<ref>), with initial datum (<ref>)
at time τ, enjoys the properties
 (<ref>), (<ref>), 
it follows that the left initial datum 
at time τ 
 must be 

    v(x,τ)=u_l(τ) 
     =  A,    ∀ x<0,
 
thus proving (<ref>).


The two conditions 

equalities (<ref>), 
(<ref>) 
and the definition (<ref>)
imply that 
the restriction of ζ 
to [0, τ[
is a (classical) genuine characteristic both for u
and u^* 
with slope f'_l(  A)>0.
Therefore we can conclude that
ζ 
satisfies the condition of an AB-gic
also for u^*
on the interval [0, τ(x)],
and hence on the whole interval [0,T] by the analysis in the point 2.
This completes the proof that ζ∈𝒞(u^*, x) ∩𝒞(u, x) when τ≥τ_-( R).
 






















 4. 
 Assume that τ<τ_-( R),
 
 with τ_-( R) as in (<ref>).
If we suppose that (<ref>) 
holds, since (<ref>) is still verified we can deduce as above 
that (<ref>) holds as well,
and then we conclude that ζ∈𝒞(u^*, x) ∩𝒞(u, x)
with the same arguments of point  3.





Therefore, let us assume that u_l(τ) ≤θ_l and that ζ(t) > 0
for all t∈ [0, τ[ . 
Observe that because of (<ref>),
and by definition (<ref>)
of the set E, we have 

    u_r(τ)<θ_r.

Then, relying on (<ref>),
we deduce with the same blow up argument of above that 

    u_r(τ) 
        =  B .

On the other hand, if we show that

    τ≤τ,

it would follow from (<ref>) that

    u^*_r(τ)=  B .

The two conditions (<ref>),
(<ref>) 
and the definition (<ref>)
imply that 
the restriction of ζ 
to [0, τ[
is a (classical) genuine characteristic both for u
and u^* 
with slope f'_r(  B)<0.
Therefore, if (<ref>)
holds, we can conclude that
ζ 
satisfies the condition of an AB-gic
also for u^*
on the interval [0, τ(x)],
and hence on the whole interval [0,T] by the analysis in the point 2.
Hence, in order to completes the proof that ζ∈𝒞(u^*, x) ∩𝒞(u, x) when τ<τ_-( R),
it remains to establish (<ref>).











By contradiction, assume that τ > τ. Define the curve


    ξ(t)  ≐ inf{ R >0   |  x-t   f_r^'(u(x,t)) ≥ 0   ∀ R>0}     t ∈ [τ, T].

Notice that ξ(τ)  =0 and that ξ(T) =  R. But now using a comparison argument between ξ(t) and the map γ(t) 
defining the shock curve at point 2,

we obtain as in <cit.>
that ξ(T)  <γ(T)= R, 
which gives a contradiction. This concludes the proof of the proposition.
    
 


























































































    





































































































































































  















































































    









In the following we introduce 

a functional that, 
for any given function v(x,t), 
measures the total amount of flux of the vector field (v(x,t), f(v(x,t),x) passing through a curve
t↦ (α(t),t), from each side
of the curve. 

Given a function v ∈𝐋^∞(ℝ× [0,T]  ;  ℝ)   that admits one-sided limits v(x±,t) at every point (t,x) ∈ ]0, T] ×ℝ, and  α∈Lip([0,T]  ;  ℝ),

we define

    ℱ_t(v,α±) ≐∫_t^T {f(v(α(t) ±,t),α(t)±) - α̇(t)v(α(t) ±,t)} t,     t∈ [0,T],

where f(u,x) is the flux (<ref>).
We also set 

    ℱ(v,α±) ≐ℱ_0(v,α±).




Notice that if u is an AB-entropy  solution of (<ref>), 
since u is in particular a distributional solution of (<ref>) on × ]0,+∞[,
it follows that 
for any curve α∈Lip
([0,T]  ;  ℝ), 

the Rankine-Hugoniot conditions yield,
for a.e. t∈ [0,T], the equality

    f(u(α(t)-,t), α(t)-)-α̇(t)u(α(t)-,t) = f(u(α(t)+,t), α(t)+)-α̇(t)u(α(t)+,t).




















Therefore, in this case we have
F(u, α+)=  F(u, α-). Hence, since there is no ambiguity, 
 whenever u is an AB-entropy  solution of (<ref>),
 we will simply write

    F_t(u, α) ≐ F_t(u, α+) ≡ F_t(u, α-)    ∀ t,      F(u, α) ≐ F(u, α+) ≡ F(u, α-).






Let u,u^* ∈𝐋^∞(ℝ× [0,T] ; ) be AB-entropy solutions to  (<ref>), and let ζ∈𝒞(u^*,x),
x ∈ℝ. Then, there holds

    ℱ_t(u,ζ) ≥ℱ_t(u^*,ζ),   ∀  t ∈ [0,T].

Moreover, one has 

    ℱ(u,ζ) = ℱ(u^*,ζ)
          ⟺  ζ∈ C (u^*,x) ∩ C(u,x).





Let 0≤τ_1≤τ_2≤ T,
be the partition of [0,T] 
for ζ∈𝒞(u^*,x) given
by Remark <ref>,
so that there holds

    ζ(t)=0,     f_l(u^*_l(t))=f_r(u^*_r(t))=γ     ∀ t∈ [τ_1, τ_2].

To fix the ideas we assume 
that 

    ζ(t)<0    ∀ t∈ [0, τ_1[ ,      ζ(t)>0    ∀ t∈ ]τ_2, T].

The cases where  ζ(t)>0 for all t∈ [0, τ_1[ , or ζ(t)<0
for all t∈ ]τ_2, T] are entirely similar.
Then, setting

    g(t)   ≐
            f_r(u(ζ(t)+,t))-ζ̇(t)  u(ζ(t)+,t)
                 if   t∈ ]τ_2, T],
            
    
            f_r(u_r(t))
                 if   t∈ [τ_1, τ_2],
            
    
            f_l(u(ζ(t)+,t))-ζ̇(t)  u(ζ(t)+,t)
                 if   t∈ [0,τ_1[ ,
        
        
    
        g^*(t)   ≐
            f_r(u^*(ζ(t),t))-ζ̇(t)  u^*(ζ(t),t)
                 if   t∈ ]τ_2, T],
            
     γ     if   t∈ [τ_1, τ_2],
            
    
            f_l(u^*(ζ(t),t))-ζ̇(t)  u^*(ζ(t),t)
                 if   t∈ [0,τ_1[ ,

we write 

    ℱ_t(u,ζ)=∫_t^T g(t) t,
              ℱ_t(u^*,ζ)=∫_t^T g^*(t) t,    ∀ t∈ [0,T].

Note that, because of (<ref>), in the definition of g we may equivalently 
take u(ζ(t)-,t) instead of u(ζ(t)+,t), while in the definition of g^* we take u^* continuous at (ζ(t),t)
since ζ is a classical genuine characteristic for u^*
when t∈ ]0,τ_1[ ∪ ]τ_2, T[ .
Observe that, since u is an AB-entropy solutions to (<ref>),
by the interface condition (<ref>) we have

    f_r(u_r(t))≥γ    ∀ t∈ [τ_1, τ_2].

On the other hand, because of the convexity of f_r there holds

    f_r(v) - f_r^'(w) v ≥ f_r(w) - f_r^'(w) w,   ∀  v,w ∈ℝ.

Moreover, since ζ is an AB-gic for u^*, and because of (<ref>), 
note that the restriction of ζ
to [0,τ_1[ is a classical genuine characteristic for u^* as solution of u_t+f_l(u)_x=0, 
and the restriction of ζ
to ]τ_2, T] is a classical genuine characteristic for u^* as solution of u_t+f_r(u)_x=0.
Hence, it follows that

    ζ̇(t)=
        
     f'_l(u^*(ζ(t),t))     if    t ∈ ]0,τ_1[ ,
     
     
     0    if    t ∈ ]τ_1, τ_2[ ,
      
     
    
     f'_r(u^*(ζ(t),t))     if    t ∈ ]τ_2,T[ .

Thus, (<ref>)-(<ref>) together imply

    f_l(u(ζ(t)+,t))-ζ̇(t)  u(ζ(t)+,t)   ≥ f_l(u^*(ζ(t),t))-
        ζ̇(t) 
        u^*(ζ(t),t)
            ∀ t∈ ]0, τ_1[ ,
        
    
        f_r(u(ζ(t)+,t))-ζ̇(t)  u(ζ(t)+,t)   ≥ f_r(u^*(ζ(t),t))-
        ζ̇(t) 
        u^*(ζ(t),t)
            ∀ t∈ ]τ_2,T[ .

Therefore, from (<ref>),
(<ref>) we deduce that

    g(t)≥ g^*(t)    ∀ t∈ [0,T],

which, because of (<ref>), yields (<ref>).

Concerning (<ref>),  if ζ∈ C (u^*,x) ∩ C(u,x), then the inequality (<ref>) is verified 
also when u and u^* switch their places, so that we have 
ℱ_t(u,ζ) ≥ℱ_t(u^*,ζ) for all t∈ [0,T],
thus proving

    ζ∈ C (u^*,x) ∩ C(u,x)
          ⟹  ℱ(u,ζ) = ℱ(u^*,ζ).

Next, given ζ∈𝒞(u^*,x),
assume that 

    ℱ(u,ζ) = ℱ(u^*,ζ).

Since, by the above analysis we have (<ref>), it follows from 
(<ref>) that g(t)=g^*(t)
for a.e. t∈ [0,T]. Because of (<ref>), (<ref>),  this in particular implies that, for a.e. t∈ [0,T], there holds

    f_r(u_r(t))=γ    if   t∈ [τ_1,τ_2] ,

and

    f_l(u(ζ(t)+,t))-f'_l(u^*(ζ(t),t)) 
        u(ζ(t)+,t)=
            
             =
            f_l(u^*(ζ(t),t))-f'_l(u^*(ζ(t),t)) 
        u^*(ζ(t),t)
                if   
            t∈ ]0,τ_1[ ,
        
       f_r(u(ζ(t)+,t))-f'_r(u^*(ζ(t),t)) 
            u(ζ(t)+,t)=
            
              =
            f_r(u^*(ζ(t),t))-f'_r(u^*(ζ(t),t)) 
            u^*(ζ(t),t)
                if   
            t∈ ]τ_2,T[ .

Since f_l, f_r are strictly convex functions, we deduce from (<ref>) that
u(ζ(t)+,t)=u^*(ζ(t),t)
for a.e. t∈ ]0,τ_1[ ∪ ]τ_2, T[ . If we repeat the same analysis taking  u(ζ(t)-,t) instead of u(ζ(t)+,t) in the definition (<ref>) of g, we find that also u(ζ(t)-,t)=u^*(ζ(t),t)
for a.e. t∈ ]0,τ_1[ ∪ ]τ_2, T[ . This shows that 
the restriction of
ζ to [0,τ_1[ 
and to ]τ_2, T] is a classical genuine characteristic for u as well,
as 
solution of u_t+f_l(u)_x=0, 
and  of u_t+f_r(u)_x=0, respectively.
Hence, by Remark <ref>
we deduce that ζ is an AB-igc
also for u, which means that ζ∈𝒞(u,x), completing the proof of

    ℱ(u,ζ) = ℱ(u^*,ζ)
          ⟹  ζ∈ C (u^*,x) ∩ C(u,x),

and thus concluding the proof of the Lemma.
















§ PROOF OF THEOREM <REF>

In this section we provide a proof of the initial data identification Theorem <ref>. To this end we first state some technical Lemmas that 
we are going to use repeatedly in the following. 




Assume u is a bounded weak solution to (<ref>), (<ref>) that admits left and right limits u(t, x±) at every point. Let t_1,t_2 ∈ [0,T] with t_1<t_2 and let α,β:[t_1,t_2] →ℝ be two Lipschitz maps with α≤β. Then it holds

    ∫_α(t_2)^β(t_2) u(t_2,x) x-∫_α(t_1)^β(t_1) u(t_1,x) x  
     = ∫_t_1^t_2f(u(t,α(t)-), α(t)-)-α̇(t)u(t,α(t)-)  x -  
    ∫_t_1^t_2f(u(t,β(t)+),β(t)+)-β̇(t)u(t,β(t)+)  x




   The Lemma is a consequence of the divergence theorem.





Fix ω∈𝒜^AB(T). Let 

    u_0^* ·=ω,    u^*(·,t) ·= u_0^*

Then 

    ℝ = ⋃_x ∈ℝ C_0(x, u^*)



   The Lemma follows immediately from a direct inspection of the solution u^* as constructed explicitly in <cit.>.











We prove that the conditions (<ref>), (<ref>)
are necessary: assume that for a solution u we have u(·, T) = ω = u^*(·,T) and u =  u_0(·). Take x̅∈ C(ω). Now, using Lemma <ref>, choose ζ_x̅∈𝒞(u^*,x̅) ∩𝒞(u,x̅) and set y̅·=ζ_x̅(0).
Choose any y < min𝒞_0(u^*,x̅).  From the structure of u^* (in particular by Lemma <ref> and by Proposition <ref>-(iv), we know that there is some x< x̅ such that y = ζ_x(0), ζ_x ∈𝒞(u^*,x). From Lemma <ref>, we deduce that 

    ℱ(u,ζ_x) ≥ℱ(u^*, ζ_x)
 
Moreover, since  ζ_x̅∈𝒞(u^*,x̅) ∩𝒞(u,x̅),  by the second part of Lemma <ref> we have

    ℱ(u^*,ζ_x̅) = ℱ(u, ζ_x̅)

Applying Lemma <ref> to the solution u^* with α = ζ_x, β = ζ_x̅ one obtains

    ∫_x^x̅ω(x)  x - ∫_y^y̅ u_0^*(ξ) ξ = ℱ(u^*,ζ_x)-ℱ(u^*,ζ_x̅)

and, again, thanks to Lemma <ref> and (<ref>), (<ref>),

    ∫_x^x̅ω(x)  x-∫_y^y̅ u_0(ξ) ξ = ℱ(u, ζ_x) -ℱ(u, ζ_x̅)≥ℱ(u^*,ζ_x)-ℱ(u^*,ζ_x̅)

Taking the difference of the two above equations one gets 
(<ref>). The proof of (<ref>) is entirely symmetric and therefore is omitted. 














Now we prove that if u_0 ∈𝐋^∞(ℝ) satisfies (<ref>), (<ref>), then 𝒮_T^ABu_0 = ω. 
What we are going to prove is the following:

    ∫_x_1^x_2ω(x) -   u_0(x)  x = 0,   ∀  x_1< x_2,    x_i ∈ C(ω),   i = 1,2

Since C(ω) is dense in ℝ, by the Lebesgue differentiation theorem this is enough to conclude that ω and 𝒮_T^ABu_0 coincide as elements of 𝐋^∞(ℝ). 
We split the proof into two parts, in which we prove respectively the two opposite inequalities needed to obtain (<ref>).  We start with proving

    ∫_x_1^x_2ω(x) -  u_0(x)  x ≥ 0,   ∀  x_1< x_2,    x_i ∈ L(ω),   i = 1,2



Take any two points x_1< x_2. We have two cases




Case 1. max𝒞_0(u,x_1) ≥min𝒞_0(u^*,x_2) (see Figure <ref>, right). Then we can choose ζ_1 ∈𝒞(u,x_1) and ζ_2 ∈𝒞(u^*,x_2) such that ζ_2(0) ≤ζ_1(0). By continuity there is a point τ∈ [0,T) in which ζ_1(τ) = ζ_2(τ). Apply Lemma <ref> to the solution u^* and the curves ζ_i_| [τ,T], i = 1,2 and Lemma <ref> to obtain

    ∫_x_1^x_2ω(x)  x = ℱ_τ(u^*,ζ_1)- ℱ_τ(u^*,ζ_2) ≥ℱ_τ(u,ζ_1)- ℱ_τ(u^*,ζ_2)

Apply again Lemma <ref> to the solution u and the curves ζ_i_| [τ,T], i = 1,2, and then Lemma <ref>, to obtain

    ∫_x_1^x_2 u_0(x)  x = ℱ_τ(u,ζ_1)- ℱ_τ(u,ζ_2) ≤ℱ_τ(u,ζ_1)- ℱ_τ(u^*,ζ_2)

Taking the difference of the two above inequalities, one obtains (<ref>). Note that in this case we did not use (<ref>), (<ref>).



Case 2. max𝒞_0(u,x_1) < min𝒞_0(u^*,x_2) (see Figure <ref>, left). Choose any ζ_1 ∈𝒞_0(u,x_1), and set y ·=ζ_1(0). Since y <min𝒞_0(u^*,x_2), the first condition of the Theorem gives us a y_2 ∈𝒞_0(u^*,x_2)(and then a ζ_2 ∈𝒞(u^*,x_2) such that ζ_2(0) = y_2) such that

    ∫_y^y_2 u_0(x)  x ≤∫_y^y_2u^*_0(x)  x

Apply Lemma <ref> to u^* with the curves ζ_i, i = 1,2 and Lemma <ref> to obtain

    ∫_x_1^x_2ω(x)  x = ℱ(u^*,ζ_1)- ℱ(u^*,ζ_2) + ∫_y^y_2u^*_0(x)  x 
    ≥ℱ(u,ζ_1)- ℱ(u^*,ζ_2)+ ∫_y^y_2u^*_0(x)  x

Apply Lemma <ref> to u with the curves ζ_i, i = 1,2 and Lemma <ref> to obtain

    ∫_x_1^x_2 u_0(x)  x = ℱ(u,ζ_1)- ℱ_τ(u,ζ_2)+∫_y^y_2u_0(x)  x 
    ≤ℱ(u,ζ_1)- ℱ(u^*,ζ_2)+∫_y^y_2u_0(x)  x

Taking the difference of the two above equations and using (<ref>), one obtains

    ∫_x_1^x_2ω(x)  x-∫_x_1^x_2 u_0(x)  x ≥∫_y^y_2u^*_0(x)  x-∫_y^y_2u_0(x)  x ≥ 0

and this proves (<ref>) also in Case 2.



The proof of the opposite inequality is entirely symmetric and is accordingly omitted. 










Let us apply the Theorem for a couple of simple profiles ω.  Let's start with

    ω(x) = 
    v_1     x < 0,
    
    v_2     x >0,

with v_1 < A and v_2= π_r,-^l(v_1). 
The solution u^* given by u^*(t) =  S_t^AB∘ S_-T^ABω in this case is

    u^*(x,t) = 
    v_1     x < 0, 
    
    v_2     x >0,

Of course, for every x> 0, 𝒞(u^*,x) is a singleton and it's unique element is precisely the unique backward characteristic with slope f_r^'∘π_r,-^l(v_1) = f_r^'(v_2). The same holds for x <L. Thanks to the fact that v_1 < A, we can say that for every L<x <0 
𝒞(u^*,x) is a singleton (and is precisely the final point of the backward genuine characteristic from x in the classical sense) so that the initial data in this case is uniquely determined.










Now we take v_1 = A, v_2 = π_r,-^l(v_1)= B̅ (see Figure <ref>). In this particular case, the initial datum is not uniquely determined. With the same arguments as above, we know that the initial datum is uniquely determined in the regions x<0 and x > -Tf_r^'(B̅). This is true essentially because 𝒞(u^*,x) is a singleton and coincides with the unique genuine backward characteristic for every x <L and x >0. Now we want to find out what 𝒞_0(u^*,x) is for x ∈ (L,0). Fix such an x. It is clear that 

    𝒞_0(u^*,x) = [0,θ_x(0)]
 with θ_x(0) being 

    θ_x(0) = -t_x f_r^'(B̅)

where t_x = T-x/f_l^'(A).
Then, with some explicit computations, what the Theorem says is that u_0 ∈ L^∞(ℝ) is such that 𝒮_Tu_0 = ω if and only if

    u_0(x) = 
    A     x <0, 
    
    v_0(x)     0<x<-Tf_r^'(B̅),
    B̅    -Tf_r^'(B̅) < x,

with v_0 ∈ L^∞(0,-Tf_r^'(B̅)) such that 

    ∫_y^-Tf_r^'(B̅) v_0(x)  x ≤B̅(y+Tf_r^'(B̅)),    y ∈ (0,Tf_r^'(B̅)),


    ∫_0^y v_0(x)  x ≥B̅y,    y ∈ (0,Tf_r^'(B̅)),

Notice that if B = B̅ = θ_r, the initial data is uniquely identified also in this second case. 








§ PROOF OF THEOREM <REF>



We prove the Theorem point by point, in order. 










Proof of (𝐢).
Follows immediately from the semigroup properties of 𝒮_T^AB. 




Proof of (𝐢𝐢).
The fact that ℐ_T^ABω is an F_σ set follows from (i). 
We prove that ℐ_T^ABω has empty interior. 

Take two points 0<x_1<x_2 in 𝒞(ω) such that the unique backward (genuine) characteristics, respectively θ_x_1, θ_x_2, from x_1, x_2, do not interact with the interface. We can also choose them in such a way that θ_x_1(0) < θ_x_2(0). Notice that in this case 𝒞_0(u^*,x_i) = {θ_x_i(0)}. Apply condition (<ref>) with x̅ = x_2, y = θ_x_1(0) to find that any element u_0 in ℐ_T^ABω satisfies

    ∫_θ_x_1(0)^θ_x_2(0)u_0(x)  x ≤∫_θ_x_1(0)^θ_x_2(0)u_0^*(x)  x

Now apply condition (<ref>) with x̅ = x_1, y = θ_x_2(0) to find that any element in ℐ_T^ABω satisfies

    ∫_θ_x_1(0)^θ_x_2(0)u_0(x)  x ≥∫_θ_x_1(0)^θ_x_2(0)u_0^*(x)  x

Inequalities (<ref>), (<ref>) together imply that every element u_0 ∈ℐ_T^ABω satisfies

    ∫_θ_x_1(0)^θ_x_2(0)u_0(x)  x = ∫_θ_x_1(0)^θ_x_2(0)u_0^*(x)  x

From condition (<ref>) one clearly sees that the set ℐ_T^ABω has always empty interior (since it shows that it is contained in an hyperplane of L^∞).









Proof of (1).
First assume that |𝒞_0(u^*,x)| = 1 for every x ∈ℝ. We will prove that any initial data u_0 that leads to ω satisfies

    ∫_y_1^y_2 u_0(x)  x = ∫_y_1^y_2 u_0^* (x)  x,   a.e.  y_1 < y_2 ∈ℝ

and this uniquely identifies u_0 as an element of L^∞(ℝ), therefore ℐ^AB_Tω is a singleton.
To prove this, notice that |𝒞_0(u^*,x)| = 1 for every x ∈ℝ immediately implies that ω is continuous outside the origin. Take any two points y_1 < y_2 ∈ℝ. By Lemma <ref>  there exist x_1, x_2 ∈ℝ such that ζ_2 ∈𝒞(u^*,x_2) such that ζ_2(0) = y_2, and by our assumption x_2 is a continuity point of ω (or zero) and 𝒞(u^*,x_2) is a singleton. Without loss of generality we can assume that neither y_1 nor y_2 are in 𝒞_0(u^*,0) (in fact C_0(u^*, 0) has zero measure since it is a singleton by assumption). Then, by (<ref>),

    ∫_y_1^y_2 u_0(x)  x ≤∫_y_1^y_2 u_0^* (x)  x

Now we exchange the role of y_1 and y_2. In particular, with the symmetrical argument (this time using (<ref>)) we find the opposite inequality

    ∫_y_1^y_2 u_0(x)  x ≥∫_y_1^y_2 u_0^* (x)  x

and this proves (<ref>).

Conversely, assume that ℐ_T^ABω = {u_0^*} is a singleton. By contradiction assume there is some x̃∈ℝ such that |𝒞_0(u^*,x̃)| ≠ 1. In order to obtain a contradiction we show that there exists an initial datum u_0 ≠ u^*_0 such that S_T^AB u_0 = ω. We do it using the characterization of Theorem <ref>.   Consider an initial datum like  u_0^* + v_0,  v_0 ∈V_0 ⊂𝐋^∞(conv  C_0(u^*, x̃)) defined as follows: v_0 ∈V_0 if and only if 

    ∫_y^max C_0(u^*, x̃) v_0(x)  x ≤ 0     ∀  y ∈conv  C_0(u^*, x̃),
    ∫_min C_0(u^*, x̃)^y v_0(x)  x ≥ 0,     ∀  y ∈conv  C_0(u^*, x̃).















The set V_0 is clearly an infinite dimensional cone. We show that 

    V ≐ u_0^* + V_0 ⊂ I^AB_Tω
 
by using Theorem <ref>. In particular, taking any u_0 ∈ V, we have to show that for every x̅ there exists y̅∈ C_0(u^*, x̅) such that (<ref>), (<ref>) hold. We verify only (<ref>), the other being entirely symmetric. 


Therefore fix any x̅, and assume first that x̅ < x̃. We choose y̅ = min C_0(u^*, x̅). Then we have, for every y < min C_0(u^*, x̅),

    ∫_y^y̅ u_0(x)  x = ∫_y^y̅ u_0^*(x)  x

because, since x̅ < x̃, we have 

    y̅ = min C_0(u^*, x̅) ≤min C_0(u^*, x̃)

so that trivially, by definition of u_0, it holds u_0 = u_0^* in every interval [y, y̅]. 
Now let x̅ > x̃, and choose y̅ = max C_0(u^*, x̅). Then we have, for every y < min C_0(u^*, x̅), that


    
  * if y ∈ (max C_0(u^*, x̃), min C_0(u^*, x̅)), then 

    ∫_y^y̅ u_0(x)  x = ∫_y^y̅ u_0^*(x)  x

because u_0 and u_0^* coincide in the interval [y, y̅];

  * if y ∈conv  C_0(u^*, x̃), then, by (<ref>),

    ∫_y^y̅ u_0(x)  x = ∫_y^max C_0(u^*, x̃) u_0(x)  x + ∫ _max C_0(u^*, x̃)^y̅ u^*_0(x)  x ≤∫_y^y̅ u_0^*(x)  x


  * if y < min C_0(u^*, x̃), notice first that by (<ref>) it holds in particular 

    ∫_min C_0(u^*, x̃)^max C_0(u^*, x̃) u_0(x)  x = ∫_min C_0(u^*, x̃)^max C_0(u^*, x̃) u_0^*(x)  x

therefore it holds

    ∫_y^y̅u_0(x)  x = ∫_y^min C_0(u^*, x̃) u_0^*(x)  x + ∫_min C_0(u^*, x̃)^max C_0(u^*, x̃) u_0(x)  x + ∫_max C_0(u^*, x̃)^y̅ u_0^*(x)  x = ∫_y^y̅u^*_0(x)  x

as wanted.

This proves (1).



Proof of (2).
To prove property (2), notice that for every u_0 ∈ℐ_T^ABω and λ≥ 0 it holds u_0^* + λ(u_0 -u_0^*) ∈ℐ_T^ABω. To see this, it's sufficient to prove that (<ref>), (<ref>) of Theorem <ref> hold. Take x̅∈𝒞(ω). Since u_0 ∈ℐ_T^ABω there is some y̅ such that (<ref>) holds. Then one has 

    ∫_y^y̅ u_0^*+λ(u_0-u_0^*)  x ≤∫_y^y̅ u_0^*+λ(u_0^*-u_0^*)  x = ∫_y^y̅ u_0^*  x

as wanted. The proof that also (<ref>) holds is entirely symmetric. 

Now we prove that u_0^* is an extremal point. Take any x̅∈𝒞(ω) such that  𝒞_0(u^*, x̅) is a singleton and call its unique element y̅. Assume by contradiction that there exists a λ∈ (0,1) such that u^*_0 = λ u_0^1 + (1-λ) u_0^2 with u_0^* ≠ u_0^i ∈ℐ_T^ABω, i = 1,2. In particular it holds

    λ∫_y̅^y u_0^1  x+ (1-λ) ∫_y̅^y u_0^2  x = ∫_y̅^y u_0^*  x,     ∀  y ∈ℝ

Then, since u_0^1, u_0^2 are different from u_0^*, there must be some y ∈ℝ such that 

    ∫_y̅^y u_0^1  x ≠∫_y̅^y u_0^*  x     and    ∫_y̅^y u_0^2  x ≠∫_y̅^y u_0^*  x.

Assume that y > y̅. Then by condition (<ref>) of Theorem <ref>, 

    ∫_y̅^y u_0^1  x > ∫_y̅^y u_0^*  x     and    ∫_y̅^y u_0^2  x  > ∫_y̅^y u_0^*  x

but this is in contradiction with (<ref>). If instead y < y̅, by condition (<ref>) of Theorem <ref>, 

    ∫_y̅^y u_0^1  x < ∫_y̅^y u_0^*  x     and    ∫_y̅^y u_0^2  x< ∫_y̅^y u_0^*  x

that is, again, a contradiction with (<ref>). This proves that u_0^* is an extremal point (and of course it is unique).



Proof of (3).
Let u_0^1, u_0^2 ∈ℐ_T^ABω and θ∈ (0,1). Let x̅∈𝒞(ω). By hypothesis there exist y̅_1, y̅_2 ∈𝒞_0(u^*,x̅) such that

    ∫_y^y̅_1 u_0^1(x)  x ≤∫_y^y̅_1u_0^*(x)  x ,     ∫_y^y̅_2 u_0^2(x)  x ≤∫_y^y̅_2u_0^*(x)  x,   ∀  y < min𝒞_0(u^*,x̅)

At this point (and only at this point) we use that 𝒞_0(u^*,x̅) is a singleton
if we assume that condition (<ref>) holds, deducing that y̅ _1 = y̅_2. Then, for y̅ = y̅_1 = y̅_2, using (<ref>),

    ∫_y^y̅θ u_0^1(x) + (1-θ)u_0^2(x) x ≤∫_y^y̅ u^*(x)  x

so that θ u_0^1+(1-θ) u_0^2 satisfies (<ref>). The proof that also (<ref>) holds is entirely similar and is accordingly omitted.













To prove that ℐ_T^[AB](ω^T)
is convex if (<ref>) is verified, 
notice the following. 
For each initial datum u_0 ∈ℐ_T^ABω, and for each x̅, Theorem <ref> gives us a y̅ that works when plugged into (<ref>), (<ref>). The possibility to prove that ℐ_T^ABω is convex with same proof that we pursued above under condition (<ref>),

depends on the ability to choose y̅ depending only on x̅ and independently from the particular initial datum u_0. Of course whenever 𝒞_0(u^*, x̅) is a singleton the choice of y̅ is forced to be the only element of 𝒞_0(u^*, x̅), so that it depends only on x̅. The problem is that when 𝒞_0(u^*, x̅) is not a singleton, different initial data u_0 ∈ℐ_T^ABω might require different y̅∈𝒞_0(u^*, x̅). However, for such x̅, if (<ref>) holds, we are always able to choose y̅(x̅)  ∈𝒞_0(u^*, x̅) ∩cl(𝒳(ω)) independently from the initial datum u_0. In fact, in order to show that (<ref>), (<ref>) hold for x̅ and y̅(x̅), it is sufficient to write (<ref>), (<ref>) for a sequence of points x̅_n, {y̅_n} = 𝒞_0(u^*, x̅_n), with x̅_n ∈𝒳(ω) and x̅_n →x̅ for n →∞. Since 𝒞_0(u^*, x̅) is closed, the limit point of the sequence {y_n}_n ∈ℕ will be in 𝒞_0(u^*, x̅), and the proof is concluded. 








Now we give an example in which the set ℐ_T^ABω is not convex. We assume f_l, f_r = u^2/2. Let

    ω_2(x) = A̅    x ≤ L, 
    
    A     x ∈ (L,0), 
    
    v     x >0,

with v < B̅ negative enough. The strategy is to find two initial data u_0^1, u_0^2 ∈ℐ_T^ABω and show that for some θ∈ (0,1), θ u_0^1+ (1-θ) u_0^2 ∉ℐ_T^ABω. We know, with some calculations, that u̅_L = -A-2√(AL/T). In order to deal with easier calculations, we choose T =1 and A,L such that  u̅_L = 0. One finds that in this case it must be A = 4L. With this choice, it also holds A̅ = -4L = B and B̅ = 4L = A. We recall that for this profile ω, the "special" initial data u_0^*, that produces the solution u^*,  has the form (see Figure <ref>)

    u_0^*(x) = A̅    x < 5L, 
    
    L-x     5L < x < L, 
    A̅    L < x < 0, 
    B̅    0 < x < -4L, 
    
    -x     -4L < x < -v, 
    
    v     -v < x,

Now we define u_0^1(x) as (see Figure <ref>)

    u_0^1(x) = A̅    x < L, 
    
    A     L < x < 0, 
    
    B     0 < x < -λ(B,v), 
    
    v     -λ(B,v) < x,

We choose v negative enough in order to have -λ(B,v) >-4L.
Finally, let, with γ >1 (see Figure <ref>),

    u_0^2(x) = A̅    x < 5L, 
    
    L-x     5L < x < L, 
    γA̅    L < x < 0, 
    γB̅    0 < x < -L, 
    B̅    -L < x < -4L, 
    
    -x     -4L < x < -v, 
    
    v     -v < x,

It's easy to see that u_0^i ∈ℐ_T^ABω, i = 1,2 (see Figures <ref>, <ref>, <ref>). 

Fix any θ∈ (0,1) and let u^θ_0 ·=θ u_0^1+ (1-θ) u_0^2. We claim that if u_0^θ∈ℐ_T^ABω then there is some y̅∈ [L,-3L] such that (<ref>) holds with y = 5L.
In fact, consider a sequence of points x̅_n ↓ L. Since they are points of continuity for ω, for each n there exists y̅_n ∈𝒞_0(u^*,x̅_n) = [L,x̅_n-4L] such that (<ref>) holds with y = 5L and y̅ = y̅_n. Since the sequence y_n is bounded we can extract a converging subsequence, that we still call y_n. Clearly the limit point (call it y̅) of the sequence must be in [L,-3L] and for this y̅ it holds 

    ∫_5L^y̅ u_0^θ(x)  x ≤∫_5L^y̅ u_0^*(x) x


Now we assume that u^θ_0 ∈ℐ_T^ABω and we see that this is in contradiction with what we have just proved. In particular we prove that 

    ∫_5L^y̅ u_0^θ(x)- u_0^*(x)  x >0,   ∀y̅∈ [L,-3L]


First assume that y̅∈ [L,0]. With some easy calculations we find

    ∫_5L^y̅ u_0^1(x)  x = 12L^2+4Ly̅,   ∫_5L^y̅ u_0^2(x)  x  = 8L^2-4Lγ(y̅-L)

and 

    ∫_5L^y̅ u_0^*(x)  x = 8L^2 -4L(y̅ -L)

so that for every y̅∈ [L,0],

    ∫_5L^y̅ u_0^θ-u_0^*  x =  θ 8L y̅+ (1-θ)[4L(L-y̅)(γ-1)] > 0.


Analogously, for every y̅∈ [0,-L],

    ∫_5L^y̅ u_0^θ-u_0^*  x =  θ( -8L y̅)+ (1-θ)[4L(L-y̅ )(γ-1)] > 0.

Finally, if y̅∈ [-L,-3L],

    ∫_5L^y̅ u_0^θ-u_0^*  x =  θ[ 8L^2 -8L(y̅ + L)]> 0.

and this is a contradiction with the fact that u^θ_0 ∈ℐ_T^ABω, proving that ℐ_T^ABω is not, in general, convex. 



































 §.§ Examples

Here we provide some examples and figures that hopefully will help to better understand how the sets 𝒞(u, x) behave and their structure. In order to do this, we use a relatively simple profile ω, but that in our opinion already captures the essence and the key points of Definition <ref>. In particular we let

    ω_3(x) = 
    v     x < L, 
    
    A     L < x < 0, 
    B̅    0 < x

with

    f_l^'(A) < L/T < f_l^'(u̅_l) < f_l^'(v)

so that ω is attainable (see the previous chapter) but it is not attainable with a locally Lipschitz solution in {x <0}, {x > 0}. 


    
  * In Figure <ref> is represented the solution u^* given by u^*(·, T) =  S^AB_t∘ S_-T^ABω.
In particular there is a compression wave that creates the shock at (L,0), a rarefaction wave that meets a shock with right state equal to A, and the left and right traces u^*_l, u^*_r are always equal to A and B̅ respectively. 
Now consider a point x̅∈ (L, 0). There is a unique backward genuine characteristic with slope f_l^'(A) until it meets the interface. At that point we can prolong it on the other side with slope f_r^'(B̅), but another possible choice is to travel along the interface for some time and then go the right at any time, or go to the left at times t ≤ t_u̅_L (the time at which the shock emerges from the interface). Therefore we will have distinct minimal and maximal lines in the set 𝒞(u^*, x̅), represented by the blue lines in Figure <ref>, while all the other blue dashed lines are the other elements in 𝒞(u^*, x̅).


  * If we consider different initial data u_0 ∈ℐ_T^ABω, we produce different solutions u with different sets 𝒞(u, x̅) and 𝒞_0(u, x̅). Consider for example a solution like u^1 as in Figure <ref>. The red line at the left of the interface is a shock with left state v and right state A. Of course such a solution is admissible if and only if L-Tλ_l(v,A) ≤ 0. For simplicity assume this condition holds (it will be sufficient to take v big enough). In this case the set C(u^1, x̅) is smaller since the left trace u_l is always equal to A. 


  * In the case of a solution like  u^2 (see Figure <ref>), it is easy to see that the set 𝒞_0(u^2, x̅) is not an interval. In fact, at time t_u̅_L the traces change their value due to two rarefaction waves arriving at the interface from both sides. 


  * Finally, we can also have solutions (for example u^3 of Figure <ref>) for which max𝒞_0(u^*, x̅) ≠max𝒞_0(u^3, x̅). Here a shock emerges from a point of the interface and is reabsorbed by the interface itself after an interval of time.
































plain






 Paper Title: An inhomogeneous p-laplacian equation with a Hardy potential.



























#1#2#1_4pt #2
Ł#1L^#1(Ω)




Department of Mathematics, Texas A&M University - San Antonio

In this work we study the existence and regularity of solutions to the following equation: 
    u + g(x) u = /|x|^p |u|^p-2u + f,
 where 1< p < N and f∈Ł m, where m≥ 1.





[MSC Classification]35J92, 35J15,35B65,35A01

An inhomogeneous p-laplacian equation with a Hardy potential
    Genival da Silva[email: gdasilva@tamusa.edu, website: <www.gdasilvajr.com>]
    August 5, 2024
===============================================================================



§ INTRODUCTION

In a recent paper <cit.>, building up on the previous work <cit.>, the authors study the existence and summability of the problem:

    -(M(x)u)+ g(x)u= /|x|^pu + f        Ω,
    
    u (x) = 0    ∂Ω,

where M(x) is positive matrix such that M(x)ξ·ξ> |ξ|^2, g(x)≥ 0, 0<< (N-2/2)^2 and f∈Ł 1.

In these notes we will study the equivalent problem for the p-Laplacian, namely:

    - u + g(x)u= /|x|^p |u|^p-2u + f        Ω,
    
    u (x) = 0    ∂Ω,

for 1<p<N. The crucial component of our analysis is the Hardy inequality:



(Hardy's inequality)
If v∈ and 1<p<N then

    ^p|v|^p/|x|^p≤ |Dv|^p,

where the constant =( N-p/p) is optimal.

The idea of all the proofs is essentially the same: truncate and obtain boundedness estimates. It's remarkable that we can apply the same methods of the linear case to this quasilinear case, despite the nonlinearities. 

The upshot is that Hardy's inequality provides a way to control the nonlinearity in terms of the Sobolev norm, which is essentially what we are trying to estimate.

The main challenges of problem (<ref>) are the low summability of the source f which puts the right hand side outside the dual of , and the nonlinearity with superlinear growth in case p>2 (for more on this type of problem see <cit.>), which poses an obstacle to existence and high regularity.

The idea of truncating an equation in order to obtain estimates is not new, it was consolidated by Stampacchia <cit.>, who proved an equivalent of Calderon-Zygmund L^p-estimates for equations with discontinuous coefficients. However, the method is not invincible, in some cases the existence of solutions is not guaranteed, even for bounded sources; nevertheless, in these notes this method will be successful due to the pivotal hole of Hardy's inequality in obtaining the estimates.

The paper is organized as follows: In section <ref>, we analyze the case with no lower order term, where we analyze two scenarios: high summability of the source, meaning f is in the dual of ; and low summability of the source, where we take the source in Ł m with m>1. In section <ref> we study the same problem with a added lower order term, we are able to obtain under certain hypothesis existence and regularity of the solutions.



 §.§ Notation



  - ⊂^N is a bounded domain.

  - The space  denotes the usual Sobolev space which is the closure of 𝒞^∞_0(), smooth functions with compact support, in the p-norm. 

  - For 1<p<∞, the p-Laplacian Δ_p is given by - (|Du|^p-2Du). 

  - For q>0, q' denotes the Holder conjugate, i.e. 1/q+1/q'=1, and q^* denotes the Sobolev conjugate, defined by q^*=qN/N-q>q, where N is the dimension of the domain ⊂. 

  - We will use the somewhat standard notation for Stampacchia's truncation functions (see <cit.>):

    T_k(s)=max{-k,min{s,k}},    G_k(s)=s-T_k(s),    k>0.


  - The letter C will always denote a positive constant which may vary from line to line. 

  - The Lebesgue measure of a set A⊆^N is denoted by |A|.

  - The symbol ⇀ denotes weak convergence. 

  - The letter  denotes the best constant in Sobolev's inequality up^*≤Dup, see <cit.> for its value.




