Global Solution of a Functional Hamilton-Jacobi Equation associated with a Hard Sphere Gas
Chenjiayue Qi



Global Solution of a Functional Hamilton-Jacobi Equation associated with a Hard Sphere Gas
    [
    September 9, 2024
==========================================================================================



§ ABSTRACT 
In recent years it has been shown for hard sphere gas that, by retaining the correlation information, dynamical fluctuation and large deviation of empirical measure around Boltzmann equation could be proved, in addition to the classical kinetic limit result by Lanford. After taking low-density limit, the correlation information can be encoded into a functional Hamilton-Jacobi equation. The results above are restricted to short time. This paper establishes global-in-time construction of a solution of the Hamilton-Jacobi equation, by analyzing a system of coupled Boltzmann equations. The global solution converges to a non-trivial stationary solution of the Hamilton-Jacobi equation in the long-time limit under proper assumptions.





  
Acknowledgements
The author is very grateful to Laure Saint-Raymond and Thierry Bodineau for their many inspiring discussions on the topic of this paper, as well as their valuable suggestions on the overall understanding of its main results.





§ INTRODUCTION

In the seminal work of Lanford <cit.>, it is shown that the average dynamics of a hard sphere gas in the low-density limit is governed by the Boltzmann equation. The proof establishes the propagation of chaos for a hard sphere gas: dynamical correlations between different hard spheres are negligible in a certain sense. 

Since the result above could be seen as a law of large numbers, one can also look at the corresponding central limit theorem and large deviation theory. In <cit.>, by retaining the correlation information between different particles, the dynamical fluctuations and large deviations of the empirical measure around Boltzmann equation are derived. 

In particular, the correlation information is encoded into the so-called cumulant generating functional ℐ_(t,g), and it is shown that after taking the low-density limit the limiting functional ℐ(t,g) satisfies a functional Hamilton-Jacobi equation. The functional Hamilton-Jacobi equation could provide a direct new proof for the convergence of the empirical measure towards the solution of Boltzmann equation. It also plays an important role in establishing the dynamical fluctuations and large deviations of a hard sphere gas. All the results in <cit.> and <cit.> mentioned above are however restricted to short times. Here we mention the recent breakthrough <cit.> on extending Lanford's argument <cit.> into long times.

The current paper is devoted to the construction of global-in-time solution ℐ(t,g) of the limiting functional Hamilton-Jacobi equation. The construction is based on the study of the Euler-Lagrange system (coupled Boltzmann equations) associated with the Hamilton-Jacobi equation. 

In subsection <ref> we recall the basic setting of the hard sphere gas, while introducing the formulation of the functional Hamilton-Jacobi equation. Then in subsection <ref>, we introduce the associated Euler-Lagrange system, i.e. the coupled Boltzmann equations. In subsection <ref> we claim the main results of our paper: the global well-posedness of the coupled Boltzmann equations, and further the existence of global-in-time bounded solutions of the functional Hamilton-Jacobi equation. In subsection <ref>, we discuss future directions based on the current results.



 §.§ A Hamilton-Jacobi Equation for Hard Sphere Gas

One approach to describe hard sphere gas at microscopic level is to fix the total number N, as well as the diameter >0 of these identical hard spheres. The evolution for the positions (𝐱_1^,...,𝐱_N^)∈T^dN and velocities (𝐯_1^,...,𝐯_N^)∈R^dN of the N particles, satisfies a system of ordinary differential equations (Newton's laws)

    d𝐱_i^/dt=𝐯_i^,  d𝐯_i^/dt=0  ,

with specular reflection at collsions if |𝐱_i^-𝐱_j^|=

    (𝐯_i^)':=𝐯_i^-(𝐯_i^-𝐯_j^)·ωω,   (𝐯_j^)':=𝐯_j^+(𝐯_i^-𝐯_j^)·ωω,  ω:=𝐱_i^-𝐱_j^/

This means after a collision with |𝐱_i^-𝐱_j^|=, the velocities (𝐯_i^,𝐯_j^) of the two particles will be changed into ((𝐯_i^)',(𝐯_j^)'). This will induce a well-defined trajectory for initial conditions of full Lebesgue measure in the canonical phase space 𝒟_N^

    𝒟_N^:={(𝐱_1^,...,𝐱_N^,𝐯_1^,...,𝐯_N^)∈T^dN×R^dN: ∀ i≠j, |𝐱_i^-𝐱_j^|>},

excluding multiple collisions and accumulation of collision times. 

This microscopic dynamics induces a Liouville equation for the probability density W_N^ of the N particles, where W_N^(t,X_N,V_N) refers to the probability density of finding N hard spheres with configuration (X_N,V_N) at time t. Here the variable X_N refers to the positions of the N particles X_N:=(x_1,...,x_N), and the variable V_N refers to the velocities of the N particles V_N:=(v_1,...,v_N). The Liouville equation for W_N^ is

    _t W_N^+V_N·∇_X_NW_N^=0,

with boundary condition corresponding to specular reflection. 

One can further consider the grand canonical formulation of a hard sphere gas: instead of fixing the total number of particles, we assume the total number 𝒩 of particles to be random with a modified Poisson distribution law. For each diameter , we fix 
a constant μ_ as the parameter for the modified Poisson distribution of the total number of particles. We assume that at initial time t=0 the probability density of having N particles and configuration (X_N,V_N) as the following with z_i:=(x_i,v_i)

    1/𝒵^μ_^N/N!∏_i=1^Nf^0(z_i)1_𝒟_N^.

Here 𝒵^ is the normalizing factor defined as

    𝒵^:=1+∑_N≥ 1μ_^N/N!∫_T^dN×R^dN∏_i=1^Nf^0(z_i)1_𝒟_N^dV_NdX_N.

It is clear that we have two sources of randomness: the number 𝒩 of particles is random, while given the total number N, the configuration (X_N,V_N) is also random. For each sample (X_N,V_N), it will follow the evolution law given by equations (<ref>) and (<ref>). Thus at each time t≥ 0 we would have the distribution law for (X_N,V_N).

In the Boltzmann-Grad scaling, we impose μ_=^-(d-1) to ensure the number of collisions per particle is of order 1 per unit time <cit.>. This scaling implies the asymptotic below for E_(𝒩), where E_ is the expectation upon the probability measure given by (<ref>)

    lim_→ 0E_(𝒩)^d-1=1.

A central object in the study of hard sphere gas is the empirical measure π_t^, defined as

    π_t^:=1/μ_∑_i=1^𝒩δ_𝐳_i^(t),

where δ_𝐳_i^(t) means the Dirac mass at 𝐳_i^(t)∈T^d×R^d. To encode the correlation information, the cumulant generating functional for hard sphere gas with diameter  is introduced

    Λ^(t,h):=1/μ_logE_[exp(μ_π_t^(h))]=1/μ_logE_[exp(∑_i=1^𝒩h(𝐳_i^(t)))],

where h is a test function with variables (x,v). By taking the Boltzmann-Grad limit → 0 with μ_^d-1=1, the functional Λ^(t,h) should converge to a limiting cumulant generating functional Λ(t,h). This functional satisfies the following functional Hamilton-Jacobi equation, with the functional derivative Λ(t,h)/ h(t) taken as a measure in x and v for each t

    _tΛ(t,h)=ℋ(Λ(t,h)/ h(t),h(t))+∫ v·∇_x h Λ(t,h)/ h(t)dvdx,  Λ(0,g)=∫ (e^g(0)-1)f^0 dvdx.

In this equation, the Hamiltonian ℋ(φ,p) is defined as

    ℋ(φ,p)=1/2∫φ(x,v)φ(x,v_*)(e^Δ p (x,v,v_*)-1)((v_*-v)·ω)_+dω dv_* dvdx,

with ω∈S^d-1 being the collision direction. The function Δ p is defined as

    Δ p(x,v,v_*):=p(x,v')+p(x,v_*')-p(x,v)-p(x,v_*).

The variables (v',v_*') is the pre-collisional configuration, defined in a way similar to (<ref>)

    v'=v-((v-v_*)·ω)ω,   v_*'=v_*+((v-v_*)·ω)ω.

This Hamilton-Jacobi equation contains a collision term represented by ℋ, and a tranport term, which resembles the Boltzmann equation. In the Hamiltonian ℋ, the term e^Δ p-1 represents the effect of collision: the p(x,v')+p(x,v_*') in Δ p has a similar role as the gain term in Boltzmann equation, and the -p(x,v)-p(x,v_*) in Δ p has the same role as the loss term in Boltzmann equation. By taking the derivative Λ(t,h)/ h(t) at h=0, the Boltzmann equation is recovered formally in a weak sense.

In fact, the Hamilton-Jacobi equation encodes much more information about the hard sphere dynamics than the usual Boltzmann equation, in particular it encodes all the dynamical correlations. For a complete justification of the contents above, the readers may read <cit.>. In <cit.>, more formal discussion with physical motivation about the meaning of this Hamiltonian is given.

One can also use test functions h on the entire trajectory during the time interval [0,t], as h(z([0,t])), which is the case in <cit.>. Particularly in this paper, we may choose the test function of the form

    h(z([0,t]))=g(t,z(t))-∫_0^tD_sg(s,z(s))ds,

where D_s refers to _s+v·∇_x, and g is a function depending on variables (t,x,v). This choice of test functions enables us to integrate the transport term in equation (<ref>). It then gives the Hamilton-Jacobi equation for ℐ(t,g):=Λ(t,h), where h is defined through g by (<ref>)

    _tℐ(t,g)=ℋ(ℐ(t,g)/ g(t),g(t)),  ℐ(0,g)=∫ (e^g(0)-1)f^0 dvdx.

This functional equation has been introduced in Theorem 7 of <cit.>. 




 §.§ Coupled Boltzmann Equations as an Euler-Lagrange System

To find the solution ℐ(t,g) of the Hamiltonian-Jacobi equation (<ref>), it is shown in the subsection 7.1.1. of <cit.> that we can look at the associated Hamiltonian system. There are two interesting equivalent formulations of the Hamiltonian system. The first formulation is for s∈[0,t]

    D_sφ_t(s)=ℋ/ p(φ_t(s),p_t(s)),  ,
       D_s(p_t-g)(s)=-ℋ/φ(φ_t(s),p_t(s)),  .

The subscript t means we are studying the coupled system in the time interval s∈[0,t], with terminal data given at time t. Given a mild solution (φ_t,p_t) of equation (<ref>) on [0,t] with initial data φ_t(0)=f^0e^p_t(0) and terminal data p_t(t)=g(t), we define the functional ℐ(t,g) as

    ℐ(t,g):=∫ (e^p_t(0)-1)f^0 dvdx+∫_0^t∫φ_t(s) D_s(p_t(s)-g(s))dvdxds+∫_0^tℋ(φ_t(s),p_t(s))ds.

It will be proved in Theorem <ref> that an equivalent form of the functional ℐ(t,g) constructed in equation (<ref>) is a mild solution of the Hamilton-Jacobi equation (<ref>). The notion of mild solution will be specified in Section <ref>.

However in this paper we do not directly deal with the coupled system given above. In <cit.> (Section 7), by performing the change of variables

    (ψ_t(s),η_t(s))=(φ_t(s)e^-p_t(s),e^p_t(s)),

an alternative equivalent formulation with better symmetry is introduced 

    D_sψ_t   =-ψ_t D_sg+∫((v_*-v)·ω)_+η_t(v_*)[ψ_t(v')ψ_t(v_*')-ψ_t(v)ψ_t(v_*)]dω dv_*,  ψ_t(0)=f^0,
    
    D_s η_t   =η_t D_sg-∫((v_*-v)·ω)_+ψ_t(v_*)[η_t(v')η_t(v_*')-η_t(v)η_t(v_*)]dω dv_*,  η_t(t)=e^g(t).

In the paper, we generalize the change of variables (<ref>) into

    (ψ_t(s),η_t(s))=(φ_t(s)e^-p_t(s)+α'|v|^2,e^p_t(s)-α'|v|^2),

for arbitrary α'∈R. The change of variables (<ref>) in <cit.> corresponds to the particular case α'=0. It will be proved in Lemma <ref> that after this generalized change of variables, the (ψ_t,η_t) satisfies the following coupled Boltzmann equations during the time interval [0,t]

    D_sψ_t=-ψ_t D_sg+∫((v_*-v)·ω)_+η_t(v_*)[ψ_t(v')ψ_t(v_*')-ψ_t(v)ψ_t(v_*)]dω dv_*,
       D_s η_t=η_t D_sg-∫((v_*-v)·ω)_+ψ_t(v_*)[η_t(v')η_t(v_*')-η_t(v)η_t(v_*)]dω dv_*,
       ψ_t(0)=f^0e^α'|v|^2,  η_t(t)=e^g(t)e^-α'|v|^2.

Under the generalized change of variables, the form of the coupled Boltzmann equation is the same as (<ref>), but with different ψ_t(0) and η_t(t). It will be explained in Section <ref> that a proper choice of α' enables us to solve the equation in a convenient functional setting. 

We define the biased collision operator 𝒬_η(ψ_1,ψ_2) as follows

    𝒬_η(ψ_1,ψ_2):=1/2∫((v_*-v)·ω)_+ η(v_*)[ψ_1(v')ψ_2(v_*')+ψ_2(v')ψ_1(v_*')
                                          -ψ_1(v)ψ_2(v_*)-ψ_2(v)ψ_1(v_*)]dω dv_*.


The function ϕ is defined as follows, related to the spatial transport

    ϕ(s,x,v):=D_sg(s,x,v)=(_s+v·∇_x)g(s,x,v)

Based on the definitions of the biased collision operator and the function ϕ, we rewrite equation (<ref>) in a more compact form 

    D_s ψ_t   =-ψ_tϕ+𝒬_η_t(ψ_t,ψ_t),  ψ_t(0)=f^0e^α'|v|^2,
    
    D_s η_t   =η_tϕ-𝒬_ψ_t(η_t,η_t),  η_t(t)=e^g(t)e^-α' |v|^2.

Since equation (<ref>) is equivalent to (<ref>) through the change of variables (<ref>), we can as well construct the functional ℐ(t,g) given in (<ref>), by solving equation (<ref>). 

We call ψ_t the 'forward component', due to its given initial data and the positive sign of collision operator. The other component η_t is called the 'backward component', due to its given terminal data at time t and also the negative sign of collision operator. Each component provides a bias for the nonlinear collision of the other component, which is transparent in the equation (<ref>). If we take g-α'|v|^2≡ 0, the coupled Boltzmann equations would degenerate to the usual Boltzmann equation, with η_t≡ 1 and ϕ≡ 0.

A pivotal tool we use in the present paper to solve equation (<ref>) is the theory of global-in-time solution for the Boltzmann equation with given initial data. There have been many works dealing with global solutions of Boltzmann equation, with different notions of solutions. Early works include <cit.> for classical solutions, <cit.> for mild solutions, and <cit.> for renormalized solutions. Specifically in the present paper, we will adapt the perturbation regime for global mild solutions in a certain weighted L^∞ space <cit.>. 

A subtle issue in the present paper is that we want to solve equation (<ref>) for those functions e^g(t) with quadratic exponential growth in the velocity variable, for example when g(t)=1/4|v|^2. If we take α'=0 in (<ref>), which corresponds to the original change of variables (<ref>) in <cit.>, it naturally requires the forward component ψ_t to have quadratic exponential decay in the velocity variable. The initial data f^0 could be assumed to have quadratic exponential decay, but it is hard to prove the propagation of this quadratic exponential decay. To overcome this difficulty, we will carry out a symmetrization procedure in Section <ref> by choosing a proper α'.




§ MAIN RESULTS



 §.§ Global-in-time Solution

As explained in subsection <ref>, to solve the Hamilton-Jacobi equation (<ref>) we will look at the associated Euler-Lagrange system, which is the coupled Boltzmann equations (<ref>). The goal is to find a certain class of functions g such that, the mild solution ℐ(t,g) could be constructed for arbitrary time t≥ 0. 

We say a pair of functions (ψ_t,η_t) is a mild solution of the coupled Boltzmann equations (<ref>), if for arbitrary s∈[0,t]

    ψ_t(s)=S_sf^0e^α'|v|^2-∫_0^s S_s-τψ_t(τ)ϕ(τ)dτ+∫_0^s S_s-τ𝒬_η_t(τ)(ψ_t(τ),ψ_t(τ))dτ,
       η_t(s)=S_-(t-s)e^g(t)e^-α'|v|^2-∫_s^t S_-(τ-s)η_t(τ)ϕ(τ)dτ+∫_s^t S_-(τ-s)𝒬_ψ_t(τ)(η_t(τ),η_t(τ))dτ.