§ THE CASE G(X)=0 AND P≥ 2.

In this section we analyze the Dirichlet problem:

    - u= /|x|^p |u|^p-2u + f        Ω,
    
    u (x) = 0    ∂Ω,

when p≥ 2, 0<< ^p and f∈Ł m for a suitable m≥1.

We  begin by considering the “truncated system” system:

    -(|D|^p-2D)  = /|x|^p+1/n ||^p-2 + f_n

where f_n=T_n(f). The classical theory of Leray-Lions operators  guarantee the existence of a unique solution ∈∩Ł∞.



 §.§ High summability: f∈Łm with m≥ (p^*)'.




Suppose f∈Ł m with (p^*)'≤ m<N/p and :

    0< < (m-1)N(N-mp)^p-1/(p-1)^p-1m^p

Then the Dirichlet problem (<ref>) has a solution u∈∩Ł s, where s=mN/N-pm.


 Take =1/-p+1||^-p as a test function in (<ref>), where ≥ p is a number to be chosen. If =p, we can easily see that Dp≤ C, so  is bounded in  and hence u up to a subsequence. On the other hand, if >p we have:

    |D |^p||^-p≤/-p+1||^/|x|^p+1/-p+1 |f(x)||u_n|^-p+1.

Rearranging using Hardy's inequality (<ref>) with v=||^/p

    p^p/^p |D ||^/p |^p≤/^p(-p+1)  |D ||^/p |^p+1/-p+1 |f(x)||u_n|^-p+1.

That is,

    (p^p/^p -  /^p(-p+1)) |D ||^/p |^p≤1/-p+1fm( ||^(-p+1)m')^1/m'

where we have used Holder's inequality in the last integral. 

Choosing  such that p^*/p=(-p+1)m'=(p-1)s, that is

    =(p-1)m(N-p)/N-pm,

we obtain:

    (p^p/^p -  /^p(-p+1))(  ||^ p^*/p)^p/p^*-1/m'=1/s≤1/-p+1fm

In order for this to be meaningful, we must require ≥ p and p^p/^p > /^p(-p+1), that is:

    (p^*)'≤ m<N/p and 0< < p^p^p(-p+1)/^p=(m-1)N(N-mp)^p-1/(p-1)^p-1m^p

We conclude that


    s≤ Cfm

for some C>0 that doesn't depend on n. Therefore, → u with u∈Łs. We can easily see that u is a weak solution by passing the limit in (<ref>).



If p=2, we recover Theorem 2.1 in <cit.>.




Notice the contrast between this case and the case u = f (treated for example in <cit.>). In the latter, we only need (p^*)'≤ m<N/p, whereas in the former an additional restriction has to be made in order to have the same regularity.




The inequality (<ref>) is actually optimal as the example in <cit.> shows. The author construct a radial example in the case p=2,=(m-1)N(N-mp)^p-1/(p-1)^p-1m^p and shows that the solution u∉Łs.




 §.§ Low summability: f∈Łm with 1<m< (p^*)'

In this scenario, the right hand side in (<ref>) is not in 𝐖^-1,p'(Ω), hence we can't apply classical existence results despite the coercivity. 



Suppose f∈Ł m with 1<m< (p^*)' and 

    0< < (m-1)N(N-mp)^p-1/(p-1)^p-1m^p.

Then the Dirichlet problem (<ref>) has a distributional solution u∈𝐖_0^1,q(), where q=pmN/-Np+m(N(p-2)+p).


 Take =[(t+||)^-p+1 - t^-p+1]sgn() as a test function in (<ref>), where ≥ 1 is a number to be chosen. We have:

    (-p+1)  |D |^p(t+||)^-p≤||^p-2/|x|^p+ |f(x)||(t+||)^-p+1 - t^-p+1|

Rearranging this we obtain:

    (-p+1)p^p/^p |D[(t+)^/p-t^/p]|^p   ≤|(t+)^/p-t^/p|^p/|x|^p+
    1/|x|^p( ||^p-2 -|(t+)^/p-t^/p|^p)
       + |f(x)||(t+||)^-p+1 - t^-p+1|

Using Hardy and Sobolev inequalities we get:

    ^p((-p+1)p^p/^p - /^p)( |(t+)^/p|^p^*)^p/p^*   ≤1/|x|^p( ||^p-2 -|(t+)^/p-t^/p|^p)
       + |f(x)||(t+||)^-p+1 - t^-p+1|

If we fix n and let t→0 we obtain:

    ^p((-p+1)p^p/^p - /^p)( ||^ p^*/p)^p/p^*≤fm( ||^(-p+1)m')^1/m'

Now we choose  such that p^*/p=(-p+1)m', that is we choose 
    =(p-1)m(N-p)/N-mp.

We conclude using (<ref>) that

    s≤ C fm

where s=mN/N-pm.

Notice that by (<ref>) we also have for a fixed t>0:

    |D |^p/(t+||)^p-≤ C,

hence, fix >1 we have:

    |D |^≤|D |^/(t+||)^(p-)α/p (t+||)^(p-)α/p,

Using Holder Inequality with p/ and p/p-:

    |D |^≤ C (  (t+||)^(p-)α/p-)^p-/p,

Let's choose  such that (p-)α/p-=s , that is ,

    =pmN/-Np+m(N(p-2)+p),

which is positive and greater than 1, since we took m<(p^*)'. 

We conclude that

    D≤ C,

so up to a subsequence, u in 𝐖_0^1,().

We can easily see that this u is a distributional solution, see for instance <cit.>. (Notice that ||^p-1/|x|^p→|u|^p-1u/|x|^p in Ł1).




If p=2, we recover Theorem 3.1 in <cit.>.




§ THE CASE G(X)≥ 0: REGULARITY GAIN

In this section we analyze the following problems:

    - u+g(x)u= /|x|^p |u|^p-2u + f        Ω,
    
    u (x) = 0    ∂Ω,

when 0<< ^p and f,g ∈Ł 1. 

Recall that the reason for choosing 0<< ^p is to ensure the coercivity of the operator T(u)=- u-/|x|^p |u|^p-2u.
In this case, the lower order term allow us to obtain not only a distributional solution but a weak solution in , where by a weak solution we mean u∈ such that g(x)u∈Ł1 and for every ∈∩Ł∞ we have:

    |Du|^p-2Du· D + g(x)u = /|x|^p |u|^p-2u  +  f



 §.§ (possibly) unbounded solutions





Suppose p≥ 2 and there is a M>0 such that

    |f(x)|≤ M g(x).

Then the Dirichlet problem (<ref>) has a weak solution u∈. Moreover, u∈Ł s for any s∈[p^*,s_), where s_ is the unique solution to the following equation:

    =^p[p(s_/p^*-1)+1]/(s_/p^*)^p.



We consider the truncated equation

    -+g(x)/1+M/ng(x)= /|x|^p+1/n||^p-2/1+||^p-1/n + f/1+|f|/n

Classical Leray-Lions theory guarantees the existence of a weak solution ∈∩Ł∞, see for example Theorem 5.1 or Example 9.12 in <cit.>.

Set g_n=g(x)/1+M/ng(x) and f_n=f/1+|f|/n, by hypothesis we have:

    |f_n(x)|≤ M g_n(x).

Now, take = as a test function in (<ref>), we have:

    |Du|^p+ g_n^2≤1/|x|^p+1/n||^p/1+||^p-1/n +  |f_n|

Using (<ref>), we have:

    |Du|^p+ g_n^2≤||^p/|x|^p + M g_n||

By Hardy's inequality:

    |Du|^p+ g_n^2≤/^p |D|^p+ M g_n||

Rearranging, we obtain:

    (1-/^p)  |Du|^p≤ g_n||(M-).

Using the trivial fact that the parabola x(M-x) has a maximum value of M^2 if 0≤ x≤ M, we conclude that:

    (1-/^p)  |Du|^p≤  M^2 g_n≤ M^2 g.

Therefore,  is bounded in  and hence u for some u∈. As before, we can easily pass the limit and conclude that u is a solution. The term g_n(x) is the tricky one, however Hardy's inequality and Vitali's convergence theorem guarantees the Ł1 convergence.

We now prove the regularity of u∈. It's enough to prove that  is bounded in Ł s. Take =||^p(-1) as a test function in (<ref>), for some ≥ 1 to be chosen later. We have:

    (p(-1)+1) |Du|^p||^p(-1)+ g_n||^p(-1)^2≤||^p/|x|^p +  |f_n|||^p(-1)+1.

Reasoning as before, we apply Hardy's inequality and use (<ref>) to obtain:

    (p(-1)+1/^p-/^p) |D||^|^p+ g_n||^p(-1)+2≤  M g_n||^p(-1)+1.

Simplifying we get:

    (p(-1)+1/^p-/^p) |D||^|^p≤ g_n||^p(-1)+1(M-||) .

As before, we can clearly see that the function x^p(-1)+1(M-x) has a maximum for 0≤ x≤ M. We have:

    (p(-1)+1/^p-/^p) |D||^|^p≤ M^p(-1)+2 g .

By Sobolev's inequality:

    (p(-1)+1/^p-/^p)1/ ||^ p^*≤ M^p(-1)+2 g .

In order for this to make sense, we must have:

    <^p[p(-1)+1]/^p

Let s= p^*, and consider the function h(s)=^p[p(s/p^*-1)+1]/(s/p^*)^p. Notice that if s=p^* then h(s)=^p and <^p which is always true. Also, h(s) is decreasing in [p^*,∞), so eventually it will hit , at a point s=s_. In conclusion, the estimate is valid for p^*≤ s < s_.



 §.§ Bounded solutions: 1<p≤ 2





Suppose there are M>0, N>1 such that

    |f(x)|≤ M g(x)  and  N/|x|^p≤ g(x)

If we assume 1<p≤ 2, then the Dirichlet problem (<ref>) has a bounded weak solution u∈∩Ł∞. 


We consider the truncated equation again (slightly modified):

    -+g_n(x)= /|x|^p+MN/n||^p-2/1+||^p-1/n + f_n

Notice that 

    N/|x|^p+MN/n≤ g_n(x).

Now, fix k>0 and choose =G_k() as a test function in (<ref>)(See the Notation section for the definition of G_k(s)). We have:

    |DG_k()|^p+ g_n(x) G_k() ≤/|x|^p+MN/n ||^p-1|G_k()| +  |f_nG_k()|.

Using (<ref>),(<ref>) and the fact that sG_k(s)≥ 0 we obtain:

    |DG_k()|^p+ g_n(x)| G_k()| ≤g_n(x)/N ||^p-1|G_k()| + M g_n(x)|G_k()|.

Simplifying:

    g_n(x)|G_k()|( ||(1-||^p-2/N)-M) ≤ 0

Choose k≫0 such that k(1-1/k^2-pN)>M, for that k we must have

    |G_k()|=0

Which is to say that ∞≤ k. On the other hand, choosing = as a test function in (<ref>):

    |D|^p≤/|x|^p+MN/n ||^p+  |f_n|.

Using Hardy's and Holder's inequalities:

    (1 - /^p)  |D|^p≤f1∞.

We conclude that   is bounded in ∩Ł∞, hence up to a subsequence u in . Arguing as in the proof of theorem <ref>, we can pass the limit in (<ref>), hence u∈∩Ł∞ is a weak solution.




Notice that if p>2, the argument in the proof the theorem <ref> fails and we believe it's possible that unbounded solutions may exist even with the strong restriction (<ref>).




We could increase the difficulty of the problem treated in this section if we had added a term of the form g(x)u^θ-1u, the problem would still be solvable but considerably more difficult since the θ would interact with the p in the estimates leading to existence estimates depending on different values of θ and p. We plan to address this question in future works.




The case p=1 remains challenging, that is, existence and regularity for the solution of the following problem:

    (Du/|Du|) + g(x) u = /|x|u/|u| + f,

where f∈Ł1. We can't use Hardy's inequality, at least, not in its usual form. See <cit.> for a related problem.




 Paper Title: Infinitesimal behavior of Quadratically Regularized Optimal Transport and its relation with the Porous Medium Equation.


[
    [
    Received 2 August 2023 / Accepted 7 May 2024
================================================

§ ABSTRACT 
 The quadratically regularized optimal transport problem has recently been considered in various applications where the coupling needs to be sparse, i.e., the density of the coupling needs to be zero for a large subset of the product of the supports of the marginals. However, unlike the acclaimed entropy-regularized optimal transport, the effect of quadratic regularization on the transport problem is not well understood from a mathematical standpoint. In this work, we take a first step towards its understanding. We prove that the difference between the cost of optimal transport and its regularized version multiplied by the ratio ε^-2/d+2
  converges to a nontrivial limit as the regularization parameter ε tends to 0. The proof  confirms a conjecture from <cit.> where it is claimed that a modification of the self-similar solution of the porous medium equation, the Barenblatt–Pattle solution, can be used as an approximate solution of the regularized transport cost for small values of ε.
  



§ INTRODUCTION AND MAIN RESULT


Optimal transport has an extremely rich history that started with Monge more than two centuries ago <cit.>. The problem is, given two  positive measures ρ_0 and ρ_1 with the same finite mass defined on Ω_0 and Ω_1 and a cost function from Ω_0×Ω_1 to [0, ∞), to find the optimal way of transporting one measure onto the other at the minimal overall cost. That is, the optimal transport cost between two probability measures having densities ρ_0 and ρ_1 with respect to the d-dimensional Lebesgue measure  ℓ_d is defined as 

    𝒲_2^2(ρ_0, ρ_1) = inf_π∈Π(ρ_0, ρ_1)∫-^2 π(,),

where Π(ρ_0, ρ_1) denotes the set of probability measures π∈𝒫(^d×^d) such that 

    ∫ f() π(, )= ∫ f() ρ_0()    and   ∫ f() dπ(, )= ∫ f() ρ_1() ,

for any bounded continuous function f∈𝒞_B(^d). In the case where ρ_1 is absolutely continuous with respect to the Lebesgue measure ℓ_d, the solution π̃ of (<ref>) is unique and of the form π̃ =( ∇ g× Id)_#ρ_1, where  g is a convex function (see <cit.>). In such a case, it is said that ∇ g pushes ρ_1 forward to ρ_0.       We refer to the two monographs by <cit.> for a detailed exposition.

We highlight that, in a common abuse of notation, we will refer—when ambiguity is absent—to the densities ρ_0 and ρ_1 and to the measures defined by them often by just speaking about the measures ρ_0 and ρ_1 or ρ_0 and ρ_1. 

After a series of fundamental theoretical developments, among which those of Kantorovich, <cit.> and many more, 
 the interest for optimal transport exploded in recent years as its usefulness for various application domains such as Machine Learning <cit.>, Statistics <cit.>, Biology <cit.>, Astronomy <cit.> or Economy <cit.>  got acknowledged. Note that it became impossible to list all the contributions to the field.

Moving away from the classical framework, a recent line of research has been inspired by new computational developments available for a particular relaxation of the optimal transport problem. The latter relaxation consists in  adding a (convex) penalty to the initial problem given in equation (<ref>) above. A particular such penalty, spearheaded by <cit.> for its easy computability on GPU, relies on the addition of a Kullback–Leibler regularization term
in the initial objective function, i.e., 

    𝒯_2, , KL(ρ_0, ρ_1)= inf_π∈Π(ρ_0, ρ_1)∫-^2dπ(,)+  KL(π| (ρ_0 ⊗ρ_1)),

where (ρ_0 ⊗ρ_1) denotes the product measure and KL(α|β)=∫log( α /β)α if α is absolutely continuous w.r.t. β  (i.e., α≪β) and KL(α|β)=+∞ otherwise. Here α /β stands for the Radon–Nikodym derivative of α w.r.t. β. Conversely, given a positive function ω we denote as ωβ the measure having Radon–Nikodym derivative ω w.r.t. β.
The addition of this penalty term  enables the use of a matrix-scaling algorithm: the Sinkhorn algorithm <cit.>. A plethora of works and applications followed, see <cit.>  for a book-long exposition.


Studying the infinitesimal behavior of the difference in the regularized cost  as the regularization parameter  tends to 0, <cit.> proved the limit 

    lim_→ 0^+𝒯_2, , KL(ρ_0, ρ_1)- 𝒲_2^2(ρ_0, ρ_1)- d/2 log(π)/= ∫log(ρ_0) dρ_0-∫log(ρ_1) dρ_1 /2

using a Gaussian approximation of the minimizers of the regularized problem. This is a natural approximation from a heuristic point of view, given that Gaussian distributions are the fundamental solutions of the heat equation, whose Wasserstein gradient flow is over the Kullback–Leibler entropy <cit.>.


Just like in the heat equation, whose  solution has maximum support for any positive time, the solutions of the optimal transport problem regularized by logarithmic entropy share the same property. That is, for every >0, the resulting coupling  is  dense, i.e., each point on the product space Ω_0 ×Ω_1 has a positive density. On the contrary, the (unregularized) optimal transport plan for the squared Euclidean distance as a cost is concentrated on the gradient of a convex function.   To remedy this, <cit.> introduced the Quadratically Regularized Optimal Transport (QOT)

    𝒯_2, ε, (·)^2(ρ_0, ρ_1) = inf_π∈Π(ρ_0, ρ_1)∫-^2π(,)+π/ (ρ_0 ⊗ρ_1) _L^2( ρ_0 ⊗ρ_1)^2,
  
where  

    π/ (ρ_0 ⊗ρ_1) _L^2( ρ_0 ⊗ρ_1)^2=∫( π/ (ρ_0 ⊗ρ_1))^2  (ρ_0 ⊗ρ_1)

in the case where π≪ (ρ_0 ⊗ρ_1), and +∞ otherwise. 

Applications of QOT have since then started blossoming. Examples include <cit.>. For some theoretical results we refer to <cit.>. <cit.> observed that the minimizer of (<ref>) is  “sparse” in the sense that its topological support seems to increase progressively as  gets larger. 
This increase of a sparse support suggests a diffusion behavior like that of the Porous Medium Equation (PME) where epsilon would play the role of the time parameter. For this reason, before setting this parallel on solid ground, we provide a short introduction to the PME in the next section.



 §.§ On the Porous Medium Equation and the QOT 


The Porous Medium Equation posed in the whole space reads, for an initial datum u_0,

    ∂_t u(t,) =Δ u(t,)^m,      t>0, ∈ℝ^d
    
        u(0,)=u_0()     x∈ℝ^d,

where m>1 is the non-linearity exponent. A mathematical study for this equation appeared first, in the case m=2, in the study of groundwater infiltration by <cit.> and since then it has attracted the attention of numerous experts. The general case m>1 can be derived from the study of an an-isotropic gas in a porous medium (hence the name) subject to Darcy's Law, see the seminal (and independent of each other) works of <cit.> and <cit.>. Since then, it has enjoyed a continuous attention from the mathematical community due to its applications in many fields, like plasma radiation, biology, population theory or even cosmology. We refer the interested reader to the book of <cit.>, broadly considered as the go-to reference on this matter. Therein, the reader can find a more detailed explanation of the main characteristics of the PME that are relevant for the study of the QOT. Even though in this article we focus on the case m=2, we will discuss the general case m>1.

The first property that leaps to the eye is the fact that the PME is non-linear, i.e., the sum of two solutions of the equation is not a solution itself. This makes it impossible for the existence of a linear description of its solutions similar to the heat equation, where the solution can be expressed as the convolution of the initial datum u_0 and the fundamental solution, the self-similar Gaussian profile (also referred as the kernel of the equation). The solution of the PME cannot be expressed as the convolution of the initial datum  with any kernel but, nevertheless, there exists a fundamental solution ℬ(t,x) of the PME, in the sense that

    ∂_t ℬ(t,) =Δℬ(t,)^m,      t>0, x∈ℝ^d
    ℬ(0,)=δ_0()    ∈ℝ^d,

where δ_0() is Dirac's Delta function located at =0 and the previous equalities should  be understood in the weak sense (or integral sense). This fundamental solution is called Barenblatt or Barenblatt–Kompaneets–Zeldovich profile due to its discoverers <cit.> and <cit.>. It reads

    ℬ(t,)=1/t^α[C- βm-1/2m^2/t^2β]_+^1/m-1,  where  α=d/d(m-1)+2, β= 1/d(m-1)+2,

and C>0 is a free constant that dictates the mass of the profile.
Contrarily to the Kullback–Leibler regularisation case, we cannot work with linear tools, i.e., semigroup theory. Still,  the profile of the solution above will play a key role in our work, see formula (<ref>) to mention but one example. 

The next property that is relevant in our study is the slow nature of the diffusion in the PME. If one writes the equation in terms of its diffusivity D(u) then we see that

    ∂_t u= div(D(u)·∇ u ),  with   D(u)=m u^m-1

and therefore the diffusivity disappears whenever u=0; in other words, our equation is degenerate parabolic at the points where u=0. This provokes the phenomena known as slow diffusion, where every solution whose initial datum has compact support will maintain a compact support that grows over time, in sharp contrast with the heat equation, where every solution becomes positive everywhere at any positive time. This characteristic of the PME relates closely to the sparse nature of the minimizer in the QOT observed by <cit.>.

Finally, there is one last connection between the PME and the QOT that we would like to mention. <cit.> explained how that the PME could be understood as a gradient flow with the 2-Wasserstein metric of the functional formed by the second moment of the solution and its free energy, namely

    E(u)=∫1/2^2u() + 1/m-1u()^m  dℓ_d(),

see also <cit.> for some applications. More or less at the same time <cit.> followed this approach, using E(u) as the natural entropy of the system, to describe the large-time behavior of the solutions of the problem. Note that  the resemblance between (<ref>) and  (<ref>) in the case m=2. Quadratically regularized optimal transport thus mimics the minimisation of the free energy just defined under the coupling constraints. As we shall see later, this last observation is where the geometry of unregularized optimal transport comes into the picture and plays a crucial role.



 §.§ Assumptions and main results


The existence of a potential link between QOT and the porous medium equation was considered folklore in the community in the last years even though never rigorously established. The first developments clearly fleshing this out were proposed by <cit.>. Their work further suggests that, by using Barenblatt profile approximations instead of Gaussian approximations (for the case of the Kullback–Leibler regularisation studied by <cit.>), one could characterize the non-trivial limit of 

    r()  (𝒯_2, ε, (·)^2(ρ_0, ρ_1) -𝒲_2^2(ρ_0, ρ_1)  )
   
for a proper choice of r(). Via a proof   based on quantization, <cit.> determined the value of the ratio r(ε) as r(ε) = ε^-2/d+2. It is worth highlighting, however, that the nonlinear nature of the penalty in (<ref>) makes impossible to utilize the linear tools present in <cit.> or most of the techniques present in the existing literature. The method we propose here is novel and relies mostly on the understanding of Barenblatt-like densities, inspired by (<ref>), and on the clever usage of mass transportation theory. 

In this work, we prove the complete (implicit) conjecture of <cit.> with this ratio under the following assumptions. 

We assume the following: 

    
  * Ω_1 and Ω_0 be open bounded sets such that 
    there exists δ_0∈ (0, π) and σ>0 such that for  every i∈{0,1} and ∈Ω_i there exists a cone
    
    𝒞_,  v,δ_0, σ={+ u:  v ucos(1/2θ) ≤⟨ v ,   u ⟩≤ vδ_0  }
 
    with vertex ,  height δ_0 and angle θ such that 𝒞_,  vδ_0, σ⊂Ω_i.
    
    
    
  * The functions ρ_0∈𝒞^β(Ω̅_̅0̅) and ρ_1∈𝒞^β(Ω̅_̅1̅) for some β>0 are densities defining  probability measures 
    
    1=∫_Ω_iρ_i() ℓ_d()=∫ dρ_i()=ρ_i(Ω),    i=1, 2.

    
  * Such  densities are upper lower bounded in their domain, i.e.,  

    λ≤ρ_0() ≤Λ  and λ≤ρ_1() ≤Λ  for all ∈Ω_0   and ∈Ω_1

for some 0< λ≤Λ < ∞.

  * The unique gradient of a convex function g pushing ρ_1 forward to ρ_0 belongs to 𝒞^2, α(Ω_1) and 

    σ_m(g)  Id≤∇^2 g ≤σ_M(g)  Id   in  Ω_1

for some 0<σ_m(g) ≤σ_M(g)< ∞.


Note that Assumption <ref>, 4. implies also that the convex conjugate of g, namely

    g^*()=sup_∈^d{⟨, ⟩ - g() },

is 𝒞^2(Ω_0). Moreover, it is well known that ∇ g^* pushes ρ_0 forward to ρ_1.  

The following is our main result. 




    Let Assumption <ref> hold and g be the unique function such that ∇ g pushes ρ_1 forward to ρ_0.   Then it holds that
    
    lim_ε→ 0^+𝒯_2, ε, (·)^2(ρ_0, ρ_1) -𝒲_2^2(ρ_0, ρ_1) /ε^2/d+2  = d^d+4/d+2(d+2)^2/d+2/(ℋ^d-1(𝒮^d-1) )^2/d+2∫_Ω_0(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0(),



 
 



As a consequence of this result, we can write explicitly the relationship between solutions of the PME and the optimizers of the QOT. Define the energy 
  
    E_(π):=   ∫1/2-^2 π(,)+ε·π^2(, )  (ρ_0 ⊗ρ_1)(,)

  and let g be the unique function such that ∇ g pushes ρ_1 forward to ρ_0. Define then, for any '∈Ω_0,
  
    ϱ('):=ρ_0( [∇^2g^*(')]^-1/2')·ρ_1(∇ g^*( [∇^2g^*(')]^-1/2')).



Choose now any '∈Ω_0 and let u(t, ; ') be the solution of 
  
    ∂_t u(t,;') =1/2(d+2)Δ_ u(t,; ' )^2,      t>0, ∈Ω_0
    
        u(0,; ')=ϱ(')^-1/d+2·δ_'()    ∈ℝ^d,

where Δ_ represents the Laplacian only in the 's variable, δ_'() denotes the Dirac's Delta function on the point =' and the previous solution must be understood in the weak sense. The constant 1/2(d+2) in front of the Laplacian operator can be considered just as a scaling factor in the  variable. The formula for u reads

    u(t,;')=1/t(C_d  t^2/d+2/ϱ(')^1/d+2 -1/2-'^2 )_+

where C_d is just a normalizing constant depending only on the dimension of the space that will be stated later.

Next, to each '∈Ω_0 and each solution u of the previous PME it corresponds a transformed function
  
    v(t,;')   := u(t, [∇^2g^*(')]^1/2; [∇^2g^*(')]^1/2')
       = 1/t(C_d  t^2/d+2/(ρ_0(')ρ_1(∇ g^*(')))^1/d+2 -1/2-'^2_∇^2 g^*('))_+,

  where
  
    -' ^2_∇^2 g()= ⟨-',  ∇^2 g()(-')⟩,
 
We are ready now to state the corollary.

    Under the assumptions of Theorem <ref> let π_() be the minimizer of (<ref>) for >0 and. Then it holds 

    lim_ε→ 0^+ε^-2/2+d| E_(π_()(,))-E_(v(, , ∇ g()))|=0




We sketch a proof of the corollary for the interested reader. The details are omitted, as they can be readily deduced by repeating the  calculations provided in the proof of Theorem <ref>.  Later on we will define a family of approximations π_ε to the true minimizer π_(ε). The key of these approximations is that, as ε→ 0,

    π_ε(,)∼ v(ε,;∇ g()).

The idea consists then in writing, adding and subtracting terms,

    ε^-2/2+d[ E_(π_()(,))-E_(v(, , ∇ g()))]    =ε^-2/2+d[ E_(π_()(,))- 𝒲^2_2(ρ_0,ρ_1)]_ℐ
       + ε^-2/2+d[ 𝒲^2_2(ρ_0,ρ_1)-E_(π_(,))]_ℐℐ
       +ε^-2/2+d[ E_(π_(,))-E_(v(, , ∇ g()))]_ℐℐℐ.

Then thanks to Theorem <ref> and the fact that π_ε approximates π_(ε) we have ℐ→ -ℐℐ, and then from (<ref>), applying the same ideas as in the proof of Theorem <ref>, we obtain that ℐℐℐ→ 0.




Define v(, , ')=u(, [∇^2 g^*()]^1/2, [∇^2 g^*()]^1/2'))
si 
    u(, , ')=1/( C_(')- 1/2-'^2 )_+

    
    E_(u(, ·, ∇ g(·)))= 1/2∫-^2 u(, , ∇ g())+ε/2 u(, , ∇ g())^2  (ρ_0 ⊗ρ_1)(,)
    
            =1/2∫-∇ g^*()^2 u(, , ')+ε/2 u(, , ')^2  (ρ_0 ⊗ρ_0)(,')




Apart from obtaining the limit conjectured by <cit.>, the main interest of this article consists on providing a constructive approach to the study of such limits for more general penalties. Our technique should work with very little changes for penalties of the form ·_L^m( ρ_0 ⊗ρ_1)^m for m>1, which corresponds to the full range of parameters of the Porous Medium Equation. We leave this generalization for further work and we highlight that we do not know if this technique could work for non-convex penalties of the form ·_L^m( ρ_0 ⊗ρ_1)^m for 0<m<1. This approach differs greatly from the one taken by <cit.>, which relies deeply on the linearity of the heat equation.   In addition, our analysis further  elucidates why the correct rate of convergence in Theorem <ref> is ^-2/d+2.  





To prove Theorem <ref>, we need to provide both an upper bound and a lower bound for 𝒯_2, ε, (·)^2(ρ_0, ρ_1). To provide the lower bound, we will use the dual formulation (see <cit.>)

    𝒯_2, ε, (·)^2(ρ_0, ρ_1) =2·sup_a,b∈ L^2(ρ_0)× L^2(ρ_1)∫{a() + b() 
    
        -1/2(a()+b()-1/2-^2)_+^2}ρ_0() ρ_1()

of (<ref>), whose solutions  are the so-called regularized potentials. 
The solutions of the primal and dual problems are related in the following way;  (a_, b_) is a maximizer of (<ref>) if and only if 

    1/(a_()+b_()-1/2-^2  )_+ρ_0() ρ_1()

is a minimizer of (<ref>). The reader should note the resemblance between (<ref>) and (<ref>) in the case m=2, with a_ε(), b_ε()∼ Cε^2/d+2 and t=ε. This relation suggests as a candidate  a pair of functions  (f̃_, g) such that 

    ∫(f̃_()+1/2^2 -g()-1/2-^2 )_+ dρ_1()=     for all ∈Ω_0.

After some calculations, 
Lemma <ref> yields 
f̃_()≍1/2^2 -g^*()+ C_(),
where 
    C_():=^2/d+2/C_d^2/d+2(ρ_0()ρ_1[∇ g^*()])^1/d+2   for    C_d:= 2^d+2/2ℋ^d-1(𝒮^d-1) 1/d(d+2).

Hence, we will find the lower bound by finding the limit as → 0 of 

    2·^-2/d+2(Γ_(f_, 1/2·^2 -g)- 1/2𝒲_2^2(ρ_0, ρ_1))

where 
    f_()=  1/2^2 -g^*()+ C_() and 
    
    Γ_ (a, b)=∫{a() + b() 
        -1/2ε(a()+b()-1/2-^2)_+^2}ρ_0() ρ_1().

For the upper bound, we will find a coupling π_∈Π(ρ_0, ρ_1) such that the functional 

    ℍ_(π)=   ∫-^2dπ(,)+dπ/  d(ρ_0 ⊗ρ_1) _L^2( ρ_0 ⊗ρ_1)^2,

at π_ achieves the same limit. The natural candidate is the measure π̃ with  π̃/ (ρ_0⊗ρ_1) (, ) = 1/(C_()-D(, ))_+, 
where 

    D(,  )= g^*()+g()-⟨, ⟩

is the  well-known  Bregman divergence. 
However, π̃ is neither a coupling nor a probability measure in the product space. 
Therefore, it must be adjusted using the following procedure: Firstly, transport ρ_0 to ρ_1 to work exclusively with the measure

    1/ε(C_ε(𝐱) - D(𝐱, ∇ g^*(𝐱')))_+ dρ_0(𝐱)  dρ_0(𝐱').

Since D(𝐱, ∇ g(𝐱')) is more difficult to control than its quadratic approximation (see Lemma <ref>)  
    1/2-' ^2_∇^2 g()=1/2⟨-',  ∇^2 g()(-')⟩,
 
we construct the symmetric version of its density

    m_ε (𝐱, 𝐱') = 1/2 ε( C_ε(𝐱) + C_ε(𝐱') - 1/2-' ^2_∇^2 g()-1/2-' ^2_∇^2 g('))_+.

Again, we highlight the resemblance between this last definition and (<ref>) in the case m=2, with C_ε(), C_ε()∼ Cε^2/d+2, also -' _∇^2 g(),-' _∇^2 g(')∼-' (see Lemma <ref> later on) and finally t=ε.

Subsequently, we normalize to create the measure with density 

    ξ_ε (𝐱, 𝐱') = m_ε(𝐱, 𝐱')/∫ m_ε(𝐱, 𝐱')  d(ρ_0 ⊗ρ_0)(𝐱, 𝐱')
 
with respect to ρ_0 ⊗ρ_0.
Within the product space, the symmetry ξ_ε is important because it implies   that both its marginals are the same and with  density ρ_(')=∫ξ_ (, ') ρ_0() with respect to ρ_0. Then, transport ρ_ερ_0 to ρ_0 and define the probability measure 

    dμ_ε (𝐱, 𝐲) = ξ_ε(∇ϕ_ε^*(𝐱), ∇ϕ_ε^*(𝐱'))/ρ_ε(∇ϕ_ε^*(𝐱)) ρ_ε(∇ϕ_ε^*(𝐱'))dρ_0(𝐱)  dρ_0(𝐱'),

where ∇ϕ_ε is the optimal transport map between  ρ_0 and ρ_ερ_0. For the rates of this coupling to be appropriate, it is necessary that the Lipschitz constant of ∇ϕ_ε tends to 1 uniformly (see (<ref>)). Using Caffarelli's regularity theory (see <cit.>), one can obtain the norm ϕ_ε_𝒞^2, α(Ω_0^δ), where 

    Ω_0^δ = {𝐱∈Ω_0 : dist(𝐱, ∂Ω_0) ≥δ}
 
and δ > 0, uniformly bounded in ε. In one dimension or in the flat torus, this holds for δ = 0, and the proof concludes at this step. In higher dimensions and with Euclidean cost, it is not entirely clear that this bound has been established in the literature—regularity up to the boundary is known for strongly convex and smooth domains  <cit.>, but the dependence on the constants is not clear to us. The same issue appears in <cit.> in a different context. Instead of attempting to prove this result, we will circumvent the issue by dividing the space Ω_0 into two parts: points that are less than δ away from the boundary and those that are farther. For the latter, we can apply interior regularity, where the dependence on the constants is explicit. For points near the boundary, we will use a different coupling—the so-called “frame coupling”—that does not achieve the appropriate limit but does not cause the functional to diverge. After taking the limit as   tends to 0, we will then take the limit as δ tends to 0 and check that it matches the lower bound.

As Caffarelli's results are only valid for convex domains, the previous discussion is applicable only to convex domains. 
The final step of the proof is to remove that assumption. It involves creating a tessellation of the domain Ω_0 into disjoint squares {I_i}_i=1^m and constructing a frame coupling in Ω_0 ∖{I_i}_i=1^m, applying the previous coupling to each of the squares. This piecewise-defined coupling is referred to as the “stained glass" coupling (see Figure <ref>).

The rest of the paper is organized as follows. In Section <ref>, we introduce the notation that will be utilized throughout the paper.  In Section <ref>, we present the necessary results from optimal transport theory that are essential for proving the main theorem. In Section <ref>, we will provide the lower bound, and in Section <ref>, we will establish the upper bound. As mentioned earlier, the upper bound will be proven first for convex supports (Section <ref>) and then for arbitrary supports (Section <ref>). 






§ NOTATION AND AUXILIARY RESULTS

In order to present the foundational concepts more seamlessly, we begin by defining a finite positive constant C(a_1, …, a_d) that depends solely on a set of parameters (a_1, …, a_d). For the sake of brevity, we denote a constant that depends only on ρ_0, ρ_1, Ω_0, Ω_1 and g by C, specifically C = C(ρ_0, ρ_1, g). A function v of ε is said to be a little-o of another function u (denoted as v = o(u)) if v(ε)/u(ε)→ 0 as ε→ 0. Occasionally, the function v(ε, 𝐱) may depend on 𝐱 within Ω_0. In such cases, we use the notation v = o(u) if sup_𝐱∈Ω_0v(ε, 𝐱)/u(ε)→ 0 as ε approaches zero. In the same manner, we will use the standard big-O notation.
 The notation x ≍_κ y is used to express an equivalence of order of convergence, considering multiplicative constants that depend solely on a parameter κ > 0, particularly in contexts approaching the limit as ε→ 0. As before κ=(ρ_0, ρ_1, Ω_0,
Ω_1, g) we just write x ≍  y.

Moreover, within a compact set Ω, the space 𝒞^p, α(Ω) is the space of Hölder continuous functions of order p with Hölder continuity parameter α in the open interval (0,1). The norm for this function space is defined as 

    f_𝒞^p, α(Ω) = ∑_k=0^p∑_|α|=ksup_x ∈Ω| d^k/dx_a_1⋯ dx_a_k f(x) | + ∑_|α|=psup_x, y ∈Ω
     x ≠ y|f(x) - f(y)|/x - y^α.


Additionally, the notation ℓ_d is employed to denote the Lebesgue measure, and g^* refers to the convex conjugate of g, calculated as g^*(x) = sup_y ∈ℝ^d (⟨ x, y ⟩ - g(y)). The Hausdorff measure of dimension d-1 is represented by ℋ^d-1.









Furthermore, for a positive definite matrix A, its square root, denoted A^1/2, is the unique positive definite matrix V such that V^2 = A.  The product measure ρ_0 ⊗ρ_1 and the space L^2(μ), which includes functions f with a finite L^2(μ)-norm given by

    f_L^2(μ) = ( ∫ f^2(x)  μ(x) )^1/2,

are also crucial. In instances where Borel measures μ and ν satisfy μ≪ν, the notation dμ/dν represents the Radon-Nikodym derivative of μ with respect to ν.  For the particular cases of ρ_i, for i=1,2, to avoid complicating the notation, we will use the following abuses of notation: dρ_i/dℓ_d= ρ_i and 

    ∫ f() dρ_i() = ∫_Ω_i f() ρ_i()  ℓ_d(),    i=1, 2.

For a Borel set A,   χ_A denotes the indicator function of A. Sometimes, when it can cause confusion, we write χ_[A]. We say that a function T pushes a measure μ forward to another measure ν if ν(A)=μ(T^-1(A)) for all Borel set A. In such a case, we write T_#μ=ν.  A set A is compactly contained in B (A⊂⊂ B)  if there exists a compact set K and an open set 𝒰 such that A⊂ K⊂𝒰⊂  B. The Euclidean Ball with center  and radius δ > 0 is denoted as 𝔹(, δ).






 §.§ Optimal transport background

The optimal transport problem  (<ref>) admits the dual formulation

    𝒲_2^2(ρ_0, ρ_1) 
       =2·sup_(a,b)∈Φ∫ a() ρ_0() + ∫ b() ρ_1(),

where 
Φ={ (a, b)∈ (𝒞(^d))^2:  a()+b()≤-^2/2,  ∀, ∈^d} .
Under Assumption <ref>, the solution of the primal problem (<ref>) is given by ( Id×∇ g)_#ρ_1 and the solution of (<ref>) by the pair 

    ( 1/2·^2 -g^*,  1/2·^2 -g ).

In general, the function g solves the Monge-Ampère equation

    (∇^2 g)=ρ_1/ρ_0(∇ g)    in Ω_1,   ∇ g (Ω_1)= Ω_0

in the so-called Brenier sense (see <cit.>). In our case, by assumption, g solves it in the strong sense. The Monge-Ampère equation is a degenerate elliptical nonlinear differential equation. The regularity of convex solutions has been studied by Caffarelli in <cit.>. Here, we highlight the result that we will use in this work.
[Caffarelli]
   Let Ω be open and convex q_1,q_2∈𝒞^0,α(Ω) be positive functions such that 
    
    ∫_Ω q_1  dℓ_d= ∫_Ω q_2   dℓ_d=1.

    Then there  exists a
    unique (up to additive constant) convex solution u of the boundary value problem 
    
    (∇^2 u  )=q_1/q_2(∇ u )   in Ω,   ∇ u (Ω)= Ω

    in 𝒞^2,α_Loc(Ω).
    Moreover, for each K⊂⊂Ω 
    
    u_𝒞^2, α(K)≤  C(K, inf_∈Ω q_1, inf_∈Ω q_2, q_1_𝒞^0,α(K), q_2_𝒞^0,α(K))


The assumption of regularity on g is key in the infinitesimal developments we are undertaking. Lemma <ref> is fundamental as it relates D(, ) with its linearized inner product 
 
    ⟨, ⟩_∇^2 g^*()= ⟨, ∇^2 g^*() ⟩.
 
 We omit its proof as it is a mere second-order Taylor expansion of the Bregman divergence 

    D(, )= g^*()+g()-⟨, ⟩

of g.


    Let Assumption <ref> hold. Then there exists σ_M(g)≥σ_m(g)>0 such that 
    σ_m(g)/2-'^2   ≤  D(, ∇ g^*('))≤σ_M(g)/2-'^2.
 
Moreover,  there exists C>0 such that 

    |    D(, ∇ g^*('))-1/2-' ^2_∇^2 g^*()|≤  C-'^2+α,
   
where 

    1/2-' ^2_∇^2 g^*()=1/2⟨-',  ∇^2 g^*()(-')⟩

and α is as in Assumption <ref>. 