Here the operators {S_τ}_τ∈R are the transport semigroup defined as S_τf(x,v)=f(x-τ v,v). For the Hamilton-Jacobi equation (<ref>), we say a functional ℐ(t,g) is a mild solution of the equation if for arbitrary t≥ 0

    ℐ(t,g)=ℐ(0,g)+∫_0^t ℋ(ℐ(s,g)/ g(s),g(s))ds,  ℐ(0,g)=∫ (e^g(0)-1)f^0dvdx.

The main result of the paper is to construct a mild solution of the Hamilton-Jacobi equation with f^0 close to the spatially homogeneous standard Maxwellian M

    M(x,v)=(2π)^-d/2exp(-1/2|v|^2),

and the function e^g close to a certain reference function ℰ. In this paper, we consider those ℰ of the form

    ℰ(x,v)=(2π)^-d/2exp(α|v|^2),  α<1/2.

To simplify notations, we define the normalization function ℬ as e^-α'|v|^2. We want to choose a proper α' that symmetrizes the forward initial data and the backward terminal data in (<ref>), with f^0e^α'|v|^2 and e^g(t)e^-α'|v|^2 being close to the same coupled equilibrium 𝒢

    𝒢:=M^1/2ℰ^1/2=(2π)^-d/2exp(-1/2(1/2-α)|v|^2).

This requires us to choose ℬ with Mℬ^-1=ℰℬ=𝒢, which yields

    ℬ=M^1/2ℰ^-1/2=exp(-1/2(1/2+α)|v|^2),

and thus

    α'=1/2(1/2+α).



Since we only consider those reference functions ℰ with α<1/2, the coupled equilibrium 𝒢 has quadratic exponential decay in v.



We will solve the coupled Boltzmann equations (<ref>) with ψ_t and η_t being perturbations around 𝒢. The perturbations should be in a L^∞ space with a polynomial weight on v, denoted by L_β^∞

    ‖ f‖_L_β^∞=sup_(x,v)∈T^d×R^d|f(x,v)(1+|v|)^β|.



For Theorem <ref>, we assume the forward initial data f^0ℬ^-1 is close to the coupled equilibrium 𝒢 with the perturbation L_x,v^2-orthogonal to a kernel 𝒦, representing the conserved quantities, to be defined in equation (<ref>) 

    ‖ f^0ℬ^-1-𝒢‖_L_β+1^∞<c,   f^0ℬ^-1-𝒢∈𝒦^⊥. H1

The kernel 𝒦 is defined as

    𝒦:={𝒢,𝒢(v)v_1,...,𝒢(v)v_d,𝒢(v)|v|^2}.

Throughout the paper, the constant c>0 is always taken to be small enough and properly tunned according to other parameters.

The terminal data e^g(t)ℬ is assumed to be close to the coupled equilibrium 𝒢 with the perturbation orthogonal to the kernel 𝒦 at any time t>0
    
    ‖ e^g(t)ℬ-𝒢‖_L_β+1^∞< c,   e^g(t)ℬ-𝒢∈𝒦^⊥.  H2

These orthogonality conditions are common in the literature for the solution of Boltzmann equations on torus to have decay in time. For example, the readers may see Theorem 2.3.1 of <cit.>.


The function g and the forward initial data f^0ℬ^-1 are assumed to have certain regularity and continuity

    ϕ(s,x,v) ≡ 0, ,   S_τf^0ℬ^-1∈(R_τ;L_β^∞)  H3.

Throughout the paper, the dimension d will be taken as d≥ 3.


For arbitrary β>4 and σ>1, we can take constants c>0 and a_*>0 depending on β,σ such that for any f^0 and g satisfying the assumptions (<ref>)-(<ref>) and any time t>0, there exists a unique mild solution (ψ_t,η_t) of the coupled Boltzmann equations (<ref>) in the function class below 

    sup_0≤ s≤ t(1+s)^σ‖ψ_t(s)-𝒢‖_L_β^∞<a_*,  sup_0≤ s≤ t(1+(t-s))^σ‖η_t(s)-𝒢‖_L_β^∞<a_*.

Furthermore, the functional ℐ(t,g) in (<ref>) is well-defined for any functions f^0 and g satisfying the assumptions (<ref>)-(<ref>), and is a global-in-time mild solution of the Hamilton-Jacobi equation (<ref>). The mild solution ℐ(t,g) is uniformly bounded for any time t≥ 0 and any functions f^0,g satisfying the assumptions (<ref>)-(<ref>). This solution ℐ(t,g) also converges to a non-trivial stationary solution as t→ +∞.


We present now a similar result (Theorem <ref>) for forcing g satisfying a different set of assumptions. The assumptions are more general in one aspect, while being more restrictive in another aspect.

For Theorem <ref>, the initial data f^0ℬ^-1 is only assumed to be close to the coupled equilibrium 𝒢, without the orthogonality condition
    
    ‖ f^0ℬ^-1-𝒢‖_L_β+1^∞<c.H4

The terminal data e^g(t)ℬ is assumed to be close to the coupled equilibrium 𝒢 without orthogonality condition, but its perturbation is assumed to decay exponentially in time with σ>0
    
    ‖ e^g(t)ℬ-𝒢‖_L_β+1^∞< e^-σ tc.H5

The function g and the forward initial data f^0ℬ^-1 are assumed to have certain regularity and continuity, where the function ϕ is defined in (<ref>) and _ϕ>0 is a small enough positive constant

    ‖ϕ‖_L_t^1(L_x,v^∞)+‖ϕ‖_C_t^0(L_x,v^∞)≤_ϕ, ,
                                  S_τf^0ℬ^-1∈(R_τ;L_β^∞) 
        H6


For arbitrary β>4 and σ>1, we can take constants c>0 and a_*>0 depending on β,σ such that for any f^0 and g satisfying the assumptions (<ref>)-(<ref>) and any time t>0, there exists a unique mild solution (ψ_t,η_t) of the coupled Boltzmann equations (<ref>) in the function class below 

    sup_0≤ s≤ t‖ψ_t(s)-𝒢‖_L_β^∞<a_*,  sup_0≤ s≤ te^σ s‖η_t(s)-𝒢‖_L_β^∞<a_*.

Furthermore, the functional ℐ(t,g) in (<ref>) is well-defined for any functions f^0 and g satisfying the assumptions (<ref>)-(<ref>), and is a global-in-time mild solution of the Hamilton-Jacobi equation (<ref>). The mild solution ℐ(t,g) is uniformly bounded for any time t≥ 0 and any functions f^0,g satisfying the assumptions (<ref>)-(<ref>).


Figure <ref> illustrates the shape of the perturbations ψ_t(s)-𝒢 and η_t(s)-𝒢 in different cases: Theorem <ref> or <ref>. 

In Theorem <ref>, for each terminal time t, the forward component ψ_t(s) decays forwards to the coupled equilibrium 𝒢, and the backward component η_t also decays backwards to the 𝒢. The polynomial decay rate σ>1, and the size ‖η_t(t)-𝒢‖_L_β^∞ of perturbation at terminal time t is uniform for all terminal time t.

In Theorem <ref>, we do not have the estimate for the convergence towards equilibrium. This is because we do not assume orthogonality condition for the initial data f^0 and the terminal data e^g(t), as well as the function D_sg could be not constant zero. These may give some (hydrodynamic) modes of constant order that will be preserved in the evolution. The price of having this generality is that, we assume that the size ‖η_t(t)-𝒢‖_L_β^∞ of the perturbation at terminal time t must decay as t→+∞.



 §.§ Future Directions


Global Solution of Forced Boltzmann Equation: Based on the results of the current paper, it would be interesting to look at the global-in-time solution φ of the forced Boltzmann equation with forcing p given by

    D_sφ=∫(φ(v')φ(v_*')exp(-Δ p)-φ(v)φ(v_*)exp(Δ p))((v_*-v)·ω)_+dω dv_*.

This type of modified equation is crucial for the large deviation theory established in <cit.> for a hard sphere gas. It is shown that for appropriate function p, we are able to look at the asymptotic probability of empirical measure π_t^ converging to an atypical density φ(t)

    π_t^→φ(t),  → 0,

when φ is a solution of (<ref>).  

Previously only local-in-time result about the forced Boltzmann equation is known. The relation between this future direction and the present results is that, the forced equation (<ref>) is exactly the forward equation in the Euler-Lagrange system (<ref>). The difference is that, in this paper we solve (<ref>) with given g, while for (<ref>) we solve it with a given forcing p.

Relation with Schrödinger Problem: In the previous paragraph we mentioned that for φ being a solution of (<ref>), we can study the large deviation cost of it. The (φ,p) is related with (ψ,η) through the change of variables (<ref>), with φ=ψη.


Based on the solution (ψ_t,η_t) given in Theorem <ref>, as t→ +∞ the corresponding density profile φ_t converges to a 'Relaxation and Anti-Relaxation' dynamics: it first relaxes to an equilibrium, and then anti-relaxes to an atypical density profile. This behaviour is related to the Schrödinger problem, namely the computation of the optimal path, given a large deviation cost function, followed by particle system from a given density at time 0 to another density at time t. The mean-field version of this relation has been investigated in <cit.>. For a survey of the Schrödinger Problem and its connection with optimal transport, see <cit.>. 

Uniform Control of the Limiting Cumulant Generating Functional:  As we have explained in Subsection <ref>, the following functional is used to encode the correlation information of a hard sphere gas with diameter 

    Λ^(t,h):=1/μ_logE_[exp(μ_π_t^(h))].

After taking the Boltzmann-Grad limit, this functional should converge to the functional Λ(t,h) encoding correlation of the limiting particle system. 

In <cit.> it has been shown the functional Λ(t,h) coincide with the solution ℐ(t,g) of the Hamilton-Jacobi equation (<ref>) in a finite time interval [0,T_c) for T_c<+∞, with h determined by g as in (<ref>). The coincidence between Λ(t,h) and ℐ(t,g) for any time t still remains to be proved.

One of the main difficulties for hard sphere gas to have global-in-time results about kinetic limit, dynamical fluctuations, and large deviations is the divergence of the upper bound for cumulant generating functionals when the time t approaches T_c. By establishing global-in-time solution ℐ, with the coincidence between Λ(t,h) and ℐ(t,g) for t∈[0,T_c), we can provide a uniform upper bound for the limiting cumulant generating functional (Figure <ref>). If this coincidence between functionals can be extended to the whole time interval, then the solution ℐ will provide a global-in-time uniform control of the limiting cumulant generating functional.

However the current result does not provide uniform control for the -cumulant generating functional. This will be left to future work. A uniform control of the cumulant generating functionals is expected to be an important step in proving long-time results about kinetic limit, dynamical fluctuations, and large deviations.








§ SYMMETRIZATION AND PERTURBATION REGIME FOR COUPLED BOLTZMANN EQUATIONS

In this section, we will perform the symmetrization procedure and define the perturbation regime, needed to solve the coupled Boltzmann equations (<ref>). 

As we have discussed in the previous sections, the symmetrization procedure consists in the change of variables 

    (ψ_t(s),η_t(s))=(φ_t(s)e^-p_t(s)+α'|v|^2,e^p_t(s)-α'|v|^2),  α'=1/2(1/2+α).

It will be proved in Lemma <ref> that the pair (ψ_t,η_t) satisfies the coupled Boltzmann equation (<ref>). 

After the symmetrization, we will perform a perturbation decomposition to (ψ,η): we look at the evolution for the perturbation of (ψ,η) from the coupled equilibrium, and we denote the perturbation as (ψ_p,η_p). The evolution equation (<ref>) of the perturbation is given as (<ref>). Finding a mild solution of equation (<ref>) is equivalent to finding a mild solution of equation (<ref>), and we will use a fixed-point method to find the solution of the latter.





A pair of functions (φ_t,p_t) is a mild solution of the coupled Boltzmann equations (<ref>), if and only if the pair of functions (ψ_t,η_t) given by the change of variables (<ref>) is a mild solution of the coupled Boltzmann equations with ℬ defined in (<ref>)

    D_s ψ(s)   =-ψ(s)ϕ(s)+𝒬_η(ψ,ψ),  ψ(0)=f^0ℬ^-1,
    
    D_s η(s)   =η(s)ϕ(s)-𝒬_ψ(η,η),  η(t)=e^g(t)ℬ.



Expanding the Hamiltonian system (<ref>), we have the following equation with Δ p_t defined in (<ref>) 

    D_sφ_t=∫(φ_t(v')φ_t(v_*')exp(-Δ p_t)-φ_t(v)φ_t(v_*)exp(Δ p_t))((v_*-v)·ω)_+dω dv_*,
       D_sp_t=ϕ(s)-∫φ_t(v_*)(exp(Δ p_t)-1)((v_*-v)·ω)_+dω dv_*,
       φ_t(0)=f^0e^p_t(0),   p_t(t)=g(t)

According to the definition of Δ p_t, a pair (φ_t,p_t) is a solution of the equation if and only if (φ_t,p_t-α'|v|^2) is a solution of the equation. Thus

    D_sφ_t=∫(φ_t(v')φ_t(v_*')exp(-Δ p_t)-φ_t(v)φ_t(v_*)exp(Δ p_t))((v_*-v)·ω)_+dω dv_*,
       D_sp_t=ϕ(s)-∫φ_t(v_*)(exp(Δ p_t)-1)((v_*-v)·ω)_+dω dv_*,
       φ_t(0)=f^0e^α'|v|^2e^p_t(0),   p_t(t)=g(t)-α'|v|^2

This concludes the proof of the lemma.




It is natural to look at the evolution of perturbations, with the hope that the smallness of the initial and terminal perturbations could imply the global well-posedness of the equation

    ψ_p:=ψ-𝒢,  η_p:=η-𝒢.


The evolution of the perturbations (ψ_p,η_p) should satisfy of the following coupled Boltzmann equations

    _sψ_p(s)
    
    =   -v·∇_x ψ_p(s)+2𝒬_𝒢(ψ_p,𝒢)_+𝒬_η_p(ψ_p,ψ_p)+2𝒬_η_p(ψ_p,𝒢)+𝒬_𝒢(ψ_p,ψ_p)_-ψ_pϕ(s)-𝒢ϕ(s),
       _sη_p(s)
    
    =   -v·∇_x η_p(s)-2𝒬_𝒢(η_p,𝒢)_-𝒬_ψ_p(η_p,η_p)-2𝒬_ψ_p(η_p,𝒢)-𝒬_𝒢(η_p,η_p)_+η_pϕ(s)+𝒢ϕ(s).


The mild solution of (<ref>) is defined as the fixed point of the fixed-point map Γ=(Γ^+,Γ^-), which will be introduced in Definition <ref>. For simplicity, we identify the mild solutions of (<ref>) with the mild solutions of (<ref>). At a rigorous level, for the solution (ψ_p,η_p) of (<ref>) considered in this paper, by performing series expansion the corresponding (ψ,η) can be shown to be a solution of (<ref>).



For the formal proof, We only detail it for the evolution of forward perturbation ψ_p, while the proof for the backward perturbation η_p is almost the same.

The definition of perturbation (<ref>) implies

    ψ=𝒢+ψ_p,  η=𝒢+η_p.

Use the equation above to replace (ψ,η) with (ψ_p,η_p) in equation (<ref>). For the evolution of the forward component, we have

    _s (𝒢+ψ_p)(s)=-v·∇_x(𝒢+ψ_p)(s)-(𝒢+ψ_p)(s)ϕ(s)+𝒬_𝒢+ψ_p(𝒢+ψ_p,𝒢+ψ_p).

Since the reference function is independent of the time t and space x variables, we get

    _s ψ_p(s)=-v·∇_xψ_p(s)-(𝒢+ψ_p)(s)ϕ(s)+𝒬_𝒢+ψ_p(𝒢+ψ_p,𝒢+ψ_p).

Now we only need the following equality to expand the third-order nonlinear collision term

    𝒬_𝒢+ψ_p(𝒢+ψ_p,𝒢+ψ_p)=2𝒬_𝒢(ψ_p,𝒢)+2𝒬_η_p(ψ_p,𝒢)+𝒬_𝒢(ψ_p,ψ_p)+𝒬_η_p(ψ_p,ψ_p).

This equality can be checked directly, using the fact that 𝒢 is the exponential of a collision invariant

    𝒬_𝒢(𝒢,𝒢)=0.


The coupled Boltzmann equations (<ref>) for the perturbation (ψ_p,η_p) will be one of our central objects in the rest of the paper. In the definition below, we define the relevant notation needed to study that equation.


We define the linear operators  and  separately as the linearized Boltzmann operators for the forward perturbation ψ_p or the backward perturbation η_p

    ψ_p:=-v·∇_x ψ_p+2𝒬_𝒢(ψ_p,𝒢),  η_p:=v·∇_x η_p+2𝒬_𝒢(η_p,𝒢).

The nonlinearity in the evolution of perturbation is denoted as

    𝒩[ψ_p,η_p]:=2𝒬_η_p(ψ_p,𝒢)+𝒬_𝒢(ψ_p,ψ_p)+𝒬_η_p(ψ_p,ψ_p).

We introduce the map Γ:(ψ_p,η_p)↦ ([ψ_p,η_p],[ψ_p,η_p]), whose fixed-point is a mild solution of the coupled Boltzmann equations (<ref>) 

    {   [ψ_p,η_p](s):=e^sψ_p(0)-∫_0^s e^(s-τ)(𝒢+ψ_p)ϕ(τ)dτ+∫_0^se^(s-τ)𝒩[ψ_p,η_p]dτ,
       [ψ_p,η_p](s):=e^(t-s)η_p(t)-∫_s^t e^(τ-s)(𝒢+η_p)ϕ(τ)dτ+∫_s^t e^(τ-s)𝒩[η_p,ψ_p]dτ.
    .


In the notations, the + sign of B^+ and  means that the operators are associated with the forward perturbation ψ_p, while the - sign of B^- and  means that the operators are associated with the backward perturbation η_p. Specifically, since the terminal data of η_p is given at time t, we will consider the evolution of η_p in the reversed time. This caused the operator B^- having a different sign from the linearized collision operator in the second line of equation (<ref>).

Sections <ref>, <ref>, and <ref> will be devoted to proving the existence and uniqueness of the fixed-point (ψ_p,η_p) of Γ,

    [ψ_p,η_p]=ψ_p,  [ψ_p,η_p]=η_p.

Specifically, we will prove that the map Γ is a contraction map in certain function spaces. In section <ref> we prove the decay estimates for the semigroups generated by  and ; In section <ref> we prove estimates of the nonlinear terms involved in the fixed-point problem; In section <ref>, we prove the contraction property of the map Γ.



§ ESTIMATE OF RELEVANT SEMIGROUPS IN L_Β^∞ NORM

The operator  (resp. ) defined in Definition <ref> generates a strongly continuous semigroup e^s (resp. e^s) on L^2(T_x^d×R_v^d). We can decompose the two semigroups as

    e^sB^+=𝒟_1^+(s)+𝒟_2^+(s),   e^sB^-=𝒟_1^-(s)+𝒟_2^-(s).

The components 𝒟_1^+(s) and 𝒟_1^-(s) have explicit expressions as

    𝒟_1^+(s)f(x,v)=e^-ν(v)sf(x-sv,v),  𝒟_1^-(s)f(x,v)=e^-ν(v)sf(x+sv,v),

where ν(v) is the frequency multiplier defined in (<ref>). For hard spheres, there are positive constants c_1,c_2>0 such that c_1(1+|v|)≤ν(v)≤ c_2(1+|v|).

The components 𝒟_2^+(s) and 𝒟_2^-(s) are respectively defined in (<ref>) and (<ref>).

Recall the definition of 𝒢 as 

    𝒢:=M^1/2ℰ^1/2=(2π)^-d/2exp(-1/2(1/2-α)|v|^2).

We define 𝒫 as the projection operator from L^2(T_x^d×R_v^d) to the subspace 𝒦 defined in (<ref>). The normalized orthogonal basis spanning 𝒦 is independent of the variable x, and is denoted as {f_i}_0≤ i≤ d+1. Notice that {f_i}_0≤ i≤ d+1 is different from the basis in (<ref>), since that basis may is not orthogonal and normalized. For a function f∈ L^2(T_x^d×R_v^d), we say f∈𝒦^⊥ if 𝒫f=0. 

Lemma <ref> is essentially a reorganization of the results in <cit.>. Its proof will be recalled in Appendix <ref>.



For β>4, there exist constants ν_*>0 and C>0 such that for any function f∈𝒦^⊥ and s≥ 0, we have

    ‖𝒟_2^+(s)f‖_L_β^∞≤ Ce^-ν_* s‖ (1+|v|)^-1f‖_L_β^∞,    ‖𝒟_2^-(s)f‖_L_β^∞≤ Ce^-ν_* s‖ (1+|v|)^-1f‖_L_β^∞,

and for any function f∈ L_β^∞ and s≥ 0, we have

    ‖𝒟_2^+(s)f‖_L_β^∞≤ C‖ f‖_L_β^∞,  ‖𝒟_2^-(s)f‖_L_β^∞≤ C‖ f‖_L_β^∞,
       ‖𝒟_1^+(s)f‖_L_β^∞≤ Ce^-ν_*s‖ f‖_L_β^∞,  ‖𝒟_1^-(s)f‖_L_β^∞≤ Ce^-ν_*s‖ f‖_L_β^∞.




For general data f∈ L_β^∞, the estimates of 𝒟_2^+(s)f and 𝒟_2^-(s)f are improved by the next Proposition.



For β>4, there exists a constant C>0 such that for arbitrary function f∈ L_β^∞ and s≥ 0 we have

    ‖𝒟_2^+(s)f‖_L_β^∞≤ C‖ (1+|v|)^-1f‖_L_β^∞,    ‖𝒟_2^-(s)f‖_L_β^∞≤ C‖ (1+|v|)^-1f‖_L_β^∞.



We detail the proof only for the forward component 𝒟_2^+ since the proof for the other component 𝒟_2^- is exactly the same.

First we consider the decomposition of f as f=𝒫f+(𝕀-𝒫)f

    𝒟_2^+(s)f=𝒟_2^+(s)𝒫f+𝒟_2^+(s)(𝕀-𝒫)f.

Using Lemma <ref> and Lemma <ref> as well as the decomposition of f, there is

    ‖𝒟_2^+(s)f  ‖_L_β^∞≤‖𝒟_2^+(s)𝒫f‖_L_β^∞+‖𝒟_2^+(s)(𝕀-𝒫)f  ‖_L_β^∞≤ C‖𝒫f  ‖_L_β^∞+C‖ (1+|v|)^-1(𝕀-𝒫)f  ‖_L_β^∞

Thus to derive the desired estimate (<ref>), it is enough to prove

    ‖𝒫f  ‖_L_β^∞≤ C‖ (1+|v|)^-1f‖_L_β^∞,  ‖ (1+|v|)^-1(𝕀-𝒫)f  ‖_L_β^∞≤ C‖ (1+|v|)^-1f‖_L_β^∞

For ‖𝒫f‖_L_β^∞, we use the fact that 𝒫f∈𝒦, and that 𝒦 is spanned by a family of orthogonal functions {f_j}_0≤ j≤ d+1 with exponential decay in v, therefore belonging to L_β^∞

    ‖𝒫f‖_L_β^∞ =    ‖∑_j=0^d+1⟨ f,f_j⟩_L_x,v^2f_j‖_L_β^∞≤∑_j=0^d+1‖⟨ f,f_j⟩_L_x,v^2f_j‖_L_β^∞.

Using Cauchy-Schwartz inequality for the inner product ⟨ f,f_j⟩ as well as the fact ‖ f_j‖_L_β^∞ is bounded, we further have

    ‖⟨ f,f_j⟩_L_x,v^2f_j‖_L_β^∞≤ C‖‖ f‖_L_x,v^2‖ f_j‖_L_x,v^2 f_j‖_L_β^∞≤ C‖ f‖_L_x,v^2.

Since β>4, the norm L_β-1^∞ is stronger than the norm L_x,v^2. This implies

    ‖⟨ f,f_j⟩_L_x,v^2f_j‖_L_β^∞≤  C‖ f‖_L_x,v^2≤ C‖ f‖_L_β-1^∞≤ C‖ (1+|v|)^-1f‖_L_β^∞.

Combining (<ref>) with (<ref>), we derive the first inequality in (<ref>)

    ‖𝒫f‖_L_β^∞≤ C‖ (1+|v|)^-1f‖_L_β^∞.

For the second inequality in (<ref>), using the triangle inequality we have

    ‖ (1+|v|)^-1(𝕀-𝒫)f  ‖_L_β^∞≤‖ (1+|v|)^-1f  ‖_L_β^∞+‖ (1+|v|)^-1𝒫f ‖_L_β^∞.

By (<ref>) there is

    ‖ (1+|v|)^-1𝒫f  ‖_L_β^∞≤‖𝒫f‖_L_β^∞≤ C‖ (1+|v|)^-1f‖_L_β^∞

This implies the second inequality in (<ref>), and thus concludes the proof of the theorem.









§ CONTROL OF THE NONLINEARITY

This section is devoted to the control of the nonlinear terms in the fixed-point map Γ in definition <ref>. Subsection <ref> proves the control of the biased collision operator. Subsection <ref> further gives the estimate of nonlinear terms needed for Theorem <ref>, while Subsection <ref> is for Theorem <ref>. The framework of the estimates in this section originates from <cit.>, with some additional analysis to handle the coupled Boltzmann equations.



 §.§ Estimates for the Biased Collision Operator


For any parameter β>4 and functions η,ψ_1,ψ_2∈ L_β^∞, the biased collision operator 𝒬_η(ψ_1,ψ_2) is bounded from above by

    ‖ (1+|v|)^-1𝒬_η(ψ_1,ψ_2) ‖_L_β^∞≤ C‖η‖_L_β^∞‖ψ_1‖_L_β^∞‖ψ_2 ‖_L_β^∞.



Recall the definition (<ref>) of the biased collision operator

    𝒬_η(ψ_1,ψ_2):=1/2∫((v_*-v)·ω)_+ η(v_*)[ψ_1(v')ψ_2(v_*')+ψ_2(v')ψ_1(v_*')
                                          -ψ_1(v)ψ_2(v_*)-ψ_2(v)ψ_1(v_*)]dω dv_*.

The operator is a summation of four terms. We will detail the proof of the following inequality

    ‖(1+|v|)^-1[∫((v_*-v)·ω)_+ η(v_*)ψ_1(v')ψ_2(v_*')dω dv_*]‖_L_β^∞≤ C‖η‖_L_β^∞‖ψ_1‖_L_β^∞‖ψ_2 ‖_L_β^∞.

The inequality above gives an upper bound for one of the four terms in the biased collision operator. The proof of upper bound for the other three terms is essentially the same. These together imply (<ref>).

By the definition (<ref>) of the L_β^∞ norm, we have for arbitrary function g∈ L_β^∞

    |g(v)|≤‖ g‖_L_β^∞ (1+|v|^β)^-1.

Using the inequality above, we deduce

    ‖(1+|v|)^-1[∫((v_*-v)·ω)_+ η(v_*)ψ_1(v')ψ_2(v_*')dω dv_*]‖_L_β^∞
    ≤   ‖∫_R^d×S^2 (1+|v|)^-1((v_*-v)·ω)_+(1+|v_*|)^-β(1+|v'|)^-β(1+|v_*'|)^-β dω dv_*‖_L_β^∞
       ×‖η‖_L_β^∞‖ψ_1‖_L_β^∞‖ψ_2 ‖_L_β^∞.

According to the definition (<ref>) of the L_β^∞ norm, this term is less than

    ‖∫_R^d×S^d-1 (1+|v|)^-1((v_*-v)·ω)_+(1+|v|)^β(1+|v_*|)^-β(1+|v'|)^-β(1+|v_*'|)^-β dω dv_*‖_L^∞
       ×‖η‖_L_β^∞‖ψ_1‖_L_β^∞‖ψ_2 ‖_L_β^∞.

To control the L^∞-norm, we write

    ((v_*-v)·ω)_+≤ |v-v_*|≤ |v|+|v_*|.

With the inequality above, we are able to control the L^∞ norm in (<ref>)

    ‖∫_R^d×S^d-1 (1+|v|)^-1((v-v_*)·ω)_+(1+|v|)^β(1+|v_*|)^-β(1+|v'|)^-β(1+|v_*'|)^-β dω dv_*‖_L^∞
    ≤   ‖∫_R^d×S^d-1 (1+|v|)^-1(|v|+|v_*|)(1+|v|)^β(1+|v_*|)^-β(1+|v'|)^-β(1+|v_*'|)^-β dω dv_*‖_L^∞.

By the definition of the pre-collisional configuration (v',v_*'), we have

    (1+|v'|)^-β(1+|v_*'|)^-β≤ (1+|v'|+|v_*'|+|v'||v_*'|)^-β≤ C(1+|v|)^-β.

Using the fact that β>4, the inequality above further implies

    ‖∫_R^d×S^d-1 (1+|v|)^-1((v-v_*)·ω)_+(1+|v|)^β(1+|v_*|)^-β(1+|v'|)^-β(1+|v_*'|)^-β dω dv_*‖_L^∞
    ≤   C‖∫_R^d×S^d-1 (1+|v|^-1)(|v|+|v_*|)(1+|v_*|)^-β dω dv_*‖_L^∞≤ C.

This concludes the proof of inequality (<ref>), and thus concludes the proof of the lemma.

Based on Lemma <ref>, we can derive the corollary below, by replacing the η or ψ_2 in Lemma <ref> with the coupled equilibrium 𝒢.

For any parameter β>4 and functions η,ψ_1,ψ_2∈ L_β^∞, we have

    ‖ (1+|v|)^-1𝒬_𝒢(ψ_1,ψ_2) ‖_L_β^∞≤ C‖ψ_1‖_L_β^∞‖ψ_2 ‖_L_β^∞,  ‖ (1+|v|)^-1𝒬_η(ψ_1,𝒢) ‖_L_β^∞≤ C‖η‖_L_β^∞‖ψ_1 ‖_L_β^∞.


From now on the terminal time t is fixed, but all the constants are independent of t.

For the forward component, we define the convolution of the semigroup e^sB^+ and the biased collision operator as

    Ψ^+[η,ψ_1,ψ_2](s):=∫_0^se^(s-τ)𝒬_η(τ)(ψ_1(τ),ψ_2(τ))dτ.

Due to the decomposition (<ref>) of e^sB^+, we also define the decomposition components of Ψ^+

    Ψ_1^+[η,ψ_1,ψ_2](s):=∫_0^s𝒟_1^+(s-τ)𝒬_η(τ)(ψ_1(τ),ψ_2(τ))dτ
       Ψ_2^+[η,ψ_1,ψ_2](s):=∫_0^s𝒟_2^+(s-τ)𝒬_η(τ)(ψ_1(τ),ψ_2(τ))dτ

For the backward component, we define the convolution of the semigroup e^sB^- and the biased collision operator as

    Ψ^-[ψ,η_1,η_2](s)=∫_s^te^(τ-s)𝒬_ψ(τ)(η_1(τ),η_2(τ))dτ.

Due to the decomposition (<ref>) of e^sB^-, we also define the decomposition components of Ψ^+

    Ψ_1^-[η,ψ_1,ψ_2](s):=∫_s^t𝒟_1^-(τ-s)𝒬_η(τ)(ψ_1(τ),ψ_2(τ))dτ
       Ψ_2^-[η,ψ_1,ψ_2](s):=∫_s^t𝒟_2^-(τ-s)𝒬_η(τ)(ψ_1(τ),ψ_2(τ))dτ


Based on the definitions of Ψ^+ and Ψ^-, we can rewrite the fixed-point map Γ in a more useful form

    [ψ_p,η_p]
    
    =   e^sB^+ψ_p(0)-∫_0^se^(s-τ)B^+(𝒢+ψ_p)ϕ  dτ
    +Ψ^+[η_p,ψ_p,ψ_p]+2Ψ^+[η_p,ψ_p,𝒢]+Ψ^+[𝒢,ψ_p,ψ_p],
       [ψ_p,η_p]
    
    =   e^(t-s)B^-η_p(t)-∫_s^te^(τ-s)B^-(𝒢+η_p)ϕ  dτ
    +Ψ^+[ψ_p,η_p,η_p]+2Ψ^+[ψ_p,η_p,𝒢]+Ψ^+[𝒢,η_p,η_p].

A proper norm must be chosen to prove that Γ is a contraction map. For this purpose we define the norm P_β^σ.

We define the norm ‖·‖_P_β^σ as

    ‖ψ_p‖_P_β^σ:=sup_0≤ s≤ t(1+s)^σ‖ψ_p(s)‖_L_β^∞.

We also define the norm ‖·‖_E_β^σ as

    ‖ψ_p‖_E_β^σ:=sup_0≤ s≤ te^σ s‖ψ_p(s)‖_L_β^∞.