As a direct consequence of Lemma <ref> we obtain the following.

 
     Let Assumption <ref> hold. Then the following estimates hold: 
     
         
  * There exists a constant C and _0>0  such that  
    
    { (, ')∈Ω_0×Ω_0:     C_()≥  D(, ∇ g^*('))  }⊂{ (, ')∈^2d:  - '≤ C^1/d+2}

        holds for every 
         ≤_0. 
         
  * For ≤_0, 
     
    sup_, '∈Ω_0|(  C_()- D(, ∇ g^*(')))_+
            - (  C_() -1/2-'_∇^2 g^*()^2)_+|
            =o(^2/d+2).

       
 



 §.§ Properties of the integrals


The following results are well-known and easy to prove passing to spherical coordinates. We state them for further reference.


For any a≥0,    it holds that 
    
    ∫( a- ^2/2)_+ ℓ_d()̆= |a|^d+2/2 C_d^(1)  ,

    where 
    
    C_d^(1) := 2^d+2/2/d(d+2)ℋ^d-1(𝒮^d-1),

    and also
    
    ∫(a- ^2/2)_+^2 ℓ_d()̆= |a|^d+4/2 C_d^(2),

    where 
    
    C_d^(2):= 2^d+6/2/d(d+2)(d+4)ℋ^d-1(𝒮^d-1).








§ LOWER  BOUND

The aim of this section is to show 

    lim inf_→ 0^+𝒯_2, ε, (·)^2(ρ_0, ρ_1) - 𝒲_2^2(ρ_0, ρ_1)/^2/d+2≥d^d+4/d+2(d+2)^2/d+2/(ℋ^d-1(𝒮^d-1) )^2/d+2∫(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0().

A sufficient condition for this to occur is given by the following result —see equation (<ref>) in the introduction, whose proof will be provided below. 
One of the key aspects of this proof is that we divide the set Ω_0 into two, the points near its boundary or far away, namely

    Ω_0^δ := {∈Ω_0:   dist  (, ∂Ω_0)> δ}   and   Ω_0^δ,c:={∈Ω_0:   dist  (, ∂Ω_0)≤δ}.

These sets will also play a fundamental role in the proofs of Section <ref>, hence their importance. As the reader will see, being at a distance δ from the boundary will provide many useful uniform estimates that are in the core of our analysis. Then, making δ→ 0 we will find that our results hold true. 



    Let Assumption <ref> hold.  Set C_d:= 2^d+2/2ℋ^d-1(𝒮^d-1) 1/d(d+2). Then 
   
    
    lim_→ 0^+Γ_(f_, 1/2·^2 -g )- 1/2𝒲_2^2(ρ_0, ρ_1)/^2/d+2=d/C_d^2/d+2∫(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0(),

    where 
    f_()=  1/2^2 -g^*()+ C_() for C_()=^2/d+2C_d^-2/d+2(ρ_0()ρ_1[∇ g^*()])^-1/d+2.



As 
 
    Γ_(f_, 1/2·^2 -g )=  ∬{1/2^2 -g^*()+ C_() + 1/2^2 -g^*()
       
     -1/2( C_() -g()-g^*()+1/2^2+1/2^2-1/2-^2)_+^2}ρ_0() ρ_1()

and, by (<ref>),
 
    𝒲_2^2(ρ_0, ρ_1)=∫(1/2·^2-g^*)ρ_0+∫(1/2·^2-g)ρ_1,

 the value of 
    L=lim_→ 0^+Γ_(f_, 1/2·^2 -g )- 𝒲_2^2(ρ_0, ρ_1)/^2/d+2
 is given by the limit (multiplied by ^-2/d+2) of
 
 
    L_   =∫ C_() ρ_0() -1/21/∬(  C_()- D(, ∇ g^*(')))_+^2 ρ_0()ρ_0(')_L'_.

Since  
 
    ^-2/d+2∫ C_() ρ_0() = ∫1/C_d^2/d+2(ρ_0()ρ_1[∇ g^*()])^1/d+2 dρ_0(),

 we only need to deal with the limit of ^-2/d+2 L'_.
 
 Set now δ>0 and define the sets Ω_0^δ and Ω_0^δ,c as in (<ref>). Then, due to Lemma <ref>, there exists _δ such that for every ≤_δ, the set Ω_0^δ, c contains the set

    𝕍_={∈Ω_0: ∃ '∈∂Ω_0:   C_()≥  D(, ∇ g^*(')) }.

 Therefore, we split 
^-2/d+2 L'_ into two terms; 

    𝕋_1=^-d+4/d+2∫_Ω_0^δ∫(  C_()- D(, ∇ g^*(')))_+^2 ρ_0(') ρ_0()

and 

    𝕋_2= ^-d+4/d+2∫_Ω_0^δ,c∫(  C_()- D(, ∇ g^*(')))_+^2 ρ_0(')ρ_0(),

and estimate each one separately. 

Estimate for (<ref>).   Since  ρ_0 is uniformly continuous over Ω_0 and the support of the function (  C_()- D(, ∇ g^*(')))_+ concentrates uniformly on the diagonal =' (see Lemma <ref>), it holds that  

    𝕋_1= ^-d+4/d+2∫_Ω_0^δ∫_Ω_0(  C_()- D(, ∇ g^*(')))_+^2 ℓ_d(') (ρ_0()+o(1 )) ρ_0(').
 
Considering that Ω_0^δ and 𝕍_ are disjoint, it follows that the inner integral can be taken over ^d, i.e., 

    𝕋_1= ^-d+4/d+2∫_Ω_0^δ∫_^d(  C_()- D(, ∇ g^*(')))_+^2 ℓ_d(') (ρ_0()+o(1 )) ρ_0(').
 
Lemma <ref> implies 

    𝕋_1≍^-d+4/d+2∫_Ω_0^δ∫_Ω_0(  C_()-1/2⟨-', ∇^2 g^*() (-') ⟩ ))_+^2 ℓ_d(') ρ_0() ρ_0().
 
We focus on the inner integral

    I()= ρ_0^2() ∫_Ω_0(  C_()-1/2⟨-', ∇^2 g^*() (-') ⟩ ))_+^2 ℓ_d(') .
 First we change variables =̆  [∇^2 g^*()]^1/2 (-'), to get 

    I()=  ρ_0^2()/(∇^2 g^*())^1/2∫(  C_()-1/2^2)_+^2 
        ℓ_d()̆.

Using Proposition <ref>, we find

    I()   =  (C_())^d+4/2 C_d^(2)ρ_0^2()/(∇^2 g^*())^1/2
       =  ^d+4/d+2 C_d^(2)ρ_0^2()/C_d^d+4/d+2(ρ_0()ρ_1[∇ g^*()])^d+4/2(d+2)(∇^2 g^*())^1/2
       = ^d+4/d+2 C_d^(2)(ρ_0() ρ_1[∇ g^*()])^1/2/C_d^d+4/d+2(ρ_0()ρ_1[∇ g^*()])^d+4/2(d+2)ρ_0()
       = ^d+4/d+2 C_d^(2)/C_d^d+4/d+2(ρ_0()ρ_1[∇ g^*()])^1/(d+2)ρ_0(),

    where 
    C_d^(2):= 2^d+6/2/d(d+2)(d+4)ℋ^d-1(𝒮^d-1).
Hence, 

    𝕋_1 ≍  C_d^(2)/C_d^d+4/d+2∫_Ω_0^δ(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0().

We simplify the constant C_d^(2) C_d^-d+4/d+2 as follows:

    C_d^(2) C_d^-1 -2/d+2   = C_d^ -2/d+22^d+6/2/d(d+2)(d+4)ℋ^d-1(𝒮^d-1) 2^-d+2/2(ℋ^d-1(𝒮^d-1))^-1 d(d+2) 
       = 4/d+4C_d^ -2/d+2,

which leads to

    𝕋_1 ≍4/d+4C_d^ -2/d+2∫_Ω_0^δ(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0().

Estimate for (<ref>). First note that under Assumption <ref>, 

    C_()≤^2/d+2/ C_d^2/d+2λ^2/d+2 .

Hence, due to Lemma <ref> and the fourth point of Assumption <ref>, we have the bound

    0≤   𝕋_2≤^-d+4/d+2∫_Ω_0^δ, c∫( ^2/d+2/C_d^2/d+2λ^2/d+2 -c-'^2)_+^2 ρ_0() ρ_0(')
       ≤Λ ^-d+4/d+2∫_Ω_0^δ, c∫( ^2/d+2/C_d^2/d+2λ^2/d+2 -c-'^2)_+^2 ℓ_d()  ρ_0(')
       ≤Λ/λ^2/d+2^-d/d+2∫_Ω_0^δ, cℓ_d({:̆ c^2≤^2/d+2/C_d^2/d+2λ^2/d+2})  ρ_0('),

for some positive constant c. As a consequence, there exists a constant c' such that 

    0≤𝕋_2≤ c' ρ_0(Ω_0^δ, c).

We can now conclude the proof by noticing that

    lim inf_→ 0^+    Γ_(f_, g)- 𝒲_2^2(ρ_0, ρ_1)/^2/d+2
       ≥d/C_d^2/d+2∫_Ω_0^δ(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0() -Cρ_0(Ω_0^δ, c)

and 

    lim sup_→ 0^+   Γ_(f_, g)- 𝒲_2^2(ρ_0, ρ_1)/^2/d+2
       ≤d/C_d^2/d+2∫_Ω_0^δ(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0() +Cρ_0(Ω_0^δ, c)

hold for any δ>0. Letting δ→ 0 we obtain the result.








§ UPPER BOUND


In this section, we establish an upper bound that is asymptotically equivalent to the lower bound.

The proof is quite technical. We will proceed as shown in Figure <ref>. First, we will assume that the set Ω_0 is convex and we will divide it between its interior Ω_0^δ and the points close to the boundary, Ω_0^δ,c. One can imagine Ω_0 as being a window where Ω_0^δ is its glass and Ω_0^δ,c its frame of width δ. In the glass part, we will define a coupling that achieves the appropriate limit, while in the frame, we will create one that tends to 0 as we let → 0 first and then  δ→ 0. For the general case, we will cover most of Ω_0 by composing a sort of stained-glass window formed by convex windows Ω_0,1, Ω_0,2, etc.... In each of the windows contained within the interior of Ω_0, we will use the same coupling as before. In those touching the boundary of Ω_0, we will use a coupling that does not blow up and decreases as the number of windows increases, albeit using many of smaller size.

Now we will present some technical results for bounding frame couplings in both the convex and general Ω_0 cases.




Let ρ_0 and g be as in Assumption <ref>. Let s:Ω_0↦ [a,b] for some 0<a<b<∞ be a given function and Ω⊂Ω_0 be any open set such that 
    there exists r_0>0 and θ∈ (0,π) such that for  every i∈{1, 2} and ∈Ω there exists a cone
    
    𝒞_,  v,r_0, σ={+ u:  v ucos(1/2θ) ≤⟨ v ,   u ⟩≤ vr_0  }
 
    with vertex ,  height r_0 and angle θ such that 𝒞_,  v,r_0, σ⊂Ω.
Then there exists  _0 and C=C(c,a,b)
such that for ≤_0 the following holds:

    
  *  There exists a function 
ψ_ such that

    s()=1/∫_Ω (ψ_ ()-D(, ∇ g^*(') )_+ρ_0(') ,   for all ∈Ω.


  *  There exists _0 and C>0 such that for every  ∈Ω and ≤_0, 

    ε^2/d+2/C≤ψ_() ≤ε^2/d+2 C,

and 

    ε^d/d+2/C≤ℓ_d(𝒮_ε, )≤ Cε^d/d+2,

where 𝒮_ε, = {'∈Ω_0:  ψ_ ()-D(, ∇ g^*(')  ≥ 0}.

  *  For every  '∈Ω, there exists a c'>0 such that

    1/∫_Ω (ψ_ ()-D(, ∇ g^*(') )_+ρ_0() ≥ c'.

 


 We prove each point separately. 

Proof of <ref>.
Set ∈Ω_0 and 

    Θ(a)=1/2 ∫_Ω (a-D(, ∇ g^*(') )_+^2ρ_0(')-a s().

Since s()>0, we have lim_a→±∞Θ(a)=+∞. Moreover, as 
Θ is   convex and differentiable,  there exists a minimizer a() which satisfies 

    1/ ∫_Ω (a()-D(, ∇ g^*(') )_+ρ_0(')= s() .

This proves the existence. 

Proof of <ref>.Again, since s()>0 for all ∈Ω_0,  ψ_  is non-negative  and Lipschitz with constant 2· diam(Ω)  (see the proof of <cit.>). This implies that 
ψ_   tends uniformly to 0 in Ω as → 0. 
Lemma <ref> implies that

    s()
           ≤∫_Ω(ψ_()- σ_m(g)/2-'^2)_+ ρ_0(')
       ≤Λ∫_Ω(ψ_()- σ_m(g)/2-'^2)_+ ℓ_d(')
       ≤Λ∫_^d(ψ_()- σ_m(g)/2-'^2)_+ ℓ_d(')
       ≤ C  (ψ_())^d+2/2,

for a certain constant C>0. Hence, the lower bound in (<ref>) follows.  Next, 

    s()    ≥∫_Ω (ψ_()- σ_M(g)/2-'^2)_+ ρ_0(')
       ≥λ∫_Ω(ψ_()- σ_M(g)/2-'^2)_+ℓ_d(').

By assumption, Ω contains the cone 𝒞_,  v,r_0, θ. As a consequence, 

    s()    ≥λ∫_𝒞_,  v,r_0, θ(ψ_()- σ_M(g)/2-'^2)_+ℓ_d(')
       =λ∫_𝒞_,  v,r_0, θ∩𝔹(,  (2ψ_()/σ_M(g))^1/2){ψ_()- σ_M(g)/2-'^2}ℓ_d(').

 Since  ψ_   tends uniformly to 0 in Ω, we can find _0>0 such that 
    ψ__𝒞(Ω)≤(2 r_0/σ_M(g))^2
 for all ≤_0. This yields 
 
    s() ≥λ∫_𝒞_,  v,ψ_(), θ∩𝔹(,  (2ψ_()/σ_M(g))^1/2){ψ_()- σ_M(g)/2-'^2}ℓ_d(').

Hence, there exists a constant C(θ,d, σ_M(g))>0 depending only on d, σ_M(g) and the ℋ^d-1-measure of the sector 

    𝒮^d-1∩{ u:  e_1 ucos(1/2θ) ≤⟨ e_1 ,   u ⟩},

where e_1 is the first element of the canonical basis of ^d,  such that 

    s()  ≥ C(d, θ) λ( ψ_())^d+2/2.

This gives the upper bound in (<ref>).  From here it is easy to obtain 
(<ref>) via the nesting

    𝔹(, 1/Cψ_())⊂𝒮_ε, ⊂𝔹(, C ψ_()),

which are consequence of Lemma <ref>.

Proof of <ref>. 
By the previous points, There exist _0 and c> 0
such that for ≤_0 

    ∫_Ω (ψ_ ()-D(, ∇ g^*(') )_+ dρ_0()   ≥∫_Ω(c^2/d+2- D(, ∇ g^*(')))_+ dρ_0() 
       ≥∫_Ω(c/Cψ_ (')- D(, ∇ g^*(')))_+ dρ_0()
    (by Lemma <ref>)    ≥ c_1 ∫_Ω(ψ_ (')- D(', ∇ g^*()))_+ dρ_0()
       ≥ c_1   inf_s().

Defining c':=c_1  inf_s() we obtain the desired result.





 §.§ Feasible coupling for the interior Ω_0^δ of a convex Ω_0


In this section we will define a possible coupling for the set Ω_0^δ. We will do so, in fact, by finding a coupling  μ_∈Π(ρ_0, ρ_0) that attains the correct limit when restricted  to (Ω_0^δ×Ω_0)∪ (Ω_0×Ω_0^δ) for δ>0. Then, we will restrict this coupling only to (Ω_0^δ×Ω_0)∪ (Ω_0×Ω_0^δ), but by doing so we will lose the part of the mass corresponding with Ω_0^δ×Ω_0^δ, which we will compensate with an extra term; see Section <ref> later on. 

We define, for ,'∈Ω_0×Ω_0,

    m_ε ( x,  x')=1/2 (C_()+C_(')-1/2-'_∇^2 g^*() ^2- 1/2-'_∇^2 g^*(') ^2 )_+ .

The following result is a direct consequence of Lemma <ref>, the definition of C_, the uniform continuity of ρ_0, ρ_1, and the transport map ∇ g, combined with the strict positivity of both ρ_0 and ρ_1. 

  Let Assumption <ref> hold. Then there exists C such that 
  
    -'≤ C ^2/d+2 ,

  for all (, ')∈ supp(m_)
and 
  
    sup_, '∈Ω_0|  m_ε ( x,  x')-1/(C_()- 1/2-'_∇^2 g^*()^2 )_+| =o(^2/d+2).


We define the 
function 

    ξ_ (, )=m_( , ')/∫ m_( , ')  (ρ_0⊗ρ_0)(, '),

which is a density (it is positive and integrates one) with respect to ρ_0⊗ρ_0. We define also 

    ρ_()=∫ξ_ (, ') ρ_0(')

as the marginal density (w.r.t. ρ_0), which by symmetry agrees with 
ρ_(')=∫ξ_ (, ') ρ_0().
Moreover it satisfies the following properties. We recall the definition of Ω_0^δ in (<ref>).















 Let Assumption <ref> hold. Then there exist two positive constants c ≤ C∈ (0, ∞) and a delta_0>0 such that for every δ<δ_0, there exists an _δ such that:


  *  ρ_ ρ_0 ∈𝒫(Ω_0) and  ξ_  (ρ_0 ⊗ρ_0)∈Π(ρ_ρ_0, ρ_ρ_0), 

  *   
    c ≤ρ_≤  C in  Ω_0,

  *   ρ_∈𝒞^0, α(Ω_0^δ) with 
ρ__𝒞^0, α(Ω_0^δ)≤ C'   for some positive constant C', and

  *  as → 0, 
    
    ρ_ -1_𝒞(Ω_0^δ)→ 0    and  ∫ m_ (·, ') ρ_0(') -1_𝒞(Ω_0^δ)→ 0 .



 Point <ref> holds by construction. We prove each of the remaining points separately. 

Proof of <ref> By definition of ξ_, it is enough to show the existence of two positive constants, c and C such that 
c ≤∫ m_(, ') ρ_0() ≤ C
for all ∈Ω_0. 
Set , '∈Ω_0.
 By Assumption <ref>, we have 
 
    C_()≤^2/d+2/C_d^2/d+2λ^2/d+2   and  1/2-'_∇^2 g^*() ^2 ≥σ_m(g)/2-'^2 .

As a consequence, 

    ∫ m_(, ') ρ_0()   ≤∫1/( ^2/d+2/C_d^2/d+2λ^2/d+2 -σ_m(g)/2-'^2 )_+ ρ_0()
       ≤Λ∫1/( ^2/d+2/C_d^2/d+2λ^2/d+2 -σ_m(g)/2-'^2 )_+ ℓ_d()
       =Λ∫1/( C_d ^2/d+2/C_d^2/d+2λ^2/d+2 -σ_m(g)/2^2 )_+ ℓ_d().

The latter quantity, call it C, does not depend on  and is finite, see Proposition <ref>.   The reverse inequality 

    ∫ m_(, ') ρ_0() ≥λ∫_Ω_01/( ^2/d+2/C_d^2/d+2Λ^2/d+2 -σ_M(g)/2-'^2 )_+ ℓ_d()

holds by the same means. From here, we conclude as in the proof  Lemma <ref>, claim <ref>. 


Proof of <ref>. Since  ∫ m_( , ')  (ρ_0⊗ρ_0)(, ') and ρ_0 are uniformly bounded away from 0 and ∞, it is enough to upper bound the 𝒞^0, α(Ω_0^δ)-norm of  the function 

    ↦∫ m_( , ) ρ_0().
  
Set 
, '∈Ω_0^δ and assume w.l.o.g. that 
-'^α≤σ_m(g)/4C_1,
 where  C_1>0 is the constant  such that 

    |_∇^2 g^*( a) ^2- _∇^2 g^*( b) ^2 |≤ C_1  ^2  a- b^α,  for all  a,  b,  ∈Ω_0

which due to the fact that ∇^2 g(') is Hölder  continuous.
Since C_d^-2/d+2 ( ρ_0()ρ_1(∇ g^*()) )^-1/d+2  is also a Hölder  continuous function, here exists C_2>0 

    |C_(')-C_()| ≤ C_2-'^α^2/d+2.

We aim at showing that 

    ∫ m_( , ) ρ_0()≤∫ m_(' , ) ρ_0() + C - '^α,

for some constant C>0. We call 

    𝒮_ := {∈Ω_0:  C_()+C_( z) -1/2-_∇^2 g^*() ^2- 1/2-_∇^2 g^*() ^2>0}

and observe that, due to (<ref>), there exists C_3>0 such that  sup_ u∈𝒮_ u- z≤ C_3 ^1/d+2.
Hence, we can find _δ>0 such that  ⋃_ u∈Ω^δ𝒮_ u⊂Ω^δ/2 for all ≤_δ. The rest of the proof always assume that ≤_δ.  Hence,

    ∫ m_( , ) ρ_0()
       = 1/2 ∫_Ω_0(C_()+C_( z) -1/2-_∇^2 g^*() ^2- 1/2-_∇^2 g^*() ^2 )_+  ρ_0()ℓ_d()
        = 1/2 ∫_^d(C_()+C_( z) -1/2-_∇^2 g^*() ^2- 1/2-_∇^2 g^*() ^2 )_+  ρ_0()ℓ_d().

 Making the change of variables '  = T_,'()=  +'-, 
one gets 

    ∫ m_( , ) ρ_0()
        
        = 1/2 ∫_^d(C_(' +  - ' )+C_( z) -1/2'-'_∇^2 g^*(' +  - ' ) ^2- 1/2'-'_∇^2 g^*() ^2 )_+ 
               ρ_0('-'+) ℓ_d(').


Via (<ref>) and (<ref>), the bound

    ∫ m_( , ) ρ_0()
       ≤1/2 ∫_^d(C_(')+C_( z') -1/2'-'_∇^2 g^*(' ) ^2-1/2'-'_∇^2 g^*(') ^2
                + 2  C_2^2/d+2-'^α+ 2 C_1'-'^2-'^α)_+
    ρ_0('-'+) ℓ_d(')

holds. 
Let 𝒮' be the set where the quantity inside the integral is positive and 

    C_5=(4 C_1/σ_m(g)(  C'σ_m(g)/4C_1+ 1/C_d^1/d+2λ^2/d+2))^1/2.
 Since -'^α≤σ_m(g)/4C_1, the nesting 
𝒮' ⊂𝔹( z',  ^1/d+2C_5)
is derived.  This combined with the bound on the regularity of the density
ρ_0('-'+)≤ρ_0(')+C_4-'^α
yield 

    ∫ m_( , ) ρ_0() 
       ≤1/2 ∫_^d(C_(')+C_( z') -1/2'-'_∇^2 g^*(' ) ^2-1/2'-'_∇^2 g^*(') ^2)_+ ρ_0(') ℓ_d(')
       +C_6 -'^α,

where 

    C_6 =2C_2C_5^dℓ_d(𝔹(0,1)) Λ +2C_1 C_5^d+2Λ +C_5 sup_'∈Ω^δ∫ m_(, ') ℓ_d(),

 which is finite (see the proof of  <ref>). 










Using (<ref>) we get 

    ∫ m_( , ) ρ_0()
        
       ≤1/2 ∫(C_()+C_( z)+ C_1/2-^2 '-^α -1/2_∇^2 g() ^2- 1/2_∇^2 g(') ^2 )_+  dρ_0() .

Using 

    ∫ m_( , ) ρ_0()
        
       ≤1/2 ∫(C_()+C_( z')+ C'^2/d+2'^α
                + C_1/2^2 '^α -1/2_∇^2 g() ^2- 1/2_∇^2 g(') ^2 )_+  dρ_0() .



Since '^α≤σ_m(g)/2C σ_M(g), it holds that ℓ_d(𝒮)≤ C ^d/d+2 and max_∈𝒮≤ C' ^1/d+2. As a consequence, 

    ∫ m_( , ) ρ_0()
           ≤1/2 ∫(C_()+C_( z')  -1/2_∇^2 g() ^2- 1/2_∇^2 g( z') ^2 )_+  dρ_0()
            + '^α(C_1 C^2σ_M(g)/2 + C' C”)

Change variables =̆T()=+'  and assume that  is small enught so that T(𝒮)∩^d∖Ω_0=∅. Then 

    1/2 ∫(C_()+C_( z')  -1/2_∇^2 g() ^2- 1/2_∇^2 g( z') ^2 )_+ dρ_0()
       =1/2 ∫(C_(-̆')+C_(') -1/2-̆'_∇^2 g(-̆') ^2- 1/2-̆'_∇^2 g( 0) ^2 )_+  ρ_0(-̆') ℓ_d()̆.

Again, using (<ref>) and (<ref>) and 
repeating the previous argument, 
we get 

    ∫ m_( , ) ρ_0()
        
       ≤1/2 ∫(C_(-̆')+C_(') -1/2-̆'_∇^2 g(-̆') ^2- 1/2-̆'_∇^2 g( 0) ^2 )_+  ρ_0(-̆') ℓ_d()̆
            + C'^α,

for some C>0. 
Recalling that ρ_0 is Hölder continuous, we also have |ρ_0(-̆')-ρ_0()̆|≤ C'^α, so that

    ∫ m_( , ) ρ_0()
        
       ≤1/2 ∫(C_(-̆')+C_(') -1/2-̆'_∇^2 g(-̆') ^2- 1/2-̆'_∇^2 g( 0) ^2 )_+  ρ_0()̆ℓ_d()̆ + C'^α
       = ∫ m_(' , ) ρ_0()+ C'^α.

The last step above wasn't clear to me; details would be enjoyable!


This yields the first part of the Hölder inequality statement. 
The reverse inequality holds due to symmetry,  exchanging  by '. 


Proof of <ref> 
By (<ref>), Lemma <ref> and the fact that the support of m_ε decreases with rate ^2/d+2, we have 



    m_ε ( x,  x')=1/( C_() -1/2-'_∇^2 g^*() ^2 +o(^2/d+2) )_+ .


Set now δ>0 and ∈Ω_0^δ. Then there exists _0>0 small enough such that 

    ∫_Ω_0^δ( C_()-1/2-'_∇^2 g^*() ^2)_+ ℓ_d(')
        = ∫( C_()-1/2-'_∇^2 g^*() ^2 )_+ ℓ_d('),

for all ≤_0. Now we finish the argument. The change of variables 
=̆  [D^2g^* ()]^1/2 (-')
 yields

    ∫ m_(, ') dρ(')   = ρ_0() (1+o(1))/(D^2g^* ())^1/2∫_Ω_0^δ( C_()-^2/2)_+ dℓ_d()̆
       = ρ_0() (1+o(1))/(D^2g^* ())^1/2 (ρ_0() ρ_1[∇ g^*()]))^1/2.

The relation (D^2g^* ())= ρ_0() /ρ_1[∇ g^*()] yields
∫ m_ (·, ') ρ_0(') -1_𝒞(Ω_0^δ)→ 0,
which in turn implies
ρ_ε -1_𝒞(Ω_0^δ)→ 0. 


<cit.> established the existence and uniqueness (up to additive constants) of a convex function ϕ_ε, for every ε, such that its gradient ∇ϕ_ε pushes forward the measure ρ_ερ_0 to ρ_0 That is, 
∫  f(∇ϕ_)  ρ_ρ_0= ∫ f  ρ_0, for all f∈𝒞(Ω_0).
In particular ϕ_ solves the boundary value problem 

    (∇^2 ϕ_)=ρ_0 ρ_/ρ_0(∇ϕ_)    in Ω_0,   ∇ϕ_ (Ω_0)= Ω_0

and its convex conjugate ϕ_^* solves 

    (∇^2 ϕ_^*)=ρ_0 /ρ_0(∇ϕ_^*) ρ_(∇ϕ_^*)   in Ω_0,   ∇ϕ_^* (Ω_0)= Ω_0.

Combining Theorem <ref> with Lemma <ref>, we can ensure the existence of _δ>0 such that

    ∇ϕ__𝒞^1, α(Ω_0^δ),  ∇ϕ_^*_𝒞^1, α(Ω_0^δ)≤ C(δ),   for all ≤_δ.

Therefore, standard stability theory (see e.g., <cit.>) and the Arzelà–Ascoli theorem yield

    ∇ϕ_^*- Id_𝒞^1(Ω_0^δ),  ∇ϕ_- Id_𝒞^1(Ω_0^δ)→ 0,   as → 0.

This limit is crucial to show the following result. 


    Let Assumption <ref> holds. There exists δ_0 such that for all  0<δ≤δ_0 there exists a function ω=ω_δ, such that ω()→ 0 as → 0, and C=C(δ)>0 such that
    
    |  D(∇ϕ_^*(),∇ g^*(∇ϕ_^*(')))- -'_∇^2 g^*(')^2 |≤   C -'^2( -'^α+ω()),

    and 
    
    |∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(∇ϕ_^*('))^2- -'_∇^2 g^*(')^2 |≤   C -'^2( -'^α+ω()),

    for all , '∈Ω^δ_0={∈Ω_0:   dist  (, ∂Ω_0)≥δ}.


We call ω()= ∇ϕ_^*- Id_𝒞^1(Ω_0^δ)^α/2, which, by (<ref>), tends to 0 as → 0. 
    Lemma <ref> first and then equation (<ref>) imply that 

    D(∇ϕ_^*(),∇ g^*(∇ϕ_^*(')))
       = 1/2∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(∇ϕ_^*('))^2 +𝒪( ∇ϕ_^*() -∇ϕ_^*('))^2+α)
       = 1/2∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(∇ϕ_^*('))^2 +𝒪( -'^2+α)

for all , '∈Ω_0^δ. Hence, the proof will be complete after showing that 

    |∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(∇ϕ_^*('))^2--'_∇^2 g^*(')^2|≤ C ω() -'^2 .

for all , '∈Ω_0^δ and some C>0. 
First we exchange ∇^2 g^*(∇ϕ_^*(')) by ∇^2 g^*('). 
Since ∇^2 g^* is Hölder continuous,  we obtain 

    |∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(∇ϕ_^*('))^2- ∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(')^2|≤ω() -'^2.

Then we exchange ∇ϕ_^* by the identity 

    ∇ϕ_^*()    -∇ϕ_^*(')_∇^2 g^*(')^2 
       =  [∇ϕ_^*- Id]() -[∇ϕ_^*- Id](')_∇^2 g^*(')^2+ -'_∇^2 g^*(')^2
            + 2⟨ [∇ϕ_^*- Id]() -[∇ϕ_^*- Id]('),   -'⟩_∇^2 g^*(')
       ≤-'_∇^2 g^*(')^2(1+3ω()+2ω() ),

which implies (<ref>)  and concludes the proof. 




The following result finds a measure μ_ with the correct marginals and provides the exact description of its density. 


 Let Assumption <ref> hold and   assume further that Ω_0 is convex. Define 
    μ_:= (∇ϕ_×∇ϕ_)_#(ξ_ (ρ_0 ⊗ρ_0))

 and
 
    u_(, '):= μ_/ (ρ_0⊗ρ_0)(, ')=    ξ_(∇ϕ_^*(), ∇ϕ_^*('))  /ρ_(∇ϕ_^*())ρ_(∇ϕ_^*(')).

 Then μ_ε belongs to Π(ρ_0, ρ_0) and, as a consequence, the measure 
π_^(1)=( Id×∇ g )_#μ_,
belongs to Π(ρ_0, ρ_1) and it
has density 

    π_^(1)/ (ℓ_d⊗ℓ_d)(, )= u_(,∇ g ()) ρ_0() ρ_1().



    We can find the shape of its density by computing the following integral for an arbitrary bounded continuous function f:

    ∫ f(∇ϕ_(), ∇ϕ_('))  ξ_(, ')   (ρ_0 ⊗ρ_0)(, ')
       = ∫
            f(, ' )  ξ_(∇ϕ_^*(), ∇ϕ_^*('))  ρ_0(∇ϕ_^*()) ρ_0(∇ϕ_^*(') )
         /[(∇^2ϕ_^*())(∇^2ϕ_^*('))]^-1ℓ_d() ℓ_d(')
       =∫f(, ')  ξ_(∇ϕ_^*(), ∇ϕ_^*('))  ρ_0(∇ϕ_^*()) ρ_0(∇ϕ_^*(') )/ρ_0(∇ϕ_^*()) ρ_0(∇ϕ_^*(')) ρ_(∇ϕ_^*())ρ_(∇ϕ_^*('))ρ_0() ρ_0(')
    
       =∫f(, ')  ξ_(∇ϕ_^*(), ∇ϕ_^*('))  /ρ_(∇ϕ_^*())ρ_(∇ϕ_^*('))ρ_0() ρ_0(').

Since the latter holds for any continuous function f,  (<ref>) holds. 

It is easy to check that by construction 

    ∫ f() u_(, ') ρ_0() ρ_0(')   =∫ f(∇ϕ_())  ξ_(, ')   (ρ_0 ⊗ρ_0)(, ')
       =∫ f(∇ϕ_())    ρ_()
       =∫ f()    ρ_0(),

so that u_(, ')  (ρ_0 ⊗ρ_0)(, ') belongs to Π(ρ_0, ρ_0). 

Now we prove some important properties of u_.



    Let Assumption <ref> hold and   assume further that Ω_0 is convex. 

Then there exists δ_0>0 such that 
for every δ∈ (0, δ_0) there exists _δ such that 

    ∫_Ω^δ_0  u_(, · ) dρ_0()_𝒞(Ω^δ,c_0)≤2/3

for all ≤_δ. 


Due to (<ref>), there exists δ_0 such that for all δ≤δ_0 there exists  _δ such that 

    {'∈Ω_0: ∃ ∈Ω_0^δ/2:   m_(, ')>0 }⊂Ω_0^δ/4

for all ≤_δ. Equation (<ref>) implies that, for a maybe smaller _δ, 

    ∇ϕ_^*( Ω_0^δ)⊂Ω_0^δ/2,     ∇ϕ_( Ω_0^δ/4)⊂Ω_0^δ/8   and  ∇ϕ_^*( Ω_0^δ/8)⊂Ω_0^δ/16

for all ≤_δ. As a consequence, 

    𝕎_,δ   :={'∈Ω_0: ∃ ∈Ω_0^δ :   m_(∇ϕ_^*(), ')>0 }
        =   {'∈Ω_0: ∃ ∈∇ϕ_^*(Ω_0^δ):   m_(, ')>0 }
       ⊂{'∈Ω_0: ∃ ∈Ω_0^δ/2:   m_(, ')>0 }
       ⊂Ω_0^δ/4

and 

    {'∈Ω_0: ∃ ∈Ω_0^δ :   m_(∇ϕ_^*(), ∇ϕ^*('))>0 }=∇ϕ (  𝕎_,δ  ) ⊂∇ϕ ( Ω_0^δ/4  ) ⊂Ω_0^δ/8.

As a consequence, 
∫_Ω^δ_0  u_(, ' ) dρ_0()=0 for every ' ∈Ω_0^δ/8,c and ≤_δ. Therefore, we can assume that ' ∈Ω_0^δ/8, ∈Ω_0^δ and ∇ϕ_^*(') , ∇ϕ_^*() ∈Ω_0^δ/16. Due to Lemma <ref>, <ref>,

    u_(, ')= ξ_(∇ϕ_^*(), ∇ϕ_^*('))  / 1+o(1) = m_(∇ϕ_^*(), ∇ϕ_^*('))  / 1+o(1) ,

where the error term is uniform for , '∈Ω_0^δ/8. On the other hand,

    | C_(∇ϕ_ε^*())-C_()|≤^2/d+2∇ϕ_^*- Id_𝒞^1(Ω_0^δ)^α =o( ^2/d+2),

where the last term is a consequence of (<ref>).  We observe that for , '∈Ω_0^δ/8 such that the inequality 
m_(∇ϕ_^*(), ∇ϕ_^*('))>0 holds,
equation (<ref>) implies
∇ϕ_^*()-∇ϕ_^*(')^2 ≤ C ^2/d+2,
and, by (<ref>), also 

    -'^2 ≤ C ^2/d+2.

 Hence, (<ref>) yields 

    -'^2_∇^2 g^*()= -'^2_∇^2 g^*(')  + o( ^2/d+2) ,

where the error term is uniform for , '∈Ω_0^δ/8 such that  m_(∇ϕ_^*(), ∇ϕ_^*('))>0. Hence, the previous display, (<ref>) and (<ref>) imply 

    m_(∇ϕ_^*(), ∇ϕ_^*(')) = 1/( C_(')-1/2-'^2_∇^2 g^*(')+ o( ^2/d+2))_+

for all , '∈Ω_0^δ/8 such that  m_(∇ϕ_^*(), ∇ϕ_^*('))>0.
This,  combined with (<ref>), yields

    ∫_Ω_0^δ  u_(, ') ρ_0()
       = ∫_Ω_0^δ  m_(∇ϕ_^*(), ∇ϕ_^*(')) ρ_0()  / 1+o(1) 
       ≤1/∫_Ω_0^δ( C_(')-1/2-'^2_∇^2 g^*(')+ o( ^2/d+2))_+ ρ_0()  / 1+o(1) 
    
       ≤1/  (ρ_0(')+o(1))∫_Ω_0^δ( C_(')-1/2-'^2_∇^2 g^*(')+ o( ^2/d+2))_+ ℓ_d()  / 1+o(1) .

Changing variables =̆ T()= [∇^2 g^*(')]^1/2 (-') to get (up to uniform o(1) terms) 

    ∫_Ω_0^δ   u_(, ') ρ_0() ≤1/ε·ρ_0(')/(∇^2 g^*('))^1/2∫_T(Ω_0^δ)( C_(')-^2/2)_+ ℓ_d().

Without losing generality we can assume that '= 0. Then 0∉ T(Ω_0^δ). Since T(Ω_0^δ) is convex, there exist a vector $̌ and a  hyperplaneH={ : ⟨, ⟩̌=0 } such thatT(Ω_0^δ)⊂H_+:={ : ⟨, ⟩̌≤0 } . As a consequence, up too(1)additive terms,

    ∫_Ω_0^δ   u_(, ') ρ_0()    ≤1/ε·ρ_0(')/(∇^2 g^*('))^1/2∫_H_+( C_(')-^2/2)_+ ℓ_d()
       = 1/ε·ρ_0(')/2 (∇^2 g^*('))^1/2∫( C_(')-^2/2)_+ ℓ_d()
    (by Proposition <ref>)    =1/ε·ρ_0(')/2 (∇^2 g^*('))^1/2|C_(')|^d+2/2 C_d 
       =1/ε·ρ_0(')/2 (∇^2 g^*('))^1/2( ^2/d+2/C_d^2/d+2(ρ_0(')ρ_1[∇ g^*(')])^1/d+2)^d+2/2 C_d
       =ρ_0(')/2 (∇^2 g^*('))^1/21/(ρ_0(')ρ_1[∇ g^*(')])^1/2.

Since,(∇^2 g^*)=ρ_0/ρ_1(∇g^*) , we get∫_Ω_0^δ   u_(, ') ρ_0()≤1/2+o(1), and the claim follows. 



 §.§ Valid coupling for the frame Ω_0^δ,c of a convex Ω_0

Forδ>0, we recall the definition (<ref>) and consider the truncated function

    u_χ_(Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0),

in an effort to avoid the problematic regions ofΩ_0×Ω_0where the candidate for coupling is not regular enough. Notice how we allow one of the variables,for example, to be close to the boundary as long as the other one,', is not. In principle, this may seem inadvisable; however, there is no issue since the support of the coupling will concentrate around the set='. Refer to the proof of Lemma <ref>  above or that of Lemma <ref> below for further details. Therefore,  the only truly problematic set is(Ω^δ, c_0×Ω^δ, c_0).

Thus, in seeking a true coupling in(Ω_0×Ω_0)we search for a densityv_such  that1=∫  v_(, ') ρ_0()= ∫  v_(, ') ρ_0('),   for all , '∈Ω_0,and we want it of the formv_= u_χ_(Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0)+  h_ε, δχ_(Ω^δ,c_0×Ω^δ,c_0 ),for some positive functionh_, δ. Irrespective of h_ε, δ, for'∈Ω^δ_0the “coupling” condition1= ∫ v_(,') dρ_0()must hold. Hence, we need only to fith_ε, δsuch that 
that1= ∫_Ω^δ,c_0   h_ε, δ(, ') dρ_0() + ∫_Ω^δ_0  u_(, ') dρ_0(),   for all '∈Ω^δ,c_0and 

    1= ∫_Ω^δ,c_0   h_ε, δ(, ') dρ_0(') + ∫_Ω^δ_0  u_(, ') dρ_0('),   for all ∈Ω^δ,c_0.

We find firsth_, δsuch that condition (<ref>) holds and then we construct a coupling based on this. In order to simplify the notation we defineq_δ():=1-(∫_Ω^δ_0  u_(, ') dρ_0('))χ_Ω^δ,c_0() ∈ (0,1].Note that due to Lemma <ref>, we can assume thatεis sufficiently small so that

    1/3≤ q_δ(𝐱) ≤ 1,   for all ∈Ω^δ, c.

Next, we show that (<ref>) holds forΩ^δ, c.