Recalling that t is the terminal time, we introduce the time reversal operator  as

    η_p^(s)=η_p(t-s).


According to the definition of P_β^σ and , we have

    ‖ψ_p(s)‖_L_β^∞≤ (1+s)^-σ‖ψ_p‖_P_β^σ,    ‖η_p(s)‖_L_β^∞≤(1+(t-s))^-σ‖η_p^‖_P_β^σ,

where the second inequality is due to

    ‖η_p^‖_P_β^σ:=sup_0≤ s≤ t(1+s)^σ‖η_p^(s)‖_L_β^∞=   sup_0≤ s≤ t(1+s)^σ‖η_p(t-s)‖_L_β^∞
    
    =   sup_0≤ s≤ t(1+(t-s))^σ‖η_p(s)‖_L_β^∞.

Similar inequalities are also true for E_β^σ

    ‖ψ_p(s)‖_L_β^∞≤ e^-σ s‖ψ_p‖_E_β^σ,    ‖η_p(s)‖_L_β^∞≤ e^-σ (t-s)‖η_p^‖_E_β^σ.





 §.§ Control of Convolutional Nonlinearity for Theorem <ref>

In this subsection, we provide the control of P_β^σ norm for Ψ^+ and Ψ^-, which is useful for Theorem <ref>.

In subsections <ref> and <ref>, all the estimates of Ψ^+ (resp. Ψ^-) will be reduced to the estimates of Ψ_1^+ and Ψ_2^+ (resp. Ψ_1^- and Ψ_2^-). 

 For any parameters β>4, σ>1, and functions η^,ψ_1,ψ_2∈ P_β^σ, we have the upper bound for the P_β^σ norm of Ψ^+[η,ψ_1,ψ_2] as

    ‖Ψ^+[η,ψ_1,ψ_2]‖_P_β^σ≤ C‖η^‖_P_β^σ‖ψ_1‖_P_β^σ‖ψ_2 ‖_P_β^σ.



Estimate of Ψ_2^+: First we would like to prove the upper bound for the L_β^∞ norm of Ψ_2^+(s), with 0≤ s ≤ t. According to the definition of Ψ^+, there is

    ‖Ψ_2^+[η,ψ_1,ψ_2](s)‖_L_β^∞= ‖∫_0≤τ≤ s𝒟_2^+(s-τ)𝒬_η(τ)(ψ_1(τ),ψ_2(τ))dτ‖_L_β^∞.

Using the boundedness of 𝒟_2^+(s) from L_β-1^∞ to L_β^∞ given by Proposition <ref>, we further have

    ‖Ψ_2^+[η,ψ_1,ψ_2](s)‖_L_β^∞≤ C∫_0≤τ≤ s‖ (1+|v|)^-1𝒬_η(τ)(ψ_1(τ),ψ_2(τ))‖_L_β^∞ dτ.

To conclude the estimate of the L_β^∞ norm, we use Lemma <ref> to control the L_β^∞ norm related to the biased collision operator

    ‖Ψ_2^+[η,ψ_1,ψ_2](s)‖_L_β^∞≤ C∫_0≤τ≤ s‖η(τ)‖_L_β^∞‖ψ_1(τ)‖_L_β^∞‖ψ_2(τ) ‖_L_β^∞dτ.

Based on the estimate (<ref>) of the L_β^∞ norm, we want to further control the P_β^σ norm of Ψ_2^+. According to the definition (<ref>) of the P_β^σ norm

    ‖Ψ_2^+[η,ψ_1,ψ_2]‖_P_β^σ   =sup_s∈[0,t](1+s)^σ‖Ψ_2^+[η,ψ_1,ψ_2](s)‖_L_β^∞
       ≤sup_s∈[0,t]C(1+s)^σ∫_0≤τ≤ s‖η(τ)‖_L_β^∞‖ψ_1(τ)‖_L_β^∞‖ψ_2(τ) ‖_L_β^∞dτ
       ≤ C(1+t)^σ∫_0≤τ≤ t‖η(τ)‖_L_β^∞‖ψ_1(τ)‖_L_β^∞‖ψ_2(τ) ‖_L_β^∞dτ.

To get the upper bound in (<ref>), we will use the upper bounds (<ref>) provided by the norm P_β^σ. The inequality (<ref>) implies

    ‖η(τ)‖_L_β^∞‖ψ_1(τ)‖_L_β^∞‖ψ_2(τ) ‖_L_β^∞≤(1+(t-τ))^-σ(1+τ)^-2σ‖η^‖_P_β^σ‖ψ_1‖_P_β^σ‖ψ_2 ‖_P_β^σ.

Combining (<ref>) with (<ref>) leads to

    ‖Ψ_2^+[η,ψ_1,ψ_2]‖_P_β^σ
    ≤    C(1+t)^σ∫_0≤τ≤ t(1+(t-τ))^-σ(1+τ)^-2σdτ‖η^‖_P_β^σ‖ψ_1‖_P_β^σ‖ψ_2 ‖_P_β^σ
    ≤    C(1+t)^σ(1+t)^-min{σ,2σ}‖η^‖_P_β^σ‖ψ_1‖_P_β^σ‖ψ_2 ‖_P_β^σ,

where the second inequality is due to Lemma <ref>. Noticing the simple fact that min{σ,2σ}=σ concludes the estimate of Ψ_2^+.

Estimate of Ψ_1^+: Using the explicit expression (<ref>) of 𝒟_1^+, we get

    |Ψ_1^+[η,ψ_1,ψ_2](s,x,v)|
    ≤    C∫_0^se^-ν(v)(s-τ)|sup_x∈T^d𝒬_η(τ)(ψ_1(τ),ψ_2(τ))(x,v)|dτ
    ≤    C∫_0^se^-ν(v)(s-τ)(1+τ)^-2σν(v)(1+|v|)^-β((1+τ)^2σ(1+|v|)^β1/ν(v)|sup_x∈T^d𝒬_η(τ)(ψ_1(τ),ψ_2(τ))(x,v)|)dτ_.

Taking the supremum over τ and v in term (I), along with the fact that ν(v) is equivalent to 1+|v| up to constants, we further have

    |Ψ_1^+[η,ψ_1,ψ_2](s,x,v)|
    ≤   ∫_0^se^-ν(v)(s-τ)(1+τ)^-2σν(v)dτ (1+|v|)^-β(sup_0≤τ≤ s(1+τ)^2σ‖1/ν(v)𝒬_η(τ)(ψ_1(τ),ψ_2(τ))‖_L_β^∞)
    ≤    C∫_0^se^-ν(v)(s-τ)(1+τ)^-2σν(v)dτ (1+|v|)^-β(sup_0≤τ≤ s(1+τ)^2σ‖η(τ)‖_L_β^∞‖ψ_1(τ)‖_L_β^∞‖ψ_2(τ)‖_L_β^∞)
    ≤   C∫_0^se^-ν(v)(s-τ)(1+τ)^-2σν(v)dτ (1+|v|)^-β‖η^‖_P_β^σ‖ψ_1‖_P_β^σ‖ψ_2 ‖_P_β^σ,

where in the second inequality we have used Lemma <ref>. For the time integral in the inequalities above, we decompose the integral and get

    ∫_0^se^-ν(v)(s-τ)(1+τ)^-2σν(v)dτ
    
    =   ∫_0^s/2e^-ν(v)(s-τ)(1+τ)^-2σν(v)dτ+∫_s/2^se^-ν(v)(s-τ)(1+τ)^-2σν(v)dτ
    ≤   e^-ν_*s/4∫_0^s/2e^-ν(v)(s-τ)/2(1+τ)^-2σν(v)dτ+(1+s/2)^-2σ∫_s/2^se^-ν(v)(s-τ)ν(v)dτ≤ C(1+s)^-σ.

The convolution inequality above implies

    ‖Ψ_1^+[η,ψ_1,ψ_2](s)‖_L_β^∞≤ (1+s)^-σ‖η^‖_P_β^σ‖ψ_1‖_P_β^σ‖ψ_2 ‖_P_β^σ.

Thus by the definition (<ref>) of the P_β^σ norm

    ‖Ψ_1^+[η,ψ_1,ψ_2]‖_P_β^σ≤ C‖η^‖_P_β^σ‖ψ_1‖_P_β^σ‖ψ_2 ‖_P_β^σ.

This along with the estimate of Ψ_2^+ concludes the proof.




 For any parameters β>4,σ>1, and functions η^,ψ_1∈ P_β^σ, we have the upper bound for the P_β^σ norm of Ψ^+[η,ψ_1,𝒢] as

    ‖Ψ^+[η,ψ_1,𝒢]‖_P_β^σ≤ C‖η^‖_P_β^σ‖ψ_1‖_P_β^σ.




Estimate of Ψ_2^+: Using exactly the same method of proving equation (<ref>) in Lemma <ref>, we have the upper bound for the L_β^∞ norm of Ψ^+[η,ψ_1,𝒢] as

    ‖Ψ_2^+[η,ψ_1,𝒢](s)‖_L_β^∞   ≤ C∫_0≤τ≤ s‖η(τ)‖_L_β^∞‖ψ_1(τ)‖_L_β^∞dτ.

This further implies 

    ‖Ψ_2^+[η,ψ_1,𝒢](s)‖_P_β^σ   ≤ Csup_0≤ s≤ t(1+s)^σ∫_0≤τ≤ s‖η(τ)‖_L_β^∞‖ψ_1(τ)‖_L_β^∞dτ
       ≤ C(1+t)^σ∫_0≤τ≤ t‖η(τ)‖_L_β^∞‖ψ_1(τ)‖_L_β^∞dτ.

Again we use the upper bounds (<ref>) provided by the norm P_β^σ, just as in the proof of Lemma <ref>,

    ‖Ψ_2^+[η,ψ_1,𝒢]‖_P_β^σ≤    C(1+t)^σ∫_0≤τ≤ t(1+(t-τ))^-σ(1+τ)^-σ‖η^‖_P_β^σ‖ψ_1‖_P_β^σdτ
    ≤    C(1+t)^σ(1+t)^-min{σ,σ}‖η^‖_P_β^σ‖ψ_1‖_P_β^σ.

Noticing the simple fact that min{σ,σ}=σ concludes the estimate of Ψ_2^+.

Estimate of Ψ_1^+: The estimate of Ψ_1^+[η,ψ_1,𝒢] is similar to the estimate of Ψ_1^+[η,ψ_1,ψ_2]. Similar to (<ref>), we have

    |Ψ_1^+[η,ψ_1,𝒢](s,x,v)|≤ C∫_0^se^-ν(v)(s-τ)(1+τ)^-σν(v)dτ (1+|v|)^-β‖η^‖_P_β^σ‖ψ_1‖_P_β^σ.

Then the time integral gives a factor (1+s)^-σ

    ‖Ψ_1^+[η,ψ_1,𝒢](s)‖_L_β^∞≤ (1+s)^-σ‖η^‖_P_β^σ‖ψ_1‖_P_β^σ.

This eventually gives

    ‖Ψ_1^+[η,ψ_1,𝒢]‖_P_β^σ≤ C‖η^‖_P_β^σ‖ψ_1‖_P_β^σ.

This along with the estimate of Ψ_2^+ conclude the proof.




 For any parameters β>4,σ>1, and functions ψ_1,ψ_2∈ P_β^σ, we have the upper bound for the P_β^σ norm of Ψ^+[𝒢,ψ_1,ψ_2] as

    ‖Ψ^+[𝒢,ψ_1,ψ_2]‖_P_β^σ≤ C‖ψ_1‖_P_β^σ‖ψ_2‖_P_β^σ.



Estimate of Ψ_2^+: The proof of this lemma is slightly different from the proof of Lemma <ref> and Lemma <ref>. Again due to the definition of Ψ^+ there is

    ‖Ψ_2^+[𝒢,ψ_1,ψ_2](s)‖_L_β^∞= ‖∫_0≤τ≤ se^(s-τ)𝒬_𝒢(ψ_1(τ),ψ_2(τ))dτ‖_L_β^∞.

The reason for the difference is that for arbitrary τ≥ 0 we have

    𝒬_𝒢(ψ_1(τ),ψ_2(τ))∈𝒦^⊥.

This equality can be verified as the following. Suppose h is a collision invariant. For each x we perform the integration over v 

    ∫𝒢h(v)𝒬_𝒢(ψ_1(τ),ψ_2(τ))(v)dvdx
    
    =   1/2∫((v_*-v)·ω)_+ h(v)𝒢(v)𝒢(v_*)
       ×(ψ_1(v')ψ_2(v_*')+ψ_2(v')ψ_1(v_*')-ψ_1(v)ψ_2(v_*)-ψ_2(v)ψ_1(v_*))dω dv_*dvdx.

Using the symmetry of the collision measure

    ((v_*-v)·ω)_+dω dv_*dv,

we can perform the change of variables (v,v_*,ω)↦ (v_*,v,-ω) or (v,v_*,ω)↦ (v',v_*',-ω). Then we would have

    ∫𝒢h(v)𝒬_𝒢(ψ_1(τ),ψ_2(τ))(v)dvdx
    
    =   1/8∫((v_*-v)·ω)_+ 𝒢(v)𝒢(v_*)(h(v)+h(v_*)-h(v')-h(v_*'))
       ×(ψ_1(v')ψ_2(v_*')+ψ_2(v')ψ_1(v_*')-ψ_1(v)ψ_2(v_*)-ψ_2(v)ψ_1(v_*))dω dv_*dvdx.

Since h is a collision invariant, we have h(v')+h(v_*')-h(v)-h(v_*)=0. This implies 

    ⟨ f_i, 𝒬_𝒢(ψ_1(τ),ψ_2(τ))⟩_L_x,v^2=0

for arbitrary 0≤ i≤ d+1, where {f_i}_0≤ i≤ d+1 is the basis of the kernel 𝒦. Thus there is 𝒬_𝒢(ψ_1(τ),ψ_2(τ))∈𝒦^⊥.

Using the orthogonality (<ref>) and Lemma <ref>, we can transform (<ref>) into

    ‖Ψ_2^+[𝒢,ψ_1,ψ_2](s)‖_L_β^∞≤ C∫_0≤τ≤ se^-ν_*(s-τ)‖ (1+|v|)^-1𝒬_𝒢(ψ_1(τ),ψ_2(τ))‖_L_β^∞dτ.

We can further control the L_β^∞ norm of the biased collision term by Lemma <ref>

    ‖Ψ_2^+[𝒢,ψ_1,ψ_2](s)‖_L_β^∞≤ C∫_0≤τ≤ se^-ν_*(s-τ)‖ψ_1(τ)‖_L_β^∞‖ψ_2(τ)‖_L_β^∞dτ.

According to the definition of the P_β^σ norm, we have the inequality

    ‖Ψ_2^+[𝒢,ψ_1,ψ_2]‖_P_β^σ≤    C(1+s)^σ∫_0≤τ≤ se^-ν_*(s-τ)(1+τ)^-2σdτ‖ψ_1‖_P_β^σ‖ψ_2 ‖_P_β^σ
    ≤    C(1+s)^σ(1+s)^-2σ‖ψ_1‖_P_β^σ‖ψ_2 ‖_P_β^σ.

This concludes the estimate of Ψ_2^+.

Estimate of Ψ_1^+: The estimate of Ψ_1^+[𝒢,ψ_1,ψ_2] is also similar to the estimate of Ψ_1^+[η,ψ_1,ψ_2]. As in (<ref>), we have

    ‖Ψ_1^+[𝒢,ψ_1,ψ_2](s)‖_L_β^∞≤ C∫_0^se^-ν(v)(s-τ)(1+τ)^-σν(v)dτ (1+|v|)^-β‖ψ_1‖_P_β^σ‖ψ_2‖_P_β^σ.

Consequently we get

    ‖Ψ_1^+[𝒢,ψ_1,ψ_2](s)‖_P_β^σ≤ C‖ψ_1‖_P_β^σ‖ψ_2‖_P_β^σ

This along with the estimate of Ψ_2^+ concludes the proof.




Since the evolution of the perturbations (ψ_p,η_p) is symmetric, we straightforwardly have the lemma below.

 For any parameters β>4, σ>1, and functions ψ,η_1^,η_2^∈ P_β^σ, we have the following upper bounds for the P_β^σ norm of various terms

    {   ‖ (Ψ^-[ψ,η_1,η_2])^‖_P_β^σ≤ C‖ψ‖_P_β^σ‖η_1^‖_P_β^σ‖η_2^‖_P_β^σ,
       ‖ (Ψ^-[ψ,η_1,𝒢])^‖_P_β^σ≤ C‖ψ‖_P_β^σ‖η_1^‖_P_β^σ,
       ‖ (Ψ^-[𝒢,η_1,η_2])^‖_P_β^σ≤ C‖η_1^‖_P_β^σ‖η_2^‖_P_β^σ.
    .






 §.§ Control of Convolutional Nonlinearity for Theorem <ref>

In this section we provide the control of the E_β^0-norm (see Definition <ref>) of Ψ^+ and the E_β^-σ norm of Ψ^-, which is useful for Theorem <ref>.

 For any parameters β>4,σ>0, functions ψ_1,ψ_2∈ E_β^0 and η^∈ E_β^-σ, we have the following upper bounds for the E_β^0 norm of various terms

    {   ‖Ψ^+[η,ψ_1,ψ_2]‖_E_β^0≤ C e^σ t‖η^‖_E_β^-σ‖ψ_1‖_E_β^0‖ψ_2 ‖_E_β^0,
       ‖Ψ^+[η,ψ_1,𝒢]‖_E_β^0≤ C e^σ t‖η^‖_E_β^-σ‖ψ_1‖_E_β^0,
       ‖Ψ^+[𝒢,ψ_1,ψ_2]‖_E_β^0≤ C ‖ψ_1‖_E_β^0‖ψ_2 ‖_E_β^0.
    .



 

The first inequality in (<ref>):

The Ψ_2^+ Term: We use the upper bound for the L_β^∞ norm of Ψ_2^+, which has been proved in (<ref>)

    ‖Ψ_2^+[η,ψ_1,ψ_2](s)‖_L_β^∞≤ C∫_0≤τ≤ s‖η(τ)‖_L_β^∞‖ψ_1(τ)‖_L_β^∞‖ψ_2(τ) ‖_L_β^∞dτ.

Since the E_β^0 norm is defined as the supremum of L_β^∞ for different time 0≤ s≤ t, we have

    ‖Ψ_2^+[η,ψ_1,ψ_2]‖_E_β^0≤ C∫_0≤τ≤ te^σ(t-τ)‖η^‖_E_β^-σ‖ψ_1‖_E_β^0‖ψ_2 ‖_E_β^0dτ≤ C e^σ t‖η^‖_E_β^-σ‖ψ_1‖_E_β^0‖ψ_2 ‖_E_β^0.

The Ψ_1^+ term: For Ψ_1^+, we have

    |Ψ_1^+[η,ψ_1,ψ_2](s,x,v)|
    ≤   C∫_0^se^-ν(v)(s-τ)e^σ (t-τ)ν(v)dτ (1+|v|)^-β(sup_0≤τ≤ se^-σ(t-τ)‖1/1+|v|𝒬_η(τ)(ψ_1(τ),ψ_2(τ))‖_L_β^∞)
    ≤   C ∫_0^se^-ν(v)(s-τ)e^σ(t-τ)ν(v)dτ (1+|v|)^-β(sup_0≤τ≤ se^-σ (t-τ)‖η(τ)‖_L_β^∞‖ψ_1(τ)‖_L_β^∞‖ψ_2(τ)‖_L_β^∞)
    ≤   e^σ t∫_0^se^-ν(v)(s-τ)e^-στν(v)dτ (1+|v|)^-β‖η^‖_E_β^-σ‖ψ_1‖_E_β^0‖ψ_2 ‖_E_β^0,

The time integral is uniformly bounded for all s and v

    ∫_0^se^-ν(v)(s-τ)e^-στν(v)dτ≤ C.

Consequently

    ‖Ψ_1^+[η,ψ_1,ψ_2]‖_E_β^0≤ C e^σ t‖η^‖_E_β^-σ‖ψ_1‖_E_β^0‖ψ_2 ‖_E_β^0.

This concludes the proof of the first inequality.

The second inequality in (<ref>): It can be derived by replacing ψ_2 with 𝒢.

The third inequality in (<ref>):

The Ψ_2^+ term: We first use the following upper bound for the L_β^∞, which has been proved as (<ref>) in Lemma <ref>

    ‖Ψ_2^+[𝒢,ψ_1,ψ_2](s)‖_L_β^∞≤ C∫_0≤τ≤ se^-ν_*(s-τ)‖ψ_1(τ)‖_L_β^∞‖ψ_2(τ)‖_L_β^∞dτ.

Then due to the definition of the E_β^0 norm, we have

    ‖Ψ_2^+[𝒢,ψ_1,ψ_2]‖_E_β^0≤ C∫_0^te^-ν_*(s-τ)‖ψ_1‖_E_β^0‖ψ_2‖_E_β^0dτ≤ C‖ψ_1‖_E_β^0‖ψ_2‖_E_β^0.

This concludes the estimate of Ψ_2^+.

The Ψ_1^+ term: Similar to (<ref>), we obtain

    |Ψ_1^+[𝒢,ψ_1,ψ_2](s,x,v)|
    ≤   C∫_0^se^-ν(v)(s-τ)ν(v)dτ (1+|v|)^-β(sup_0≤τ≤ s‖1/1+|v|𝒬_𝒢(ψ_1(τ),ψ_2(τ))‖_L_β^∞)
    ≤   C∫_0^se^-ν(v)(s-τ)ν(v)dτ (1+|v|)^-β‖ψ_1‖_E_β^0‖ψ_2 ‖_E_β^0,

The time integral gives a uniformly bounded constant. As a consequence

    ‖Ψ_1^+[𝒢,ψ_1,ψ_2]‖_E_β^0≤ C ‖ψ_1‖_E_β^0‖ψ_2 ‖_E_β^0.

This, along with the estimate of Ψ_2^+, concludes the proof.





 For any parameters β>4,σ>0, any functions ψ∈ E_β^0 and η_1^,η_2^∈ E_β^-σ, we have the following upper bounds for the norm of various terms

    {   ‖ (Ψ^-[ψ,η_1,η_2])^‖_E_β^-σ≤ Ce^σ t‖ψ‖_E_β^0‖η_1^‖_E_β^-σ‖η_2^‖_E_β^-σ,
       ‖ (Ψ^-[ψ,η_1,𝒢])^‖_E_β^-σ≤ C‖ψ‖_E_β^0‖η_1^‖_E_β^-σ,
       ‖ (Ψ^-[𝒢,η_1,η_2])^‖_E_β^-σ≤ Ce^σ t‖η_1^‖_E_β^-σ‖η_2^‖_E_β^-σ.
    .



 

The first inequality in (<ref>): 

The Ψ_2^- term: Similar to (<ref>), we have

    ‖Ψ_2^-[ψ,η_1,η_2](s)‖_L_β^∞≤ C∫_s≤τ≤ t‖ψ(τ)‖_L_β^∞‖η_1(τ)‖_L_β^∞‖η_2(τ) ‖_L_β^∞dτ.

According to the definition (<ref>) of the relevant norms, the equation above implies

    ‖ (Ψ_2^-[ψ,η_1,η_2])^‖_E_β^-σ   :=sup_0≤ s≤ te^-σ (t-s)‖Ψ_2^-[ψ,η_1,η_2](s)‖_L_β^∞
       ≤ Csup_0≤ s≤ te^-σ (t-s)∫_s≤τ≤ t‖ψ(τ)‖_L_β^∞‖η_1(τ)‖_L_β^∞‖η_2(τ) ‖_L_β^∞dτ
       ≤ Csup_0≤ s≤ te^-σ (t-s)∫_s≤τ≤ te^2σ(t-τ)‖ψ‖_E_β^0‖η_1‖_E_β^-σ‖η_2 ‖_E_β^-σdτ

After the integrating over time s≤τ≤ t, there is

    ‖ (Ψ_2^-[ψ,η_1,η_2])^‖_E_β^-σ   ≤ Csup_0≤ s≤ te^σ (t-s)‖ψ‖_E_β^0‖η_1‖_E_β^-σ‖η_2 ‖_E_β^-σ
       ≤ Ce^σ t‖ψ‖_E_β^0‖η_1^‖_E_β^-σ‖η_2^‖_E_β^-σ.

The Ψ_1^- term: For Ψ_1^-, we have

    |Ψ_1^-[η,ψ_1,ψ_2](s,x,v)|
    ≤   C∫_s^te^-ν(v)(τ-s)e^2σ (t-τ)ν(v)dτ (1+|v|)^-β(sup_0≤τ≤ se^-2σ(t-τ)‖1/1+|v|𝒬_ψ(τ)(η_1(τ),η_2(τ))‖_L_β^∞)
    ≤   C ∫_s^te^-ν(v)(τ-s)e^2σ(t-τ)ν(v)dτ (1+|v|)^-β(sup_0≤τ≤ se^-2σ (t-τ)‖ψ(τ)‖_L_β^∞‖η_1(τ)‖_L_β^∞‖η_2(τ)‖_L_β^∞)
    ≤   C∫_s^te^-ν(v)(τ-s)e^2σ(t- τ)ν(v)dτ (1+|v|)^-β‖ψ‖_E_β^0‖η_1^‖_E_β^-σ‖η_2^‖_E_β^-σ,

For the time integral, there is

    ∫_s^te^-ν(v)(s-τ)e^2σ(s- τ)ν(v)dτ=e^2σ(t-s)∫_0^se^-(ν(v)+2σ)(s-τ)ν(v)dτ≤ C  e^2σ(t-s).

Consequently

    ‖Ψ_1^-[ψ,η_1,η_2](s)‖_L_β^∞≤ Ce^2σ(t-s)‖ψ‖_E_β^0‖η_1^‖_E_β^-σ‖η_2^‖_E_β^-σ.

According to the definition (<ref>) of the E_β^-σ norm, we further have

    ‖Ψ_1^-[ψ,η_1,η_2]‖_E_β^-σ≤ Ce^σ t‖ψ‖_E_β^0‖η_1^‖_E_β^-σ‖η_2^‖_E_β^-σ.

This concludes the proof of the first inequality in (<ref>). 

The second inequality in (<ref>): 

The Ψ_2^- term: In a way similar to (<ref>), there is

    ‖Ψ_2^-[ψ,η_1,𝒢](s)‖_L_β^∞≤ C∫_s≤τ≤ t‖ψ(τ)‖_L_β^∞‖η_1(τ)‖_L_β^∞dτ,

which implies

    ‖ (Ψ_2^-[ψ,η_1,𝒢])^‖_E_β^-σ   ≤ Csup_0≤ s≤ te^-σ (t-s)∫_s≤τ≤ te^σ(t-τ)‖ψ‖_E_β^0‖η_1^‖_E_β^-σdτ≤ C‖ψ‖_E_β^0‖η_1^‖_E_β^-σ.

The Ψ_1^- term: Similar to (<ref>), we have

    |Ψ_1^-[η,ψ_1,𝒢](s,x,v)|≤ C∫_s^te^-ν(v)(τ-s)e^σ(t- τ)ν(v)dτ (1+|v|)^-β‖ψ‖_E_β^0‖η_1^‖_E_β^-σ.

Consequently

    ‖ (Ψ_1^-[ψ,η_1,𝒢])^‖_E_β^-σ≤ C‖ψ‖_E_β^0‖η_1^‖_E_β^-σ.

This concludes the proof of the second inequality.            

The third inequality in (<ref>): It can be derived by replacing ψ with 𝒢.






§ SOLVING THE FIXED-POINT PROBLEM

In this section, we will prove the fixed-point map (,) defined in (<ref>) is a contraction map. Thus it has a unique fixed point in a certain function class, while a fixed point of (Γ^+,Γ^-) is consequently a mild solution of the coupled Boltzmann equations (<ref>).  Under the assumptions of Theorem <ref>, it is a contraction w.r.t. ψ_p∈ P_β^σ and η_p^∈ P_β^σ with β>4,σ>1; Under the assumptions of Theorem <ref>, it is a contraction w.r.t. ψ_p∈ E_β^0 and η_p^∈ E_β^-σ with β>4,σ>0.


 §.§ Fixed Point for Theorem <ref>



For any parameters β>4,σ>1 and assuming (<ref>), we have the following estimates of the fixed point map Γ: for the forward component  there is

    ‖[ψ_p,η_p]‖_P_β^σ   ≤ C(‖ψ_p(0)‖_L_β^∞+‖η_p^‖_P_β^σ‖ψ_p‖_P_β^σ^2+‖η_p^‖_P_β^σ‖ψ_p‖_P_β^σ+‖ψ_p‖_P_β^σ^2),

and for the backward component  there is

    ‖([ψ_p,η_p])^‖_P_β^σ   ≤ C(‖η_p(t)‖_L_β^∞+‖η_p^‖_P_β^σ^2‖ψ_p ‖_P_β^σ+‖η_p^‖_P_β^σ‖ψ_p‖_P_β^σ+‖η_p^‖_P_β^σ^2).



We will detail the proof for the forward component . The proof for the backward component  is almost exactly the same.


According to the decomposition of [ψ_p,η_p] in (<ref>) and the assumption (<ref>) of ϕ≡ 0, we can use the triangle inequality for the P_β^σ norm

    ‖[ψ_p,η_p]‖_P_β^σ
    ≤   ‖ e^sψ_p(0)‖_P_β^σ+‖Ψ^+[η_p,ψ_p,ψ_p]‖_P_β^σ+‖Ψ^+[η_p,ψ_p,𝒢]‖_P_β^σ+‖Ψ^+[𝒢,ψ_p,ψ_p]‖_P_β^σ.

For the term associated with the initial perturbation ψ_p(0) in (<ref>), we have

    ‖ e^sψ_p(0)‖_P_β^σ=sup_0≤ s≤ t(1+s)^σ‖ e^sψ_p(0)‖_L_β^∞≤ Csup_0≤ s≤ t(1+s)^σe^-ν_*s‖ψ_p(0)‖_L_β^∞≤ C‖ψ_p(0)‖_L_β^∞,

where the first inequality is according to the assumption ψ_p(0)∈𝒦^⊥ and Lemma <ref>.
To control the other three terms involving the P_β^σ norm of Ψ^+ in (<ref>), we use Lemmas <ref>, <ref>, and <ref>

    ‖[ψ_p,η_p]‖_P_β^σ   ≤ C‖ψ_p(0)‖_L_β^∞+C‖η_p^‖_P_β^σ‖ψ_p‖_P_β^σ^2+C‖η_p^‖_P_β^σ‖ψ_p‖_P_β^σ+C‖ψ_p‖_P_β^σ^2.

This concludes the estimate of [ψ_p,η_p]. The estimate of [ψ_p,η_p] is the same, which concludes the proof of the lemma.


To use the contraction principle to find the fixed-point, we work in the following function class Ω where a_* is a positive constant

    Ω:={(ψ_p,η_p)|‖ψ_p‖_P_β^σ≤ a_*, ‖η_p^‖_P_β^σ≤ a_*, ψ_p(0)=(f^0-M)ℬ^-1, η_p(t)=(e^g(t)-ℰ)ℬ}.

It is equipped with the norm ‖ (ψ_p,η_p)‖_Ω:=‖ψ_p‖_P_β^σ+‖η_p^‖_P_β^σ. The goal is to prove Γ maps the region Ω into itself, and is a contraction map with respect to ‖·‖_Ω. 


For the constant c in assumption (<ref>) and (<ref>) being small enough, there exists a constant a_*>0 such that the fixed-point map Γ=(,) maps the region Ω to itself.



Using Lemma <ref>, we have

    ‖[ψ_p,η_p]‖_P_β^σ≤ C(c+a_*^3+a_*^2+a_*^2)  ‖([ψ_p,η_p])^‖_P_β^σ≤ C(c+a_*^3+a_*^2+a_*^2),

since (ψ_p,η_p)∈Ω. Thus to make sure Γ maps Ω into itself, we only need 

    C(c+a_*^3+a_*^2+a_*^2)≤ a_*.

It is convenient to assume C>1. If c is small enough, for example c<1/10C^-4, we can choose a_*=1/5C^-3. This concludes the proof.





For arbitrary β>4 and σ>1, suppose the functions f^0 and g satisfy the assumptions (<ref>)-(<ref>). Then for arbitrary terminal time 0<t<+∞, there exists a unique fixed point (ψ_p,η_p) of Γ, thus a mild solution of the coupled Boltzmann equations (<ref>), in the function class below with constant a_*>0 depending on c

    ‖ψ_p‖_P_β^σ<a_*,  ‖η_p^‖_P_β^σ<a_*.



With Lemma <ref>, we only need to verify Γ is a contraction map on Ω.