    Let Assumption <ref> hold and Ω_0 be convex.  there exist δ_0>0 and θ∈ (0,π) such that for all δ≤δ_0 and ∈Ω^δ,c there exists a cone
    𝒞_,  v,r, θ
    with vertex ,  height r=r_δ>0 and angle θ such that 𝒞_,  v,r, σ⊂Ω^δ,c.  

    Let δ_0 be such that Ω_0^δ exists and is convex and nonempty,  δ≤δ_0 and  r'_δ= δ/4. Then, for ∈Ω^δ,c_0 and r' ≤ r_δ', the ball 𝔹(𝐱, r') intersects at most one of ∂Ω_0^δ or ∂Ω_0, but not both simultaneously. In the last case, the claim holds by Assumption <ref>. Therefore, we can assume that 𝔹(𝐱, r') ∩∂Ω_0=∅. 
 Since Ω_0^δ is convex and ∉Ω_0^δ, there exists a separating half-space H_+={: ⟨,  v⟩≥ a} such that ∈ H_+ and Ω_0^δ∩  H_+=∅. Therefore, 
 𝔹(𝐱, r'  ) ∩ H_+⊂𝔹(𝐱, r  ) ∩Ω^δ,c_0,
for every r'≤ r_δ. Setting θ=π/4 and 
 r=max{ x_1:  (x_1, …, x_d)≤ r_δ'    and  (x_1, …, x_d)cos(1/2θ) ≤ 
x_1 }>0
the result follows. 


  

Due to Lemma <ref> and (<ref>), the conditions of Lemma <ref> hold fors=q_δandΩ=Ω^δ,c. As a consequence, forsmall enough there existsc_,δ:Ω^δ,c →such thatq_δ()=1/∫_Ω^δ,c_0 (c_,δ ()-D(, ∇ g^*(') )_+ρ_0(') ,   for all ∈Ω^δ,c_0.Moreover,
there existsC=C(δ_0)such that for everyδ≤δ_0there exists_δ>0such that for≤_δ, the functionc_, δexists,  

     
    ε^2/d+2 C^-1≤  c_, δ() ≤ε^2/d+2 C,    ε^d/d+2 C^-1≤ℓ_d(𝒮_ε,δ, )≤ Cε^d/d+2, 
       and
       1/∫_Ω^δ,c_0 (c_, δ ()-D(, ∇ g^*(') )_+ρ_0() ≥ C^-1,

     for every'∈Ω_0^δ,cwhere𝒮_ε,δ, = { '∈Ω_0:  c_, δ ()-D(, ∇g^*(')  ≥0}.To simplify the exposition, we first introduce a few recurrent quantities. Define

    M_ε, δ (𝐚, 𝐛) := (c_ε, δ(𝐚) - D(𝐚, ∇ g^*(𝐛))_+/ε,

and the density 

    h_ε, δ(𝐱, 𝐱') := ∫_Ω_0^δ,cM_ε, δ(𝐱', 𝐳) M_ε, δ(𝐱, 𝐳)/∫_Ω_0^δ,c M_ε, δ(𝐯', 𝐳)   dρ_0(𝐯')  dρ_0(𝐳),

with the associated measure given by

    dπ_ε^(2)(𝐱, 𝐱') = h_ε, δ(𝐱, 𝐱')   dρ_0(𝐱)   dρ_0(𝐱').

 Let Assumption <ref> hold. Then  
     q_δ()= ∫_Ω^δ,c_0  h_ε, δ(, ')ρ_0(')    and    q_δ(')= ∫_Ω^δ,c_0   h_ε, δ(, ')ρ_0(). 
    As a consequence, 
    v_ (ρ_0 ⊗ρ_0)∈Π(ρ_0, ρ_0),
    where
    
    v_= [u_χ_(Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0)+ h_, δχ_(Ω^δ,c_0×Ω^δ,c_0 ) ]
    
    Set 
    and compute 
    
    ∫_Ω^δ,c_0  h_ε, δ(, ')ρ_0(')   = ∫_Ω^δ,c_0∫_Ω^δ,c_0 M_, δ (', )  M_, δ (, )  /∫_Ω^δ,c_0 M_, δ('̌, ) ρ_0('̌) ρ_0()  ρ_0(')
       =   ∫_Ω^δ,c_0∫_Ω^δ,c_0M_, δ (', )  ρ_0(') M_, δ (, )  /∫_Ω^δ,c_0 M_, δ('̌, ) ρ_0('̌) ρ_0()  
       = ∫_Ω^δ,c_0 M_, δ (, )  ρ_0() 
       =q_δ().

The same holds for the other variable by symmetry. 

We give some estimates on the function h_, δ(, '), which will be useful for the sequel.  

    Let assumption <ref> hold. Then there there exists  δ_0>0 and C>0 such that for every δ∈ (0, δ_0)  there exist _δ
such that  

    ∫_Ω^δ,c_0×Ω^δ,c_0 D(,∇ g^*(')) h_, δ(, ')   (ρ_0⊗ρ_0)(, ') ≤ C  ℓ_d(Ω_0^δ,c ) ^2/d+2 ,

and 

    /2∫_Ω^δ,c_0×Ω^δ,c_0   (h_(, '))^2 (ρ_0⊗ρ_0)(, ')  ≤ C ℓ_d(Ω_0^δ,c ) ^2/d+2 ,

for all ≤_δ.
 

By (<ref>),  it is sufficient to verify (<ref>) and (<ref>) by substituting  h_, δ by  M_, δ. 



First we give some estimates on the support of M_,δ. The set of (, ') such that  ∫_Ω^δ,c_0 M_, δ (', )  M_, δ (, )  ρ_0()>0 is called 
A_δ, . It can be also described as the set of (, ') such that  there exists ∈Ω_0^δ,c belonging to supp(M_, δ (, · ))∩ supp(M_, δ (', · )), i.e.,  satisfying
M_, δ (', )  M_, δ (, )>0.
Recall that 
∫_Ω^δ,c_0 M_, δ (', )  M_, δ (, )  ρ_0()>0. 
(<ref>) implies that there exists C>0 such that 
supp(M_, δ (, · ))⊂𝔹(, C^2/d+2),
so that 
A_δ,⊂{ (, ')∈ (Ω^δ,c_0)^2:  𝔹(, C^2/d+2)∩𝔹(', C^2/d+2)≠∅}.
Therefore, by increasing C, we have 

    A_δ,⊂{ (, ')∈ (Ω^δ,c_0)^2:  '∈𝔹(, C^2/d+2)}.

Now we give some estimates on the maximum value that h_, δ can take. 
According to  (<ref>),  there exists δ_0>0 and C(δ_0)>0 such that for every δ∈ (0, δ_0)  there exist _δ
such that for every ≤_δ, it holds that 
M_, δ (', )  ≤   C(δ_0)^-d/d+2. 
Hence,  

    ∫_Ω^δ,c_0 M_, δ (', ) M_, δ (, ) dρ_0()   ≤ C(δ_0)^-d/d+2∫_Ω^δ,c_0 M_, δ (, ) ρ_0()
        = C(δ_0)^-d/d+2 q_δ()
       ≤  C(δ_0)^-d/d+2,

which implies 

    max_h_, δ(', )≤ C(δ_0) ^-d/d+2.

At this point, we have all the ingredients to prove (<ref>) and (<ref>). Firstly,  
we find an upper bound for (<ref>) as follows: 

    /2∫_Ω^δ,c_0×Ω^δ,c_0     (h_(, '))^2 (ρ_0⊗ρ_0)(, ') 
       ≤/2∫_Ω^δ,c_0max_h_, δ(', )∫_Ω^δ,c_0  h_, δ(, ')ρ_0() ρ_0(')  
       = /2∫_Ω^δ,c_0max_h_, δ(, ) q_δ()ρ_0(') 
       ≤/2∫_Ω^δ,c_0max_h_, δ(', )ρ_0(') 
    (by (<ref>))    ≤C(δ_0) ^2/d+2/2Λℓ_d(Ω^δ,c_0).

Hence, (<ref>) holds. To prove (<ref>), we use first Lemma <ref> and then (<ref>)  to obtain 

    ∫_Ω^δ,c_0×Ω^δ,c_0    D(,∇ g^*(')) h_, δ(, ')  (ρ_0⊗ρ_0)(, ') 
       ≤σ_M(g)/2∫_Ω^δ,c_0×Ω^δ,c_0-'^2 h_, δ(, ')   (ρ_0⊗ρ_0)(, ')
       ≤ C(δ_0, g) ^2/d+2∫_Ω^δ,c_0×Ω^δ,c_0  h_, δ(, ')   (ρ_0⊗ρ_0)(, ').

Since, via Lemma <ref>, we have
∫_Ω^δ,c_0 h_, δ(, ')  ρ_0() =q_δ(') ≤ 1,
wich yields

    ∫_Ω^δ,c_0×Ω^δ,c_0    D(,∇ g^*(')) h_, δ(, ')  (ρ_0⊗ρ_0)(, ') ≤ C(δ_0, g) ^2/d+2Λℓ_d(Ω^δ,c_0 ).

The result follows. 


 §.§ Limit for convex supports


Let us define the measure ν_ε(,') = v_(, ')   (ρ_0⊗ρ_0)(, '),
meaning that if we defineπ_ε:=(Id ×∇g^*)_# ν_then we readily obtain

    π_(x,y) =(Id ×∇ g^*)_#ν_=v_(, ∇ g())   (ρ_0⊗ρ_1)(, ).


In this section, we find the limit of ^-2/d+2 (ℍ_( π_)-𝒲_2^2(ρ_0, ρ_1)), whereℍ_ ( π)= ∫-^2 π(,)   (ρ_0⊗ρ_1)(,) +  π/ (ρ_0⊗ρ_1)_L^2(ρ_0⊗ρ_1)^2,in the case whereΩ_0is a convex domain, which will be provided by the following asymptotic bound.  

    Let Assumption <ref> hold and Ω_0 be convex. Then 
    lim sup_→ 0^+𝒯_2, ε, (·)^2(ρ_0, ρ_1) - 𝒲_2^2(ρ_0, ρ_1)/^2/d+2≤d^d+4/d+2(d+2)^2/d+2/(ℋ^d-1(𝒮^d-1) )^2/d+2∫(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0(). 
First, as in the proof of Lemma <ref>, there exists δ_0 such that for all δ≤δ_0 there exists  _δ such that  

    {'∈Ω_0: ∃ ∈Ω_0^δ :   m_(∇ϕ_^*(), ∇ϕ^*('))>0 }⊂Ω_0^δ/8, 
       ∇ϕ_^*( Ω_0^δ)⊂Ω_0^δ/2,     ∇ϕ_^*( Ω_0^δ/4)⊂Ω_0^δ/8   and  ∇ϕ_^*( Ω_0^δ/8)⊂Ω_0^δ/16

for all ≤_δ. Therefore, from now on we assume that ≤_δ for 0<δ≤δ_0. 
We analyze first the linear term, where π_0 is the minimizer of (<ref>). By duality (<ref>), it holds that 

    1/2∫-^2  (π_-π_0)(,)
    =  ∫1/2-^2 π_(,) - ∫(1/2^2-g^*()) ρ_0()-∫(1/2^2-g())  ρ_1().

Since π_∈Π(ρ_0, ρ_1), we get 

    1/2∫-^2  (π_-π_0)(,)   =∫ D(, ) π_(,)
        
       =∫ D(, ∇ g^*(')) ν_ (,').

We observe that 

    ∫ D(,∇ g^*(')) ν_(,') 
       = ∫_(Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0) D(,∇ g^*(')) u_(, ')   (ρ_0⊗ρ_0)(, ')_𝕃_1
            + ∫_(Ω^δ,c_0×Ω^δ,c_0) D(,∇ g^*(')) h_, δ(, ')   (ρ_0⊗ρ_0)(, ')_𝕃_2 .

We denote the first term by 𝕃_1 and the second by 𝕃_2. To bound 𝕃_2, Lemma <ref> ensures that there exist δ_0 > 0 and C > 0 such that for every δ∈ (0, δ_0), there is an ε_δ satisfying   

    𝕃_2=∫_(Ω^δ,c_0×Ω^δ,c_0) D(,∇ g^*(')) h_, δ(, ')   (ρ_0⊗ρ_0)(, ')≤ Cρ_0(Ω_0^δ) ^2/d+2

for every ≤_δ.
For 𝕃_1 we observe that Lemma <ref> (point  <ref>), (<ref>) and (<ref>), imply  that
𝕃_1≤1/1+o(1)∫_(Ω^δ'_0×Ω^δ'_0) D(∇ϕ_^*(),∇ g^*(∇ϕ_^*('))) m_(, ')   (ρ_0⊗ρ_0)(, '),  
for some 0<δ'< δ. 
By 
Lemma <ref>, we have 
|  D(∇ϕ_^*(),∇ g^*(∇ϕ_^*(')))- 1/2-'_∇^2 g^*(')^2 |≤   C -'^2( -'^α+ω()),  
for all , '∈Ω_0^δ',
where ω()→ 0 as → 0.



 
  

As a consequence, up to o(1) terms, we have the bound

    𝕃_1   ≤∫_(Ω^δ'_0)^21/2-'_∇^2 g^*(')^2 m_(, ')   (ρ_0⊗ρ_0)(, ')
            + C ∫1/2-'^2 ( -'^α+ω())  m_(, ')   (ρ_0⊗ρ_0)(, '),

where, due to (<ref>), the last term is of order  o(ε^2/d+2). Therefore, it holds that

    𝕃_1  ≤∫_(Ω^δ'_0)^21/2-'_∇^2 g^*(')^2 m_(, ')   (ρ_0⊗ρ_0)(, ') + o(ε^2/d+2).






 Lemma <ref> yields the estimate
 
    m_(, ')=1/( C_(') -1/2-'_∇^2 g^*(')^2)_+ + ω() ^-d/d+2,

which implies

    𝕃_1 
        ≤^-1∫_(Ω^δ'_0)^21/2-'_∇^2 g^*(')^2 ( C_(') -1/2-'_∇^2 g^*(')^2)_+   (ρ_0⊗ρ_0)(, ')
    + ω() ^-d/d+2∫_ supp(m_)1/2-'_∇^2 g^*(')^2   (ρ_0⊗ρ_0)(, ')
        +o(^2/d+2).

The bound (<ref>) implies 

    𝕃_1 
        ≤^-1∫_(Ω^δ'_0)^21/2-'_∇^2 g^*(')^2 ( C_(') -1/2-'_∇^2 g^*(')^2)_+   (ρ_0⊗ρ_0)(, ')
    
        +o(^2/d+2).

By adding and subtracting C_(') inside the integral, the bound

    𝕃_1    ≤    -^-1∫_(Ω^δ'_0)^2( C_(') -1/2-'_∇^2 g^*(')^2)_+^2  (ρ_0⊗ρ_0)(, ')
           
         + ^-1∫_(Ω^δ'_0)^2 C_(') ( C_(') -1/2-'_∇^2 g^*(')^2)_+  (ρ_0⊗ρ_0)(, ')_𝔻 +o( ^2/d+2)

follows. 
By using   (<ref>) and Lemma <ref> (point  <ref>) again we get 

    𝔻≤∫ C_(') u_(, ')  (ρ_0⊗ρ_0)(, ')
    
        + w() ^-d/d+2∫_C(')≥1/2-'^2 C_(')  (ρ_0⊗ρ_0)(, ') + o( ^2/d+2),

which yields 
𝔻≤∫ C_(') ρ_0(')+ o( ^2/d+2)
and  a fortiori
    𝕃_1 ≤    -^-1∫_(Ω^δ'_0)^2( C_(') -1/2-'_∇^2 g^*(')^2)_+^2  (ρ_0⊗ρ_0)(, ')
    
        + ∫ C_(') ρ_0(')  +o( ^2/d+2)



Now we deal with the penalty term. As before, we divide 
 π/ (ρ_0⊗ρ_1)_L^2(ρ_0⊗ρ_1)^2=∫ (v_(, '))^2 (ρ_0⊗ρ_0)(, ') 

into two additive terms
ℚ_1= ∫_(Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0)  (u_(, '))^2 (ρ_0⊗ρ_0)(, ') 
and  

    ℚ_2= ∫_Ω^δ,c_0×Ω^δ,c_0  (h_, δ(, '))^2 (ρ_0⊗ρ_0)(, ')≤   C^2/d+2ℓ_d(Ω^δ,c_0),

where the bound is derived from Lemma <ref>, and the constant C is independent of δ.
 We bound now ℚ_1.  
 Since C_() is Hölder continuous with constant C^2/d+2 and owing to (<ref>), it holds that 
 
    C_(∇ϕ_^*()) - C_()≤ C ^2/d+2ω() ,    with ω()→ 0.

  By 
Lemma <ref>, we also have 
|∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(∇ϕ_^*('))^2- -'_∇^2 g^*(')^2 |≤ C -'^2( -'^α+ω()) 
and 
|∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(∇ϕ_^*())^2- -'_∇^2 g^*()^2 |≤ C -'^2( -'^α+ω()). 
 Since ξ_(∇ϕ_^*(), ∇ϕ_^*('))>0 implies
 
    C^1/d+2   ≥∇ϕ_^*()-∇ϕ_^*(')
       ≥-'-[∇ϕ_^*- Id]()-[∇ϕ_^*- Id](')
       ≥-'(1-∇ϕ_^*- Id_𝒞^1(Ω_0^δ)),


 were we used the lower bound on ∇^2 g^*() and the fact that all norms on ℝ^d are equivalent (which is hidden in the constant).
 Now, 
 using (<ref>) we get
 
    |∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(∇ϕ_^*('))^2- -'_∇^2 g^*(')^2 |≤ C ω() ^2/d+2

and 

    |∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(∇ϕ_^*())^2- -'_∇^2 g^*()^2 |≤  Cω() ^2/d+2,

for all (, ')∈ (Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0) such that ξ_(∇ϕ_^*(), ∇ϕ_^*('))>0 and
a function ω=ω_δ such that ω()→ 0 as → 0.  Since ∇^2 g^* is uniformly continuous, 
by the same arguments as before, 
|-'_∇^2 g^*(')^2 - -'_∇^2 g^*()^2 |≤ω() ^2/d+2,  
for all (, ')∈ (Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0) such that ξ_(∇ϕ_^*(), ∇ϕ_^*('))>0. Therefore, this last fact combined with (<ref>), (<ref>) and (<ref>) implies 
ξ_(∇ϕ_^*(), ∇ϕ_^*(')) 
    ≤1/(C_()-1/2-'_∇^2 g^*(')^2 
    +ω() ^2/d+2)_+.  

for some  function ω=ω_δ such that ω()→ 0 as → 0. 
This combined with the point <ref> of
Lemma <ref> and (<ref>) yields

    u_(, ')   =ξ_(∇ϕ_^*(), ∇ϕ_^*('))  /ρ_(∇ϕ_^*())ρ_(∇ϕ_^*('))
       ≤1/(1+o(1))(C_()-1/2-'_∇^2 g^*()^2 
        +ω() ^2/d+2)_+.  
       ≤1/(1+o(1))(C_()-1/2-'_∇^2 g^*()^2 
         )_+ +ω() ^-d/d+2,

where the the error terms are uniform on the set of (, ')∈ (Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0) such that 
ξ_(∇ϕ_^*(), ∇ϕ_^*('))>0
and a fixed 0<δ≤δ_0. 
As a consequence,  it holds — up to o(^2/d+2) additive terms, that

    ℚ_1   = ∫_(Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0)  (u_(, '))^2 (ρ_0⊗ρ_0)(, ') 
       ≤∫_(Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0)(C_()-1/2-'_∇^2 g^*()^2 
         )_+^2 (ρ_0⊗ρ_0)(, ')+ o(^2/d+2)
       ≤∫_(Ω^δ'_0)^2(C_()-1/2-'_∇^2 g^*()^2 
         )_+^2 (ρ_0⊗ρ_0)(, ')+ o(^2/d+2),
 
for 0<δ'<δ. 
The last display and (<ref>) yield 

    2 𝕃_1 +ℚ_1 ≤    -^-1∫_(Ω^δ'_0)^2( C_(') -1/2-'_∇^2 g^*(')^2)_+^2  (ρ_0⊗ρ_0)(, ')
    +
    2∫ C_(') ρ_0(') +o( ^2/d+2),

while  (<ref>) and (<ref>) give 
2 𝕃_2+ℚ_2≤ C^2/d+2ℓ_d(Ω^δ,c_0).
From the relation 

    ℍ_ ( π_)- 𝒲_2^2(ρ_0, ρ_1)=2 𝕃_1+2 𝕃_1+ℚ_1+ℚ_2
 we get the estimate

    ^-2/d+2((ℍ_ ( π_)- 𝒲_2^2(ρ_0, ρ_1) )
    ≤ 2^-2/d+2∫  C_(') ρ_0(') - ^d+4/d+2∫_(Ω^δ_0)^2(C_()-1/2-'_∇^2 g^*()^2 
         )_+^2   (ρ_0⊗ρ_0)(, ')_from 2𝕃_1+ℚ_1
    
       + C ℓ_d(Ω^δ,c_0)_from 2𝕃_2+ℚ_2+o(1)

and we conclude as in the proof of the lower bound. 

We now use the bound

    ^d+4/d+2/2∫_(Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0)(C_()-1/2-'_∇^2 g^*()^2 
         )_+^2   (ρ_0⊗ρ_0)(, ')
    ≥^d+4/d+2/2∫_(Ω^δ_0×Ω_0)(C_()-1/2-'_∇^2 g^*()^2 
         )_+^2   (ρ_0⊗ρ_0)(, ')

and observe that for fixed δ>0 and   small enough the latter is equal to
^d+4/d+2/2∫_Ω^δ_0×^d (C_()-1/2-'_∇^2 g^*()^2 
     )_+^2   (ρ_0⊗ρ_0)(, ').
Solving the integral as in (<ref>), we obtain 

    ^d+4/d+2∫_Ω^δ_0×^d (C_()-1/2-'_∇^2 g^*()^2 
         )_+^2   (ρ_0⊗ρ_0)(, ') 
    ≍ 2/d+4C_d^ -2/d+2∫_Ω_0^δ(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0().
As a consequence, 

    ^-2/d+2(ℍ_ ( π_)- 𝒲_2^2(ρ_0, ρ_1)≤4/d+4C_d^ -2/d+2∫_Ω_0^δ(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0()
    
       +^2/d+2+ C ℓ_d(Ω^δ,c_0)+o(1),


 §.§ General case

The aim of this section is to prove Theorem <ref> in the general case, namely, to eliminate the assumption of convex support. To this end, we will construct the aforementioned “stained glass” structure. Let T_δ={I_1}_i∈  be a regular tessellation of squaresI_iof same length β>0 of^d.  Set𝒢(β)={ i∈:  I_i⊂Ω_0}and note that it is finite due to the compactness ofΩ_0. It is clear that we can writeΩ_0as the union ofΩ^(β)_0=  ⋃_i∈𝒢(β) I_i andΩ^(β,c)_0=Ω∖⋃_i∈𝒢(β) I_i. We define the measureπ_β, =∑_i∈𝒢(β)π_i^χ_(I_i×∇ g^* (I_i)) + γ_β, χ_(Ω_0^(β,c)×Ω_0^(β,c)),where π_i^ ∈Π(χ_I_iρ_0, χ_[∇g^*(I_i)] ρ_1) is the true minimizer of1/2∫-^2dπ(,)+ε/2dπ/  d(ρ_0 ⊗ρ_1) _L^2(χ_I_iρ_0, χ_[∇ g^*(I_i)]ρ_1)^2,among allπ∈Π(χ_I_iρ_0, χ_[∇g^*(I_i)] ρ_1) , and γ_β,  ∈Π(χ_Ω_0^(β,c)ρ_0, χ_∇^2 g^*(Ω_0^(β,c)) ρ_1) to be determined. By construction, π_β, ∈Π(ρ_0, ρ_1) .Moreover, since∇gis the gradient of a convex function and  pushes χ_[∇g^*(I_i)]ρ_1forward toχ_[I_i] ρ_0, it holds that𝒲_2^2(ρ_0,ρ_1)=𝒲_2^2(χ_[Ω_0^(β,c)]ρ_0,  χ_[∇ g^*(Ω_0^(β,c))]ρ_1)+∑_i∈𝒢(β)𝒲_2^2(χ_[I_i]ρ_0, χ_[∇ g^*(I_i)]ρ_1).It also holds that 

    ℍ_ ( π_β, )= ∑_i∈𝒢(β)𝒯_2, , (·)^2(χ_[I_i]ρ_0, χ_[∇ g^*(I_i)]ρ_1)+ 
    ∫-^2 γ_β, (,) + ·γ_β, / (ρ_0⊗ρ_1)_L^2(ρ_0⊗ρ_1)^2.

We define γ_β, following the same steps as π_^(2) in the previous section. Note thatΩ_0^(β,c)satisfies the assumptions of    Lemma <ref>.  Denote 

    W_ = ∫ - ^2  γ_β, (, ) + ·γ_β, / (ρ_0 ⊗ρ_1)_L^2(ρ_0 ⊗ρ_1)^2,

and repeat the calculations yielding (<ref>) and (<ref>). As a consequence,

    lim sup_→ 0W_ - 𝒲_2^2(χ_[Ω_0^(β,c)]ρ_0, χ_[∇ g^* (Ω_0^(β,c))]ρ_1)/^2/d+2≤ C ℓ_d (Ω_0^(β,c)).

Applying Lemma <ref> for each i ∈𝒢(β) yields

    lim sup_→ 0^+ℍ_ (π_β, ) - 𝒲_2^2(ρ_0, ρ_1)/^2/d+2
           ≤d^d+4/d+2(d+2)^2/d+2/(ℋ^d-1(𝒮^d-1) )^2/d+2∑_i ∈𝒢(β)∫_I_i(ρ_0() ρ_1[∇ g^*()]))^-1/d+2ρ_0()+ C ℓ_d (Ω_0^(β,c)) 
           ≤d^d+4/d+2(d+2)^2/d+2/(ℋ^d-1(𝒮^d-1) )^2/d+2∫_Ω∖Ω_0^(β,c)(ρ_0() ρ_1[∇ g^*()]))^-1/d+2ρ_0() + C ℓ_d (Ω_0^(β,c)).
 Theorem <ref> follows by letting β→ 0. 

§ CONCLUSION AND FURTHER PERSPECTIVES

The entropy-regularized transport, without logarithmic terms, has recently been applied to problems such as manifold learning, economics, and biology. In these problems, the goal is generally to find a non-exact but sparse solution to the optimal transport problem (<ref>). In this work, we take a first step towards understanding this problem for quadratic entropy. Additionally, the methodology presented (the stained glass method) can be generalized to more general entropies, unlike the Gaussian approximation method given in <cit.>, which is only valid for logarithmic penalization. We have also observed that, similarly to the problem regularized by logarithmic entropy—where there exists a relationship between the fundamental solution of the heat equation and the solutions of the regularized problem—with quadratic penalization, the solutions can be approximated by a fundamental solution of the porous media equation.

To conclude this work, we will highlight a series of open problems related to entropy-regularized transport with quadratic entropy:


  *  It is known that the solution of (<ref>) has support that contracts as ε decreases, however, the convergence rates of the support are unknown. It is also not known whether the support decreases monotonically.
    
  *  It is known that the optimal potentials of the problem regularized by logarithmic entropy are C^∞ irrespective of ρ_i. For the quadratic case, we conjecture that this may not be the case, and at most, the potentials (solutions of (<ref>)) will have two degrees more of regularity than ρ_i.
    
  *  In the discrete case, it is known that the effect of quadratic norm penalization in a linear program is null for a sufficiently small regularization parameter. However, this value is not specifically characterized in the literature. Indeed, from a mathematical perspective, it would be interesting to study this point regarding the shape of a given polytope.







Set∈∂Ωand the family of balls{𝔹(, 1/n)}_n∈with centerand radium1/n. Set∈Ωand recall that ψ_tends uniformly to0inΩ. Set_0>0such thatψ__𝒞(Ω)≤δ_0. 


Ifdist(, ∂Ω)≥1/n_0, there exists_0=(n_0)such that for≤_0small enough,𝒮_ε,  ⊂Ωand(1-q()) ≥λ∫(ψ_()- C -'^2)_+ℓ_d(')= 1/C  (ψ_())^d+2/2.If dist(𝐱, ∂Ω) < 1/n_0, we denote 𝐱^* ∈min_𝐳∈∂Ω𝐱 - 𝐳. The segment [𝐱, 𝐱^*] is contained in 𝔹(𝐱^*, 1/n_0) ∩Ω; otherwise, 𝐱^* would not be the closest point on ∂Ω to 𝐱. Change coordinates in order to have^*=0and=λe_1withe_1being the first element of the canonical basis.

     Let Ω be an open bounded and set with 𝒞^0,1 boundary with Lipschitz constant L.  There exists n_0∈ and c=c(L)>0 such that for every n≥ n_0 and ∈Ω, 
     ℓ_d( 𝔹(𝐱, 1/n) ∩Ω)≥c/n^d.
It is enough to prove it for ∈∂Ω. 
Assume that 

Let {V_i}_i ∈ I be the finite covering of ∂Ω and φ_i:U_i→ V_i the charts.  For i≠ j, V_i∩ V_j is either empty or there exists an open ball 𝔹(_0, δ_i,j) of radius δ_i,j>0. Let δ^*>0 be the smallest of all of them. Take n_0≥ 1/δ^* and  such that -^*≤ 1/n_0. By construction 𝔹(^*,1/n_0)∩∂Ω⊂ V_i  for some i∈ I.
Set  ^*∈min_𝐳∈∂Ω𝐱 - 𝐳.  Note that  the  segment [𝐱, 𝐱^*] is contained in Ω̅; otherwise, 𝐱^* would not be among the closest points on ∂Ω to 𝐱.  Change coordinates in order to have ^*= 0 and =λ e_1 with e_1 being the first element of the canonical basis. For  t∈ (0, 1/n_0] we define the hyperplane 
    H_t={∈^d: ⟨,  e_1⟩ =t}.
Set _̆t∈∂Ω∩ H_t. Pythagoras theorem applied to the right-angled triangle with vertexes (_̆1,t  e_1, λ e_1) and hypotenuse the segment [_̆1, λ e_1] implies
_̆1 -λ e_1^2=(λ-t)^2+_̆1 -t  e_1^2.  
Since  ^*∈min_𝐳∈∂Ω𝐱 - 𝐳, we have 
2 tλ≤_̆1 -t  e_1^2,
so that 
 dist(t  e, ∂Ω∩ H_t  )≥( 2 tλ)^1/2
The cone 
𝒞_={: arccos(⟨,  e_1⟩/) ≤θ}
We claim that the set
    A={ t∈ (0, λ]: ^*∈min_𝐳∈∂Ω t  e_1 - 𝐳}
   is open and closed in (0, λ) and non-empty. To prove this claim we take a sequence { t_n}_n∈⊂ (0, λ) such that t_n → t and ^*∈min_𝐳∈∂Ω t_n   e_1 - 𝐳.  Then 
    t_n   e_1 - ^*≤ t_n   e_1 - 𝐳  for all ∈∂Ω.
   Taking limits, as n→∞, yields the closeness. 
   
   
   Therefore (t e_1)^*=^* for all t∈ (0,
   Set ∈̆∂Ω∩ H_t and notice that -̆t e_1≥^*-t e_1
    
Take n_0≥ 1/δ^* and  such that -^*≤ 1/n_0. By construction 𝔹(^*,1/n_0)∩∂Ω⊂ V_i  for some i∈ I. 
   Note that  the  segment [𝐱, 𝐱^*] is contained in 𝔹(𝐱^*, 1/n_0) ∩Ω; otherwise, 𝐱^* would not be the closest point on ∂Ω to 𝐱. 
   
   
   
  
   
   
   Set ,̆'̆∈ U_iφ_i()̆
   
   
   Since ∂Ω is Lipschitz and ψ_   tends uniformly to 0 in Ω, we can find _0>0 and a cone 
𝒞_={: arccos(⟨,  e_1⟩/) ≤θ} containing the segment [ 0, λ e_1],  whose angle θ depends only on L and such that
𝒞_∩𝒮_, ⊂𝒮_, ∩Ω, 
for all ≤_0. 


   We know thatV_ε=ε^-d+4/d+2 (f_ε()+g()-⟨, ⟩)_-^2 ρ_1() ρ_0()∫ (f_ε()-g^*()+ ⟨-1/2∇ g^*(),    [D^2g^* ()]^-1 (-∇ g^*())⟩)_-^2 ρ_0() ρ_1() .∫ (f_ε()-g^*()+1/2⟨-∇ g^*(),    [D^2g^* ()]^-1 (-∇ g^*()))_-^2 ℓ_d() ρ_1[∇ g^*()]) ρ_0()Change=̆F()=[D^2g^* ()]^-1/2 (-∇g^*())∫([D^2g^* ()])^1/2∫(f_ε()-g^*()+ ^2/2)_-^2 ℓ_d()̆ρ_1[∇ g^*()]) ρ_0()  .C_d^(2)∫([D^2g^* ()])^1/2| f_ε()-g^*()|^d+4/2ρ_1[∇ g^*()]) ρ_0()  .ε^d+4/d+2 C_d^(2) C_d^-d+4/d+2∫([D^2g^* ()])^1/2 ((ρ_0()ρ_1[∇ g^*()]))^-d+4/2(d+2))ρ_1[∇ g^*()]) ρ_0()  .Since(D^2g^* ())=ρ_0()/ρ_1(∇g()), 



Let Assumption <ref> hold. There exists  δ_0>0 such that for every δ∈ (0, δ_0)  there exist _δ
such that for ≤_δ the following holds: 

  *  There exists a function 
c_, δ such that
q_δ()=1/∫_Ω^δ,c_0 (c_,δ ()-D(, ∇ g^*(') )_+ρ_0(') ,   for all ∈Ω^δ,c_0.   
  *  There exist positive constants C=C(δ_0) and  c=c(δ_0) such that for every  ∈Ω^δ,c_0
    ε^2/d+2 c ≤  c_, δ() ≤ε^2/d+2 C,

and 

    ε^d/d+2 c≤ℓ_d(𝒮_ε,δ, )≤ Cε^d/d+2,

where 𝒮_ε,δ, = {'∈Ω_0:  c_,δ ()-D(, ∇ g^*(')  ≥ 0}.
  *  There exists a positive constant  c=c(δ_0) such that for every  '∈Ω^δ,c_01/∫_Ω^δ,c_0 (c_,δ ()-D(, ∇ g^*(') )_+ρ_0() ≥ c.  We prove each point separately. 

 Proof of <ref>.
Set ∈Ω_0. Since the function 
Θ(a)=1/2 ∫_Ω^δ,c_0 (a-D(, ∇ g^*(') )_+^2ρ_0(')-a q_δ()  
is differentiable,  convex and lim_a→±∞Θ(a)=+∞, there exists a minimizer a() which satisfies 
1/ ∫_Ω^δ,c_0 (a()-D(, ∇ g^*(') )_+ρ_0(')= q_δ() . 
This proves the existence. 

 Proof of <ref>. First we notice that c_, δ() must be positive  and 
c_, δ  tends uniformly to 0 in Ω_0^δ,c. 
Lemma <ref> implies that

    (1-q_δ())
           ≤∫_Ω_0^δ,c(c_, δ()- λ/2-'^2)_+ ρ_0(')
       ≤ C(Λ, c, d) (c_, δ())^d+2/d

and 

    (1-q_δ())   ≥∫_Ω_0^δ,c (c_, δ()- C-'^2)_- ρ_0(')
       ≥λ∫_Ω_0^δ,c(c_, δ()- C -'^2)_+ℓ_d(').

Denote for ∈Ω_0^δ,c the ball 𝔹(,c_, δ()) of center  and radius  c_, δ(). Note that since  ∂Ω_0^δ,c is 𝒞^0,1 with uniformly bounded (for δ≤δ_0 with δ_0>0 small enught) Lipschitz constant, there exist a constant C(δ)=C(δ_0) and _δ such  that 
 C(δ )ℓ_d(𝔹(,c_, δ())∩Ω_0^δ,c)≥  (c_, δ())^d/2. 
As a consequence, using Lemma <ref>, 
   C(λ, δ, g, g^*) ≥    (c_, δ())^d+2/2
and (<ref>) holds. (<ref>) holds due to the nesting 
𝔹(,c  c_, δ())⊂𝒮_ε,δ, ⊂𝔹(,C  c_, δ()),  
which are consequence of Lemma <ref>.

 Proof of <ref>. 
By the previous points, There exists  δ_0>0 such that for every δ∈ (0, δ_0)  there exist _δ and c=c(δ_0)> 0
such that for ≤_δ
    ∫_Ω^δ,c_0 (c_,δ ()-D(, ∇ g^*(') )_+ dρ_0()   ≥ c∫_Ω^δ,c_0 (c_,δ(')- D(', ∇ g()))_+ dρ_0() 
       =c  inf_q_δ()

By Lemma <ref> and letting _δ smaller if necessary,  we can ensure that inf_q_δ()≥ 1/2 for all ≤_δ

 Paper Title: Feedback stabilization for entropy solutions of a 2x2 hyperbolic system of conservation laws at a junction.



Feedback stabilization for 2 × 2  system at a junction]Feedback stabilization for entropy solutions of a 2 × 2 hyperbolic system of conservation laws at a junction 


G. M. Coclite]Giuseppe Maria Coclite
[G. M. Coclite]Polytechnic University of Bari, Department of Mechanics, Mathematics and Management, Via E. Orabona 4, 70125 Bari,  Italy.
giuseppemaria.coclite@poliba.it


N. De Nitti]Nicola De Nitti
[N. De Nitti]EPFL, Institut de Mathématiques, Station 8, 1015 Lausanne, Switzerland.
[]nicola.denitti@epfl.ch


M. Garavello]Mauro Garavello
[M. Garavello]Università di Milano Bicocca, Dipartimento di  Matematica e Applicazioni, via R. Cozzi 55, 20125 Milano,  Italy.
mauro.garavello@unimib.it

F. Marcellini]Francesca Marcellini
[F. Marcellini]Università di Brescia, Dipartimento di Ingegneria dell'Informazione, via Branze 38, 25123 Brescia,   Italy.
francesca.marcellini@unibs.it




[2020]35L65, 35L45, 35B25.

§ ABSTRACT 
  We consider the p-system in Eulerian coordinates on a star-shaped
  network. Under suitable transmission conditions at the junction and
  dissipative boundary conditions in the exterior vertices, we show
  that the entropy solutions of the system are exponentially
  stabilizable. Our proof extends the strategy by Coron
  et al. (2017) and is based on a front-tracking algorithm used to
  construct approximate piecewise constant solutions whose BV norms
  are controlled through a suitable exponentially-weighted Glimm-type
  Lyapunov functional.



[
    [
    August 5, 2024
==================







§ INTRODUCTION



We aim to stabilize the entropy solutions to the p-system (or Euler equations)
in a star-shaped network around a given equilibrium by using boundary
feedback controls. The p-system models a fluid in a rectilinear 1D pipe
through the evolution of some macroscopic quantities, namely its density
and linear momentum; see <cit.> and the references therein for
a more complete description.

Let us consider N ∈∖{0, 1} rectilinear tubes,
 modeled by the real interval I  (0, 1), exiting a junction J,
which is located at the position x = 0.
For
ℓ∈{1, …, N}, the direction and section of the
ℓ-th tube are described, respectively, by the direction and the
norm of a vector ν_ℓ∈ℝ^3\{0 } .
All tubes are filled with the same compressible, inviscid and
isentropic (or isothermal)
fluid, and we assume that friction along the walls is neglected.
Hence the fluid dynamics 
can be modeled through n copies of the one-dimensional
p-system in Eulerian coordinates:

    ∂_tρ_ℓ+∂_x q_ℓ=0,
            t >0,   x ∈ [0,1),
        
    ∂_t q_ℓ+∂_x(q_ℓ^2/ρ_ℓ
          +p(ρ_ℓ))=0,             t>0,  x ∈[0,1),
          ℓ∈{1, …, N}.

Here, t is the time, and, along the ℓ-th tube,
x is the abscissa, ρ_ℓ = ρ_ℓ(t, x) is the fluid density,
and q_ℓ = q_ℓ(t,x) is its linear momentum density. 
We assume that the pressure law p=p(ρ) is the same for all the tubes;
it plays the role of the equation of state of the fluid under
consideration and it is assumed to satisfy the following hypothesis: 

    P
      p ∈ C^2(^+;^+),     p'>0,    p”>0.


We supplement <ref> with a set of initial conditions, of coupling
conditions at the junction J, and of boundary conditions at the position
x=1 of each pipe.
As initial conditions, we consider

    (ρ_ℓ, q_ℓ)(0, x)
      =(ρ_0,ℓ, q_0,ℓ)(x),    x ∈ I,   ℓ∈{1, …, N},

where ρ_0,ℓ and q_0,ℓ are given functions
in L^1(I) with finite total variation.

In this paper, at the junction J, we consider
the notion of P-solutions, introduced in <cit.>,
which prescribes that 


  * mass is conserved;


  * the trace  of the linear momentum flux
  
  (sometimes called dynamic pressure) at J
  is the same for all tubes;


  * entropy may not decrease at the junction.

This concept of solution yields the well-posedness in
L^1; see <cit.> for further details.
Finally, at the exterior boundary we impose a feedback control, described in terms of the Riemann invariants of the p-system.