We need to prove for arbitrary (ψ_p,η_p)∈Ω and (ψ_p^*,η_p^*)∈Ω, there is

    ‖[ψ_p,η_p]- [ψ_p^*,η_p^*]‖_P_β^σ≤1/4(‖ψ_p-ψ_p^*‖_P_β^σ+‖ (η_p-η_p^*)^‖_P_β^σ),
       ‖([ψ_p,η_p]- [ψ_p^*,η_p^*])^‖_P_β^σ≤1/4(‖ψ_p-ψ_p^*‖_P_β^σ+‖ (η_p-η_p^*)^‖_P_β^σ).


Since the two pair of functions share the same initial and terminal data, the difference between [ψ_p,η_p] and [ψ_p^*,η_p^*] would be

    [ψ_p,η_p]- [ψ_p^*,η_p^*]=   ∫_0^se^(s-τ)(𝒬_η_p(ψ_p,ψ_p)-𝒬_η_p^*(ψ_p^*,ψ_p^*))dτ
    
    +   ∫_0^se^(s-τ)(𝒬_η_p(ψ_p,𝒢)-𝒬_η_p^*(ψ_p^*,𝒢))dτ
    
    +   ∫_0^se^(s-τ)(𝒬_𝒢(ψ_p,ψ_p)-𝒬_𝒢(ψ_p^*,ψ_p^*))dτ.

Here we have ignored the variable τ for simplicity in notation. Now we consider the three terms in (<ref>) separately. For the first line in the RHS of (<ref>)

    𝒬_η_p(ψ_p,ψ_p)-𝒬_η_p^*(ψ_p^*,ψ_p^*)   =𝒬_η_p-η_p^*(ψ_p,ψ_p)+(𝒬_η_p^*(ψ_p,ψ_p)-𝒬_η_p^*(ψ_p^*,ψ_p^*))
       =𝒬_η_p-η_p^*(ψ_p,ψ_p)+𝒬_η_p^*(ψ_p+ψ_p^*,ψ_p-ψ_p^*).

For the second line in the RHS of (<ref>)

    𝒬_𝒢(ψ_p,ψ_p)-𝒬_𝒢(ψ_p^*,ψ_p^*)=𝒬_𝒢(ψ_p+ψ_p^*,ψ_p-ψ_p^*),

For the third line in the RHS of (<ref>)

    𝒬_η_p(ψ_p,𝒢)-𝒬_η_p^*(ψ_p^*,𝒢)=𝒬_η_p-η_p^*(ψ_p,𝒢)+𝒬_η_p^*(ψ_p-ψ_p^*,𝒢).

According to the equations above as well as the definition <ref> of Ψ^+, we have

    [ψ_p,η_p]-[ψ_p^*,η_p^*]   =Ψ^+[η_p-η_p^*,ψ_p,ψ_p]+Ψ^+[η_p,ψ_p+ψ_p^*,ψ_p-ψ_p^*]
       +Ψ^+[𝒢,ψ_p+ψ_p^*,ψ_p-ψ_p^*]
       +Ψ^+[η_p-η_p^*,ψ_p,𝒢]+Ψ^+[η_p^*,ψ_p-ψ_p^*,𝒢].

To consider the P_β^σ norm of this difference, we first use the triangle inequality for the P_β^σ norm, and then use Lemmas <ref>, <ref>, and <ref>. These would imply

    ‖[ψ_p,η_p]-[ψ_p^*,η_p^*]‖_P_β^σ
    ≤    C(‖ (η_p-η_p^*)^‖_P_β^σ ‖ψ_p‖_P_β^σ^2+‖ (η_p^*)^‖_P_β^σ ‖ψ_p+ψ_p^*‖_P_β^σ ‖ψ_p-ψ_p^*‖_P_β^σ
    
    +   ‖ψ_p+ψ_p^*‖_P_β^σ ‖ψ_p-ψ_p^*‖_P_β^σ
    
    +   ‖ (η_p-η_p^*)^‖_P_β^σ ‖ψ_p‖_P_β^σ
    +‖ (η_p^*)^‖_P_β^σ ‖ψ_p-ψ_p^*‖_P_β^σ).

Notice that each term in the RHS of (<ref>) consists of either ‖ψ_p-ψ_p^*‖_P_β^σ or ‖ (η_p-η_p^*)^‖_P_β^σ. Since the constant c in assumptions (<ref>)-(<ref>) is small enough, the constant a_* constructed in Lemma <ref> could also be small enough. This means the various terms ‖ψ_p‖_P_β^σ,‖η_p^‖_P_β^σ,‖ψ_p^*‖_P_β^σ,‖η_p^*,‖_P_β^σ are also small enough. Consequently

    ‖[ψ_p,η_p]-[ψ_p^*,η_p^*]‖_P_β^σ≤1/4(‖ψ_p-ψ_p^*‖_P_β^σ+‖ (η_p-η_p^*)^‖_P_β^σ).

Through exactly the same argument as above, we can show for c small enough and thus a_* small enough, there is

    ‖([ψ_p,η_p]-[ψ_p^*,η_p^*])^‖_P_β^σ≤1/4(‖ψ_p-ψ_p^*‖_P_β^σ+‖ (η_p-η_p^*)^‖_P_β^σ).

This shows Γ is a contraction map from Ω to Ω, which concludes the proof.




 §.§ Fixed Point for Theorem <ref>


For any parameters β>4,σ>0 and assuming (<ref>), we have the following estimates of the fixed point map Γ. If ψ_p∈ E_β^0 and η_p^∈ E_β^-σ, then for  there is

    ‖[ψ_p,η_p]‖_E_β^0
    ≤    C(‖ψ_p(0)‖_L_β^∞+‖ψ_p‖_E_β^0^2+_ϕ‖ψ_p‖_E_β^0)+Ce^σ t(‖η_p^‖_E_β^-σ‖ψ_p‖_E_β^0^2+‖η_p^‖_E_β^-σ‖ψ_p‖_E_β^0),

and for  there is

    ‖([ψ_p,η_p])^‖_E_β^-σ
    ≤    C(‖η_p(t)‖_L_β^∞+_ϕ‖η_p^‖_E_β^-σ+‖η_p^‖_E_β^-σ‖ψ_p‖_E_β^0)+Ce^σ t(‖η_p^‖_E_β^-σ^2‖ψ_p ‖_E_β^0+‖η_p^‖_E_β^-σ^2).

Here _ϕ is the small positive constant introduced in assumption (<ref>).


The proof of this lemma is very similar to the proof of Lemma <ref>.

The Forward Component : First according to the decomposition (<ref>) of [ψ_p,η_p] and the triangle inequality for E_β^0 norm, we get

    ‖[ψ_p,η_p]‖_E_β^0
    ≤   ‖ e^sψ_p(0)‖_E_β^0+‖Ψ^+[η_p,ψ_p,ψ_p]‖_E_β^0+‖Ψ^+[η_p,ψ_p,𝒢]‖_E_β^0+‖Ψ^+[𝒢,ψ_p,ψ_p]‖_E_β^0
    
    +   ‖∫_0^se^(s-τ)B^+ψ_p(τ)ϕ(τ)dτ‖_E_β^0.

Here we have the extra term involving ϕ since the assumption does not require the function to be constant 0.

To control the term involving ϕ, we write

    ‖∫_0^se^(s-τ)B^+ψ_p(τ)ϕ(τ)dτ‖_E_β^0≤sup_0≤ s≤ t(C∫_0^s‖ψ_p(τ)ϕ(τ)‖_L_β^∞dτ)   ≤ C∫_0^t‖ψ_p(τ)ϕ(τ)‖_L_β^∞dτ
       ≤ C‖ψ_p‖_E_β^0∫_0^t‖ϕ(τ)‖_L^∞dτ
       ≤ C_ϕ‖ψ_p‖_E_β^0.

Then we use Lemma <ref> to control the E_β^0 norms of various Ψ^+ terms. This yields the desired estimate of .

The backward component : the proof for the backward component  is quite similar. First we control the term in  involving ϕ

    ‖(∫_s^te^(τ-s)B^+η_p(τ)ϕ(τ)dτ)^‖_E_β^-σ   ≤sup_0≤ s≤ te^-(t-s)σC∫_s^t ‖η_p(τ)‖_L_β^∞‖ϕ‖_L^∞dτ
       ≤ e^-σ t_ϕsup_0≤ s≤ te^sσC∫_s^t e^(t-τ)σ‖η_p^‖_E_β^-σdτ
       ≤ Ce^-σ t_ϕsup_0≤ s≤ te^sσe^(t-s)σ‖η_p^‖_E_β^-σ≤ C_ϕ‖η_p^‖_E_β^-σ.

Next by Lemma <ref>, we can control the E_β^-σ norms of various Ψ^- terms in . This concludes the estimate of  as well as the proof of the lemma.



To use the contraction principle to find the fixed-point, we will work in the function class Ω where a_* is a positive constant

    Ω:={(ψ_p,η_p)|‖ψ_p‖_E_β^0≤ a_*, e^σ t‖η_p^‖_E_β^-σ≤ a_*, ψ_p(0)=(f^0-M)ℬ^-1, η_p(t)=(e^g(t)-ℰ)ℬ}.

It is equipped with the norm

    ‖ (ψ_p,η_p)‖_Ω:=‖ψ_p‖_E_β^0+e^σ t‖η_p^‖_E_β^-σ

In terms of the estimate of norms, the two components ψ_p and η_t are no longer symmetric w.r.t. the time-reversal. A factor e^σ t is needed before ‖η_p^‖_E_β^-σ in the definition of ‖·‖_Ω.

For the constants c and _ϕ in assumption (<ref>)-(<ref>) being small enough, there exists a constant a_*>0 such that the fixed-point map Γ=(,) maps the region Ω to itself.


According to Lemma <ref>, there is

    ‖[ψ_p,η_p]‖_E_β^0≤ C(c+a_*^2+_ϕa_*)+Ce^σ t(e^-σ ta_*^3+e^-σ ta_*^2)=C(c+a_*^2+a_*^3+(_ϕ+1)a_*),

which is the same case as in the proof of Lemma <ref> if _ϕ is small enough. Thus the forward component  can be estimated in a similar way. 

For the backward component , using Lemma <ref> we have

    ‖ ([ψ_p,η_p])^‖_E_β^-σ   ≤ C(e^-σ tc+_ϕe^-σ ta_*+e^-σ ta_*^2)+Ce^σ t((e^-σ ta_*)^2a_*+(e^-σ ta_*)^2)
       ≤ Ce^-σ t(c+a_*^2+a_*^3+(_ϕ+1)a_*),

or equivalently

    e^σ t‖ ([ψ_p,η_p])^‖_E_β^-σ≤ C(c+a_*^2+a_*^3+(_ϕ+1)a_*).

Consequently the estimate of e^σ t‖ ([ψ_p,η_p])^‖_E_β^-σ is the same as that of ‖[ψ_p,η_p]‖_E_β^0. This concludes the proof.


For any parameters β>4,σ>0, suppose the functions f^0 and g satisfy the assumptions (<ref>)-(<ref>). Then for arbitrary terminal time 0<t<+∞, there exists a unique fixed point (ψ_p,η_p) of Γ, thus a mild solution of the coupled Boltzmann equations (<ref>), in the function class below with constant a_*>0 depending on c

    ‖ψ_p‖_E_β^0<a_*,   e^σ t‖η_p^‖_E_β^-σ<a_*.



With Lemma <ref>, we only need to verify Γ is a contraction map on Ω.

The method is to prove for arbitrary (ψ_p,η_p)∈Ω and (ψ_p^*,η_p^*)∈Ω that the following holds

    ‖[ψ_p,η_p]- [ψ_p^*,η_p^*]‖_E_β^0≤1/4(‖ψ_p-ψ_p^*‖_E_β^0+e^σ t‖ (η_p-η_p^*)^‖_E_β^-σ),
       e^σ t‖([ψ_p,η_p]- [ψ_p^*,η_p^*])^‖_E_β^-σ≤1/4(‖ψ_p-ψ_p^*‖_E_β^0+e^σ t‖ (η_p-η_p^*)^‖_E_β^-σ).


The Forward Component : Thus the difference between [ψ_p,η_p] and [ψ_p^*,η_p^*] is 

    [ψ_p,η_p]-[ψ_p^*,η_p^*]   =Ψ^+[η_p-η_p^*,ψ_p,ψ_p]+Ψ^+[η_p,ψ_p+ψ_p^*,ψ_p-ψ_p^*]
       +Ψ^+[𝒢,ψ_p+ψ_p^*,ψ_p-ψ_p^*]
       +Ψ^+[η_p-η_p^*,ψ_p,𝒢]+Ψ^+[η_p^*,ψ_p-ψ_p^*,𝒢]
       +∫_0^se^(s-τ)B^+(ψ_p-ψ_p^*)ϕ dτ.

Modifying equation (<ref>), we obtain

    ‖∫_0^s e^(s-τ)B^+(ψ_p-ψ_p^*)ϕ dτ‖_E_β^0≤ C_ϕ‖ψ_p-ψ_p^*‖_E_β^0.

For the first three lines on the RHS of (<ref>), their E_β^0 norms can be controlled using Lemma <ref>. This implies

    ‖[ψ_p,η_p]   -[ψ_p^*,η_p^*]‖_E_β^0
    ≤    C(e^σ t‖ (η_p-η_p^*)^‖_E_β^-σ ‖ψ_p‖_E_β^0^2+e^σ t‖η_p^*,‖_E_β^-σ ‖ψ_p+ψ_p^*‖_E_β^0 ‖ (ψ_p-ψ_p^*)^‖_E_β^0
    
    +   ‖ψ_p+ψ_p^*‖_E_β^0 ‖ψ_p-ψ_p^*‖_E_β^0
    
    +   e^σ t‖ (η_p-η_p^*)^‖_E_β^-σ ‖ψ_p‖_E_β^0
    +e^σ t‖η_p^*,‖_E_β^-σ ‖ψ_p-ψ_p^*‖_E_β^0+_ϕ‖ψ_p-ψ_p^*‖_E_β^0),

which verifies the first equation in (<ref>) if we take a_*>0 small enough, similar to what is done in the proof of Theorem <ref>.

The Backward Component : Similar to the estimate of , the difference between [ψ_p,η_p] and [ψ_p^*,η_p^*] is 

    [ψ_p,η_p]-[ψ_p^*,η_p^*]   =Ψ^-[ψ_p-ψ_p^*,η_p,η_p]+Ψ^-[ψ_p,η_p+η_p^*,η_p-η_p^*]
       +Ψ^-[𝒢,η_p+η_p^*,η_p-η_p^*]
       +Ψ^-[ψ_p-ψ_p^*,η_p,𝒢]+Ψ^-[ψ_p^*,η_p-η_p^*,𝒢]
       +∫_s^te^(τ-s)B^-(η_p-η_p^*)ϕ dτ.

For the E_β^-σ norm for the fourth line of (<ref>), modifying equation (<ref>) we obtain

    ‖∫_s^t e^(τ-s)B^-(η_p-η_p^*)ϕ dτ‖_E_β^-σ≤ C_ϕ‖η_p-η_p^*‖_E_β^-σ.

For the first three lines on the RHS of (<ref>), their E_β^-σ norms can be controlled using Lemma <ref>. This implies

    ‖([ψ_p,η_p]-[ψ_p^*,η_p^*])^‖_E_β^-σ
    ≤    C(e^σ t‖ψ_p-ψ_p^* ‖_E_β^0 ‖η_p^‖_E_β^-σ^2+e^σ t‖ψ_p^*‖_E_β^0 ‖ (η_p+η_p^*)^‖_E_β^-σ ‖ (η_p-η_p^*)^‖_E_β^-σ
    
    +   e^σ t‖ (η_p+η_p^*)^‖_E_β^-σ ‖ (η_p-η_p^*)^‖_E_β^-σ
    
    +   ‖ψ_p-ψ_p^*‖_E_β^0 ‖η_p^‖_E_β^-σ
    +‖ψ_p^*‖_E_β^0 ‖ (η_p-η_p^*)^‖_E_β^-σ+_ϕ‖(η_p-η_p^*)^‖_E_β^-σ).

This inequality can be reorganized by multiplying an e^σ t over the two sides

    e^σ t‖([ψ_p,η_p]-[ψ_p^*,η_p^*])^‖_E_β^-σ
    ≤    C(‖ψ_p-ψ_p^* ‖_E_β^0 (e^σ t‖η_p^‖_E_β^-σ)^2+‖ψ_p^*‖_E_β^0 e^σ t‖ (η_p+η_p^*)^‖_E_β^-σ e^σ t‖ (η_p-η_p^*)^‖_E_β^-σ
    
    +   e^σ t‖ (η_p+η_p^*)^‖_E_β^-σ e^σ t‖ (η_p-η_p^*)^‖_E_β^-σ
    
    +   ‖ψ_p-ψ_p^*‖_E_β^0 e^σ t‖η_p^‖_E_β^-σ
    +‖ψ_p^*‖_E_β^0 e^σ t‖ (η_p-η_p^*)^‖_E_β^-σ+_ϕe^σ t‖(η_p-η_p^*)^‖_E_β^-σ),

which verifies the second inequality in (<ref>) if we take a_* small enough. As a consequence Γ is a contraction from Ω to Ω, having a unique fixed point. This concludes the proof.