The main result of the present paper is the existence of a feedback type
control, acting at the exterior boundary of the star-shaped network, which
asymptotically stabilizes the p-system <ref>
around an equilibrium configuration.
Roughly speaking, with an equilibrium over the network we mean a vector

    ((ρ̅_1, q̅_1), …,
        (ρ̅_N, q̅_N)) ∈(^+ ×)^N

of constant states, which provide an equilibrium solution for <ref>.
The stabilization's proof is done through a Lyapunov-type functional, similar
to the one considered in <cit.>, which is decreasing in time,
on the wave-front tracking construction, and eventually vanishes.
We strongly use the fact that, locally around the equilibrium, one
eigenvalue of the linearized system is strictly negative and one
strictly positive.
It is important to notice that, in this paper, we do not work with strong solutions,
but with weak solutions, which
possibly have discontinuities. 

Existence and well-posedness of entropy admissible solutions
for the Cauchy problem at junctions
was established in <cit.>, and, with a similar technique, can be also extended in the case of the initial-boundary value problem. The problem of the stabilization is, instead, more difficult. While our Lyapunov functional is similar
to the one considered in <cit.>, we have to face additional challenges. Indeed, in <cit.>, the authors considered a 2×2 hyperbolic system
of conservation laws with positive characteristic speeds on a bounded segment; as a consequence, the feedback boundary control acts on the left boundary according to the trace of the solution at the right boundary. On the other hand, in our work, the presence of a strictly positive eigenvalue and a strictly negative one complicates the analysis of the feedback mechanism. 

Further recent results inspired by <cit.> are contained in <cit.>. Previous to <cit.>, the stabilization of a scalar
conservation law through a stationary feedback mechanism was achieved
in <cit.>; in the case of open-loop controls, results on
asymptotic stabilization and on controllability for a general hyperbolic system of
conservation laws with either genuinely nonlinear or linearly
degenerate characteristic fields (in the sense of Lax) and
characteristic speeds strictly separated from 0, were obtained in
<cit.> by suitably acting on both sides of the interval or
in <cit.> by acting on a single boundary point. In <cit.>, an entropy-based Lyapunov functional was used for the stability analysis of equilibria in networks of scalar conservation laws. For the problem of exact controllability of both classical and entropy solutions for systems of conservation laws on an interval, we refer to <cit.>.
Instead, few results about the stabilization of hyperbolic systems on networks in the context of entropy solutions are currently available in the literature (see
for example <cit.> for networked wave equations). 


On the other hand, for the stabilization of classical solutions, many more works have appeared throughout the years (for both scalar equations and systems, on a single segment or on networks). The exponential stabilization of gas flow governed by the isothermal Euler equations in fan-shaped networks in the L^2-sense has been studied in <cit.>. For a single pipe, a strict H^1-Lyapunov functional and feedback stabilization for the quasilinear isothermal Euler equations with friction have been studied in <cit.>. Similar results have been obtained for H^2-Lyapunov functions in <cit.> and for C^1-Lyapunov functions in <cit.>. The finite-time stabilization of a network of strings is studied in <cit.>. For the problem of exact controllability of classical solutions on networks, we refer to <cit.>. Finally, we refer to the book <cit.> for further results. 


The paper is organized as follows. In <Ref>, we
collect several preliminary notions on the solutions of the p-system
at a junction. In <Ref>, we present our main
stabilization result. The proof, based on a front-tracking approximation and on the
study of an exponentially weighted Glimm functional, is contained
in <Ref>.






§ BASIC DEFINITIONS


In this section, we recall the key features of the p-system
in Eulerian coordinates,
we introduce the concept of solution of <ref>
with initial conditions <ref> and with coupling and feedback type
boundary conditions, and we state the main result of this paper.





 §.§ Quantities of interest for the p-system



Following <cit.>, let us recall here some quantities
of interest for the p-system (<ref>).
First, let us introduce the notation 

    u   (ρ, q)^⊤,
           f(u) (q, q^2/ρ
      +p(ρ))^⊤,

so that the p-system <ref> can be written in the compact form
u+ f(u)=0.
The Jacobian matrix of flux function f is

    ∇ f(u)=(
      [              0              1; -q^2/ρ^2+p'(ρ)          2 q/ρ ])

and so, by assumption <ref>, the system is strictly hyperbolic and
its eigenvalues λ_1, λ_2
and the corresponding right eigenvectors r_1, r_2 are given by

    [                             λ_1 (ρ, q) = q/ρ-c(ρ),                                λ_2(ρ, q)=q/ρ+c(ρ),;       r_1(ρ, q)=[
        [         -ρ; ρ c(ρ) - q ]], r_2(ρ, q) =
          [
          [       ρ,; q+ρ c(ρ) ]], ]

where c(ρ) := √(p'(ρ)) is the sound speed.
Moreover, we have that

    ∇λ_1(ρ, q) · r_1(ρ, q)
      = c(ρ)+ρ c^'(ρ) > 0,
          ∇λ_2(ρ, q)
      · r_2(ρ, q)=c(ρ)+ρ c^'(ρ) >0,

so that both characteristic fields are genuinely nonlinear.
The Riemann invariants v_1 and v_2
of the p-system (<ref>) are

    v_1 (ρ, q) :=
      q/ρ + ∫_1^ρ√(p'(r))/r r,
          
      v_2 (ρ, q):=
      q/ρ - ∫_1^ρ√(p'(r))/r r.

Note that

    ∇ v_1(ρ, q) · r_1(ρ, q) 
                = ∇ v_2(ρ, q) · r_2(ρ, q) = 0,    
            
    ∇ v_1(ρ, q) · r_2(ρ, q) 
                = ∇ v_2(ρ, q) · r_1(ρ, q) 
            = 2 √(p'(ρ)),

for every (ρ, q) ∈^+ ×; v_1 is constant along the rarefaction curves of the first family,
while v_2 is constant along the rarefaction curves of the second family. Moreover, v_1 is monotonically increasing along the 
rarefaction curves of the second family and v_2 is monotonically increasing along the 
rarefaction curves of the first family.

The shock and the rarefaction curves through a state u ∈^+ × can be
parameterized as

    σ↦ S_k(σ)(u),   σ↦ R_k(σ)(u),

where k ∈{1,2} and the parametrization is chosen so that

    λ_k(S_k(σ)(u))
      =λ_k(R_k(σ)(u))=λ_k(u)+σ;

see <Ref> (left). Consequently we use the following 
parametrization for the Lax curve

    σ↦ℒ_k(σ)(u) = {[ R_k(σ)(u),   if σ≥ 0,; S_k(σ)(u),  if σ < 0. ].


We introduce the regions 

    A_-={ℝ^+×ℝ: λ_2(ρ, q)<0}, 
       A_0^-={ℝ^+×ℝ: λ_2(ρ, q) ≥ 0, q ≤ 0}, 
       A_0^+={ℝ^+×ℝ: λ_1(ρ, q) ≤ 0, q ≥ 0}, 
       A_+={ℝ^+×ℝ: λ_1(ρ, q)>0}, 
       A_0=A_0^-∪ A_0^+ .

Here A_0 is known as subsonic region; see <Ref> (right).

Finally, we introduce the functions

    P(ρ, q)
          = q^2/ρ + p(ρ),
      
    
    E(ρ, q)  
          =  q^2/2ρ
        + ρ∫_1^ρp(r)/r^2 r,
      
    
      F(ρ, q)
          =   q/ρ(E(ρ,q) + p(ρ)),

which are respectively the dynamic pressure, the energy, and
the energy flow; see <Ref> (right) for the plot of a level curve of P.













 §.§ Initial-Boundary Valued Problem



Assigning at time t=0 an initial state
(ρ_o, ℓ, q_o,ℓ) 
∈ L^1 (I; ℝ^+×ℝ) in each of the N ducts exiting 
the junction (ℓ∈{1, …, N})
and N boundary data B_ℓ∈ L^1_loc(^+; ),
we consider with the following initial-boundary value problem:

    ∂_tρ_1+∂_x q_1=0,
            t >0,   x ∈ I,
    	
    ∂_t q_1 + ∂_x P(ρ_1, q_1) = 0,
            t >0,   x ∈ I, 
    	
    ⋮
    ∂_tρ_N+∂_x q_N=0,
            t >0,   x ∈ I,
    	
    ∂_t q_N + ∂_x P(ρ_N, q_N) = 0,
            t >0,   x ∈ I, 
    	
    
        (ρ_1, q_1)(0, x) = (ρ_0,1, q_0,1)(x),
            x ∈ I, 
    	
    ⋮
    
        (ρ_N, q_N)(0, x) = (ρ_0,N, q_0,N)(x),
            x ∈ I, 
    	
    
    	v_2(ρ_1(t, 1), q_1(t, 1)) = B_1 (t),
    	    t > 0,
    	
    ⋮
    
    	v_2(ρ_N(t, 1), q_N(t, 1)) = B_N (t),
    	    t > 0.



  
  The tuple ((ρ_1, q_1), …, (ρ_N, q_N)) is an entropy solution to
  the Cauchy problem <ref> if the following conditions
  hold.
  
  
  * For every ℓ∈{1, …, N},
    (ρ_ℓ, q_ℓ) ∈ C^0([0, +∞); L^1(I; ^+ ×)).
		
  
  * For every ℓ∈{1, …, N} and for a.e. t > 0,
    the map x ↦ (ρ_ℓ(t, x), q_ℓ(t,x)) has finite
    total variation.

  
  * For every ℓ∈{1, …, N} and for every
    Φ∈ C^∞_c((0, +∞) × I; ), it holds
    
    ∫_0^+∞∫_0^1 [∂_t Φ(t, x)
            (
              [ ρ_ℓ(t,x); q_ℓ(t,x) ]) + ∂_x Φ(t,x) f(ρ_ℓ(t, x), q_ℓ(t,x))
          ]  x  t = (
            [ 0; 0 ]).

    
  
  * For every ℓ∈{1, …, N} and for every
    Φ∈ C^∞_c((0, +∞) × I; ^+), it holds
    
    ∫_0^+∞∫_0^1 [∂_t Φ(t, x)
            E(ρ_ℓ(t,x), q_ℓ(t,x))
            + ∂_x Φ(t,x) F(ρ_ℓ(t, x), q_ℓ(t,x))
          ]  x  t ≥ 0.

		
  
  * For every ℓ∈{1, …, N},
    (ρ_ℓ(0,x), q_ℓ(0,x)) = (ρ_0,ℓ(x), q_0,ℓ(x))
    for a.e. x ∈ I.

  
  * For a.e. t > 0, the following coupling conditions hold:
    
    
  * ∑_ℓ = 1^N ν_ℓ q_ℓ (t, 0^+)
      = 0;
    
  * there exists P_∗ = P_∗(t) > 0 (possibly depending on
      t) such that, for every ℓ∈{1, …, N},
      
    P(ρ_ℓ(t, 0^+), q_ℓ(t, 0^+)) = P_∗(t);

    
  * ∑_ℓ = 1^N ν_ℓ F(ρ_ℓ(t,
      0^+), q_ℓ (t, 0^+)) ≤ 0.
    
    
  
  * For every ℓ∈{1, …, N} and for a.e. t > 0,
    
    v_2(ρ_ℓ(t, 1^-), q_ℓ(t, 1^-)) = B_ℓ(t).

  



Given N constant states u̅_1, …, u̅_N ∈^+ × and the vector
k⃗ = (k_1, …, k_N) ∈ [0, 1]^N, we consider the
system with feedback boundary control:

    ∂_tρ_1+∂_x q_1=0,     t >0,   x ∈ I,
        
    ∂_t q_1 + ∂_x P(ρ_1, q_1) = 0,     t >0,   x
        ∈ I,
        
    ⋮
    ∂_tρ_N+∂_x q_N=0,     t >0,   x ∈ I,
        
    ∂_t q_N + ∂_x P(ρ_N, q_N) = 0,     t >0,   x
        ∈ I,
        
    
        (ρ_1, q_1)(0, x) = (ρ_0,1, q_0,1)(x),     x ∈
        I,
        
    ⋮
    
        (ρ_N, q_N)(0, x) = (ρ_0,N, q_0,N)(x),     x ∈
        I,
        
    
        v_2(ρ_1(t, 1), q_1(t, 1)) = k_1 v_1(ρ_1(t,
          1^-), q_1(t, 1^-)) 
                         - k_1 v_1(u̅_1) +
        v_2(u̅_1),   t>0,
    
    ⋮
    
        v_2(ρ_N(t, 1), q_N(t, 1)) = k_N v_1(ρ_N(t,
          1^-), q_N(t, 1^-)) 
                          - k_N v_1(u̅_N) +
        v_2(u̅_N),   t>0.

Formally system <ref> reduces to <ref>
with the position

    {[ B_1(t) = k_1 v_1(ρ_1(t, 1^-), q_1(t, 1^-)) 
          - k_1 v_1(u̅_1) +
          v_2(u̅_1),;                                                                                    ⋮;  B_N(t) = k_N v_1(ρ_N(t, 1^-), q_N(t, 1^-))
          - k_N v_1(u̅_N) +
          v_2(u̅_N). ].



We now define first the notion of solution and then that of equilibrium
solution for the system with feedback 
boundary control.


  
  The tuple ((ρ_1, q_1), …, (ρ_N, q_N)) is an entropy solution to
  the problem <ref> if the following conditions
  hold.
  
  
  * For every ℓ∈{1, …, N},
    (ρ_ℓ, q_ℓ) ∈ C^0([0, +∞); L^1(I; ^+ ×)).
		
  
  * For every ℓ∈{1, …, N} and for a.e. t > 0,
    the map x ↦ (ρ_ℓ(t, x), q_ℓ(t,x)) has finite
    total variation.

  
  * For every ℓ∈{1, …, N} and for every
    Φ∈ C^∞_c((0, +∞) × I; ), it holds
    
    ∫_0^+∞∫_0^1 [∂_t Φ(t, x)
            (
              [ ρ_ℓ(t,x); q_ℓ(t,x) ]) + ∂_x Φ(t,x) f(ρ_ℓ(t, x), q_ℓ(t,x))
          ]  x  t = (
            [ 0; 0 ]).

    
  
  * For every ℓ∈{1, …, N} and for every
    Φ∈ C^∞_c((0, +∞) × I; ^+), it holds
    
    ∫_0^+∞∫_0^1 [∂_t Φ(t, x)
            E(ρ_ℓ(t,x), q_ℓ(t,x))
            + ∂_x Φ(t,x) F(ρ_ℓ(t, x), q_ℓ(t,x))
          ]  x  t ≥ 0.

		
  
  * For every ℓ∈{1, …, N},
    (ρ_ℓ(0,x), q_ℓ(0,x)) = (ρ_0,ℓ(x), q_0,ℓ(x))
    for a.e. x ∈ I.

  
  * For a.e. t > 0, the following coupling conditions hold:
    
    
  * ∑_ℓ = 1^N ν_ℓ q_ℓ (t, 0^+)
      = 0;
    
  * there exists P_∗ = P_∗(t) > 0 (possibly depending on
      t) such that, for every ℓ∈{1, …, N},
      
    P(ρ_ℓ(t, 0^+), q_ℓ(t, 0^+)) = P_∗(t);

    
  * ∑_ℓ = 1^N ν_ℓ F(ρ_ℓ(t,
      0^+), q_ℓ (t, 0^+)) ≤ 0.
    
    
  
  * For every ℓ∈{1, …, N} and for a.e. t > 0,
    
    v_2(ρ_ℓ(t, 1), q_ℓ(t, 1))
          = k_ℓ v_1(ρ_ℓ(t,
            1^-), q_ℓ(t, 1^-)) - k_ℓ v_1(u̅_ℓ) +
          v_2(u̅_ℓ).

  




  
  We say that the tuple (ρ̅, q̅)
  = ((ρ̅_1, q̅_1), …, (ρ̅_N, q̅_N))
  ∈(^+ ×)^N
  is a equilibrium solution to the Cauchy problem <ref>
  if the tuple 
  ((ρ_1, q_1), …, (ρ_N, q_N)), defined by
  (ρ_ℓ(t, x), q_ℓ(t, x)) = (ρ̅_ℓ,
    q̅_ℓ) for every  
  ℓ∈{1, …, N}, x ∈ I, and t > 0,
  provides, in the sense of <Ref>,
  a solution to the Cauchy problem <ref> with initial
  conditions
  
    (ρ_0, ℓ(x), q_0, ℓ(x))
        = (ρ̅_ℓ, q̅_ℓ)     for x ∈ I.








§ MAIN RESULTS: WELL-POSEDNESS AND FEEDBACK STABILIZATION VIA A LYAPUNOV FUNCTIONAL



The main result of this paper deals with the 
well posedness result for the 
Cauchy
problem <ref>
and with a stabilization result
for the solutions of the same Cauchy
problem <ref>
with feedback control acting at the external boundary, according
to <Ref>. More precisely,
following <cit.>,
we prove existence of entropy-admissible
solutions; then we consider a suitable feedback mechanism to
achieve exponential stability in a similar way as in <cit.>.


  
  Fix an N-tuple of subsonic states
  (ρ̅, q̅) = ((ρ̅_1, q̅_1),
    …, (ρ̅_N, q̅_N))
  ∈(A_0)^N,
  giving a
  equilibrium solution to the Cauchy problem <ref>
  in the sense of <Ref>
  and such that
  ∑_ℓ=1^N ν_ℓ F(ρ̅_ℓ, q̅_ℓ) < 0.

  Then, there exist the constants k̅ > 0, δ_0 >0, L>0,
  C > 0, ν > 0,  
  a domain 𝒟, and,
  for every k = (k_1, …, k_N) ∈ [0, k̅]^N,  a semigroup
  S:[0,+∞[×𝒟→𝒟, with
  the following properties.
  
  
  * Domain:
    𝒟⊇{(ρ, q) ∈(ρ̅,
      q̅)+L^1(I ;(ℝ^+×ℝ)^N):  TV(ρ,
      q) ≤δ_0}.
		
  
  *  Semigroup property: For
    (ρ, q) ∈𝒟, S_0(ρ, q)=(ρ, q) and, for
    s, t ≥ 0, S_s S_t(ρ, q)=S_s+t(ρ, q).
		
  
  *  Lipschitz type estimate: For
    (ρ, q),(ρ^', q^') ∈𝒟 and s, t ≥ 0, it holds
    
    S_t(ρ, q)-S_s(ρ^', q^')
          _L^1≤ L ·((ρ, q)-(ρ^',
          q^')_L^1+|t-s|).


  
  *  Entropy admissibility: For
    every (ρ, q) ∈𝒟, the map
    (t, x) ↦ S_t(ρ, q)(x) is a weak entropy solution to the
    Cauchy problem <ref> in the sense of <ref>.

    
  *  Exponential stabilization: 
  For every
  (ρ_0,q_0) ∈𝒟  and t≥ 0,
  
    TV(S_t(ρ_0, q_0))
        ≤ Ce^-ν tTV(ρ_0, q_0).

  







§ PROOF OF THEOREM 2.3



This section contains the proof of <Ref>, which is based
on the wave-front tracking technique with the use of a
specific weighted
Glimm-type functional, inspired by the one introduced in <cit.>.

First, in <ref>, we construct a Riemann solver. Then, in <ref>, we introduce a suitable Glimm-type functional. These preliminaries allow us to construct an approximate wave-front tracking solution in <ref>. 

The interaction estimates in <ref> yield the existence of an entropy admissible solution in  <ref>. Moreover, in  <ref>, we actually show that there exist a L^1-contracting semigroup of solutions. 

Finally, in <ref>, we obtain the exponential stabilization result; thus concluding the proof of <ref>.



 §.§ Wave-front tracking approximation and Riemann solvers



In this subsection, we construct piecewise constant approximations
via the wave-front tracking method; see <cit.>
for the general theory. 
Note that here we can avoid the use of non-physical waves; see <cit.> or <cit.>.

At first, we give the following definition of an
-approximate wave-front tracking solution to <ref>.

  
  Given > 0, the map 𝐮_ = (u_1,, …,
    u_N,) is an -approximate
  wave-front tracking solution to <ref>
  if the following conditions hold.
  
  
  * For every ℓ∈{1, …, N},
    u_ℓ,  = (ρ_ℓ, , q_ℓ, ) ∈
    C^0 ( [0, +∞[; L^1(I; ^+ ×) ).
    
  
  * For every ℓ∈{1, …, N},
    (ρ_ℓ, , q_ℓ, )
    is piecewise constant, with discontinuities along finitely
    many straight lines in (0, +∞) × I.
    Moreover, the jumps can be of the first family or
    of the second family.
    
  
  * For ℓ∈{1, …, N},
    along each jump x = x(t) of the first family (resp. second family),
    the values u^- = u_ℓ, (t, x(t)^-)
    and u^+ = u_ℓ, (t, x(t)^+) are related by
    
    u^+ = ℒ_1(σ_1)(u^-)
            (resp. 
            u^+ = ℒ_2(σ_2)(u^-) )

    for some wave size σ_1 (resp. σ_2).
    
    Moreover, if σ_1 < 0 (resp. σ_2 < 0), then the discontinuity
    is a shock wave and
    
    ẋ(t) - λ(u^-, u^+)≤

    where λ(u^-, u^+) denotes the velocity associated to the
    Rankine-Hugoniot condition.

    Finally, if σ_1 > 0 (resp. σ_2 > 0), then
    σ_1 ≤ (resp. σ_2 ≤), 
    the discontinuity
    is a part of a rarefaction fan and
    
    ẋ(t) - λ_1(u^-)≤  (ẋ(t) - λ_2(u^-)≤).

    
 
  
  * For every ℓ∈{1, …, N}, it holds that
    
    {[ (ρ_ℓ, (0,·), q_ℓ, (0,·) )
                - (ρ_0, ℓ, q_0, ℓ)_L^1 (I_ℓ)
              < ,;          TV(ρ_ℓ, (0, ·), q_ℓ, (0, ·)
              )
              ≤TV( ρ_0, ℓ, q_0, ℓ). ].

    
  
  * For a.e. t ∈^+,
    
    {[                                                     ∑_ℓ = 1^N ν_ℓ q_ℓ, (t, 0^+) = 0,;     P(ρ_1,  (t, 0^+), q_1,  (t, 0^+))
              = P(ρ_2,  (t, 0^+), q_2,  (t, 0^+)),;                                                                                    ⋮; P(ρ_N-1,  (t, 0^+), q_N-1,  (t, 0^+))
              = P(ρ_N,  (t, 0^+), q_N,  (t, 0^+)),;                         ∑_ℓ = 1^N ν_ℓ
              F(ρ_ℓ, (t, 0^+), q_ℓ, (t, 0^+)) ≤ 0. ].


  
  * For every ℓ∈{1, …, N} and for a.e.
    t > 0
    
    v_2(ρ_ℓ,  (t, 1), q_ℓ, (t, 1))
          = k_ℓ v_1(ρ_ℓ, (t, 1^-), q_ℓ, (t, 1^-))
          - k_ℓ v_1(u̅_ℓ) +
          v_2(u̅_ℓ).

  


We briefly review how to construct a wave-front tracking approximate solution
in the sense of <Ref>.

Define u̅_ℓ = (ρ̅_ℓ, q̅_ℓ) for every
ℓ = 1, …, N and choose δ̅> 0 such that
B(u̅_ℓ, δ̅) ⊆ A_0 for every
ℓ = 1, …, N.
Given > 0, for every ℓ,
approximate the initial condition with piecewise constant
functions (ρ_0, ℓ, , q_0, ℓ, )
with a finite number of discontinuities such that

    {[ (ρ_0, ℓ, , q_0, ℓ, )
          - (ρ_0, ℓ, q_0, ℓ)_L^1 (0,1)
          < ,;                TV(ρ_0, ℓ, , q_0, ℓ, )
          ≤TV( ρ_0, ℓ, q_0, ℓ). ].

Then, at the junction, at the exterior boundary, and at each point of jump
along the pipes we solve the corresponding Riemann problems.


  * At each discontinuity inside a pipe
  we use the accurate Riemann solver, described
  in <Ref>.



  * At the boundary x=1, we use the Riemann solver ,
  described in <Ref>.



  * At the junction x=0, we use the Riemann solver , described
  in <Ref>.
  


We approximate each rarefaction wave by means of rarefaction fans.

This construction can be extended up to a first time t̅_1 at which
two waves interact in a duct or a wave hits the junction or the external
boundary.
Since at time t̅_1 the approximation functions are piecewise
constant with a finite number of discontinuities we can repeat the previous
construction up to e second time t̅_2 of interaction and so on.
In this construction we impose that any rarefaction fan is not split
any further and, without loss of generality, we assume that no more
of two waves interact at the same point in a pipe and no more of one wave
interacts at same time with the junction or with the external boundary.







  §.§.§ The classical Riemann problem


Assuming, without loss of generality, that a pipe is modeled by the real
line , we consider the Riemann problem

    {[                                                                           ρ +  q = 0,                                                                           t>0,   x ∈,;                                                                     q +  P(ρ, q) = 0,                                                                           t>0,   x ∈,; (ρ(0,x), q(0,x))
          =
          {[ (ρ_l, q_l),      x < 0,; (ρ_r, q_r),      x > 0; ]. ].

see <cit.> for more details. We denote with

    [                    ^acc:                 (^+ ×)^2                        ⟶                       ^2;                          ((ρ_l, q_l), (ρ_r, q_r))                        ⟼               (σ_1, σ_2) ]

the accurate Riemann solvers, which is well defined provided (ρ_r, q_r) sufficiently closed
to (ρ_l, q_l);
see <cit.> for a complete construction.




  §.§.§ The Riemann problem at the junction


Assuming that we have N pipes, modeled by the semiline
line (0, +∞), we consider the Riemann problem

    {[                                    ρ_1 +  q_1 = 0,                                      t>0,   x > 0,;                            q_1 +  P(ρ_1, q_1) = 0,                                      t>0,   x > 0,;                                                  ⋮;                                    ρ_N +  q_N = 0,                                      t>0,   x > 0,;                            q_N +  P(ρ_N, q_N) = 0,                                      t>0,   x > 0,; (ρ_1(0,x), q_1(0,x))
          =
          (ρ_0,1, q_0,1),                                             x > 0,;                                                  ⋮; (ρ_N(0,x), q_N(0,x))
          =
          (ρ_0,N, q_0,N),                                             x > 0. ].

We denote with

    [                                         :                                  (^+ ×)^N                                         ⟶                                        ^N;                                           ((ρ_0,1, q_0,1), …,
          (ρ_0,N, q_0,N))                                         ⟼                        (σ_2,1, …, σ_2, N) ]

the Riemann solver for <ref>. More precisely, the map 
gives, in each pipe, the strengths of the waves of the second family
generated by <ref>; see <cit.>.




  §.§.§ The Riemann problem at the external boundary


Assuming, without loss of generality, that a pipe is modeled by the semiline
line (-∞, 1), we consider the Riemann problem

    {[                                ρ +  q = 0,                              t>0,   x < 1,;                          q +  P(ρ, q) = 0,                              t>0,   x < 1,; (ρ(0,x), q(0,x))
          =
          (ρ_l, q_l),                                     x < 1. ].

For k > 0 and u̅∈^+ ×,
we denote with

    [                   :    (^+ ×)^2 × (0,1)                   ⟶                    ;                     ((ρ_l, q_l), u̅, k)                   ⟼                 σ_1 ]

the Riemann solver for <ref>. More precisely, the map 
gives the strength of the wave of the first family
generated by <ref>, such that
(ρ_r, q_r)
= ℒ_1(((ρ_l, q_l),
    u̅, k)) (ρ_l, q_l) is the trace of the solution
to <ref> at x=1 and

    v_2(ρ_r, q_r)
      = k v_1(ρ_r, q_r)
      - k v_1(u̅) +
      v_2(u̅).

Note that the Riemann solver  is well
defined, since the following result holds.


  Fix k > 0 and u̅∈^+ ×.
  
  There exists a neighborhood
  𝒰⊆^+ × of u̅ with the following property.
  For every u_l ∈𝒰,
  there is a unique u_r ∈𝒰 such that the relations
  
    u_r = ℒ_1((u_l, u̅, k)) (u_l)
             and     
        v_2 (u_r ) =
        v_2(u̅)
        + k v_1(u_r)
        - k v_1(u̅)

  hold.



  Consider the function
  
    Ψ(u, σ)   v_2(ℒ_1(σ)(u))
        - k v_1(ℒ_1(σ)(u))
        - v_2(u̅) + k v_1(u̅).

  We clearly have that Ψ(u̅, 0) = 0. Moreover,
  
    ∂_σΨ(u, σ)
        = (∇ v_2(ℒ_1(σ)(u))
        - k ∇ v_1(ℒ_1(σ)(u)) )
        ·d/dσℒ_1(σ)(u)

  and so, by <ref>,
  
    ∂_σΨ(u̅, 0)
        = (∇ v_2(u̅)
        - k ∇ v_1(u̅) )
        · r_1 (u̅)
        = ∇ v_2(u̅)
        · r_1 (u̅)  0.


  The Implicit Function Theorem permits to conclude.
  The C^1-regularity of Ψ follows from the regularity result
  established in <cit.>.





 §.§ Glimm functionals



First, let us introduce the definition of approaching waves in the case
of the p-system. For the general definition
see <cit.> or <cit.>.


  
  The waves σ_ℓ,i,α and σ_ℓ, j, β are
  said to be approaching if x_α < x_β and i>j or
  if i=j and 
  min{σ_ℓ,i,α, σ_ℓ, j,β}<0.


For a given γ > 0, we introduce the following weighted
functionals

    V_γ(t) = ∑_ℓ=1^n∑_α∈𝒥_ℓ( 2 K_J|σ_ℓ, 1, α|e^γ x_α
        +|σ_ℓ, 2, α|e^-γ x_α),

where x_α denotes the position of the discontinuity.
Note that, for every ℓ∈{1, …, n} and
α∈𝒥_ℓ, either σ_ℓ, 1, α = 0
or σ_ℓ, 2, α = 0, since every discontinuity corresponds to a wave of the first family or to a wave of the second family.
Moreover, we define the quadratic weighted functionals

    Q_γ = Q_γ^1,1 + Q_γ^2,2 + Q_γ^1,2,
 
where

    Q_γ^1,1(t)
          =∑_ℓ=1^n∑{|σ_ℓ, 1, αe^γ x_ασ_ℓ, 1, βe^γ x_β|:
        (σ_ℓ, 1, α, σ_ℓ, 1, β)
        ∈𝒜_ℓ},
      
    
      Q_γ^2,2(t)
          =∑_ℓ=1^n∑{|σ_ℓ, 2, α
        e^-γ x_ασ_ℓ, 2, βe^-γ x_β|:
        (σ_ℓ, 2, α, σ_ℓ, 2, β)
        ∈𝒜_ℓ},
      
    
      Q_γ^1,2(t)
          =∑_ℓ=1^n∑{|σ_ℓ, 1, α
        e^γ x_ασ_ℓ, 2, βe^-γ x_β|
        :(σ_ℓ, 1, α, σ_ℓ, 2, β)
        ∈𝒜_ℓ},

where, as in <cit.>,
𝒜_ℓ denotes the set of approaching waves in
the ℓ-th pipe, and 𝒥_ℓ is the set of jumps of
the solution in the in the ℓ-th pipe.
Finally, for κ > 0, we define

    J_γ(t) = V_γ(t)+ κ  Q_γ(t).



    The functional J_γ is a Glimm-type functional
    with exponential weights inspired by the one proposed
    in <cit.> for the stabilization of a 2×2
    system with strictly positive velocities.
    It is essentially composed by a linear part,
    namely V_γ, and by the quadratic part Q_γ.
    
    The term V_γ measures the total strength of the waves of
    a wave-front tracking approximate solution.
    The strengths of the waves of the first family are weighted
    by the constant 2K_J for controlling the total variation
    increment due to waves' interactions with the junction.
    Moreover, the exponential terms produce an exponential decay
    of V_γ along a wave-front tracking approximate solution.
    Note that V_γ is equivalent to the total variation
    of the approximate solution;
    see for example <cit.>.

    Finally, Q_γ is a quadratic interaction potential
    and is composed by three parts, namely Q_γ^h_1, h_2
    for h_1, h_2 ∈{1, 2 }.
    The term Q_γ^h_1, h_2 considers all the possible
    interactions between waves of the h_1 and h_2 families.







 §.§ Interaction estimates



First, we recall the interaction estimates inside a duct;
see <cit.>
and <cit.>.


  
  There exists a constant K with the following property.
  
  
  * If there is an interaction in a duct between two waves
    σ_1^-and σ_2^-, respectively of the first
    and the second family, producing the waves σ_1^+and
    σ_2^+, then
    
    |σ_1^+-σ_1^-|+|σ_2^+-σ_2^-| ≤ K ·|σ_1^-σ_2^-|;

    see <Ref> (left).

  
  * If there is an interaction in a duct between two waves
    σ_i^' and σ_i^'' of the same
    i-th family producing waves of total size σ_1^+and
    σ_2^+, then
    
    [         |σ_1^+-(σ_1^''+σ_1^')|+|σ_2^+| ≤ K ·|
            σ_1^'σ_1^''|                                                           if  i=1,; |σ_1^+|+|σ_2^+-(
            σ_2^''+σ_2^')|
            ≤ K ·|σ_2^'σ_2^''|                                                          if  i=2 ; ]

    see <Ref> (middle and right).

  

As a consequence we deduce the monotonicity of the functional
J_γ, defined in <ref>.





    
    Consider κ > 4 K K_J (e^γ + e^3γ).
    Assume that at time t̅ there is an interaction
    in a tube between two waves. Then
    
    Δ J_γ(t̅) < 0.




    Assume first that the interaction happens
    at the location x̅ between a wave of the first family
    with strength σ_1^- with a wave of the second family with
    strength σ_2^-. By <Ref> the emerging waves σ_1^+ and 
    σ_2^+ satisfy
    
    σ_1^+ - σ_1^- + σ_2^+ - σ_2^-≤ K σ_1^- σ_2^-.

    Then, since V_γ is sufficiently small,
    
    Δ J_γ(t̅) 
                 = Δ V_γ(t̅) + κΔ Q_γ(t̅)
            
       ≤ 2K_J σ_1^+ e^γx̅
            + σ_2^+ e^-γx̅ 
            - 2K_J σ_1^- e^γx̅
            - σ_2^- e^-γx̅
         
            - κσ_1^- σ_2^-
            +κ K σ_1^- σ_2^- e^2 γ V_γ(t̅^-) 
            
       ≤ 2 K_J (σ_1^+ - σ_1^- e^γx̅ + σ_2^+ - σ_2^- e^-γx̅)
            - κ/2σ_1^- σ_2^-
       ≤ 2 K_J K σ_1^- σ_2^- (e^γx̅
            + e^-γx̅)
            - κ/2σ_1^- σ_2^-
       ≤(2 K_J K (e^γx̅
            + e^-γx̅) - κ/2) 
            σ_1^- σ_2^-
            < 0.


    Assume now that the interacting waves, σ_1' and σ_1”, are both of the first family
    and that the interaction happens at the location x̅.
    By <Ref> the emerging waves σ_1^+ and 
    σ_2^+ satisfy
    
    σ_1^+ - (σ_1' + σ_1”) + σ_2^+≤ K σ_1' σ_1”.

    Then, since V_γ is sufficiently small,
    
    Δ J_γ(t̅) 
                 = Δ V_γ(t̅) + κΔ Q_γ(t̅)
            
       ≤ 2K_J σ_1^+ e^γx̅ 
            + σ_2^+ e^-γx̅ 
            - 2K_J σ_1' e^γx̅
            - 2 K_Jσ_1” e^γx̅
         
            - κσ_1' σ_1” e^2γx̅
            +κ Kσ_1' σ_1” e^2γx̅  V_γ(t̅^-) 
            
       ≤ 2K_J (σ_1^+ - σ_1' - σ_1”
            e^γx̅ + σ_2^+ e^-γx̅)
            - κ/2σ_1' σ_1” e^2 γx̅
       ≤ 2 K_J K σ_1' σ_1” (e^γx̅+ e^-γx̅)
            - κ/2σ_1' σ_1” e^2γx̅
       ≤( 2 K_J K (e^γx̅+ e^-γx̅) - κ/2e^2γx̅) σ_1' σ_1”
            < 0.


    Assume now that the interacting waves, σ_2' and σ_2”, are both of the second family
    and that the interaction happens at the location x̅.
    By <Ref>, the emerging waves σ_1^+ and 
    σ_2^+ satisfy
    
    σ_1^+ + σ_2^+ - (σ_2' + σ_2”)≤ K σ_2' σ_2”.

    Then, since V_γ is sufficiently small,
    
    Δ J_γ(t̅) 
                 = Δ V_γ(t̅) + κΔ Q_γ(t̅)
            
       ≤ 2K_J σ_1^+ e^γx̅ 
            + σ_2^+ e^-γx̅ 
            - 2K_J σ_2' e^-γx̅
            - 2 K_Jσ_2” e^-γx̅
         
            - κσ_2' σ_2” e^-2γx̅
            +κ Kσ_2' σ_2” e^-2γx̅  V_γ(t̅^-) 
            
       ≤ 2K_J (σ_1^+ e^γx̅ + σ_2^+ - σ_2' - σ_2”
            e^-γx̅)
            - κ/2σ_2' σ_2” e^-2 γx̅
       ≤ 2 K_J K σ_2' σ_2” (e^γx̅+ e^-γx̅)
            - κ/2σ_1' σ_1” e^-2γx̅
       ≤( 2 K_J K (e^γx̅+ e^-γx̅) - κ/2e^-2γx̅) σ_2' σ_2”
            < 0,
   
    concluding the proof.

    
The following result deals with the interactions of waves with the junction.

  
  There exist δ_J>0 and K_J ≥ 1 with the following property.
    For any u̅∈(A_0)^N that yields a equilibrium solution
    to the Riemann problem, for any 1-waves σ_l^-∈ (-δ_J, δ_J) hitting the junction and producing the 2-waves
    σ_l^+, it holds that
    
    ∑_ℓ=1^N|σ_ℓ^+| ≤ K_J ·∑_ℓ=1^n|
          σ_ℓ^-|;

    see <Ref>.

For the proof, see <cit.>.








    Assume that at time t̅ there is an interaction
    between a wave and the junction J. Then
    
    Δ J_γ(t̅) < 0.




    Assume, without loss of generality, that the interaction is 
    due to a wave of the first family of strength σ_1^- from the pipe
    ℓ = 1.
    Denote with σ_ℓ^+ the strength of the emerging waves of the second
    family in the pipe ℓ=1, …, n.
    By <Ref>, we obtain that
    
    ∑_l=1^n|σ_l^+| ≤ K_J 
          ·| σ_1^-|.

    
    Then, since V_γ is sufficiently small,
    
    Δ J_γ(t̅) 
                 = Δ V_γ(t̅) + κΔ Q_γ(t̅)
            
       ≤∑_ℓ=1^n σ_ℓ^+ - 2K_J σ_1^-
            +κ∑_l=1^n σ_ℓ^+ V_γ(t̅^-) 
            
       ≤ K_J σ_1^- - 2K_J σ_1^-
            + κ K_J σ_1^- V_γ(t̅^-)
            
       ≤ - K_J σ_1^-
            + κ K_J σ_1^- V_γ(t̅^-)
            
       ≤ - K_J/2σ_1^- < 0,

    concluding the proof.







Finally, we deal with the interaction of a wave with the external boundary.

  
  There exists a constant C > 0 with the following property.
  Assume that a wave of the second family
  with strength σ_2^- interacts with the external boundary 
  x=1 at a time t̅ from the ℓ-th pipe,
  ℓ∈{1, …, N}; see <Ref>.
  Then, 
  the emerging wave of the first family
  has a strength σ_1^+ satisfying the estimate
  
    σ_1^+≤ C k_ℓσ_2^-.







  Let us denote by u_l and u_m the states on the left and on the
  right of the interacting wave, respectively, so that
  u_m = ℒ_2(σ_2^-) (u_l).

  Since u_m is an equilibrium at the external boundary, then
  
    v_1(u_m) = k_ℓ v_2(u_m)
        - k_ℓ v_2(u̅_ℓ) + v_1(u̅_ℓ).

  The emerging wave (u_l, u_r) satisfies
  u_r = ℒ_1(σ_1^+) (u_l)
  and
  
    v_1(u_r) = k_ℓ v_2(u_r)
        - k_ℓ v_2(u̅_ℓ) + v_1(u̅_ℓ).


  Consider the function
  
    ψ(σ_1, σ_2) :=
        v_2(ℒ_1(σ_1)(
            u_l))
        - k_ℓ v_1 (ℒ_1(σ_1)
            (u_l))
        -  v_2(ℒ_2(σ_2)(
            u_l))
        + k_ℓ v_1 (ℒ_2(σ_2)
            (u_l)),

  whose zeros represent the possible strengths of the interacting
  and generated waves at the right boundary.
  The functions v_2 and v_1, defined in (<ref>),
  are of class C^2
  thanks to (<ref>). Also the Lax curves have
  the same regularity; see <cit.>.
  Therefore, ψ is of class C^2. Clearly,
  
    ψ(0, 0) = v_2(u_l) - k_ℓ v_1(u_l) - v_2(u_l)
        + k_ℓ v_1(u_ℓ) = 0.