§ JUSTIFICATION OF THE MILD FUNCTIONAL SOLUTION

In this section, we show in Theorem <ref> that we can construct mild solutions of the Hamilton-Jacobi equation, using the mild solution of the coupled Boltzmann equations. The notion of a mild solution for the Hamilton-Jacobi equation (<ref>) has also been defined in (<ref>). Theorem <ref> has been proved in <cit.> under some analyticity assumptions, in the framework of the Cauchy-Kovalevskaya Theorem. In this paper we do not have these analyticity conditions, and our proof is a modification of the proof in <cit.>, with some additional analysis.

To construct a mild solution of the functional Hamilton-Jacobi equation (<ref>), we will consider the Hamiltonian system characterized by the associated Euler-Lagrange equation. Specifically given a terminal time t> 0, we consider the following Hamiltonian system defined on [0,t]

    D_sφ_t=ℋ/ p(φ_t,p_t),  
       D_s(p_t-g)=-ℋ/φ(φ_t,p_t),  .

Here the Hamiltonian ℋ has been defined in (<ref>) as

    ℋ(φ,p)=1/2∫φ(x,v)φ(x,v_*)(e^Δ p (x,v,v_*)-1)((v_*-v)·ω)_+dω dv_* dvdx.

Given a mild solution of equation (<ref>) on [0,t], we define the functional ℐ(t,g) as

    ℐ(t,g):=-1+⟨ f^0,e^p_t(0)⟩+ D_s(p_t(s)-g(s)),φ_t(s)+∫_0^tℋ(φ_t(s),p_t(s))ds.

Here the notation ⟨·,·⟩ refers to the inner product in L^2(T_x^d×R_v^d), while ·,· refers to the inner product in L^2([0,t]×T_x^d×R_v^d), with given terminal time t.

However as the readers have seen in this paper, we do not directly deal with the Euler-Lagrange system (<ref>). Instead, we have performed the change of variables in (<ref>) to make the system more symmetric

    (ψ_t,η_t):=(φ_te^-p_t+α'|v|^2,e^p_t-α'|v|^2).

This change of variables provides a new Hamiltonian ℋ' from the original ℋ

    ℋ'(ψ,η):=-1/4∫(ψ(v')ψ(v_*')-ψ(v)ψ(v_*))(η(v')η(v_*')-η(v)η(v_*))((v_*-v)·ω)_+dω dv_*dvdx.

The Hamiltonian ℋ'(ψ,η) can also be written equivalently as

    ℋ'(ψ,η)   =1/2∫ψ(v)ψ(v_*)(η(v')η(v_*')-η(v)η(v_*))((v_*-v)·ω)_+dω dv_*dvdx
       =1/2∫(ψ(v')ψ(v_*')-ψ(v)ψ(v_*))η(v)η(v_*)((v_*-v)·ω)_+dω dv_*dvdx.

Replacing (φ_t,η_t) by (ψ_t,η_t) by the change of variables (<ref>), we have the evolution equation for (ψ_t,η_t) during the time interval [0,t]

    D_s ψ_t   =-ψ_tϕ+𝒬_η_t(ψ_t,ψ_t),  ψ_t(0)=f^0ℬ^-1
    
    D_s η_t   =η_tϕ-𝒬_ψ_t(η_t,η_t),  η_t(t)=e^g(t)ℬ.

This equation for (ψ_t,η_t) can also be written as

    D_s ψ_t(s)   =-ψ_t(s)ϕ(s)+ℋ'(ψ_t(s),η_t(s))/η,  ψ_t(0)=f^0ℬ^-1,
    
    D_s η_t(s)   =η_t(s)ϕ(s)-ℋ'(ψ_t(s),η_t(s))/ψ,  η_t(t)=e^g(t)ℬ.

In this case, the functional ℐ(t,g) constructed in equation (<ref>) is equivalent to 

    ℐ(t,g):=-1+⟨ f^0ℬ^-1,η_t(0) ⟩+ D_sη_t(s),ψ_t(s)-ϕ(s),ψ_t(s)η_t(s)+∫_0^tℋ'(ψ_t(s),η_t(s))ds.

Theorem <ref> shows that if the (ψ_t,η_t) in the definition of ℐ(t,g) is the mild solution given in Theorem <ref> or <ref>, then the functional ℐ(t,g) is a mild solution of the functional Hamilton-Jacobi equation.

Some ingredients are needed to prove Theorem <ref>, for example some continuity estimates. In Lemma <ref>, we will prove that the solution (ψ_t(s),η_t(s)) of the coupled Boltzmann equations, is continuous in 0≤ s≤ t under the L_β^∞ norm. This enables us to give a precise definition of D_sψ_t(s) and D_sη_t(s). Using Lemma <ref>, we are going to define D_sψ_t(s) and D_sη_t(s) as the following limits in L_x,v^2

    D_sψ_t(s)=lim_τ→ 0(ψ_t(s+τ)-ψ_t(s)/τ-S_τψ_t(s)-ψ_t(s)/τ),
       D_sη_t(s)=lim_τ→ 0(η_t(s+τ)-η_t(s)/τ-S_τη_t(s)-η_t(s)/τ).

Using the mild formulation of (ψ_t(s),η_t(s)), for each  the difference is equal to a time integral. For example according to the mild formulation of the coupled Boltzmann equations

    ψ_t(s+τ)=S_τψ_t(s)-∫_s^s+τS_u-sψ_t(u)ϕ(u)du+∫_s^s+τS_u-sℋ'(ψ_t(u),η_t(u))/ηdu.

This implies

    lim_τ→ 0(ψ_t(s+τ)-ψ_t(s)/τ   -S_τψ_t(s)-ψ_t(s)/τ)
    
    =   lim_τ→ 0(-τ^-1∫_s^s+τS_u-sψ_t(u)ϕ(u)du+τ^-1∫_s^s+τS_u-sℋ'(ψ_t(u),η_t(u))/ηdu).

Thus by the continuity of (ψ_t(s),η_t(s)) under L_β^∞ proved in Lemma <ref>, the integral is continuous with respect to τ and the limits in (<ref>) exist

    D_s ψ_t(s)=-ψ_t(s)ϕ(s)+ℋ'(ψ_t(s),η_t(s))/η, D_s η_t(s)=η_t(s)ϕ(s)-ℋ'(ψ_t(s),η_t(s))/ψ.

Now the definition of ℐ(t,g) is rigorous. 

 
If we take (ψ_t(s),η_t(s)) to be the mild solution of the coupled Boltzmann equations given in Theorem <ref> or <ref>, then the functional ℐ(t,g) defined in equation (<ref>) is a mild solution of the Hamilton-Jacobi equation (<ref>).


Take an arbitrary time t. We want to consider the difference ℐ(t+τ,g)-ℐ(t,g) with τ being a small positive number. We use δ_τ to refer to the variation with respect to the terminal time

    δ_τψ_t(s)=ψ_t+τ(s)-ψ_t(s),  δ_τη_t(s)=η_t+τ(s)-η_t(s).


Time differential of ℐ:
according to the definition (<ref>) of ℐ, there is

    ℐ(t+τ)-ℐ(t,g)
    
    =   ⟨ f^0,δ_τη_t(0)⟩+ D_sη_t,δ_τψ_t+ D_sδ_τη_t,ψ_t+ D_sδ_τη_t,δ_τψ_t_+∫_t^t+τ⟨ D_sη_t+τ(s),ψ_t+τ(s)⟩ ds
       -ϕ,δ_τψ_tη_t-ϕ,ψ_tδ_τη_t-ϕ,δ_τψ_tδ_τη_t_-∫_t^t+τ⟨ϕ(s),ψ_t+τ(s)η_t+τ(s)⟩ ds
       +δ_τψ_t,ℋ'(ψ_t,η_t)/ψ+δ_τη_t,ℋ'(ψ_t,η_t)/η+∫_t^t+τℋ'(ψ_t+τ(s),η_t+τ(s))ds
       +[∫_0^tℋ'(ψ_t+τ(s),η_t+τ(s))-∫_0^tℋ'(ψ_t(s),η_t(s))ds-δ_τψ_t,ℋ'(ψ_t,η_t)/ψ-δ_τη_t,ℋ'(ψ_t,η_t)/η]_

In the equation above, the first line on the RHS is the variation of ⟨ D_sη_t,ψ_t ⟩ with respect to t, and the second line is the variation of ⟨ϕ,ψ_tη_t⟩ with respect to t. The third and the fourth lines are the variation of ∫_0^t ℋ'(ψ_t(s),η_t(s))ds.

It will be proved in Lemma <ref> that for arbitrary 0≤ s≤ t, the variation is uniformly of order τ

    ‖δ_τψ_t(s)‖_L_β^∞=O(τ), ‖δ_τη_t(s)‖_L_β^∞=O(τ),

since each component of (ψ_t(s),η_t(s)) is continuous in 0≤ s≤ t under the L_β^∞ norm. Combine (<ref>) with (<ref>), we can show those higher-order remainders in (<ref>) are of order o(τ).

Performing integration by parts according to Lemma <ref>, we have

    D_sδ_τη_t,ψ_t=-δ_τη_t,D_sψ_t+⟨δ_τη_t(t),ψ_t(t)⟩-⟨δ_τη_t(0),ψ_t(0)⟩.

Combine (<ref>) with (<ref>) and Lemma <ref>, the difference ℐ(t+τ,g)-ℐ(t,g) becomes

    ℐ(t+τ,g)-ℐ(t,g)
    
    =   δ_τη_t,-D_s ψ_t(s)-ψ_t(s)ϕ(s)+ℋ'(ψ_t(s),η_t(s))/η
       +δ_τψ_t,D_s η_t(s)-η_t(s)ϕ(s)+ℋ'(ψ_t(s),η_t(s))/ψ+∫_t^t+τℋ'(ψ_t+τ(s),η_t+τ(s))ds+o(τ).

Using the fact that (ψ_t,η_t) is also the mild solution of (<ref>) and equation (<ref>), there is

    ℐ(t+τ,g)-ℐ(t,g)=  ∫_t^t+τℋ'(ψ_t+τ(s),η_t+τ(s))ds+o(τ).

Since each component of (ψ_t(s),η_t(s)) is continuous in 0≤ s≤ t under the L_β^∞ norm, the Hamiltonian ℋ'(ψ_t(s),η_t(s)) is also continuous in 0≤ s≤ t. Consequently the functional ℐ(t,g) is differentiable in time

    _tℐ(t,g)=ℋ'(ψ_t(t),η_t(t))=ℋ(φ_t(t),p_t(t)).

Differential of ℐ with respect to g: 
Now we want to fix t and differentiate ℐ(t,g) against g(t). The rigorous proof is essentially the same as the proof of the differentiability in t. It is because if we have a variation of g, the terminal data η_t(t)=e^g(t)ℬ will be changed accordingly. This is the same case as the change of terminal data when we are considering the differentiability in t. Here we give the formal proof for simplicity. Using δ to denote the variation, we differentiate ℐ(t,g) against g(t)

    ℐ(t,g)=   ⟨ f^0ℬ^-1,δη_t(0)⟩+ D_sδη_t,ψ_t+ D_sη_t,δψ_t-ϕ,δψ_tη_t-ϕ,ψ_tδη_t
       +δψ_t,ℋ'(ψ_t,η_t)/ψ+δη_t,ℋ'(ψ_t,η_t)/η.

Again we perform an integration by parts

    D_sδη_t,ψ_t=-δη_t,D_sψ_t+⟨δη_t(t),ψ_t(t)⟩-⟨δη_t(0),ψ_t(0)⟩.

Using the same technique as in the study of _tℐ, with (ψ_t,η_t) being the mild solution of (<ref>), we get

    ℐ(t,g)=⟨δ e^g(t)ℬ,ψ_t(t)⟩-δϕ,ψ_tη_t   =⟨δ g(t) ,η_t(t)ψ_t(t)⟩-δϕ,ψ_tη_t
       =⟨δ g(t) ,φ_t(t)⟩-δϕ,φ_t(t).

This shows that ℐ(t,g)/ g(t)=φ_t(t). Along with p_t(t)=g(t), (<ref>) is equivalently

    _tℐ(t,g)=ℋ(ℐ(t,g)/ g(t),g(t)).

This concludes the proof.




 Consider the mild solution (ψ_t(s),η_t(s)) derived in Theorem <ref> or Theorem <ref> of the coupled Boltzmann equations (<ref>). Each component is continuous w.r.t. s and t in the region 0≤ s≤ t under the L_β^∞ norm.


We only detail the proof for (ψ_t(s),η_t(s)) in Theorem <ref>. The proof for (ψ_t(s),η_t(s)) in Theorem <ref> is essentially the same. In the proof, we will use C(g) as a positive constant depending on g, C(f^0) as a positive constant depending on f^0, and C(f^0,g) as a positive constant dependent on f^0 and g.

Continuity in s: suppose we want to consider the difference between (ψ_t,p(s),η_t,p(s)) and (ψ_t,p(s+τ),η_t,p(s+τ)) for τ>0. We use the notation 

    (ψ_t,p^τ(s),η_t,p^τ(s)):=(ψ_t,p(s+τ),η_t,p(s+τ)),   0≤ s≤ t-τ.

The fixed-point (ψ_t,p,η_t,p) is derived as the limit of (ψ_t,p^(n),η_t,p^(n))_n≥ 0 under the iteration n→ +∞

    ψ_t,p^(0):=e^s(f^0ℬ^-1-𝒢), η_t,p^(0):=e^(t-s)(e^g(t)ℬ-𝒢),
       ψ_t,p^(n+1)=_t[ψ_t,p^(n),η_t,p^(n)],   η_t,p^(n+1)=_t[ψ_t,p^(n),η_t,p^(n)],   n≥ 0.

It also satisfies the following iteration relation

    ψ_t,p^(n+1)(s)=e^sB^+(f^0ℬ^-1-𝒢)+∫_0^s e^(s-u)B^+𝒩[ψ_t,p^(n),η_t,p^(n)]du,
       η_t,p^(n+1)(s)=e^(t-τ-s)B^-e^τ B^-(e^g(t)ℬ-𝒢)+∫_s^t-τ e^(u-s)B^-𝒩[η_t,p^(n),ψ_t,p^(n)]du+∫_t-τ^te^(u-s)B^-𝒩[η_t,p^(n),ψ_t,p^(n)]du_.

The fixed-point (ψ_t,p^τ,η_t,p^τ) can be derived as the limit of (ψ_t,p^τ,(n),η_t,p^τ,(n))_n≥ 0 

    (ψ_t,p^τ,(n)(s),η_t,p^τ,(n)(s)):=(ψ_t,p^(n)(s+τ),η_t,p^(n)(s+τ)),   0≤ s≤ t-τ.

It satisfies the iteration relation

    ψ_t,p^τ,(n+1)(s)=e^sB^+e^τ B^+(f^0ℬ^-1-𝒢)+∫_0^s e^(s-u)B^+𝒩[ψ_t,p^τ,(n),η_t,p^τ,(n)]du+∫_0^τ e^(s-u)B^+𝒩[ψ_t,p^(n),η_t,p^(n)]du_,
       η_t,p^τ,(n+1)(s)=e^(t-τ-s)B^-(e^g(t)ℬ-𝒢)+∫_s^t-τ e^(u-s)B^-𝒩[η_t-τ,p^*,(n),ψ_t-τ,p^*,(n)]du.

For now we use P_β^σ as the norm on the time interval [0,t-τ]. For the forward component, there is 

    ‖ψ_t,p^τ,(n+1)-ψ_t,p^(n+1)‖_P_β^σ
    ≤   ‖∫_0^se^(s-u)𝒩[ψ_t,p^τ,(n),η_t,p^τ,(n)]du-∫_0^se^(s-u)𝒩[ψ_t,p^(n),η_t,p^(n)]du‖_P_β^σ+C(f^0)(1+t)^στ
    ≤   1/4(‖ψ_t,p^(n)-ψ_t,p^τ,(n)‖_P_β^σ+‖(η_t,p^(n)-η_t,p^τ,(n))^‖_P_β^σ)+C(f^0)(1+t)^στ.