  Moreover,
  
    ∂_σ_1ψ(σ_1, σ_2)
            =
        ∇ v_2(ℒ_1(σ_1)
            (u_l)) ·ℒ_1'(σ_1)(u_l)
              - k_ℓ∇ v_1 (ℒ_1(σ_1)
          (u_l))
          ·ℒ_1'(σ_1)(u_l),
          
    ∂_σ_2ψ(σ_1, σ_2)
            =
        -∇ v_2(ℒ_2(σ_2)
            (u_l)) ·ℒ_2'(σ_2)(u_l)
          + k_ℓ∇ v_1 (ℒ_2(σ_2)
          (u_l))
          ·ℒ_2'(σ_2)(u_l),

  and so, by (<ref>),
  
    ∂_σ_1ψ(0, 0)
            =
        ∇ v_2(u_l) · r_1 (u_l)
        - k_ℓ∇ v_1 (u_l)
        ·
        r_1 (u_l)
        = ∇ v_2(u_l)
        ·
        r_1 (u_l) > 0,
        
    ∂_σ_2ψ(0, 0)
            =
        - ∇ v_2(u_l) · r_2 (u_l)
        + k_ℓ∇ v_1 (u_l)
        ·
        r_2 (u_l)
        = k_ℓ∇ v_1(u_l)
        ·
        r_2 (u_l) > 0.

  Since ∂_σ_1ψ (0,0)  0, by the implicit function theorem, there exists a
  C^2 function σ_1 = σ_1(σ_2) with bounded derivative
  such that σ_1(0) = 0 and ψ(σ_1(σ_2), σ_2) = 0
  for every σ_2 in a suitable neighborhood of 0.
  Moreover
  
    σ_1'(0) = - k_ℓ∇ v_1(u_l)
        ·
        r_2 (u_l)/∇ v_2(u_l)
        ·
        r_1 (u_l) < 0,

  which implies the existence of a constant C>0, depending
  only on u̅ and δ, such that
  
    σ_1(σ_2)≤ C k_ℓσ_2

  for every σ_2 in a suitable neighborhood of 0.
  This permits to prove (<ref>).



    Note that in the previous proof we deduce that the implicit
    function σ_2 ↦σ_1(σ_2)
    is strictly decreasing since its derivative is strictly 
    negative. By the choice of the parametrization of the Lax curves,
    see (<ref>),
    we deduce that if the interacting wave is a shock (resp.,
    a rarefaction), then the emerging wave is a rarefaction
    (resp., a shock).





    Let K_J be as in <Ref> and C > 0 as in <Ref>.
    Fix k_ℓ≤e^-2γ/4 C K_J for every ℓ = 1, …, n.
    Assume that at time t̅ there is an interaction
    between a wave and the external boundary. Then
    
    Δ J_γ(t̅) < 0.




    Assume, without loss of generality, that the interaction is 
    due to a wave of the second family of strength σ_2^- from the ℓ-th pipe.
    Denote with σ_1^+ the strength of the emerging waves of the first
    family in that pipe.
    By <Ref>, we obtain that
    
    |σ_1^+| ≤ C k_ℓ| σ_2^-|.

    
    Then, since V_γ is sufficiently small,
    
    Δ J_γ(t̅) 
                 = Δ V_γ(t̅) + κΔ Q_γ(t̅)
            
       ≤ 2 K_J σ_1^+ e^γ - σ_2^- e^-γ
            +κσ_1^+ e^γ V_γ (t̅^-)
            
       ≤ 2 C K_J k_ℓ e^γσ_2^- 
            - σ_2^- e^-γ
            + κσ_1^+ e^γ V_γ(t̅^-)
            
       ≤ (2C K_J k_ℓ e^γ - e^-γ) σ_2^-
            + C κ k_ℓσ_2^- e^γ V_γ(t̅^-)
            
       ≤ - e^-γ/2σ_2^-
            + C κ k_ℓσ_2^- e^γ V_γ(t̅^-)
            
       ≤ - e^-γ/4σ_2^- < 0,

    concluding the proof.







 §.§ Existence of an approximate wave-front tracking solution




In this subsection, we deal with the existence of a wave-front tracking approximate
solution, in the sense of <Ref>.
In the wave-front tracking approximate solution, we do not consider the so-.called non-physical waves, which is used 
to control the total number of waves and interactions in the case of systems (see <cit.>).
Indeed for a 2× 2 system with suitable assumptions, it is possible to avoid such technicality (see <cit.> and <cit.>).


    
    Fix > 0 and consider a wave-front tracking approximate solution u_
    as in <Ref>.

    Then, inside every pipe and except for a finite number of interactions,
    there is at most one outgoing wave of
    each family for every interaction.



    Inside a pipe, by construction, only the interactions of
    waves of the same family can produce
    a rarefaction fan at positive times.
    More precisely, if two waves of the k_1
    family (k_1 ∈{1, 2}) interacts at
    some positive time t̅ inside a pipe,
    then the emerging wave of the k_2 family
    (k_2 ∈{1, 2}∖{k_1})
    can be a rarefaction of strength bigger than  and so it is split in a rarefaction fan.
    Without loss of generality, let us assume that at time t̅ > 0
    two waves of the first family of strength σ_1' and σ_1” interact together producing a rarefaction of the second family of strength σ_2 with σ_2 >. Then, using <Ref> and <Ref>, we deduce that
    
    < σ_2≤ K σ_1' σ_1”≤K/( 2 K_J K (e^γx̅+ e^-γx̅) - κ/2e^2γx̅)Δ J_γ(t̅).

    This implies that
    
    Δ J_γ(t̅) ≤( 2 K_J K (e^γx̅+ e^-γx̅) - κ/2e^2γx̅)/K
             < 0.

    Since the functional J_γ is non increasing, then we deduce that such interactions can happen at most a finite
    number of times.



  
  For every > 0,
  the construction illustrated in <Ref> produces a wave-front
  tracking approximate solution, defined for every time t ≥ 0.


    Fix > 0 and consider a wave-front tracking approximate solution u_.
    We need to prove that the total number of 
    interactions (and consequently of waves) remain finite.
    
    
    First, assume, by contradiction, that there
    exists an infinite number of
    interactions.
    Define T such that, for t < T there are a finite number of waves and interactions, but at time t = T there are infinitely many, accumulating
    at a point (T, x̅).
    

    Case 1: x̅∈ (0, 1). This case is completely
    similar to the one of <cit.>.
    We report here for completeness.
    
    Fix α > 0 such that
    x̅ - 2 α > 0 and
    x̅ + 2 α < 1. Fix Δ t > 0 such that
    Δ t < α/Λ_max. Consider
    the rectangle
    
    R = [T - Δ t , T] × [x̅ - α, x̅ + α].

    There exists a sequence (t_i, x_i) of interaction points belonging to R such that
    
    (t_i, x_i) → (T, x̅)

    and t_1 < t_2 < ⋯.
    Define ℐ as the set
    of all the points (t_i, x_i).
    
    Define ℱ the sets of all the waves which can be joined, forward in time, to some points on ℐ and which intersects the set R.
    We split ℱ = ℱ_1 ∪ℱ_2, where ℱ_1, resp. ℱ_2, is the subset of ℱ of waves of the first family, resp. of the second family.

    Consider the following sets.
    
        
  * ℐ_1:
        the set of all interaction points of R in which there are exatcly two outgoing waves, one of the first family and one of the second family.

        
  * ℐ_2:
        the set of all interaction points of R in which the
        two interacting waves belongs to ℱ, but
        there is at most one ougoing waves belonging to ℱ.
        
        
  * ℐ_3:
        the set of all interaction points of R in which the
        two interacting waves do not belong to ℱ.

        
  * ℐ_4:
        the set of all interaction points of R in which the
        two interacting waves  belong to ℱ and
        there are at least two outgoing waves of the same family belonging to ℱ.        
    

    By <Ref>,
    the set ℐ_4 is finite.

    Define, for t ∈ [T - Δ t, T], the functional 𝒱(t) as the number of waves, that at time t belongs to ℱ.
    Note that, for interactions in ℐ_1 and ℐ_3 the functional 𝒱 does not change, while 𝒱 strictly decreases by 1 or 2 for interactions in ℐ_2. Finally, 𝒱 can increase for interactions in ℐ_4.

    Moreover, 𝒱(T - Δ t) is finite, 𝒱(t) ≥ 0 for all t, therefore,
    since ℐ_4 is finite,
    also ℐ_2 is finite.

    Note that all the points in ℐ_3 do not belong to
    ℐ.

Starting from (t_1, x_1),  we go forward in time with two continuous lines: the first one made of waves of ℱ_1 (first family) and the second one using waves of ℱ_2 (second family). When we reach an interaction point (t̃, x̃) belonging to ℐ_2 or ℐ_4, we stop and start over from a point (t_j, x_j), with t_j>t̃.

Since ℐ_2 and ℐ_4 are finite sets, there exists a point (t^*, x^*) ∈ℐ from which we can construct two lines γ_1(t), γ_2(t) until the time T. 

The bounds on the velocities imply

    γ_1(t) ⩽ x^*-c(t-t^*),   γ_2(t) ⩾ x^*+c(t-t^*) .


Choose T̅∈ (t^*, T)
such that

    (T - T̅) Λ_max < c (T̅ - t^*)

and fix n̅ such that
t_n̅ > T̅.


Since γ_1 is composed of segments of ℱ, the point (t_n̅, γ_1(t_n̅)) can be joined to some point (t_h, x_h) of ℐ. Moreover, the bounds on the velocities imply that 

    x_h ⩽γ_1(t_n̅)+ Λ_max(t_h-t_n̅),    h ⩾n̅ .


Analogously, 
since γ_2 is composed of segments of ℱ, the point (t_n̅, γ_2(t_n̅)) can be joined to some point (t_k, x_k) of ℐ. 
The bounds on the velocities imply that 

    x_k ⩾γ_2(t_n̅) - Λ_max(t_h-t_n̅),    k ⩾n̅ .



Putting these estimates together, we conclude 

    x_k-x_h    ⩾γ_2(t_n̅)-Λ_max(t_k-t_n̅)-[γ_1(t_n̅) + Λ_max(t_h-t_n̅)] 
       ≥ 2c (t_n - t^*) -Λ_max(t_k - t_n̅ + t_h - t_n̅)
       ≥ 2c (T̅ - t^*) - 2 Λ_max (T - T̅) 
        > 0.

This contradicts the fact that 
the sequence x_i tends to x̅.


Case 2: x̅ = 1. First, we suppose T < 1/Λ_max. Once we obtain the result under this restriction, we can reproduce the same argument in the time-intervals [T, T +  1/Λ_max], [T +  1/Λ_max, T +  2/Λ_max ], and so on. 

First, note that the number of interactions at x=1 is finite. The assumption
T < 1/Λ_max
implies that a wave of the second family generated at the junction
does not reach the boundary x=1 within time T.
Moreover, at positive times, new
waves of the second family can be
generated only when two waves
of the first family interact together, but this, by <Ref>, can happen at most a finite number of times. 

    Therefore, there exists a sequence (t_i, x_i) of interaction points such that
    
    (t_i, x_i) → (T, 1)

    and t_1 < t_2 < ⋯
    and x_1 < x_2 < ⋯.
    Define ℐ as the set
    of all the points (t_i, x_i).
    
    Define ℱ the sets of all the waves which can be joined, forward in time, to some points on ℐ.
    We split ℱ = ℱ_1 ∪ℱ_2, where ℱ_1, resp. ℱ_2, is the subset of ℱ of waves of the first family, resp. of the second family.

    Consider the following sets.
    
        
  * ℐ_1:
        the set of all interaction points in which there are exactly two outgoing waves, one of the first family and one of the second family.

        
  * ℐ_2:
        the set of all interaction points in which the
        two interacting waves belongs to ℱ, but
        there is at most one ougoing waves belonging to ℱ.
        
        
  * ℐ_3:
        the set of all interaction points in which the
        two interacting waves do not belong to ℱ.

        
  * ℐ_4:
        the set of all interaction points in which the
        two interacting waves  belong to ℱ and
        there are at least two outgoing waves of the same family belonging to ℱ.        

        
  * ℐ_5:
        the set of all interaction points at x = 1.        

    

    By <Ref>,
    the set ℐ_4 is finite.
    Also ℐ_5 is finite
    as already noted.

    Define, for t ∈ [0, T], the functional 𝒱(t) as the number of waves, that at time t belongs to ℱ.
    Note that, for interactions in ℐ_1 and ℐ_3 the functional 𝒱 does not change, while 𝒱 strictly decreases by 1 or 2 for interactions in ℐ_2. Finally, 𝒱 can increase for interactions in ℐ_4 and ℐ_5.

    Moreover, 𝒱(0) is finite, 𝒱(t) ≥ 0 for all t, therefore,
    since ℐ_4  and ℐ_5 are finite,
    also ℐ_2 is finite.

    Note that all the points in ℐ_3 do not belong to
    ℐ.

Starting from (t_1, x_1),  we go forward in time with two continuous lines: the first one made of waves of ℱ_1 (first family) and the second one using waves of ℱ_2 (second family). When we reach an interaction point (t̃, x̃) belonging to ℐ_2, ℐ_4, or ℐ_5, we stop and start over from a point (t_j, x_j), with t_j>t̃.

Since ℐ_2, ℐ_4, and ℐ_5 are finite sets, there exists a point (t^*, x^*) ∈ℐ from which we can construct two lines γ_1(t), γ_2(t) until the time T. 
We conclude now exactly as in Case 1. 


Case 3: x̅ = 0. We proceed as in Case 2, by noticing that there are at most a finite
number of wave interactions with
the junction at x=0, provided
Λ_max T < 1.





 §.§ Existence of a solution




Let us consider a -wave-front tracking  solution u_.
By the interaction estimates, using the functional J_0 on
u_ (i.e., J_γ with γ = 0), we deduce, owing to
<Ref>,
<Ref>, and <Ref>

    J_0(t) ≤ J_0(0^+)

since J_0 varies
only at times of interaction.
Therefore, we have 

    TV(u_(t)) ≤ J_0(t) ≤ J_0(0^+).

Since it is standard to prove that there exists L > 0 such that

    ∑_ℓ=1^N
        (ρ_ℓ, (t), q_ℓ, (t)) - (ρ_ℓ,  (s), q_ℓ, (s))_L^1(I_ℓ)≤ L t-s

for every t, s ≥ 0, then
we conclude,
by Helly's compactness theorem (see <cit.>), that u_ converges strongly in L^1 to a limit point u. Moreover, it is standard to deduce that u satisfies <ref> as well as an entropy condition, i.e., <ref>.




 §.§ Existence of a semigroup




Let us consider two -wave-front tracking solutions u and ũ, and define the functional

    Φ(u, ũ) ∑_ℓ = 1^N ∑_i=1^2
        ∫_I_ℓs_ℓ, i(x) W_ℓ, i(x)  x̣.

Here, s_ℓ, i(x) measures the strengths of the i-th shock wave in the ℓ pipe at the point x; the weights
W_ℓ, i(x) are defined by

    W_ℓ, i(x)   1 + κ_1 A_ℓ, i(x) 
        + κ_1 κ_2 (J_0(u) + J_0(v))

for suitable κ_1 > 0 and κ_2 > 0; 
A_ℓ, i(x) = A_ℓ, i^1(x) + A^2_ℓ, i(x), with 

    A_ℓ, i^1(x)    = ∑_α∈𝒥(u) ∪𝒥(ũ) 
     x_α < x 
     k_α≠ i 
     k_α = 2σ_ℓ, k_α, α
        + ∑_α∈𝒥(u) ∪𝒥(ũ) 
     x_α > x 
     k_α≠ i 
     k_α = 1σ_ℓ, k_α, α, 
    
        A_ℓ, i^2(x)    = {[ ∑_α∈𝒥(u) 
     x_α < x 
     k_α = iσ_ℓ, k_α, α + ∑_α∈𝒥(ũ) 
     x_α > x 
     k_α = iσ_ℓ, k_α, α,                                                                 if  s_ℓ, i(x) < 0,; ∑_α∈𝒥(ũ) 
     x_α < x 
     k_α = iσ_ℓ, k_α, α + ∑_α∈𝒥(u) 
     x_α > x 
     k_α = iσ_ℓ, k_α, α,                                                                 if  s_ℓ, i(x) ≥ 0, ].

(see <cit.> and <cit.>).
We set κ_1, κ_2 to satisfy 1 ≤ W_ℓ, i(x) ≤ 2
for every ℓ∈{1, …, N} and i=1, 2. In this way,
the functional Φ is equivalent to the L^1-distance.

Exactly the same calculations as in <cit.>
imply that, for every t > 0 when no interaction happens,

    /ṭΦ(u(t), ũ(t)) ≤ C .

If t > 0 is an interaction time for u or ũ, then
<Ref>, <Ref>, and 
<Ref> imply that

    Δ [J_0(u(t)) + J_0(ũ(t))] < 0,

and so, choosing κ_2 large enough, we obtain

    ΔΦ(u(t), ũ(t)) < 0.

Thus, for every 0 ≤ s ≤ t, we obtain that

    Φ(u(t), ũ(t)) - Φ(u(s), ũ(s))
        ≤ C  (t-s),

proving <ref>.

In particular, from this Lipschitz dependence result, it is standard to deduce the semigroup property claimed in <ref>.



 §.§ Decay of the Glimm-type functional



In this part we prove that the Glimm-type functional J_γ, for γ > 0,
has an exponential-in-time decay, similarly as in <cit.>.
Let a time t̅ be fixed; we consider the variation of the functional J_γ locally around t̅ according to four cases:

	
  * there are no waves' interactions at t = t̅ inside
        the pipes nor an interaction of a wave with the junction or the
        external boundary;
        
	
  * there is an interaction between two waves inside a pipe
        at t = t̅;
        
	
  * there is a wave hitting the external boundary at time 
 t = t̅;
	
    
  * there is a wave hitting the junction at time t = t̅.

In the latter three cases, the estimates in <ref> imply that Δ J_γ(t̅) ≤ 0 provided
V_γ sufficiently small. Hence we focus only on the first case,
where no interaction happens.
 


    Assume that at the time t̅ no interactions of any type
    happen. Then the functional J_γ, defined in (<ref>),
    is differentiable and it holds
    
    /ṭ J_γ(t̅) ≤ -c γ J_γ(t̅).




    Assume that there is no any interactions at time t = t̅. Since there is a finite number of fronts
    at time t̅, then there exists a neighborhood 𝒰 of t̅ in which no interaction happens. Hence, in 𝒰, V_γ(t) and Q_γ are smooth functional.
    
    First compute 
    
    /ṭV_γ(t) 
               = ∑_ℓ=1^n ∑_α∈𝒥_ℓ(2K_J|σ_ℓ, 1, α|e^γ x_α (t)γẋ_α (t) - |σ_ℓ, 2, α|e^-γ x_α(t)γẋ_α(t)   )
            
       ≤∑_ℓ=1^n ∑_α∈𝒥_ℓ(-2c γ K_J|σ_ℓ, 1, α|e^γ x_α (t)  - c γ |σ_ℓ, 2, α|e^-γ x_α(t))
            
        = - c γ∑_ℓ=1^n ∑_α∈𝒥_ℓ(2K_J|σ_ℓ, 1, α|e^γ x_α (t) + |σ_ℓ, 2, α|e^-γ x_α(t))
            = - c γ V_γ(t).

    Moreover, we have
    
    /ṭ Q_γ^1,1(t)
                = ∑_ℓ=1^n ∑_(σ_ℓ, 1, α,
            σ_ℓ, 1, β) ∈𝒜_ℓσ_ℓ, 1, ασ_ℓ, 1, β
            e^γ(x_α(t) + x_β(t))γ(ẋ_α(t) + ẋ_β(t))
            
       ≤ - c γ∑_ℓ=1^n ∑_(σ_ℓ, 1, α,
            σ_ℓ, 1, β) ∈𝒜_ℓσ_ℓ, 1, ασ_ℓ, 1, β
            e^γ(x_α(t) + x_β(t))
            = - c γ Q_γ^1,1(t),
            
    /ṭ Q_γ^2,2(t)
                = -∑_ℓ=1^n ∑_(σ_ℓ, 2, α,
            σ_ℓ, 2, β) ∈𝒜_ℓσ_ℓ, 2, ασ_ℓ, 2, β
            e^-γ(x_α(t) + x_β(t))γ(ẋ_α(t) + ẋ_β(t))
            
       ≤ - c γ∑_ℓ=1^n ∑_(σ_ℓ, 2, α,
            σ_ℓ, 2, β) ∈𝒜_ℓσ_ℓ, 2, ασ_ℓ, 2, β
            e^-γ(x_α(t) + x_β(t))
            = - c γ Q_γ^2,2(t),
            
    /ṭ Q_γ^1,2(t)
                = ∑_ℓ=1^n ∑_(σ_ℓ, 1, α,
            σ_ℓ, 2, β) ∈𝒜_ℓσ_ℓ, 1, ασ_ℓ, 2, β
            e^γ(x_α(t) - x_β(t))γ(ẋ_α(t) - ẋ_β(t))
            
       ≤ - c γ∑_ℓ=1^n ∑_(σ_ℓ, 1, α,
            σ_ℓ, 2, β) ∈𝒜_ℓσ_ℓ, 1, ασ_ℓ, 2, β
            e^γ(x_α(t) - x_β(t))
            = - c γ Q_γ^1,2(t),

    and so
    
    /ṭ Q_γ(t) ≤ -c γ Q_γ(t).

    Therefore, we deduce that
    
    /ṭ J_γ(t) ≤ -c γ J_γ(t),

    concluding the proof.       



    Fix γ > 0.
    Then the functional J_γ,
    defined in (<ref>),
    satisfies the inequality
    
    J_γ(t) ≤ e^- c γ
            J_γ(0^+).


The proof is immediate and so we omit it. This completes the proof of <ref>. 

Putting together all the results obtained in the previous sections, we conclude the proof of <ref>.




§ ACKNOWLEDGMENTS


We thank J. M. Coron, M. Gugat, and E. Zuazua for their helpful comments. 

GMC and NDN are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

GMC has been partially supported by the Project  funded  under  the  National  Recovery  and  Resilience  Plan  (NRRP),  Mission  4, Component  2,  Investment  1.4 (Call  for  tender  No.  3138  of  16/12/2021), of  Italian  Ministry  of  University and Research funded by the European Union (NextGenerationEU Award,  No.  CN000023,  Concession  Decree  No.  1033  of  17/06/2022)  adopted  by  the  Italian Ministry of University and Research (CUP D93C22000410001), Centro Nazionale per la Mobilità Sostenibile.  He has also been supported by the 
Italian Ministry of Education, University and Research under the  Programme “Department of Excellence” Legge 232/2016 (CUP D93C23000100001), and by the Research Project of National Relevance “Evolution problems involving interacting scales” granted by the Italian Ministry of University and Research (MUR PRIN 2022, project code 2022M9BKBC, CUP D53D23005880006).

NDN has been funded by the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract number MB22.00034 through the project TENSE.

MG and FM were partially supported by the GNAMPA-INdAM project
Modellizzazione ed Analisi attraverso Leggi di Conservazione,
(CUP E53C23001670001).

MG was partially supported by the Project  funded  under  the
National  Recovery  and  Resilience  Plan  (NRRP)
of  Italian  Ministry  of  University and Research
funded by the European Union (NextGenerationEU Award, No.  CN000023,  Concession  Decree  No.  1033  of  17/06/2022  adopted  by  the  Italian Ministry of University and Research, CUP: H43C22000510001, Centro Nazionale per la Mobilità Sostenibile).

MG was also partially supported by the Project  funded  under  the
National  Recovery  and  Resilience  Plan  (NRRP)
of  Italian  Ministry  of  University and Research
funded by the European Union (NextGenerationEU Award, No.  
ECS_00000037, CUP H43C22000510001,
MUSA – Multilayered Urban Sustainability Action).

MG and FM were partially supported by the PRIN 2022 project Modeling, Control and Games through Partial Differential Equations (D53D23005620006),
funded by the European Union (NextGenerationEU).
















 Paper Title: On the mean-field limit of the Cucker-Smale model with Random Batch Method.




    
        
        
     
        a]Yuelin Wang sjtu_wyl@sjtu.edu.cn
        a]Yiwen Lin linyiwen@sjtu.edu.cn
[a]School of Mathematical Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, P.R.China 


On the mean-field limit of the Cucker-Smale model with Random Batch Method
    [
    
==========================================================================



§ ABSTRACT 
    In this work, we focus on the mean-field limit of the Random Batch Method (RBM) for the Cucker-Smale model. Different from the classical mean-field limit analysis, the chaos in this model is imposed at discrete time and is propagated to discrete time flux. We approach separately the limits of the number of particles N→∞ and the discrete time interval τ→ 0 with respect to the RBM, by using the flocking property of the Cucker-Smale model and the observation in combinatorics. The Wasserstein distance is used to quantify the difference between the approximation limit and the original mean-field limit. Also, we combine the RBM with generalized Polynomial Chaos (gPC) expansion and proposed the RBM-gPC method to approximate stochastic mean-field equations, which conserves positivity and momentum of the mean-field limit with random inputs.

    
    Keywords: Random Batch Method, mean-field limit, Cucker-Samle model, stochastic Galerkin.






§ INTRODUCTION


    Collective behaviors of many-body systems are ubiquitous in the real world, like  flocking of birds <cit.>, swarming of fishes <cit.>, synchronicity of fireflies <cit.> and pacemaker cells <cit.>. We use the jargon “flocking" to describe the phenomenon in which self-propelled particles organize into an ordered motion using only limited environmental information and simple rules <cit.>. Since there is a great deal of literature on collective behaviors and related models, we refer the readers to <cit.> and the references therein.
    
     
    The Cucker-Smale model, proposed by Cucker and Smale <cit.>, is a famous model of collective behaviors that models phenomenologically  the flocking phenomenon. It is in the form of an N-body second-order system of ordinary differential equations or particle systems  for position and velocity, which resemble Newton's equations.  
    

     In addition, the mean-field limit, as the number of particles N →∞,  is also a prevalent tool for simplifying large particle systems, which makes use of continuum models derived within the framework of mathematical kinetic theory, to approximate collective behaviors. In these years, there has been extensive research on the mean-field limit of particle systems and we refer to <cit.> for survey articles.
     
    As a classical type of N-body systems, in the Cucker-Smale model, each particle interacts with N-1 particles, and thus the computational cost to solve it is 𝒪(N^2) per time step. An efficient algorithm to reduce computational complexity to 𝒪(N) is the Random Batch Method (RBM) proposed by Jin et al. <cit.> in 2020. The RBM indeed constructs a randomly decoupled system with subsystems of interaction between p particles, where the constant p≪ N. In this system, at each time step, the interaction only occurs within small batches with p particles. The choice of batches is random and thus the time-average effect makes it  a good approximation of the original system <cit.>.
    
     
    
    
    An interesting problem is to understand the mean-field limit of the RBM. In 2022, Jin and Li considered the mean-field limit of the RBM on first-order systems with Gaussian noise in <cit.>. 
        
        
        
        
    Inspired by <cit.>, in this paper, we investigate the mean-field limit of the RBM on the Cucker-Smale model as the number of particles tends to infinity. Unlike the first-order system considered in <cit.>, the Cucker-Smale model is a second-order system with interaction coupled with both velocity and position. Technically, moment control needs to be  derived by the asymptotic flock estimate instead of using the contraction property in first-order systems in <cit.>. Specifically, we mainly analyze the path from (b) to (c), and briefly discuss the path from (a) to (d) in the following figure (<ref>). On the upper left corner, Ha et al. give an estimate on (a) in <cit.> and a proof of the existence of mean-field limit on (d) in <cit.>. Their analysis on (d) is uniform in time but implicit in N. In later work <cit.>, an explicit estimate on N is given but only works on finite time. The four models in (<ref>) are represented in Section <ref>.
    
    
    Cucker-Smale system (<ref>) @ > (d)  N→∞ >> mean-field limit (<ref>)
    
             @A (a) τ→ 0 AA  @AA (c) τ→ 0 A
    Random Batch Cucker-Smale model (<ref>) @>> (b)  N→∞ > mean-field limit of RBM (<ref>)


    
   With the justification of (b) and (c),   we employ the RBM, which is an efficient particle-based numerical scheme for the approximation of stochastic mean-field equations of the Cucker-Smale model, inspired by <cit.>. We consider such an application in the stochastic setting, where the interacting force depends on a random variable modeling uncertainties.  This is an important problem in Uncertainty Quantification <cit.>. The method combines the RBM with a generalized Polynomial Chaos (gPC) expansion in the random space and thus we call it the RBM-gPC. It avoids loss of positivity like MCgPC in <cit.>, and moreover preserves the mean velocity during evolution.

    
    The rest of the paper is organized as follows. In Section <ref>, we give a concise introduction to the Cucker-Smale model including notations, assumptions and prerequisites. Section <ref> presents our main results on the analysis of limits (b) and (c) in (<ref>) with helpful discussions on the limits from (a) to (d). Section <ref> and Section <ref> provide the proof details. In Section <ref>, we show some numerical experiments with the RBM-gPC. Finally, Section <ref> is devoted to a summary of our main results and some remaining issues to be explored in the future.
    
    
    
    


  




§ PRELIMINARIES

In this section, we introduce the main models in (<ref>) with assumptions and notations. Also, some basic properties of the models are given.



 §.§ The Cucker-Smale model

Let X_i and V_i be the position and velocity of the i-th particle with unit mass, and ψ(| X_j -X_i|) be the communication weight between the j-th and i-th particles. The Cucker-Smale model reads as the following

    {d/dt X_i(t) =    V_i(t),   i=1,⋯, N,
    d/dt V_i(t) =   κ/N-1∑_j=1^N 
            ψ(|X_j(t)-X_i(t)|)(V_j(t)-V_i(t)),
        .

where κ is the nonnegative coupling strength and ψ satisfies  positivity, boundedness, Lipschitz continuity and mononticity conditions, i.e., there exist positive constants ψ_0, ψ_M  > 0  such that 

    0< ψ_0 ≤ψ(r) ≤ψ_M, ∀ r≥0;   ψ_Lip<∞;   (ψ(r_1)-ψ(r_2))(r_1-r_2)≤ 0,  r_1,r_2 ∈ℝ_+.

Without loss of generality, we set ψ_M = 1 in this paper for convenience. 
Under the condition (<ref>) of ψ, it is observed in <cit.> that the total momentum is conserved as a constant and the total energy is nonincreasing along the Cucker-Smale flow. Actually,

    supposing that {(X_i,V_i)} is the solution of  system (<ref>), then for any t>0, one has
    
    d/dt∑_i=1^N V_i (t) = 0,   d/dt∑_i=1^N |V_i(t)|^2 = -κ/N-1∑_i,jψ(|X_j(t) - X_i(t)|) |V_j (t) - V_i (t) |^2.




A key observation of the Cucker-Smale model, given by Ha et al. in <cit.>, is that the standard deviations of particle phase-space positions are dominated by SDDI (the system of dissipative differential inequalities):

    | d𝒳/dt| ≤𝒱,   d𝒱/dt≤ -ϕ (𝒳) 𝒱,

where (𝒳,𝒱) are nonnegative functions and ϕ is a nonnegtive measurable function. This is a useful tool to analyze the Cucker-Smale model. In this paper, we use a variant  of (<ref>) (Lemma <ref>) in the proof of our main result. One can find its proof in Lemma 3.1 in <cit.>.

    Suppose that two nonnegative Lipschitz functions 𝒳 and 𝒱 satisfy the coupled differential inequalities:
    
    | d𝒳/dt| ≤𝒱,   d𝒱/dt≤ -α𝒱 + γ e^-α t𝒳,    a.e.  t>0,

    where α and γ are positive constants. Then, 𝒳 and 𝒱 satisfy the uniform bound and decay estimates:
    
    
    𝒳(t) ≤2M/α (𝒳(0) + 𝒱(0)),   𝒱(t) ≤ M (𝒳(0) + 𝒱(0)) e^-α t/2,    t≥ 0,

    where M is given by
    
    M:= max{1,2γ/α e} + 8γ/α^3 e^3.





 §.§ The mean-field limit of the Cucker-Smale model


The corresponding evolution for the distribution function f(x, v, t) was first derived by Tadmor, etc. in <cit.> by the BBGKY hierarchy (e.g. <cit.>), which follows

    ∂_t f + v·∇_x f+ ∇_v · [ξ[f]f]=0,

where the operator ξ is defined by

    ξ[f](x,v,t) = κ∫∫ψ(|x-y|)(w-v)f(y,w,t)  dwdy.

It is clear that 

    ∂/∂ t∫∫ vfdxdv =0.

Similar with energy-decreasing in (<ref>) in the microscopic scale, the mean-field limit has the following property: 
 
Let f_0 be compactly supported in ℝ^2d with x = ∫ x f_0 (x,v) dxdv and v = ∫ v f_0 (x,v) dxdv. Define C_x0:=inf{X≥0  |  supp(f_0)⊂ B(x,X) ×ℝ^d } and C_v:=inf{V≥0  |  supp(f_0)⊂ℝ^d × B(v,V)}, where B(x,X) designates the ball of center x and radius X in ℝ^d. Let f_t be the unique solution of  (<ref>) with f(0) = f_0. If the initial data satisfies

    C_v < κ∫_C_x0^∞ψ (2s) ds,

then there exists C_x>0 such that 
    
    supp[f_t]⊂ B(x + tv, C_x) × B(v, C_v e^-κψ(2C_x) t).



    Under  assumption (<ref>) of ψ,  one has ψ(2C_x)≥ψ_0. Actually, the assumption of the lower bound ψ_0 is for the convenience of the proof and can be relaxed if (<ref>) is satisfied by both discrete and continuous systems, guaranteeing the flocking in the Cucker-Smale model.




 §.§ The Wasserstein distance


To introduce the topology of the space of probability measures, here we recall the definitions of the Wasserstein space <cit.>. For a metric space 
 (ℝ^d,ρ) and q≥1, define the Wasserstein space of order q by

    𝒫_q,ρ (ℝ^d):= {μ∈𝒫(ℝ^𝕕); ∫_ℝ^dρ(x,0)^q ν (dx)< ∞},

endowed with the Wasserstein metric 

    W_q,ρ(μ,ν) := ( inf_π∈Π (μ,ν){∫_ℝ^d ×ℝ^dρ(x,y)^q π (dx,dy)})^1/q,

where μ,ν∈𝒫_q (ℝ^d) and Π (μ,ν) is the set of probability measures on ℝ^d ×ℝ^d with marginals μ and ν respectively. The W_q metric, or distance, induces a kind of weak topology and measures the closeness between distributions. We use the notation W_q,p when ρ(x,y):= x-y _p and typically abbreviate W_q = W_q,2. 
For convenience in notation, we simplify · as |·| without ambiguity.


Recall the Jordan decomposition for a signed measure |μ| := μ^+ + μ^-. One can control the W_q distance by the weighted total variation by the following lemma (see e.g. Theorem 6.15 in <cit.>), which helps the proof of our main theorems, i.e., Theorem <ref> and Theorem <ref>, in Section <ref>.

    Let μ and ν be two probability measures on a Polish space (𝔛, d). Let q∈ [1,∞) and x_0 ∈𝔛. Then 
    
    W_q (μ,ν) ≤ 2^1-1/q( ∫ d(x_0,x)^q d |μ-ν|(x) )^1/q.





 §.§ The Random Batch method and its mean-field dynamics


In each sub-time interval [t_k-1,t_k), we set [i]_k as the batch containing the particle i. The RBM-(approximated) system of (<ref>) becomes:

    {∂_t X_i^R(t) =    V_i^R(t),   i=1,⋯, N,
    ∂_t V_i^R(t) =   κ/p-1∑_j∈[i]_kψ(|X_j^R(t)-X_i^R(t)|)(V_j^R(t)-V_i^R(t)),
            
        .

with initial data (X_i^R(0),V_i^R(0)) = (X_i^in,V_i^in), i = 1,⋯,N. See Algorithm <ref>.









In the N→∞ limit, the N-particle system is then reduced to a p-particle
subsystem described by the following system for t∈ [t_k,t_k+1):

    {∂_t X_i(t) =    V_i(t),   i=1,⋯, N,
    ∂_t V_i(t) =   κ/p-1∑_j=1^p 
            ψ(|X_j(t)-X_i(t)|)(V_j(t)-V_i(t)),
        .

with Z_i:=(X_i,V_i) and Z_i (t_k) being i.i.d drawn from f(·; t_k). Here f(·; t_k) is the law of Z_1(t_k^-) for k≥ 1 and f(·;0) equals the initial distribution f_0 . We impose Z_1 (t_k^-) = Z_1 (t_k^+).  Except the individual particle 1, the rest p-1 particles are sampled from an infinite pool of independent particles from particle 1 at each time step t_k. This process becomes the mean-field limit model of the RBM system (<ref>), and one may write the following mean-field limit for the RBM in terms of the probability distribution as shown in Algorithm <ref>, while (<ref>) becomes its microscopic description.










§ THE MAIN RESULT


In this paper, we show the conservation law and the mean-field limit of the Random-Batch Cucker-Smale model. First, we show the conservation of momentum and dissipation of kinetic energy of the  mean-field limit system of RBM in Section <ref>. Then, we analyze the procedure (b) and (c) in Section <ref>. 


Assume that the initial data of Algorithm <ref> and Algorithm <ref> obey the same law f_0(x,v)dxdv. For notational convenience, recall x:= ∫ x f_0(x,v) and v:=∫ vf_0 (x,v); and set z_i:=(x_i,v_i) for each i and 

    f(x,v;t):= ∫_ℝ^2d(p-1) f^(p)(⋯ ; t^-)dx_2 ⋯ dx_p dv_2⋯ dv_p = f_t(x,v).

Note that f is defined on ℝ^2d in Algorithm <ref> and f^(p) is defined on ℝ^2dp. Also, the definition of f is consistent with that in Algorithm <ref> at t=t_k.

In the following, we suppose that f_0 is compactly supported and then there exists a constant R>0 such that supp[f_0]⊂ B(0,R), where B(0,R) is a ball with radius R centered in the origin. Without loss of generality, we set v=0 as a standardization of the initial momentum.




 §.§ Basic characteristics of the mean-field limit system of RBM


Here, we show that the two basic characteristics: conservation of momentum and dissipation of kinetic energy, as observed in the Cucker-Smale model (<ref>) and the RBM system (<ref>), also holds for the RBM mean-field system described in Algorithm <ref>.

 
For t∈ (t_k,t_k+1), it holds
    
    ∂/∂ t∫_ℝ^2d v f(x,v;t) dx dv = 0.

In particular, since v=0, ∫ vf(x,v;t) = 0 for any t∈ [0,T].



By the definition of f, one has

    ∂/∂ t∫_ℝ^2d vf(x,v;t)dxdv =    ∫_ℝ^2d∫_ℝ^2d(p-1) v ∂/∂ t f^(p) (x,·, v, ·;t^-)dxdx_2⋯ dx_p dvdv_2⋯ dv_p
    
            =    ∫_ℝ^2dp v_1 [ -∑_i=1^p ∇_x_i· v_i f^(p)(⋯ ; t) - ∑_i=1^p ∇_v_i·ξ_i f^(p)(⋯ ;t)]dz_1⋯ dz_p
    
            =    κ/p-1∑_j=2^p∫_ℝ^2dpψ (|x_j - x_1|)(v_j-v_1) f^(p)_t(z_1,⋯ ,z_p)dz_1⋯ dz_p
    
            =     0.

The last equality is derived from the fact that ψ (|x_j - x_1|)(v_j-v_1)f^(p)_t is an odd function on (z_j,z_1).





Note that f(x,v;t_k^-) = f(x,v;t_k) = f(x,v;t_k^+). Then by induction one has ∫ vf(x,v;t) = v.


 
For any t∈ (t_k,t_k+1),
    
    ∂/∂ t∫_ℝ^2d |v|^2 f(x,v;t)dxdv ≤ 0.

In particular, M_2^v f_t ≤ M_2^vf_0, where M_2^v f_t:=∫_ℝ^2d |v|^2 f(x,v;t)dxdv. Moreover, since 0<ψ_0≤ψ, it holds
    
    M_2^v f_k+1≤ M_2^v f_0(e^-2κψ_0 τ + 8κ^2 ψ_0/p-1τ^2 )^k .



    
    For any t∈ (t_k,t_k+1),
        
    ∂/∂ t M_2^v f_t =    ∫_ℝ^2dp |v_1|^2 ( -∑_i=1^p ∇_x_i· (v_i f^(p)_t) - ∑_i=1^p ∇_v_i· (ξ_i f^(p)_t) )
    
                    =   ∫_ℝ^2dpκ/p-1∑_j=1^pψ(|x_1-x_j|)(v_j-v_1)· 2v_1 f^(p)_t
    
                    =    -∫_ℝ^2dpκ/p-1∑_j=1^pψ(|x_1-x_j|)|v_j-v_1|^2 f^(p)_t
    ≤    0.

        The last equality is derived by the symmetry of ψ and f_t^(p). By the continuity of f, one has M_2^v f_t ≤ M_2^vf_0. Since ψ has a positive lower bound ψ_0, furthermore, it holds
        
    ∂/∂ t M_2^v f_t ≤    -∫_ℝ^2dpκ/p-1∑ _j=1^p ψ_0 |v_j-v_1|^2 f^(p)_t
    
                    =    -2κψ_0 ∫_ℝ^2d |v|^2 f_t(x,v) dxdv + 2κψ_0 ∫_ℝ^2dp v_1· v_2 f^(p)_t,

        by the symmetry of f^(p)_t. For the second term in (<ref>), one has
        
    ∂/∂ t   ∫_ℝ^2dp v_1 · v_2 f^(p)_t
    
                    =    κ∫_ℝ^2dpv_2/p-1·∑_k 1^p ψ(|x_1-x_k|)(v_k-v_1)f^(p)_t + v_1/p-1·∑_k 2^p ψ(|x_2-x_k|)(v_k-v_2) f^(p)_t
    
                    =    2κ∫_ℝ^2dpv_2/p-1·∑_k 1^p ψ(|x_1-x_k|)(v_k-v_1)f^(p)_t.

        For the last integration in (<ref>), by the indistinguishability of the particles, one has  
        
    ∫_ℝ^2dp v_2·∑_k 1^p ψ(|x_1-x_k|)(v_k-v_1)f^(p)_t
       = ∫_ℝ^2dp v_2·∑_k 1,2^p ψ(|x_1-x_k|)(v_1-v_k)f^(p)_t + ∫_ℝ^2dp v_2·ψ(|x_1-x_2|)(v_2-v_1)f^(p)_t
       = ∫_ℝ^2dp v_2·∑_k 1^p ψ(|x_1-x_k|)(v_1-v_k)f^(p)_t 
                    +∫_ℝ^2dp v_2·ψ(|x_1-x_2|)(v_2-v_1)f^(p)_t
       + ∫_ℝ^2dp v_2·ψ(|x_1-x_2|)(v_2-v_1)f^(p)_t
       = -∫_ℝ^2dp v_2·∑_k 1^p ψ(|x_1-x_k|)(v_k-v_1)f^(p)_t +∫_ℝ^2dp |v_2-v_1|^2 ψ(|x_1-x_2|) f^(p)_t.

        Hence, recalling ψ≤ 1, it holds
        
    ∫_ℝ^2dp v_2·∑_k 1^p ψ(|x_1-x_k|)(v_k-v_1)f^(p)_t
            = 1/2∫_ℝ^2dp |v_2-v_1|^2 ψ(|x_1-x_2|)f^(p)_t≤ 2 M_2^v f_t.

        Then it yields from (<ref>) that 
        
    ∂/∂ t∫_ℝ^2dp v_1 · v_2 f^(p)_t≤4κ/p-1 M_2^vf_t_k-1.

        By combining (<ref>) and (<ref>), it holds
        
    ∂/∂ tM_2^vf_t    ≤ -2κψ_0 M_2^vf_t + 2κψ_04κτ/p-1M_2^vf_t_k-1 +  2κψ_0∫ v_1 · v_2 f_t_k-1^(p)
       =  -2κψ_0 M_2^vf_t + 2κψ_04κτ/p-1M_2^vf_t_k-1
       ≤  -2κψ_0M_2^v f_t + 8κ^2ψ_0/p-1τ M_2^vf_t_k-1.

        By Grönwall's inequality, one has
        
    M_2^v f_k≤ M_2^v f_t_k-1(e^-2κψ_0 τ + 8κ^2ψ_0/p-1τ^2 ) ≤ M_2^v f_0(e^-2κψ_0 τ + 8κ^2 ψ_0/p-1τ^2 )^k .

    
    


    
    
    
    
    
    
    
    
    
    
    
    
    





 §.§ The main theorems



First we discuss the Wasserstein distance between the distribution of the RBM of the N-particle system (<ref>) and the distribution of the N→∞ system (<ref>). 

Define the operator 𝒬_N^(k) on the probability measure space 𝒫(ℝ^2d) as follows. Let Z_i^R(0) be i.i.d drawn from f_0. Corresponding to Algorithm <ref>, we define 

    𝒬_N^(k)(f_0):= Law(Z_1^R(t_k)),

where Law(Z_1^R) means the law of Z_1^R.  Conditioning on a specific sequence of random batches, the particles are not exchangeable. However, when considering the mixture of all possible sequences of random batches, the laws of Z_i^R(t_k) are identical.

In addition, we define 𝒬_∞ as one time step evolution in Algorithm <ref>. Thus one arrives at 

    𝒬_∞^k (f_0) = 𝒬_∞∘⋯∘𝒬_∞(f_0),  (k copies),

which is expected to be the mean field limit of the RBM after k steps. Note that the 𝒬_∞ dynamics can fully determine the probability transition, knowing the marginal distribution of Z_1, while knowing only the marginal distribution is not enough for the dynamics of 𝒬_N^(k). The joint distribution must be known in the latter case. 


Theorem <ref> gives an estimate between 𝒬_∞^k (f_0) and 𝒬_N^(k)(f_0) on the W_q distance. In this theorem, we introduce a quantity ϵ_k∈ℝ, which was first introduced in <cit.>. Plainly speaking, the ϵ_k means the probability that the particle i is not “clean", where a particle i is clean at t_k^- means its batchmates at t≤ t_k were mutually independent and independent to the particle i when they interacted. For better organization, we give the definition of "clean"   and more details in Section <ref>.


    For f_0 with the compact support supp [f_0], it holds that 
    
    W_q(𝒬_∞^k (f_0),𝒬_N^(k) (f_0))≤ C (1+t_k) ϵ_k^1/q,

    with q ∈ [1,∞) and C=C(κ,ψ_0,R). For fixed k, it holds
    
    lim_N→∞ϵ_k =0,

    with ϵ_k≤𝒪(N^-1).







    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    



    Then, we consider the limit dynamics given by the operator 𝒬_∞ and how 𝒬_∞^n (f_0) approximates the dynamics of the Fokker-Planck equation (<ref>), i.e. the procedure (c) in (<ref>). We prove that the W_q distance can be controlled by τ with a smooth initial distribution f_0 with a compact support uniformly in time. 
    
     
        Let f be a solution to the Fokker-Planck  equation (<ref>) with initial condition f(0) = f_0. 
        It holds
        
        
        
        
        the uniform-in-time estimate
        
    sup_n W_q (𝒬_∞^n (f_0), f(nτ))≤ Cτ,

        with q∈[1,∞)and C=C(κ,ψ_0,R).
    

    







 §.§ Some helpful discussions



In addition, we give some  discussions on Lemma <ref> that will be used for the proof of Theorem <ref>, and on the commutation of (<ref>).





  §.§.§ Approximation using PDE analysis


Similar with [<cit.>, Lemma 4.4], one can obtain an 𝒪(τ^2/q) bound on the W_q distance between 𝒬_∞ (f_t_k) and S(f_t_k) in Proposition <ref>. Here S(f_t_k) denotes the law of (<ref>) in Section <ref>.


     Let f be a solution to the Fokker-Planck  equation (<ref>) with initial condition f(0) = f_0 ∈ (C^2∩ W^2,∞)(ℝ^2d). Suppose that the interaction kernel ψ satisfies 
    sup_r≥ 0 (|ψ^'(r)| + |ψ^''(r)|) ≤ϵ_ψ
 with a constant ϵ_ψ. It holds
    
    W_q (𝒬_∞ (f_t_k),S(f_t_k)) ≤ C(κ,ϵ_ψ , R, d) τ^2/q,    q∈ [1,∞).


The proof relies on the PDE analysis on the evolution equations on 𝒬_∞ (f_t_k) and S(f_t_k), using the corresponding operator groups. There are two limitations on this method. One is that more regular assumptions on both the interaction kernel and initial data are needed to guarantee the existence of a unique classical solution f_t ∈ C^2([0,T)×ℝ^2d) for (<ref>). In addition, if one uses the estimate obtained by Lemma <ref> with the total variance bound, the order of τ is 2/q. 
A detailed proof of Proposition <ref> is given in Appendix <ref>. 





  §.§.§ The mean-field limit of the Cucker-Smale system
 

Up to now, we have given a brief analysis of the path ((b),(c)) in the lower right corner of (<ref>). Here, we discuss the upper left corner path in the sense of the Wasserstein distance as well. Denote by 

    μ_t^N:=∑_i=1^N 1/Nδ_z_i(t)
 the empirical measure and μ_t^R the empirical measure derived by the RBM with μ_0^R = μ_0^N.



 
    For the path from (a) to (d) in (<ref>), it holds 
    
    𝔼 W_2 (f_t,μ_t^R)     ≤ C(κ, R,ψ)√(τ/p-1) +
            C(R)·{   N^-1/4     if  d<4,
       N^-1/4√(log(1+N))     if  d=4,
       N^-1/d     if  d>4.
            .



    We first consider the procedure (a). For the first τ→ 0 limit, by the definition of the W_q distance, one has
    
    W_2^2 (μ_t^R, μ_t^N) 
            ≤1/N∑_i=1^N (|X_i^R - X_i|^2 + |V_i^R - V_i|^2).

    Based on [<cit.>, Theorem 3.2], one has
    
    𝔼(1/N∑_i-1^N |V_i^R (t) - V_i (t)|^2)
            ≤ C(1/p-1-1/N-1)τ + Cτ^2 + C(1+τ) exp(-κψ_0 t),

    for some constant C depending on κ, R and ψ.
    Note that
    
    d/dt∑_i=1^N |X_i^R - X_i|^2 ≤
        2√(∑_i=1^N |X_i^R - X_i|^2)√(∑_i=1^N |V_i^R - V_i|^2),

    by the Cauchy-Schwarz inequality and
    
    √(∑_i=1^N |X_i^R - X_i|^2)≤√(∑_i=1^N |V_i^R - V_i|^2) T.

    It thus holds
    
    𝔼 W_2 (μ_t^N,μ_t^R) ≤𝔼1/√(N)√(∑_i=1^N |X_i^R - X_i|^2 + ∑_i=1^N |V_i^R - V_i|^2)
    ≤√(C τ(1/p-1-1/N-1))(1+T).
.

    Next, for the procedure (d), by the estimate of [<cit.>, Corollary 1], one has W_q (μ_t^N,f_t) ≤ C(κ,R)W_q(μ_0^N,f_0). 
    In addition, according to the convergence rate of the empirical measure in [<cit.>, Theorem 1], with f_0 compactly supported, one has 
    
    𝔼 W_q^q (f_0,μ_0^N) ≤ C(R)·{   N^-1/2     if  q>d/2,
       N^-1/2log(1+N)      if  q=d/2,
       N^-q/d     if  q<d/2.
        .


    Therefore, under the same assumption of Theorem <ref> and Theorem <ref>, for the path from (a) to (d), one has 
    
    𝔼 W_2 (f_t,μ_t^R)    ≤𝔼[W_2 (μ_t^N,μ_t^R) + W_2 (f_t,μ_t^N)]    
             ≤ C(κ,R,ψ)√(τ/p-1) + C(κ,R) 𝔼 W_2 (f_0,μ_0^N)
       ≤ C(κ,R,ψ)√(τ/p-1) + C(κ,R)(𝔼 W_2^2 (f_0,μ_0^N))^1/2.











§ PROOF OF THEOREM <REF>


    In this section, we prove Theorem <ref> by an observation in combinatorics. 
    Recalling the critical quantity ϵ_k we mentioned above Theorem <ref>, we first give the Lemmas <ref>-<ref>, for the proof of Theorem <ref>, and show the mathematical definition of ϵ_k in Section <ref>.
    The proof of Lemmas <ref>-<ref> are shown in Sections <ref>-<ref> respectively.
    
    Finally, Theorem <ref> will be proved in Section <ref> by combining Lemma <ref>, Lemma <ref> and an estimate on the diameters of the compact support in space and velocity, shown in Lemma <ref>.


    
    
        Under the assumption and notation of Theorem <ref>, it holds
            
    |𝒬_∞^k (f_0) - 𝒬_N^(k) (f_0)|_TV≤ 2ϵ_k,

        where ϵ_k is a positive constant  with respect to N.
    
    
    
    
        For any fixed k, with the assumption and notation of Theorem <ref>, it holds that ϵ_k ≤𝒪(N^-1).
    


    




 §.§ Prerequisites

    To give a detailed explanation of ϵ_k in Theorem <ref>, we introduce the concept “clean” in Definition <ref>. First, for each particle i, we define a sequence of lists {L_i^(k)} associated with i, given as follows:
    
        
  * L_i^(0) = {i};
        
  * For k≥ 1, let 𝒞_q^(k-1) be the batch that particle i stays in for t∈[t_k-1,t_k), then
        
    L_i^(k) = ∪_j∈𝒞_q^(k-1) L_j^(k-1).

    
    Here, L_i^(k) can be viewed as the particles that have impacted i for t < t_k. Clearly, a particle i_1∈ L_i^(k) might not have been a batchmate of i. It could have been a batchmate of i_2, and then i_2 was a batchmate of i at some time. The important observation is that if L_i^(k) and L_j^(k) do not intersect for a given sequence of random batches, then particles i and j are independent at t_k^-. Note that we are not claiming all particles in L_j^(k) are independent of those in L_j^(k) at t_k^-. In fact, it is possible that some i_1 ∈ L_i^(k) and j_1 ∈ L_j^(k) are in the same batch on [t_k-1, t_k). However, i_1 and j_1 must be independent at the times when they were added to the batches that eventually impact i,j at t_k^-. Then we are motivated to define the following.
    
    
        We say the particle i is clean on [t_k,t_k+1) if the batch 𝒞_q^(k) that contains i at t_k^+ satisfies the following:
        
            
  * any j∈𝒞_q^(k) is clean at t_k^-;
            
  * any j,ℓ∈𝒞_q^(k) with jℓ, L_j^(k) and L_ℓ^(k) do not intersect.
        
    
    In other words, a particle i is “clean" at t_k^- if its batchmates at t<t_k were mutually independent and independent to i when they interacted.
    
   
    Let A_k denote the set of particles that are clean at t_k^-. Then
    
    A_0 = {1,⋯, N}.

    For k≥ 1, one has 
    
    A_k = { i ∈ A_k-1: i∈𝒞_q^(k-1), ∀ j,ℓ∈𝒞_q^(k-1),  jℓ ,
    
            j∈ A_k-1, ℓ∈ A_k-1,  L_j^(k-1)∩ L_ℓ^(k-1) = ∅}.

    Now we define
    
    ϵ_k := ℙ(1∉ A_k).

    Note that by symmetry, ϵ_k is also the probability that particle i is not clean.









 §.§ Proof of Lemma <ref>


    We first introduce Lemma <ref>.
     
        Consider a fixed sequence of divisions of random batches {𝒞^(ℓ)}_ℓ≤ k-1.
        
            
  * It holds that
                
    |L_i^(k)|≤ p^k,

                and the particle i is clean at t_k^- if and only if the equality holds.
            
  * The distribution of Z_i (Recall Z_i := (X_i,V_i).) for a clean particle i at t_k^- is 𝒬_∞^k (f_0).
        
        Here the symbol |A| means the cardinality of a set A.
    
    
        The proof is a straightforward induction. Here, let us just briefly mention the proof of the second claim. For k=0, the statement is trivial. Now, suppose the statement is true for all k ≤ m-1. We now consider k = m.
        
        For the given sequence of random batches {𝒞^(ℓ)}_ℓ≤ k-1, that a particle i is clean at t_m^- means that on [t_m-1, t_m), the particles in the batch for i are independent of i at t_m-1. By the induction assumption, the distribution of one particle at t_m-1 is given by 𝒬_∞^m-1(f_0). By the independence, the joint distribution of them at t_m-1 is therefore
        
    f^(p) (⋯, t_m-1) = 𝒬_∞^m-1(f_0)^⊗ p.

        From t_m-1 to t_m, the evolution of the joint distribution obeys  equation (<ref>). Hence, at t_m^-, the distribution of particle i is given by 𝒬_∞^m(f_0). 
    



    Now we begin to prove Lemma <ref>.

    
        By Lemma <ref>, one has
        
    𝒬_N^(k) (f_0) = ℙ(1∈ A_k)𝒬_∞^k (f_0)
                + ℙ(1∉ A_k)ν_k,

        for some probability measure ν_k. To see this, consider all possible sequences of random batches. Only the first k divisions of batches (i.e. ones at t_0,⋯, t_k-1) will affect the distribution at t_k. This subsequence (the first k divisions) can take only finitely many values, and let {c^ℓ}_ℓ≤ k-1 be such values. Then, for any E⊂ℝ^d that is Borel measurable,
        
    𝒬_N^(k) (f_0)[E] = ∑_{𝒞^ℓ = c^ℓ , ℓ≤ k-1}ℙ(𝒞^ℓ = c^ℓ ,  ℓ≤ k-1)
                ℙ(Z_1 ∈ E  | 𝒞^ℓ = c^ℓ ,  ℓ≤ k-1 ).

        Lemma <ref> tells us if {c^ℓ}_ℓ≤ k-1 is a value such that particle 1 is clean, then
        
    ℙ(Z_1 ∈ E  | 𝒞^ℓ = c^ℓ ,  ℓ≤ k-1 ) = 𝒬_∞^k (f_0)[E].

        Hence,
        
    𝒬_N^(k) (f_0)[E]    =  ℙ(1∈ A_k)𝒬_∞^k (f_0)[E] 
         + ∑_{𝒞^ℓ = c^ℓ , ℓ≤ k-1},  1∉ A_kℙ(𝒞^ℓ = c^ℓ ,  ℓ≤ k-1)
                ℙ(Z_1 ∈ E  | 𝒞^ℓ = c^ℓ ,  ℓ≤ k-1 )
       = ℙ(1∈ A_k)𝒬_∞^k (f_0)[E] + ℙ(1∉ A_k)ν_k (E),

        
        
        
        
        
        
        with
        
    ν_k (E) = ∑_{𝒞^ℓ = c^ℓ , ℓ≤ k-1},  1∉ A_kℙ(𝒞^ℓ = c^ℓ ,  ℓ≤ k-1)/ℙ(1∉ A_k)ℙ(Z_1 ∈ E  | 𝒞^ℓ = c^ℓ ,  ℓ≤ k-1 ).

        Clearly, ν_k is a convex combination of some conditional marginal distributions of Z_1, each being Law(Z_1) conditioning on a particular sequence of batches for {1∉ A_k}. Hence ν_k is a probability measure. 
        
        By (<ref>), it holds that
        
    |𝒬_∞^k(f_0) - 𝒬_N^(k) (f_0)| ≤ 
                (1-ℙ(1∈ A_k))𝒬_∞^k (f_0) + ℙ(1∉ A_k)ν_k
                = ϵ_k (𝒬_∞^k (f_0) + ν_k).

        Therefore, the total variation distance between the two measures is controlled by
        
    |𝒬_∞^k(f_0) - 𝒬_N^(k) (f_0)|_TV≤ 
                (1-ℙ(1∈ A_k)) + ℙ(1∉ A_k) = 2ϵ_k.

    




    
    
    




 §.§ Proof of Lemma <ref>

    Furthermore, we provide the proof of Lemma <ref> which is a simple variant of [<cit.>, Theorem 3.2].

    
        First of all, clearly, one has
        
    ϵ_0  = 1-1 = 0.

        Now, one can do induction on k. Assume
        
    lim_N→∞ϵ_k = 0,

        and then consider the batches for t_k → t_k+1^-. Assume the batch for particle 1 is 𝒞_q^(k). Denote
        
    B_k = {∀ j, ℓ∈𝒞_q^(k),  jℓ,   L_j^(k)∩ L_ℓ^(k) = ∅}.

        Let ℬ := 𝒞_q^(k)\{1} be the set of other particles that share the same batch with particle 1. Then, by definition of A_k+1, it holds
        
    ℙ (1∈ A_k+1) = ∑_j_1,⋯, j_p-1ℙ(ℬ={j_1,⋯, j_p-1})×
    ℙ(B_k ∩{1∈ A_k}∩_ℓ =1^p-1{j_ℓ∈ A_k} | ℬ= {j_1,⋯, j_p-1}).

         Denote E:= {ℬ={j_1,⋯, j_p-1}}, where we omit the dependence in j_ℓ, 1≤ℓ≤ p-1 for notational convenience. Conditioning on B_k ∩ E (i.e., provided that the event B_k ∩ E happens), whether the particles are clean or not is independent. Hence,
         
    ℙ(B_k ∩{1∈ A_k}∩_ℓ =1^p-1{j_ℓ∈ A_k} |  E) 
    
                 = ℙ(B_k | E))ℙ({1∈ A_k}∩_ℓ = 1^p-1{j_ℓ∈ A_k}|E,B_k)
    
                 = ℙ(B_k|E)Π_ℓ=1^pℙ(j_ℓ∈ A_k|E,B_k),

         where one has set j_p =1. Moreover,
         
    ℙ(j_ℓ∈ A_k|E,B_k) = ℙ({ j_ℓ∈ A_k }∩ E∩ B_k)/ℙ(E∩ B_k) = ℙ({1 ∈ A_k }∩ E∩ B_k)/ℙ(E∩ B_k) = ℙ({1 ∈ A_k }∩ B_k)/ℙ(B_k).

         The second and the last equalities are due to symmetry. For the last equality, ℙ({ℬ = {j_1⋯,j_p-1}}∩ B_k) should be equal for all possible j_1,⋯,j_p-1, and the same  is true for the numerator. This actually is a kind of independence. Hence, eventually due to the fact
         
    ∑_j_1,⋯, j_p-1ℙ (ℬ = {j_1,⋯, j_p-1}) ℙ(B_k | ℬ = {j_1,⋯, j_p-1}) = ℙ (B_k),

        one has
        
    1-ϵ_k+1 = ℙ (1∈ A_k+1) ≥ℙ (B_k) (1-ϵ_k 
                )^p.

        Hence, it suffices to show
        
    lim_N→∞ℙ (B_k) =1.

        To get an estimate for this, we consider the following equivalent way to construct L_i^(k): one starts with L_i ←{i} and repeat the following k times:
        
            
  * Set L_tmp← L_i and A=∅.
            
  * Loop the following until L_tmp is empty.
            
                
  (a) Pick a particle i_1 ∈ L_tmp, then choose p-1 particles from {1,⋯,N}\ A ∪{i_1} denoted by { i_2,⋯ , i_p}.
                
  (b) Set L_i← L_i ∪{ i_2, ⋯, i_p}.
                
  (c) Set A← A ∪{ i_1, i_2, ⋯, i_p}.
                
  (d) Set L_tmp← L_tmp\{ i_1, i_2, ⋯, i_p}.
            
        
        In the above, we are actually looking back from t_k-1. In the j-th iteration, we are constructing batches at t_k-j. Hence, this is an equivalent way to construct L_i^(k).

        Now, we estimate ℙ(B_k) by constructing the lists L_j_ℓ^(k): 1≤ℓ≤ p for j_ℓ∈𝒞_q^(k) using the above procedure. Consider that the lists for j_1,⋯,j_ℓ -1 have been constructed, which have included at most (ℓ -1)p^k particles. Now, for L_j_ℓ^(k) not to intersect with the previous lists, one has to choose particles from {1,⋯,N}\ [∪_z=1^ℓ -1 L_j_z^(k)∪ A ∪{i_1}] in 2.(a) step. Conditioning on the specific choices of L_j_ℓ^(k): 1≤ℓ≤ p and A with 
        
    N_1:= |L_j_ℓ^(k)∪ A∪{i_1}|,   N_2:= |A|,

        this probability is controlled from below by
        
    N- N_1p-1/N-1- N_2p-1≥N- 1-ℓ p^kp-1/N-1p-1.

        Hence, as N→∞,
        
    ℙ(B_k)≥Π_ℓ=1^p [N- 1-ℓ p^kp-1/N-1p-1]^k = 1- 𝒪(N^-1).

        Hence, lim_N→∞ϵ_k+1=0 and the claim follows.
    
    





 §.§ Proof of Theorem <ref>

    


    
    
    Now we can prove Theorem <ref>.
    
    Recall that the TV distance is defined by
    
    |μ - ν|_TV := sup_A∈ℱ |μ(A) - ν(A)|,

    for two probability measures μ, ν defined on certain sample space Ω with events ℱ. 


    We denote the diameters of the compact support in the spatial and velocity variables at time t by 𝒟_X(t) and 𝒟_V(t), respectively, i.e.
    
    𝒟_X(t) := max_i,j| X_i-X_j|,   𝒟_V(t) := max_i,j| V_i-V_j|.

    
    For the RBM system (<ref>), Ha et al. <cit.> gave an estimate on 𝒟_X^R and 𝒟_V^R shown in Lemma <ref>. This estimate works on any fixed sequence of batches and we provide the proof in Appendix <ref> for completeness.
    




        
        Let X^R,V^R be a solution of the RBM model (<ref>) with initial data f_0 satisfying 
        
    𝒟_X(0) + 𝒟_V(0)<∞.

        Then one has
        
    𝒟_V^R (t) ≤𝒟_V(0),   𝒟_X^R (t)≤𝒟_X(0) + 𝒟_V(0)t,   ∀ t ≥ 0.

    



    
    

    
        Denote z:= (x,v), Z^R_i:=(X_i^R,V_i^R), and |z|^q:=|x|^q + |v|^q.
        By Lemma <ref>, taking μ = 𝒬_∞^k (f_0) and ν = 𝒬_N^(k)(f_0), one has
        
    W_q (𝒬_∞^k (f_0),𝒬_N^(k)(f_0)) ≤ 2^1-1/q(∫_ℝ^2d |z|^q d | 𝒬_∞^k (f_0)-𝒬_N^(k)(f_0)|(z))^1/q.


       Note that Lemma <ref> degenerates into the original Cucker-Smale model when p=N. Hence for t∈[0,t_1), any {Z_i^(0)(t)}_i=1^p satisfying (<ref>), one obtains
       
    𝒟_V^(0)(t)≤𝒟_V(0),   𝒟_X^(0) (t) ≤𝒟_X (0) + 𝒟_V(0)t.

       Also, by the assumption of f_0, one has v=0, and therefore x=0. Since (<ref>) holds independent of the choice Z_i, defining 
       
    𝖷(t) :=   max{ |x|  |∃ v,  s.t.  f(x,v;t) >0},
    𝖵(t) :=   max{ |v|  |∃ x,  s.t.  f(x,v;t) >0},

       one has for t∈[0,t_1],
       
    𝖵(t)≤ C,   𝖷(t)≤ C(1+t),

       where C is a constant depending on R. 
       Then for t∈[t_1,t_2], for the new subsystem {Z_i^(1)}_i=1^p with initial distribution f_t_1, one can use Lemma <ref> again to get
       
    𝒟_V^(1)(t)   ≤𝒟_V^(1)(0)≤ 2𝖵(t_1) ≤ C, 
    𝒟_X^(1) (t)    ≤𝒟_X^(1) (0) + 𝒟_V^(1)(t-τ)
               ≤ 2𝖷(t_1) + 2𝖵(t_1)(t-τ)≤ C(1+t),

       and (<ref>) holds for t∈[t_1,t_2].
       By induction, for any t, (<ref>) holds. Therefore |𝖷(t_k)|^q + |𝖵(t_k)|^q ≤ C(1+t_k)^q.       
       
       Also, by Lemma <ref>, it holds |Z^R_i|^q≤ C(1+t_k)^q.
       Then for any z∈supp[| 𝒬_∞^k (f_0)-𝒬_N^(k)(f_0)|], |z|^q≤ C(1+t_k)^q. Therefore, 
       
    ∫_ℝ^2d |z|^q d | 𝒬_∞^k (f_0)-𝒬_N^(k)(f_0)|(z) 
               
               
               
               ≤  C(1+t_k)^q | 𝒬_∞^k (f_0)-𝒬_N^(k)(f_0)|_TV.

       
       By Lemma <ref>, one has 
       
    ∫_ℝ^2d |z|^q d | 𝒬_∞^k (f_0)-𝒬_N^(k)(f_0)|(z) ≤ C(1+t_k)^q ϵ_k.

       Hence W_q (𝒬_∞^k (f_0),𝒬_N^(k)(f_0)) ≤ C(1+t_k)ϵ_k^1/q. Then by Lemma <ref>, we finish the proof.
    


    
        The above analysis is not enough to get the mean-field limit independent of τ and for fixed N, ϵ_k→ 1 as k→∞. This is a limitation of the technique in the paper.
    














§ PROOF OF THEOREM <REF>


    Here gives an outline of the proof of Theorem <ref>. First, we show the stability of 𝒬_∞ in Section <ref>. Second, an auxiliary system is introduced in Section <ref> as a bridge to estimate the W_q distance between 𝒬_∞(f_t_k) and f_t_k+1. 
    
    Last,  the proof of Theorem <ref> is given in Section <ref>.




 §.§ Stability of 


In Lemma <ref> and Lemma <ref>, we give the stability analysis on the operator 𝒬_∞.
 
    Suppose two probability measures μ_1,μ_2 satisfying ∫ vdμ_1=∫ vdμ_2, with compact supports supp[μ_1] , supp[μ_2] respectively, where supp[μ_i]⊂ℝ^d× B(0,R_μ), i=1,2, and the constant R_μ >0. Then it holds
    
    W_q (𝒬_∞ (μ_1) ,𝒬_∞ (μ_2)) ≤ CW_q(μ_1,μ_2),    q=[1,∞),

    where C(κ,ψ,R_μ) is a positive constant.



    Set (X_i,V_i)(0)∼μ_1, and (X_i,V_i)(0)∼μ_2, for i=1, ⋯, p. The key to the proof is to make use of Lemma <ref> to get the contraction property of the velocity difference. 
    
    The two steps are as follows.
    

  
Step 1.
    First, we show an auxiliary estimate on 𝒟_V(t). By following the formulation of [<cit.>, Lemma 2.2], denote
    
    Ψ_ij:= ψ(|X_i-X_j|)/p + ( 1- ∑_j=1^p ψ(|X_i-X_j|)/p)δ_ij,

    where δ_ij is the Kronecker delta. Then Ψ_ij is nonnegative and satisfies
    
    Ψ_ij≥ψ(|X_i -X_j |)/p,   ∑_j=1^p Ψ_ij =1,  ∑_j=1^pΨ_ij(V_j-V_i) = ∑_j=1^pψ(|X_i -X_j |)/p(V_j-V_i).

    We denote V_i = (V_i^k) and obtain the following equality
    
    1/2d/dt   |V_i - V_j|^2
            = 1/2d/dt∑_k=1^d|V_i^k - V_j^k|^2
    
            =   ∑_k=1^d(V_i^k - V_j^k)(V_i^k - V_j^k)
    
            =   ∑_k=1^d κ p/p-1 (V_i^k - V_j^k)[
            ∑_m=1^p Ψ_im(V_m^k - V_i^k) - ∑_m=1^p Ψ_jm(V_m^k - V_j^k)]
    
            =    -∑_k=1^d κ p/p-1 (V_i^k - V_j^k)^2
            +  ∑_k=1^d κ (V_i^k - V_j^k)( ∑_m=1^p (Ψ_im-Ψ_jm)V_m^k)
    
            =    -∑_k=1^d κ p/p-1 (V_i^k - V_j^k)^2
       +  ∑_k=1^d κ p/p-1 (V_i^k - V_j^k)[ ∑_m=1^p (Ψ_im-min{Ψ_im,Ψ_jm} + min{Ψ_im,Ψ_jm} - Ψ_jm)V_m^k].

    For a given time t, one can choose indices i_t and j_t satisfying the relation 
    
    𝒟_V(t) = |V_i_t(t) -V_j_t(t)|,

    and thus for each m=1,2,⋯,p,
    
    ∑_k=1^d (V_i_t^k -V_j_t^k)V_j_t^k ≤∑_k=1^d (V_i_t^k -V_j_t^k)V_m^k ≤∑_k=1^d (V_i_t^k -V_j_t^k)V_i_t^k.

    Combining (<ref>) and (<ref>), one obtains
    
    1/2d/dt|V_i_t - V_j_t|^2       
            ≤     -∑_k=1^d κ p/p-1 (V_i_t^k - V_j_t^k)^2
       +  ∑_k=1^d κ p/p-1 (V_i_t^k - V_j_t^k)V_i_t^k ∑_m=1^p (Ψ_im-min{Ψ_im,Ψ_jm}) 
       -  ∑_k=1^d κ p/p-1 (V_i_t^k - V_j_t^k)V_j_t^k ∑_m=1^p (Ψ_jm -min{Ψ_im,Ψ_jm})
    
            =   -∑_m=1^pκ p/p-1min{Ψ_im,Ψ_jm}|V_i_t - V_j_t|^2
    ≤    - κ p/p-1ψ_0 |V_i_t - V_j_t|^2.

    Note that 𝒟_V is Lipschitz continuous with respect to t and thus it is differentiable almost everywhere. Consider the instant t at which 𝒟_V is differentiable. Then,
    
    d/dt𝒟_V^2 ≤d/dt|V_i_t - V_j_t|^2.

    Combining with (<ref>) and (<ref>), one obtains
    
    |d/dt𝒟_X| ≤𝒟_V,   d/dt𝒟_V ≤ -κ p/p-1ψ_0 𝒟_V,

    which is an SDDI. By Lemma <ref>, there exists a positive constant R_μ such that 
    
    𝒟_V(t)≤ 2e^-κψ_0 tR_μ.




    

  
Step 2.
    
    It can be obtained that 
    
    d/dt𝔼 | V_1 - V_1 |^q =  
               1/pd/dt𝔼∑_i=1^p (∑_k=1^d | V_i^k - V_i^k |^2)^q/2,

    with
    
    d/dt𝔼   (∑_k=1^d | V_i^k - V_i^k |^2)^q/2 
            =  𝔼[ q/2(∑_k=1^d | V_i^k - V_i^k |^2)^q/2-1d/dt∑_k=1^d 
    |V_i^k - V_i^k|^2 ]
    
            =    qκ/p-1𝔼[ |V_i - V_i|^q-2∑_j=1^p ψ(|X_j - X_i|)∑_k=1^d(V_j^k - V_i^k - V_j^k + V_i^k)( V_i^k - V_i^k). 
       .  + ∑_j=1^p 
            |V_i - V_i|^q-2(ψ(|X_j - X_i|) - ψ(|X_j - X_i|))
           ∑_k=1^d(V_j^k - V_i^k)( V_i^k - V_i^k) 
           ].

    Rewrite the right-hand side of (<ref>) as
   
    d/dt𝔼 | V_1 - V_1 |^q
           := I_21 + I_22,

   where
   
    I_21:= qκ/p-1𝔼∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) |V_i - V_i|^q-2 (V_j^k - V_i^k - V_j^k + V_i^k)(V_i^k - V_i^k),

   and 
   
    I_22:= qκ/p-1𝔼∑_i,j=1^p ∑_k=1^d
            (ψ(|X_j - X_i|) - ψ(|X_j - X_i|))
    ·|V_i - V_i|^q-2
           ( V_i^k - V_i^k )(V_j^k - V_i^k).

   First estimate I_21 using the index changing trick:
   
    I_21 =    qκ/p-1𝔼∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) |V_i - V_i|^q-2 [(V_j^k -V_j^k)-( V_i^k - V_i^k )](V_i^k - V_i^k)
    
               =    qκ/p-1𝔼[ ∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) |V_i - V_i|^q-2 (V_j^k -V_j^k)(V_i^k - V_i^k).
       .- ∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) |V_i - V_i|^q-2(V_i^k - V_i^k)^2]
    
               =    qκ/p-1𝔼[ ∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) |V_j - V_j|^q-2 (V_j^k -V_j^k)(V_i^k - V_i^k).
       .- ∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) |V_i - V_i|^q-2(V_i^k - V_i^k)^2]
    
               =     -qκ/p-1𝔼[ ∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|). 
       ·[|V_i - V_i|^q-2 (V_i^k -V_i^k) - |V_j - V_j|^q-2 (V_j^k -V_j^k)]
               .(V_i^k - V_i^k)]
    
               =    qκ/2(p-1)𝔼[ ∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) .
       .·[|V_i - V_i|^q-2 (V_i^k -V_i^k) - |V_j - V_j|^q-2 (V_j^k -V_j^k)][(V_j^k - V_j^k)-(V_i^k - V_i^k)]].

    One can use the relation 
    
    ∑_k=1^d[|V_i - V_i|^q-2 (V_i^k -V_i^k) - |V_j - V_j|^q-2 (V_j^k -V_j^k)]
            ·.[(V_j^k - V_j^k)-(V_i^k - V_i^k)]] ≤ 0,

    and the condition (<ref>) of ψ to obtain
    
    I_21≤   qκ/2(p-1)ψ_0 𝔼{∑_i,j=1^p ∑_k=1^d 
            [|V_i - V_i|^q-2 (V_i^k -V_i^k) - |V_j - V_j|^q-2 (V_j^k -V_j^k)]
       · [(V_j^k - V_j^k)-(V_i^k - V_i^k)]}
    
            =    qκ/2(p-1)ψ_0 𝔼[∑_i=1^p ∑_k=1^d 
            |V_i - V_i|^q-2 (V_i^k -V_i^k)∑_j=1^p (V_j^k -V_j^k).
       -  ∑_i=1^p ∑_k=1^d 
             p|V_i - V_i|^q-2 (V_i^k -V_i^k)^2
            -  ∑_j=1^p ∑_k=1^d 
             p|V_j - V_j|^q-2 (V_j^k -V_j^k)^2
       +  .∑_j=1^p ∑_k=1^d 
             |V_j - V_j|^q-2 (V_j^k -V_j^k)∑_i=1^p (V_i^k -V_i^k)].

    By the momentum condition 𝔼∑_i=1^p (V_i^k - V_i^k) = 0, it holds
    
    I_21≤  -p^2qκ/p-1ψ_0 𝔼|V_1 - V_1|^q.


   For I_22, one has
    
    I_22≤   qκ/p-1ψ_Lip𝔼[
                ∑_i,j=1^p ∑_k=1^d | |X_j - X_i| - 
                |X_j - X_i | | V_i - V_i|^q-2|V_i^k - V_i^k| |V_j^k - V_i^k|
                ]
    ≤   qκ/p-1ψ_Lip𝔼[
                ∑_i,j=1^p| |X_j - X_i| - 
                |X_j - X_i | | |V_i - V_i|^q-1 |V_j - V_i|
                ],

    by the Cauchy inequality. Then by the inequality | |a-b| - |c-d| | ≤ |a-c| + |b-d|, one has
   
    I_22≤   qκ/p-1ψ_Lip𝔼[
                ∑_i,j=1^p| |X_i - X_i| + 
                |X_j - X_j | | |V_i - V_i|^q-1 |V_j - V_i|
                ]
    ≤   qκ/p-1ψ_Lip𝒟_V(t) 𝔼[p
                ∑_i=1^p |X_i - X_i| | V_i - V_i |^q-1 + 
                ∑_i,j=1^p |X_j - X_j| | V_i - V_i |^q-1]
    ≤   2q p^2κ/p-1ψ_Lip𝒟_V(t) 𝔼[
                 |X_1 - X_1| | V_1 - V_1 |^q-1],

    by Hölder's inequality. Now by combining (<ref>) and (<ref>) and (<ref>), one has
    
    d/dt𝔼| V_1 - V_1 |^q ≤    -p/p-1qκψ_0 
                𝔼|V_1 - V_1|^q 
       + 2p/p-1qκψ_Lip𝒟_V(t)
                𝔼[|X_1-X_1| |V_1 - V_1|^q-1]
    ≤    -p/p-1qκψ_0 
                𝔼|V_1 - V_1|^q 
       + 2p/p-1qκψ_Lip𝒟_V(t)
                (𝔼[|X_1-X_1|^q])^1/q (𝔼|V_1 - V_1|^q)^1-1/q,

    similarly by Hölder's inequality. Since
    
    d/dt (𝔼   |V_1 - V_1|^q)^1/q = 
                1/q(𝔼|V_1 - V_1|^q)^1/q-1d/dt𝔼|V_1 - V_1|^q
    ≤    -pκ/p-1ψ_0 
                (𝔼[|V_1 - V_1|^q_q])^1/q 
                + 2pκ/p-1ψ_Lip𝒟_V(t)
                (𝔼[|X_1-X_1|^q])^1/q.

    and 
    
    d/dt𝔼|X_1 - X_1|^q =    𝔼q/2(∑_k=1^d |X_1^k - X_1^k|^2)^q/2-1∑_k=1^d(X_1^k - X_1^k)· 2(V_1^k - V_1^k)
    ≤   𝔼 q |X_1 - X_1|^q-1 |V_1 - V_1 |
    ≤    q (𝔼|X_1 - X_1|^q)^1-1/q (𝔼|V_1 - V_1|^q)^1/q,

    thus it holds 
    
    d/dt(𝔼|X_1 - X_1|^q)^1/q = 
            1/q(𝔼|X_1 - X_1|^q)^1/q-1d/dt𝔼|X_1 - X_1|^q
            ≤ (𝔼|V_1 - V_1|^q)^1/q.

    Note that 𝒟_V≤ 2 e^-κψ_0 t  R_μ by Step 1. 
    Now one knows that both (𝔼|X_1 - X_1|^q)^1/q and (𝔼|V_1 - V_1|^q)^1/q satisfy Lemma <ref> and then there exists a constant C such that
    
    (𝔼|X_1 - X_1|^q)^1/q(t)
            + (𝔼|V_1 - V_1 ^q)^1/q(t)
            ≤ C [(𝔼|X_1 - X_1 |^q)^1/q(0)
            + (𝔼|V_1 - V_1 |^q)^1/q(0)].

    Then, 
    
    W_q (𝒬_∞ (μ_1) ,𝒬_∞ (μ_2)) ≤   (𝔼 |X_1 -X_1|^q (t) + 𝔼 |V_1 -V_1|^q(t))^1/q
    ≤   (𝔼|X_1 - X_1|^q (t))^1/q
             + (𝔼|V_1 - V_1|^q (t))^1/q
    ≤    C [(𝔼|X_1 - X_1|^q)^1/q(0)
            + (𝔼|V_1 - V_1|^q)^1/q(0)]
    ≤   2 C [𝔼 |X_1 -X_1|^q (0) + 𝔼 |V_1 -V_1|^q(0)]^1/q,

    since a^1/q+ b^1/q≤ 2^1-1/q(a+b)^1/q for a≥0, b≥ 0 with q≥ 1. For any ϵ>0, the initial data can be chosen by a coupling such that 
    
    [𝔼 |X_1 -X_1|^q (0) + 𝔼 |V_1 -V_1|^q(0)]^1/q≤ϵ + W_1(μ_1,μ_2).

    Hence by the arbitrariness of ϵ, it holds W_q (𝒬_∞ (μ_1) ,𝒬_∞ (μ_2)) ≤ C W_q(μ_1,μ_2),







    
    For f defined in (<ref>) with R<∞, one has
    
    W_q(𝒬_∞^n-m+1(f_m-1),𝒬_∞^n-m(f_m))
            ≤ C W_q(𝒬_∞(f_m-1), f_m),    q=[1,∞),

    where C is relevant to ψ,κ,R and d.
    
    
        
        By the flocking property shown in Step 1 in Lemma <ref>, one has supp^v[𝒬_∞(f_m-1)]⊂supp^v[f_m-1]. Also, supp^v[f_m-1]⊂supp^v[f_0] by Lemma <ref>. Furthermore, for n-m-1≥ k≥ 0, 𝒬^k+1_∞ (f_m-1) and 𝒬^k_∞ (f_m) have compact supports, where 𝒬^0_∞ (f_m):=f_m.
        
        Similarly with the proof of Proposition <ref>, note that (X_1(kτ),V_1(kτ) and (X_1(kτ), V_1(kτ)) obey 𝒬^k+1_∞ (f_m-1) and 𝒬^k_∞ (f_m) respectively, which satisfy the momentum equality assumption of Lemma <ref>. 
    
        
        Denote 
        
    |Δ X_1|:=(𝔼|X_1 - X_1|^q)^1/q,   |Δ V_1|:=(𝔼|V_1 - V_1|^q)^1/q,   
            |Δ Z_1|:=|Δ X_1|+|Δ V_1|,

        for q≥ 1.
        Hence, by the analysis in Lemma <ref>, (<ref>) and (<ref>) still hold for t∈ [0,(n-m)τ].  
        By Lemma <ref>, one obtains
        
    |Δ Z_1 ((n-m)τ)| ≤ C |Δ Z_1 (0)|.

        By choosing a suitable coupling of the initial data, the proof is finished.
    


    

 §.§ An auxiliary system

    
    We introduce an auxiliary system (<ref>) with law S(f_t_k) to get the estimate on 
    W_q (𝒬_∞ (f_t_k), f_t_k+1) ≤ W_q(S(f_t_k),f_t_k+1) + W_q(S(f_t_k),𝒬_∞ (f_t_k) ).
 

    Rewrite (<ref>) into the following particle system
    
    {d/dtX_i(t) =   V_i(t),  i=1,⋯, N,
    d/dtV_i(t) =   ∫κψ(|x-X_i|)(v-V_i) f_t(x,v)dxdv.
            .


    In order to show the law of (X,V) at time nτ is close to 𝒬_∞^n, we consider the transient system
    
    {d/dtX̂_i(t) =   V̂_i(t),  i=1,⋯, N,
    d/dtV̂_i(t) =   ∫κψ(|x-X̂_i|)(v-V̂_i) f_t_k(x,v)dxdv,
            .

    with (X̂(0),V̂(0))∼f_t_k.
    Denote S(f_t_k):=Law(X̂(τ),V̂(τ)).

    
    Under the same assumption of Lemma <ref>, it holds
        
    supp[S(f_t_k)]⊂
                B(0,C)× B(0,C e^-κψ_0 t_k+1).

    
    
           For S(f_t_k), define
        
    𝖷_k (t) :=   max{ |x(t,x^k,v^k)|  | (x^k,v^k)∈supp[f_t_k]},
    𝖵_k (t) :=   max{ |v(t,x^k,v^k)|  | (x^k,v^k)∈supp[f_t_k]},

        where the pair (x(t,x^k,v^k), v(t,x^k,v^k)) denotes (x(t),v(t)) being the solution of (<ref>) with (x(t_k),v(t_k)) = (x^k,v^k). Note that the maximum is reached because it is assumed that supp[f_t_k] is compact. Denote K_t^x ⊂supp^x [f_t_k] (resp. K_t^v ⊂supp^v [f_t_k]) the set of points (x^k,v^k) such that the maximum is reached in 𝖷_k(t) (resp. in 𝖵_k(t)). By definition, one has |𝖷_k(t)|^2 = |x(t,x^k,v^k)|^2 for every (x^k,v^k)∈ K_t^x. It follows from Danskin's theorem <cit.> and from the fact that ∂_t x(t,x^k,v^k) = v(t,x^k,v^k) that
        
    𝖷_k(t)·𝖷_k(t) = max{ x(t,x^k,v^k) · v(t,x^k,v^k) | (x^k,v^k)∈ K_t^x }.

        Using the Cauchy-Schwarz inequality, one obtains that 
        
    𝖷_k(t) ≤𝖵_k(t).

        Similarly, by  Danskin's theorem again, one has
        
    𝖵_k(t)·𝖵_k(t) = max{ v(t,x^k,v^k) ·∫κψ(|x-x(t,x^k,v^k)|)(v-v(t,x^k,v^k)) f_t_kdxdv
    | (x^k,v^k)∈ K_t^v }.

        Note that by the definition of K_t^v, v(t,x^k,v^k) · (v-v(t,x^k,v^k))≤ 0. It holds that 
        
    𝖵_k(t)·𝖵_k(t)  ≤
                -κψ_0 ∫ v(t,x^k,v^k) · v(t,x^k,v^k) f_t_k dxdv,   (x^k,v^k)∈ K_t^v ,

        since ∫ v f_t_k dxdv = 0 and ∫f_t_kdxdv =1. Then one has 
        
    𝖵_k(t) ≤ -κψ_0 𝖵_k(t),

        and thus (<ref>) holds.
    


    
    Under the same assumption of Lemma <ref>, there exists a positive constant α such that
        
    W_q(S(f_t_k),f_t_k+1)≤ C e^-κψ_0 t_kτ^2,    q∈[1,∞),

        with C=C(κ,ψ_0,ψ_Lip, R).
    
    
    
    
    
    
        Let ℛ(x,v,t) = ∫ψ(|y-x|)(w-v)(f_t(y,w)-f_t_k(y,w))dydw. For t∈ [t_k,t_k+1), by straightforward calculus, it holds
        
        
    | ∂ℛ/∂ t (x,v,t)| =    | ∫ψ(|y-x|)(w-v) ∂f_t/∂ t(y,w)dydw|
    
                    =    | ∫ψ(|y-x|)(w-v) [ - w·∇
                    _y f_t(y,w) ..
       + ..∇_w ·(f_t(y,w)κ∫ψ (|y-y|)(w-w)f_t(y,w)dydw)]dydw |
    
                    =    | ∫[w·∇_y ψ(|y-x|)(w-v) f_t(y,w)..
        - . .dκψ(|y-x|)f_t(y,w)∫ψ (|y-y|)(w-w)f_t(y,w)dydw]dydw |
    ≤   ψ_Lip∫ |w| |v-w| f_t (y,w) dydw 
                    + dκ∫ |w-w| f_t(y,w) f_t (y,w) dydwdydw
    ≤   ψ_Lip∫ (|w| |v| + |w|^2) f_t (y,w) dydw 
                    + dκ∫ 2|w| f_t(y,w) dydw.

        
         By Lemma <ref>, 
         
    |∂ℛ/∂ t(X_1,V_1,t)| ≤ C e^-κψ_0 t.

        
        
        
        
        We first consider q to be an even integer.
        For (X_1,V_1) obeying (<ref>) and (X̂_1,V̂_1) obeying (<ref>), with (X_1,V_1) (t_k) = (X̂_1,V̂_1) (t_k) ∼f_t_k, it holds
        
    d/dt   |V_1 -V̂_1|^2 =  2∑_k=1^d(V^k_1 -V̂^k_1)d/dt(V^k_1 -V̂^k_1)
    
                    =    2κ∑_k=1^d(V^k_1 -V̂^k_1)
                     [∫ψ (|x-X_1|) (v - V_1) f_t_k(x,v) dxdv .
       .- ∫ψ (|x-X̂_1|) (v - V̂_1) f_t_k(x,v) dxdv + ℛ(X_1,V_1)]^(k)
    
                    =     2κ∑_k=1^d(V^k_1 -V̂^k_1)( ∫ψ (|x-X_1|) [(v - V_1)-(v-V̂_1)] f_t_k(x,v) dxdv)^(k)
        + 2κ∑_k=1^d(V^k_1 -V̂^k_1) (∫ [ψ (|x-X̂_1|)- ψ (|x-X_1|)](v - V̂_1) f_t_k(x,v) dxdv ) ^(k)
        + 2κ∑_k=1^d(V^k_1 -V̂^k_1) ℛ(X_1,V_1)^(k)
    ≤     -qψ_0 | V_1 - V̂_1|^2 
                    +2κψ_Lip   |X_1-X̂_1 | ∑_k=1^d
                    (V^k_1 -V̂^k_1)(∫ (v-V̂_1)f_t_k)^(k)
       + 2κ |V_1 - V̂_1 | |ℛ(X_1,V_1)|,
    ≤     -2κψ_0 | V_1 - V̂_1|^2 
                    +2κψ_Lip   |X_1-X̂_1|  |V_1 - V̂_1 |
                    |V̂_1|
       + 2κ |V_1 - V̂_1 | |ℛ(X_1,V_1)|,

        where the superscript (k) means the k-th dimension of vectors. Since ∫ vf_t dxdv = v =0 by (<ref>), and |V̂_1|≤ C e^-κψ_0 t_k by Lemma <ref> and Lemma <ref>, it holds
        
    d/dt |V_1 -V̂_1|^2   
                    ≤   -2κψ_0 | V_1 - V̂_1|^2 
                    +Cκψ_Lip   e^-κψ_0 t_k |X_1-X̂_1|  |V_1 - V̂_1 |
    
                    + 2κ |V_1 - V̂_1 | |ℛ(X_1,V_1)|,

        and
        
    d/dt|V_1 - V̂_1| ≤ -ψ_0 κ |V_1 - V̂_1|
                + Cκψ_Lip   e^-κψ_0 t_k |X_1-X̂_1|
                + κ  |ℛ(X_1,V_1)|.

        Also, with
        
    d/dt | X_1 - X̂_1| ≤ |V_1-V̂_1|,

        one has
        
    d/dt(|X_1-X̂_1| + |V_1-V̂_1| ) ≤ C(|X_1-X̂_1| + |V_1-V̂_1| ) + C|ℛ|.

        By Grönwall's inequality, it holds
        
    |X̂_1-X_1| + |V̂_1-V_1| ≤ C e^-κψ_0 t_kτ^2.

        Then
        
    [𝔼[|X̂_1-X_1|^q + |V̂_1-V_1|^q]]^1/q≤[𝔼[|X̂_1-X_1| + |V̂_1-V_1|]^q]^1/q≤ C e^-κψ_0 t_kτ^2,

        which implies
        
    W_q(S(f_t_k),f_t_k+1)≤ C(κ,ψ_0,ψ_Lip, R ) e^-κψ_0 t_kτ^2.

    



    
    
    
    The estimate on W_q(S(f_t_k),𝒬_∞ (f_t_k) ) is given by Lemma <ref>.
    
    
    
    
        Under the same assumption of Theorem <ref>, it holds
        
    W_q (𝒬_∞ (f_t_k),S(f_t_k)) ≤ C(κ, R) τ^2,    q∈ [1,∞).

    
    
    
    Set (X̂_1,V̂_1) and {(X_i,V_i)} to be the solutions of (<ref>) and (<ref>) respectively. The initial data (X̂_1(0),V̂_1(0)) and {(X_i(0),V_i(0))} follow f_t_k. Take the coupling (X̂_1(0),V̂_1(0)) = (X_1(0),V_1(0)). We introduce {(X_i^E, V_i^E)} as the approximation of {(X_i,V_i)} by the forward Euler method with time step τ. Then, taking the expectation of the selection of {(X_j(0),V_j(0)}_j 1, one has 
    
    d/dt1/2𝔼 |V_1^E - V̂_1|^2 =   ∫κψ (| x- X_1(0)|) (v- V_1(0))·𝔼(V_1^E(t)-V̂_1(t))f_t_k
        - ∫κψ (| x- X̂_1(t)|) (v- V̂_1(t))·𝔼(V_1^E(t)-V̂_1(t))f_t_k
    ≤   κ∫ψ (|x-X_1(0)|) (V̂_1(t)-V_1(0))·𝔼(V_1^E(t)-V̂_1(t))f_t_k
       + κ∫ψ_Lip |X_1(0)-X̂_1(t)| (v-V̂_1(t))·𝔼(V_1^E(t)-V̂_1(t))f_t_k.

    By Lemma <ref> and Lemma <ref>, one has 
    
    | v-V̂_1(t)|≤ Ce^-κψ_0 t_k.

    Note that 
    
    |X_1(0)-X̂_1(t)|≤ Ce^-κψ_0 t_kτ

    and 
    
    |V̂_1(t)-V_1(0)|≤ Ce^-κψ_0 t_kτ.

    Take the expectation of (X_1(0),V_1(0)), one has
    
    d/dt𝔼 |V_1^E - V̂_1|^2
             ≤ Ce^-κψ_0 t_kτ√(𝔼|V_1^E - V̂_1|^2).

    Hence
    
    𝔼 |V_1^E - V̂_1|^2 ≤ Ce^-2κψ_0 t_kτ^4.

    Similar with (<ref>), one has
    
    𝔼 |X_1^E - X̂_1|^2 ≤ Ce^-2κψ_0 t_kτ^6.

    Now consider
    
    d/dt1/2𝔼|V^E_1(t) -V_1(t)|^2 =   𝔼(V^E_1(t) -V_1(t))κ/p-1∑_j=1^p[ψ(X_j(0)-X_1(0))(V_j(0)-V_1(0)) 
       - ψ(X_j(t) - X_1(t))(V_j(t) - V_1(t))]
    ≤   𝔼κ(V^E_1 -V_1)[ψ_Lip|X_2(0)-X_1(0)- X_2(t) + X_1(t))(V_2(0)-V_1(0)) 
       + ψ(X_2(t) - X_1(t))(V_2(0) - V_1(0) - V_2(t) + V_1(t))].

    By Lemma <ref> and Lemma <ref>, one knows that |V_j(t)|≤ e^-κψ_0 t_k, for any j and t∈ [0,τ]. Hence it's easy to obtain  
    
    d/dt𝔼 |V_1^E - V_1|^2
             ≤ Ce^-κψ_0 t_kτ√(𝔼|V_1^E - V_1|^2),

    and 
    
    𝔼 |V_1^E - V_1|^2 ≤ Ce^-2κψ_0 t_kτ^4,   𝔼 |X_1^E - X̂_1|^2 ≤ Ce^-2κψ_0 t_kτ^6.

    Then, by the triangular inequality of Wasserstein distance, one has
    
    W_2 (𝒬_∞ (f_t_k),S(f_t_k)) ≤ C(κ, R)e^-κψ_0 t_kτ^2.

    The estimate of W_q is similar by calculating 
    
    d/dt(𝔼|V_1^E - V̂_1|^q)^1/q=1/q(𝔼|V_1^E - V̂_1|^q)^1/q-1d/dt𝔼|V_1^E - V̂_1|^q ≤  C(κ, R)e^-κψ_0 t_kτ.

        
    
    


    

 §.§ Proof of Theorem <ref>

    Now we prove Theorem <ref> by combining the above lemmas.
    
    
    By the triangular inequality, it holds
    
    W_q(𝒬_∞^n (f_0), f(nτ))  ≤∑_m=1^n W_q (𝒬_∞^n-m+1 (f_m-1),𝒬_∞^n-m(f_m)).

    And then, by the stability result in Lemma <ref>, it holds
    
    ∑_m=1^n W_q (𝒬_∞^n-m+1 (f_m-1),𝒬_∞^n-m(f_m))
            ≤∑_m=1^n C W_q(𝒬_∞ (f_m-1), f_m).

    Again by the triangular inequality, one has
    
    W_q(𝒬_∞ (f_m-1), f_m) ≤ W_q(𝒬_∞ (f_m-1), S(f_m-1))
            + W_q(S(f_m-1), f_m).

    Then, by Lemma <ref> and Lemma <ref>, we have
    
    ∑_m=1^n W_q (𝒬_∞^n-m+1 (f_m-1),𝒬_∞^n-m(f_m))
            ≤∑_m=1^n  C e^-κψ_0 (m-1)ττ^2 ≤  C τ,

    where the constant C is independent of n.
        One completes the proof.
    
    
    
    
        Although our analysis is based on W_q,2 distance, the main results in Section <ref> also work on W_q,p, p∈ [1,∞), by noting that
    
    x _p≤ d^1-1/q+1/px _q,    x∈ℝ^d,    1<p,q<∞.

    







§ APPLICATION: THE RBM-GPC


In this section, we show a simple application for the mean-field limit analysis for the Cucker-Smale model combined with the RBM. 

Based on the above analysis, and inspired by Carrillo's work <cit.>, we can construct the numerical scheme for the approximation of stochastic mean-field models of swarming which preserve the nonnegativity of the solution. 

Let us consider the stochastic mean-field equation for the distribution function f(θ, x, v; t):

    ∂_t f + v·∇_x f = ∇_v · [ℋ[f]f],

where

    ℋ[f](θ,x,v;t) = κ∫∫ψ(|x-y|,θ)(v-w)f(θ,y,w;t)dwdy,

with ψ satisfying  positivity, boundedness, Lipschitz continuity and mononticity conditions: there exists a positive constant ψ_M  > 0  such that 

    0≤ψ(r,θ) ≤ψ_M, ∀ r≥0,   ψ_Lip<∞,


    (ψ(r_1,θ)-ψ(r_2,θ))(r_1-r_2)≤ 0,   r_1,r_2 ∈ℝ_+.

Here, θ is a random variable from a measurable space (Ω,ℱ) to (I_Θ,ℬ_ℝ), with a probability distribution π (θ), with I_Θ⊂ℝ and ℬ_ℝ the Borel set.

It is easy to see that the total mass and momentum of (<ref>) is conserved in time.
Note that if we directly apply the classical stochastic Galerkin method to (<ref>), the solution might lose its positivity. However, we can make use of the microscopic dynamics to approximate f. Our method is to consider the corresponding Cucker-Smale model with random input θ on the interaction kernel ψ:

    {∂_t x_i(t,θ) =    v_i(t,θ),  i=1,⋯, N,
    ∂_t v_i(t,θ) =   κ/N-1∑_j=1^N 
            ψ(|x_j(t,θ)-x_i(t,θ)|,θ)(v_j(t,θ)-v_i(t,θ)).
        .


Denote {Φ_k(θ)}_j=0^∞ to be the gPC basis with respect to π, which means that Φ_k is a polynomial of degree k, and
these polynomials form an orthonormal basis of L^2(Ω,ℱ,π). 
We approximate the position and the velocity of i-th agent as follows:

    x_i(θ,t)≈ x^K_i = ∑_k=0^K^kx̂_i(t)Φ_k(θ),
           v_i(θ,t)≈ v^K_i = ∑_k=0^K^kv̂_i(t)Φ_k(θ),

where

    ^kx̂_i=∫ x_i(θ,t)Φ_k(θ)dπ,   ^kv̂_i=∫ v_i(θ,t)Φ_k(θ)dπ.

We obtain the following system

    {d/dt^ℓx̂_i (t) =   ^ℓv̂_i (t),
    d/dt^ℓv̂_i (t) =   κ/N-1∑_j=1^N∑_k=0^K e_ℓ k^ij(^kv̂_j(t)-^kv̂_i(t)),
           l = 0,1,…,K..

where

    e_ℓ k^ij=1/|Φ_ℓ|^2∫ψ(|x_j-x_i|,θ) Φ_k(θ)Φ_ℓ(θ)dπ.

Then, we design Algorithm <ref>, on the generalized Polynomial Chaos expansion in the random space based on the RBM:




    Although being derived from different numerical methods (the RBM and the Monte Carlo method), Algorithm <ref> is similar with the Monte Carlo gPC (MCgPC) proposed in <cit.> at first glance. Both can reduce the total cost of (<ref>) to 𝒪(KN). The main difference between them is that the RBM-gPC preserves the momentum during evolution. Actually, since the RBM updates information in the unit of batches, (instead, the MCgPC updates information of each particle independently at per time step.) for any t∈[t_m,t_m+1], the total sum of velocities is conserved in each batch by calculus:
    
    d/dt∑_i∈𝒞_q^ℓv̂_i (t) = κ/p-1∑_i,j∈𝒞_q∑_k=0^K e_ℓ k^ij(^kv̂_j(t)-^kv̂_i(t))=0,

    for l=0,⋯,K. Furthermore, it holds 
    
    ∑_i=1^N ^ℓv̂_i (t) = ∑_q=1^N/p∑_i∈𝒞_q^ℓv̂_i (t) = ∑_q=1^N/p∑_i∈𝒞_q^ℓv̂_i (t_m) = ∑_q=1^N/p∑_i∈𝒞_q^ℓv̂_i (t_0) = ∑_i=1^N ^ℓv̂_i (0).

    










    We use 
    x_i^K,R(t,θ):= ∑_k=0^K^kx̂_i(t)Φ_k(θ),   v_i^K,R(t,θ):=∑_k=0^K^kv̂_i(t)Φ_k(θ)
 to denote the approximation of position and velocity obtained by Algorithm <ref>. The kinetic distribution f^R_N,K(θ,x,v,t) is recovered from the empirical density function 
    f^R_N,K(θ,x,v,t) = 1/N∑_i=1^N δ(x-x_i^K,R(t,θ))⊗δ(v-v_i^K,R(t,θ)).
 Common approximations methods for Dirac delta distributions can be used when needed. For example, we fix an upper and lower bound for the (x,v)-space, to restrict our computational domain in a cube space and discretize it in uniform cells {C_l}_l=1^N_l of size |C_l|=h^2d, where h is the edge length of each cell. One can reconstruct the distribution f^R_N,K by histograms counting the number of particles belonging to each cell. Then the discrete distribution function f^R_N,K reads
    
    f^R_N,K(θ,x,v,t) = 1/|C_l|(1/N∑_i=1^N 𝕀(x_i^K,R(t,θ),v_i^K,R(t,θ)∈ C_l))·(∑_l=1^N_l𝕀((x,v)∈ C_l)),

    where 𝕀(·) is the indicator function.
    Clearly, the positivity of the distribution function is preserved. One can also refer to a brief review, by Lee et al. <cit.>, on the numerical methods for regularized Dirac delta functions.








    With respect to the stochastic variable θ, the dynamics of the N-particle system achieves spectral accuracy as we adopt the generalized polynomial chaos (gPC) basis <cit.>, provided that we have a smooth dependence of the particle solution from the random field.

    For any given θ and K, denote the distribution function of the N-particle system (<ref>) by f_N,K. 
    
    If one uses the reconstruction (<ref>), it holds that
        
    𝔼 W_2 (f^R_N,K,f^R_N,K) ≤(𝔼1/N∑_i=1^N |(x_i^R,v_i^R) - c_l(i)|^2 )^1/2,

        where c_l(i) is the center of the cell C_l containing (x_i^R,v_i^R). Since h is the uniform edge length, |(x_i^R,v_i^R) - c_l(i)|^2 ≤∑^d (h/2)^2 for any i. Combining with the estimate of W_2 (f_N,K,f^R_N,K) in Section <ref>, it holds that
    
    
    
    
        
    𝔼 W_2 (f_N,K,f^R_N,K)     ≤𝒪(√(τ/p-1)) +
           {   𝒪(N^-1/4)     if  d<2,
       𝒪(N^-1/4√(log(1+N)))     if  d=2,
       𝒪(N^-1/d)     if  d>2,
            .
            + 𝒪(√(d)h).







 §.§ Numerical experiments

Here we present some numerical tests for the RBM-gPC. In the following, we take the Legendre gPC base.



  
The space homogeneous case
We consider the space-independent case in the one-dimensional setting:

    ∂_t f(θ,v;t) = ∂_v [ K(θ)(v-u)f(θ,v;t)],

with u = ∫ vf(v)dv and K(θ)=0.5 + 0.01θ, θ∼𝒰(0,1). The long time solution is a Dirac delta δ(v-u) provided K(θ)>0, see <cit.>. We take the Legendre polynomial basis. The gPC approximation is given for h=0,⋯,M by 

    ∂_t f̂_h (v;t) = 1/|Φ_h|^2∂_v [ ∑_k=0^M (v-u) ℋ_hkf̂_t_k(v;t)],

where

    ℋ_hk = ∫ K(θ)Φ_h (θ) Φ_k (θ) d g (θ),   f̂_h (v;t) = ∫ f(θ,v;t) Φ_h (θ) d g (θ).

We consider an initial density function f_0(v) by

    f_0(v) = β[exp(-(v-μ)^2/2σ^2) + exp(-(v+μ)^2/2σ^2)],   σ^2=0.1,  μ = 0.5,

with β>0 a normalization constant. Discrete samples of the initial data of the ODE system are obtained from f_0(v). Denote the expected temperature of the system 𝒯 by 

    𝒯(f) = ∫ (v-u)^2 f(θ,v;t)dvdg(θ).

In Figure <ref>, we show the evolution of the expected density toward the Dirac delta function δ(v-u), u=0 for the RBM-gPC scheme at different times with p=2 and N=10^4.

In Figure <ref>, we take T=0.5, K=3, dt=10^-2 and dv=10^-2. Time integration is performed through a 4th order Runge-Kutta method. In Figure <ref>, we evaluate the mean square error (MSE) for the expected temperature 

    MSE_T := 1/n_m∑_i=1^n_m|𝒯(f_ref) - 𝒯(f_rbm^(i)) |^2,

as N grows with p=2, where 𝒯(f_ref) denotes the reference expected temperature obtained by (<ref>) through a central difference scheme, and 𝒯(f_rbm^(i)) denotes the expected temperature obtained by the RBM-gPC in the i-th turn. We have applied n_m=100 for random initialization and batch splits. 
Figure <ref> shows the difference of TV distance between the expected reference distribution and the expected distribution by the RBM-gPC:

    Err_TV := | 𝔼_θ f_ref - 𝔼_θ1/n_m∑_i=1^n_m f_rbm^(i)|_L^1,

with n_m =100 as well.












Also, Figure <ref> and Figure <ref> show the decay of MSE_T as the batch size p and the time step dt, respectively. The reference distribution of Figure <ref> is obtained by the RBM-gPC with N=2^8, p = 2^8, and dt = 10^-2, and of Figure <ref> is obtained with N=2^8, p = 2, and dt = 10^-2. We still take n_m=100.






  
Stochastic 1D Cucker-Smale dynamics
In this test, we consider the 1D Cucker-Smale dynamics. Let us consider the initial distribution as the following bivariate and bimodal distribution

    f_0(x,v) = 1/2πσ_xσ_vexp(-x^2/2σ_v ^2)[exp(-(v+1)^2/2σ_x ^2)+ exp(-(v-1)^2/2σ_x^2)],

with σ_x^2 =0.5, σ_v^2=0.2. Our initial data for particle positions and velocities are
sampled from f_0. The stochastic interactions are given by

    H(θ; |x_i - x_j|) = 1/(1+|x_i - x_j|^2)^γ(θ),

with γ (θ) = 0.1+ 0.05 θ, θ∼𝒰[-1,1]. The results are obtained through the RBM-gPC scheme with dt = 10^-2, N=10^4, batch size p =2, and K=3. The expected distribution is reconstructed by  in Matlab in the domain [-3,3]×[-3,3] with 100 gridpoints in both space and velocity. Figure <ref> presents the evolution over the time interval t∈[0,4] of the expected distribution. 






  
Stochastic 2D Cucker-Smale dynamics
Figure <ref> shows the evolution over the time interval t∈ [0,4] of the 2D Cucker-Smale nonhomogeneous mean-field model. For the initial distribution, we consider uniformly distributed N=10^4 particles on a 2D annulus with a circular counterclockwise motion

    f_0 (x,v) = 1/|C|χ (x∈ C)δ( v - k∧ x/|x|),

being C: = {x∈ℝ^2: 0.5≤ |x_1-x_2|≤ 1 } and k the fundamental unit vector of the z-axis, where |C| means the cardinality of set C.
The interaction kernel shares the same form with (<ref>).
The evolution shows the flocking phenomenon in the top row and that the mean velocity is conserved in the bottom row of Figure <ref>. The results are obtained through the RBM-gPC scheme with dt = 10^-2, N=10^4, batch size p =2, and K=3. The reconstruction follows the same method as in the 1D case.








§ CONCLUSION

In conclusion, we give an analysis on the mean-field limit of the RBM for the Cucker-Smale model and obtain a long-time approximation. In addition, as a simple application, we propose a generalized Polynomial Chaos based Random Batch Method (RBM-gPC) as a numerical method to simulate the mean-field limit dynamics of the Cucker-Smale model with random inputs.

This paper focuses on the separate limits of the number of particles N and the time step τ but not a uniform limit of (N,τ). For future work, it will be interesting to investigate the mixture of particles as time evolves, and to study on the uniform limit. Also, numerically, an error estimate on this model applied with the particle method (corresponding to a new model) can be an interesting topic to study.







§ ACKNOWLEDGEMENT

We thank Prof. Shi Jin and Prof. Lei Li for detailed and valuable comments and advice on the manuscript. This work is supported by the NSFC grant (No. 12031013, 12201404), the Shanghai Municipal Science and Technology Key Project (No. 22JC1402300), and Project supported by the National Science Foundation for International Senior Scientists (No. 12350710181).



    
    

§ PROOF OF LEMMA <REF>

    The proof is given by Lemma 2.4 in <cit.>.
    
        For the first assertion, we claim that the relative velocities are non-increasing in time. Let t∈ [t_m-1, t_m) be given. Then one can choose time-dependent indices k and ℓ such that 
        
    |V_k^R (t) - V_ℓ^R (t)| = 𝒟_V^R(t).

        Then for such k and ℓ, one has 
        
    d/dt | V_k^R(t) -V_ℓ^R(t)|^2 =    
                2(V_k^R(t) - V_ℓ^R (t)) ·d/dt(V_k^R(t) - V_ℓ^R (t))
    
                =    2κ/p-1∑_j∈ [k]_mψ (|X_j^R - X_k^R|)(V_j^R - V_k^R)·(V_k^R - V_ℓ^R)
       - 2κ/p-1∑_j∈[i]_mψ (|X_j^R - X_ℓ^R|)(V_j^R - V_ℓ^R)·(V_k^R - V_ℓ^R).

        In order to show that the right-hand side of (<ref>) is not positive, we use the maximality of | V_k^R(t) -V_ℓ^R(t)| at time t. Since
        
    | V_k^R(t) -V_ℓ^R(t)| ≥ | V_j^R(t) -V_ℓ^R(t)|,    j = 1,⋯,N,

        one has
        
    (V_j^R(t)-V_k^R (t))·(V_k^R(t)-V_ℓ^R (t)) =
                - ((V_k^R-V_ℓ^R) - (V_j^R-V_ℓ^R))· (V_k^R - V_ℓ^R)
    ≤ - | V_k^R(t) -V_ℓ^R(t)|^2 + | V_j^R(t) -V_ℓ^R(t)|| V_k^R(t) -V_ℓ^R(t)|≤ 0,    j= 1,⋯,N.

        Similarly, one has 
        
    (V_j^R(t)-V_ℓ^R (t))· (V_k^R(t)-V_ℓ^R (t)) ≥ 0,    j=1,⋯,N.

        Hence, | V_k^R(t) -V_ℓ^R(t)|^2 is non-increasing in time.

        Then by the definition od 𝒟_X^R and 𝒟_V^R, one has 
        
    d/dt𝒟_X^R(t)≤𝒟_V^R(t) ≤𝒟_V^R(0),    a.e.  t>0.

        This yields the second assertion on 𝒟_X^R.
    
    
    
    

    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
        
    
    
    
    
    
    

    
    
    
    
    
    
    
    
    
    
    
    
    
    

    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
        
    
    
    
    
    
    
    
    
    
    
    
    


    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    

    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    


    




    
    
    
    
    

§ DETAILS ON PROPOSITION <REF>


    

  
Fixed-time regularity of f. First we check the regularity of f by Lemma <ref>, which is needed in the proof of Proposition <ref>.
    
    
    For the flocking kinetic model (<ref>), suppose that the initial probability measure f_0 ∈ (C^2∩ W^2,∞)(ℝ^2d) is compactly supported, and the interaction kernel ψ satisfies 
    sup_r≥ 0 (|ψ^'(r)| + |ψ^''(r)|) ≤ϵ_ψ
 with a constant ϵ_ψ. Then, for any T∈ (0,∞), there exists a unique classical solution f_t ∈ C^2([0,T)×ℝ^2d) for (<ref>). 

    The proof stem of Lemma <ref> is generally a smoothness check for f inspired by in [<cit.>, Theorem 3.1]. The main difference with <cit.> is that here we need better regularity for the operator (ℒ^*)^2 and (ℒ_k^*)^2 in Lemma <ref>.
    
    Local existence follows from [<cit.> Theorem 3.1] by the standard fixed point argument. One only needs to obtain a C^2 bound of f to conclude the global existence of classical solutions. One can express the non-conservative kinetic model (<ref>) in terms of the non-linear transport operator 𝒯:=∂_t + v·∇_x + ξ [f] ·∇_v,
    
    𝒯f = -f∇_v ·ξ[f].

    We claim that there exists a positive constant C(d,κ,ϵ_ψ, R) such that
    
    | 𝒯f| ≤ C |f|,

    
    | 𝒯(∂_x_if) | ≤ C (|f| + |∇_v f| + |∂_x_if)|,

    
    | 𝒯(∂_v_i f) | ≤ C (|∇_v f| + |∂_x_if|),

    
    | 𝒯(∂_x_i x_jf) | ≤ C (  |∂_x_if| + |∂_x_jf| + |∂_x_i x_jf| +|∇_v f| + |∇_v∂_x_if| + |∇_v∂_x_jf|),

    
    | 𝒯(∂_x_i v_jf) | ≤ C ( |∂_x_i x_jf| +|∇_v f| + |∂_v_j∂_x_if| + |∇_v∂_v_jf| + |∇_v∂_x_if|),


    
    | 𝒯(∂_v_i v_jf) | ≤ C ( |∂_x_i v_jf| + |∂_x_j∂_v_if| +  |∂_v_j∂_v_if|).


    
    To verify these inequalities, note that -∇_v·ξ [f] = κ d ∫ψ (|y-x|)f(y,w,t)dydw and thus |∇_v·ξ [f]| ≤κ d. By direct calculus, one obtains the following estimates:
 

    For (<ref>),
    
    | 𝒯f| ≤| -κf∇_v ·∫ψ(|y-x|) (w-v) f(y,w,t) dydw| ≤κ d |f|;

    for (<ref>),
    
    | 𝒯(∂_x_if) |    ≤| - (∂ _x_iξ [f]) ·∇_v f -  (∂_x_i∇_v ·ξ [f])f  -  (∇_v ·ξ [f])∂ _x_if|
       ≤κϵ_ψ (√(M_2^v f_0) + 2R) |∇_v f| + κ d ϵ_ψ |f| + κ d |∂_x_if|)
       ≤ C (|f| + |∇_v f| + |∂_x_if)|;

    For (<ref>),
    
    |𝒯∂_v_if | ≤ |-∂_x_if| + κ |∇_v f| + κ d |∂_v_if|
            ≤ C (|∇_v f| + |∂_x_if|).

    The estimate on (<ref>)-(<ref>) is similar and we omit it for brevity.
    
    Now, let F(t) measure the W^2,∞-norm of f,
    
    F := f(·,t) _W^2,∞ .

    The above inequalities imply
    
    d/dtF(t) ≤ CF(t).

    Then we end up with the energy bound by Grönwall's inequality
    F(t)≤ F(0) e^Ct.
    Equipped with this a priori W^2,∞ estimate, standard continuation principle yields a global extension of local classical solutions. 




The following is a simple prerequisite for Proposition <ref>.

    
For any g_1∈ L^1(ℝ^2d),g_2∈ L^1(ℝ^2dp), the operators ℒ_k^* and ℒ^* defined by (<ref>) and (<ref>) respectively, it holds
    
    ∫ | e^τℒ_k^*g_1| dz ≤∫ | g_1| dz,

    
    ∫|  e^τℒ^*  g_2  | dz_1 ⋯ dz_p ≤∫|  g_2  | dz_1 ⋯ dz_p.


     
    
    One only needs to consider g_1≥ 0 since there is composition g_1 = g_1^+ -g_1^-. Also, since e^τℒ_k^*cg_1 = ce^τℒ_k^*g_1 and e^τℒ^*cg_2 = ce^τℒ^*g_2 for any constant c, one can set g_1 be a probability density function. Then,
    
    ∫ |e^τℒ_k^*g_1| dz = ∫ e^τℒ_k^*g_1 dz,

    since e^τℒ_k^*g_1 is another probability density function derived from g_1 by the dynamics (<ref>). It's clear that
    
    ∫ e^τℒ_k^*g_1 dz = 
        ∫∫_0^τ∂/∂ tg_1(t)dtdz + ∫ g_1 dz = ∫ g_1dz,

    where g_1(t) denotes the solution of (<ref>) with initial datum g_1. For the general g_1, 
    
    ∫ | e^τℒ_k^*g_1| dz ≤∫ | e^τℒ_k^*g_1^+| dz
             + ∫ | e^τℒ_k^*g_1^-| dz
             =  ∫ | g_1^+| dz + ∫ | g_1^-| dz =  ∫ | g_1| dz.

    The proof of (<ref>) is similar.





    
    

  
Proof of Proposition <ref>.
    It's clear that S = e^τℒ_k^*f_t_k, where 
    
    ℒ_k^* := -  (v ·∇_x) · -
        ∇_v ·( ∫κψ(|y-x|)(w-v)f_t_k(y,w)dydw ·).

    Define the operator of (<ref>) as
    
    ℒ^* := -∑_i=1^p (v_i∇_x_i)·  - ∑_i=1^p∇_v_i· (ξ_i ·).

    Then one has
    
    𝒬_∞ (f_t_k) =   ∫ e^τℒ^*Π_i=1^p f_t_k(x_i,v_i)dx_2⋯ dx_pdv_2⋯ dv_p 
    
            =   f_t_k (x_1,v_1) + τ∫ℒ^* Π_i=1^p f_t_k (x_i,v_i)dx_2⋯ dx_pdv_2⋯ dv_p 
       + ∫_0^τ (τ -s) ∫ (ℒ^*)^2 e^sℒ^*Π_i=1^p f_t_k (x_i,v_i)dx_2⋯ dx_pdv_2⋯ dv_p ds,

    where
    
    
    ∫ℒ^* Π_i=1^p f_t_k (z_i) dz_2⋯ dz_p 
                =     -∑_i=1^p ∫ (v_i ∇_x_i)·Π_j=1^p f_t_k (z_j) - ∇ _v_i· (ξ_i Π_j=1^p)f_t_k (z_j)dz_2⋯ dz_p)
    
                =     - (v_1 ∇_x_1) ·f_t_k (z_1) - ∇_v_1·∫κψ (|y-x_1|)(w-v_1) f_t_k (y,w) dydw.

    
    Note that S(f_t_k) = f_t_k + τℒ_k^*f_t_k + ∫_0^τ (τ -s)(ℒ_k^*)^2 e^sℒ_k^*f_t_k ds, then
    
        
    | S(f_t_k) - 𝒬_∞ (f_t_k) |(z_1)≤∫ _0^τ (τ-s) [ |(ℒ_k^*)^2 e^sℒ_k^*f_t_k(z_1)|  +  | ∫ (ℒ^*)^2 e^sℒ^*Π_i=1^p f_t_k (z_i) dz_2⋯ dz_p | ]ds.

    
    
    As shown in Lemma <ref>, f_t_k∈ C^2(z), hence it holds 
    
    ∫ |(ℒ_k^*)^2 f_t_k| dz≤ C ,  ∫| (ℒ^*)^2 Π _i=1^p f_t_k (z_i)| dz_2 ⋯ dz_p≤ C,
 
    where C=C(κ,ϵ_ψ ,R,d). By Lemma <ref>, taking g_1 = (ℒ_k^*)^2  f_t_k and g_2 = (ℒ^*)^2 Π _i=1^p f_t_k (z_i), one has
    
    ∫ |(ℒ_k^*)^2 e^τℒ_k^*f_t_k| dz ≤∫ |(ℒ_k^*)^2 f_t_k| dz,

    and
    
    ∫| (ℒ^*)^2 e^τℒ^*Π _i=1^p f_t_k (z_i)| dz_1 ⋯ dz_p ≤∫| (ℒ^*)^2 Π _i=1^p f_t_k (z_i)| dz_1 ⋯ dz_p,

    Using the exchangability of ℒ^* and e^τℒ^*, with ℒ^*=ℒ_k^* or ℒ^*. 
    
    Then one has |S(f_t_k) - 𝒬_∞ (f_t_k)|_TV≤ C τ^2. 
    
    For S(f_t_k), by Lemma <ref>, it holds
    
    supp[S_∞(f_t_k)]⊂
            B(0,C)× B(0,C e^-κψ_0 t).


    For the support of 𝒬_∞(f_t_k), by Lemma <ref>, one knows that 
    
    supp[𝒬_∞(f_t_k)]⊂
            B(0,C)× B(0,C e^-κψ_0 t_k).

    
    Hence for any z ∈supp|S(f_t_k) - f_t_k|, it holds |z|≤ C.
    Then, by Lemma <ref> and applying similar analysis on Lemma <ref> Step 1, one has 
    
    W_q (𝒬_∞ (f_t_k),S(f_t_k)) ≤ C(κ,ϵ_ψ , f_0,d) τ^2/q.















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