Here the term C(f^0)(1+t)^στ is due to assumption (<ref>) and Lemma <ref>

    ‖ e^sB^+e^τ B^+(f^0ℬ^-1-𝒢)-e^sB^+(f^0ℬ^-1-𝒢)‖_L_β^∞   ≤ C‖ e^τ B^+(f^0ℬ^-1-𝒢)-(f^0ℬ^-1-𝒢)‖_L_β^∞
       ≤ C(f^0)(1+t)^στ.

Similarly for the backward component, we get

    ‖(η_t,p^τ,(n+1)-η_t,p^(n+1))^‖_P_β^σ≤1/4(‖ψ_t,p^(n)-ψ_t,p^τ,(n)‖_P_β^σ+‖(η_t,p^(n)-η_t,p^τ,(n))^‖_P_β^σ)+C(g)(1+t)^στ.

Here the term C(g)(1+t)^στ is due to assumption (<ref>) and Lemma <ref>, in a way similar to (<ref>).

Equations (<ref>) and (<ref>) together imply the contraction relation with a C(f^0,g)(1+t)^στ error

    ‖ψ_t,p^(n+1)-ψ_t,p^τ,(n+1)‖_P_β^σ+‖(η_t,p^(n+1)-η_t,p^τ,(n+1))^‖_P_β^σ
    ≤   1/2(‖ψ_t,p^(n)-ψ_t,p^τ,(n)‖_P_β^σ+‖(η_t,p^(n)-η_t,p^τ,(n))^‖_P_β^σ)+C(f^0,g)(1+t)^στ.

Taking n→ +∞, we have shown (ψ_t(s),η_t(s)) is right continuous in s

    ‖ψ_t,p-ψ_t,p^τ‖_P_β^σ+‖(η_t,p-η_t,p^τ)^‖_P_β^σ≤ C(f^0,g)(1+t)^στ→ 0,  τ→ 0.

The same analysis also holds true if we take τ<0, which proves (ψ_t(s),η_t(s)) is left continuous in s. This concludes the proof of the continuity in s.

Continuity in t: to prove the lemma, we first notice

    δ_τψ_t(s)=δ_τψ_t,p(s)=ψ_t+τ,p(s)-ψ_t,p(s),  δ_τη_t(s)=δ_τη_t,p(s)=η_t+τ,p(s)-η_t,p(s).

Here the (ψ_t,p,η_t,p) is the fixed point of the map (<ref>) with terminal time t. Recalling Theorem <ref>, we assume ϕ≡ 0

    {   _t[ψ_p,η_p](s):=e^s(f^0ℬ^-1-𝒢)+∫_0^se^(s-u)𝒩[ψ_p,η_p]du,
       _t[ψ_p,η_p](s):=e^(t-s)(e^g(t)ℬ-𝒢)+∫_s^t e^(u-s)𝒩[η_p,ψ_p]du,
    .

while (ψ_t+τ,p,η_t+τ,p) is the fixed point of the map (<ref>) with terminal time t+τ

    {   _t+τ[ψ_p,η_p](s):=e^s(f^0ℬ^-1-𝒢)+∫_0^se^(s-u)𝒩[ψ_p,η_p]du,
       _t+τ[ψ_p,η_p](s):=e^(t+τ-s)(e^g(t+τ)ℬ-𝒢)+∫_s^t+τ e^(u-s)𝒩[η_p,ψ_p]du.
    .

The fixed-point (ψ_t,p,η_t,p) is derived as the limit of (ψ_t,p^(n),η_t,p^(n))_n≥ 0 under the iteration n→ +∞

    ψ_t,p^(0):=e^s(f^0ℬ^-1-𝒢), η_t,p^(0):=e^(t-s)(e^g(t)ℬ-𝒢),
       ψ_t,p^(n+1)=_t[ψ_t,p^(n),η_t,p^(n)],   η_t,p^(n+1)=_t[ψ_t,p^(n),η_t,p^(n)],   n≥ 0.

The fixed-point (ψ_t+τ,p,η_t+τ,p) is derived as the limit of (ψ_t+τ,p^(n),η_t+τ,p^(n)) under the iteration n→ +∞

    ψ_t+τ,p^(0):=e^s(f^0ℬ^-1-𝒢), η_t+τ,p^(0):=e^(t+τ-s)(e^g(t+τ)ℬ-𝒢),
       ψ_t+τ,p^(n+1)=_t+τ[ψ_t+τ,p^(n),η_t+τ,p^(n)],   η_t+τ,p^(n+1)=_t+τ[ψ_t+τ,p^(n),η_t+τ,p^(n)],   n≥ 0.

Regarding the involved functions and the norm P_β^σ as defined on the time interval [0,t], we want to prove

    ‖ψ_t,p^(n+1)-ψ_t+τ,p^(n+1)‖_P_β^σ+‖(η_t,p^(n+1)-η_t+τ,p^(n+1))^‖_P_β^σ≤ O(τ)+1/2(‖ψ_t,p^(n)-ψ_t+τ,p^(n)‖_P_β^σ+‖(η_t,p^(n)-η_t+τ,p^(n))^‖_P_β^σ).

Once (<ref>) is proved, we will have

    ‖ψ_t,p-ψ_t+τ,p‖_P_β^σ+‖(η_t,p-η_t+τ,p)^‖_P_β^σ≤ O(τ).

Now we prove (<ref>). For the forward component, according to (<ref>) there is

    ‖ψ_t+τ,p^(n+1)-ψ_t,p^(n+1)‖_P_β^σ   =‖∫_0^se^(s-u)𝒩[ψ_t+τ,p^(n),η_t+τ,p^(n)]du-∫_0^se^(s-u)𝒩[ψ_t,p^(n),η_t,p^(n)]du‖_P_β^σ
       ≤1/2(‖ψ_t,p^(n)-ψ_t+τ,p^(n)‖_P_β^σ+‖(η_t,p^(n)-η_t+τ,p^(n))^‖_P_β^σ)

For the backward component, we have

    η_t+τ,p^(n+1)(s)-η_t,p^(n+1)(s)=   e^(t+τ-s)(e^g(t+τ)ℬ-𝒢)-e^(t-s)(e^g(t)ℬ-𝒢)
       +∫_s^t+τe^(u-s)𝒩[η_t+τ,p^(n),ψ_t+τ,p^(n)]du-∫_s^te^(u-s)𝒩[η_t,p^(n),ψ_t,p^(n)]du.

To control the difference between these different terminal data, we decompose it as

    ‖ e^(t+τ-s)(e^g(t+τ)ℬ-𝒢)-e^(t-s)(e^g(t)ℬ-𝒢)‖_L_β^∞
    ≤   ‖(e^(t+τ-s)(e^g(t+τ)ℬ-𝒢)-e^(t+τ-s)(e^g(t)ℬ-𝒢))‖_L_β^∞  
       +‖(e^(t+τ-s)(e^g(t)ℬ-𝒢)-e^(t-s)(e^g(t)ℬ-𝒢))‖_L_β^∞

The term (I) is controlled using the boundedness of e^(t+τ-s)B^- in L_β^∞ (see Lemma <ref>) and assumption (<ref>)

    ≤ C‖(e^g(t+τ)ℬ-𝒢)-(e^g(t)ℬ-𝒢)‖_L_β^∞≤ C(g)τ.
 

According to Lemma <ref> and the boundedness of e^(t-s)B^- in L_β^∞, the term (II) is controlled as

    =‖ e^(t-s)(e^τ B^-(e^g(t)ℬ-𝒢)-(e^g(t)ℬ-𝒢))‖_L_β^∞
       ≤ C‖ e^τ B^-(e^g(t)ℬ-𝒢)-(e^g(t)ℬ-𝒢)‖_L_β^∞≤ C(g)τ.

These imply

    ‖(η_t+τ,p^(n+1)-η_t,p^(n+1))^‖_P_β^σ   ≤‖(_t[ψ_t+τ,p^(n),η_t+τ,p^(n)]-_t[ψ_t,p^(n),η_t,p^(n)])^‖_P_β^σ+C(g)(1+t)^στ,
       ≤ C(g)(1+t)^στ+1/2(‖ψ_t,p^(n)-ψ_t+τ,p^(n)‖_P_β^σ+‖(η_t,p^(n)-η_t+τ,p^(n))^‖_P_β^σ).

Now equation (<ref>) is proved, which concludes the proof of the continuity in t.


We have performed two integrations by parts in the formal proof, which are (<ref>) and (<ref>). Here we will justify the first one, while the second one can be justified in the same way. 

 Consider the solution (ψ_t,η_t) derived in Theorem <ref> or Theorem <ref> of the coupled Boltzmann equations (<ref>). For arbitrary t>0 and τ>0, we have

    D_sδ_τη_t,ψ_t=-δ_τη_t,D_sψ_t+⟨δ_τη_t(t),ψ_t(t)⟩-⟨δ_τη_t(0),ψ_t(0)⟩.



Since the operator D_s is rigorously defined as the limit in (<ref>) whose convergence is uniform in s, we have

    ∫_0^t⟨ D_sδ_τη_t(s),ψ_t(s)⟩ ds
    
    =   lim_→ 0(∫_0^t ⟨δ_τη_t(s+)-δ_τη_t(s)/,ψ_t(s)⟩ ds-∫_0^t ⟨S_δ_τη_t(s)-δ_τη_t(s)/,ψ_t(s)⟩ ds).

For each >0, the first term in the RHS of (<ref>) equals to

    ∫_0^t ⟨δ_τη_t(s), ψ_t(s-)-ψ_t(s)/⟩ ds+∫_t^t+⟨δ_τη_t(s), ψ_t(s-)/⟩ ds-∫_0^⟨δ_τη_t(s), ψ_t(s-)/⟩ ds.

The second term in the RHS of (<ref>) equals to

    ∫_0^t ⟨S_δ_τη_t(s)-δ_τη_t(s)/,ψ_t(s)⟩ ds=∫_0^t ⟨δ_τη_t(s),S_-ψ_t(s)-ψ_t(s)/⟩ ds.

These equations together imply

    ∫_0^t⟨ D_sδ_τη_t(s),ψ_t(s)⟩ ds
    
    =   lim_→ 0[∫_0^t ⟨δ_τη_t(s), ψ_t(s-)-ψ_t(s)/-S_-ψ_t(s)-ψ_t(s)/⟩ ds
                    +∫_t^t+⟨δ_τη_t(s), ψ_t(s-)/⟩ ds-∫_0^⟨δ_τη_t(s), ψ_t(s-)/⟩ ds],

Each component of (ψ_t(s),η_t(s)) is continuous in 0≤ s≤ t under the L_β^∞ norm. As a consequence the limit exists and implies

    D_sδ_τη_t(s),ψ_t(s) =-δ_τη_t(s),D_sψ_t(s) +⟨δ p_t(t),φ_t(t) ⟩-⟨δ p_t(0),ψ_t(0) ⟩.

This concludes the proof of the lemma.

To conclude this section, we give the postponed proof of a technical lemma used in the proof of Theorem <ref>.

Consider the mild solution (ψ_t(s),η_t(s)) derived in Theorem <ref> or Theorem <ref> of the coupled Boltzmann equations (<ref>), the following estimate holds

    ⟨δ_τη_t(t),ψ_t(t)⟩+∫_t^t+τ⟨ D_sη_t+τ(s),ψ_t+τ(s)⟩ ds-∫_t^t+τ⟨ϕ(s),ψ_t+τ(s)η_t+τ(s)⟩ ds=o(τ)



By the equivalent expression (<ref>) of D_sη, we have

    ∫_t^t+τ⟨ D_sη_t+τ(s),ψ_t+τ(s)⟩ ds-∫_t^t+τ⟨ϕ(s),ψ_t+τ(s)η_t+τ(s)⟩ ds   =-∫_t^t+τ⟨𝒬_ψ_t+τ(η_t+τ,η_t+τ),ψ_t+τ⟩ ds.

Using the continuity (Lemma <ref>) of (ψ_t(s),η_t(s)) under the L_β^∞ norm, we further have

    lim_τ→ 0τ^-1(-∫_t^t+τ⟨𝒬_ψ_t+τ(η_t+τ,η_t+τ),ψ_t+τ⟩ ds)=-⟨𝒬_ψ_t(t)(η_t(t),η_t(t)),ψ_t(t) ⟩.


The variation δ_τη_t(t) is decomposed as

    δ_τη_t(t)   =(η_t+τ(t)-η_t+τ(t+τ))+(η_t+τ(t+τ)-η_t(t))
       =(η_t+τ(t)-η_t+τ(t+τ))+(e^g(t+τ)ℬ-e^g(t)ℬ).

Using the mild formulation of the coupled Boltzmann equations, there is

    η_t+τ(t)-η_t+τ(t+τ)=   S_-τe^g(t+τ)ℬ-e^g(t+τ)ℬ-∫_t^t+τS_-(s-t)η_t+τϕ ds
       +∫_t^t+τ S_-(s-t)𝒬_ψ_t+τ(η_t+τ,η_t+τ) ds.

Based on the continuity of (ψ_t(s),η_t(s)) under the L_β^∞ norm (Lemma <ref>), the continuity of ϕ under the L_x,v^∞ norm (assumption (<ref>) or (<ref>)), as well as the continuity of the transport semigroup in L_x,v^2, we obtain the following convergence in L_x,v^2

    lim_τ→ +∞η_t+τ(t+τ)-η_t+τ(τ)/τ
              =lim_τ→ +∞(v·∇_xe^g(t+τ)ℬ-τ^-1∫_t^t+τη_t+τϕ ds+τ^-1∫_t^t+τ𝒬_ψ_t+τ(η_t+τ,η_t+τ) ds)
             =v·∇_xe^g(t+τ)ℬ-η_t(t)ϕ(t)+𝒬_ψ_t(t)(η_t(t),η_t(t)).

Combine the equation above with (<ref>), we have

    lim_τ→ 0δ_τη_t(t)/τ=D_se^g(t)ℬ-D_sg(t)e^g(t)ℬ+𝒬_ψ_t(t)(η_t(t),η_t(t))=𝒬_ψ_t(t)(η_t(t),η_t(t)).

Equations (<ref>) and (<ref>) together conclude the proof of the lemma.




§ UNIFORM BOUNDEDNESS, LONG-TIME BEHAVIOUR, AND STATIONARY SOLUTIONS

In Theorem <ref> we have constructed a solution ℐ(t,g) of the Hamilton-Jacobi equation. In this section we will give a description of its various properties.  Recall that ⟨·,·⟩ refers to the inner product in L^2(T_x^d×R_v^d), and ·,· refers to the inner product in L^2([0,t]×T_x^d×R_v^d), with given terminal time t.  

According to Theorem <ref>, the functional ℐ(t,g) defined as

    ℐ(t,g):=⟨ f^0,η_t(0)-1 ⟩+ D_sη_t(s),ψ_t(s)-η_t(s),ψ_t(s)ϕ(s)+∫_0^tℋ'(ψ_t(s),η_t(s))ds.

is a mild solution of the Hamilton-Jacobi equation. The Hamiltonian ℋ' is defined as

    ℋ'(ψ,η):=1/2∫η(v)η(v_*)(ψ(v')ψ(v_*')-ψ(v)ψ(v_*))((v_*-v)·ω)_+dω dv_*dvdx,

Here (ψ_t(s),η_t(s)) is the mild solution of the coupled Boltzmann equations (<ref>) during the time interval [0,t]. 


 §.§ Functional Solution for Theorem <ref>


Under the assumptions (<ref>)-(<ref>), the functional solution ℐ(t,g) constructed in Theorem <ref> of the Hamilton-Jacobi equation is uniformly bounded for arbitrary time t≥ 0 and function g satisfying the assumptions (<ref>)-(<ref>).


First we perform an integration by parts

    D_sη_t(s),ψ_t(s)   =η_t(s),-D_sψ_t(s)+⟨η_t(t),ψ_t(t) ⟩-⟨η_t(0),ψ_t(0)⟩
       =η_t(s),-D_sψ_t(s)+⟨η_t(t),ψ_t(t) ⟩-⟨η_t(0),f^0⟩,

where in the second equality we have used the fact that ψ_t(0)=f^0. The integration by parts can be justified in the spirit of Lemma <ref>.

This integration by parts, along with the fact that ∫ f^0dxdv=1, implies 

    ℐ(t,g)=-1+⟨η_t(t),ψ_t(t) ⟩+η_t(s),-D_sψ_t(s) -η_t(s),ψ_t(s)ϕ_t(s) +∫_0^tℋ'(ψ_t(s),η_t(s))ds.

According to the definition of ℋ', we have

    ℋ'(ψ_t(s),η_t(s))=1/2⟨η_t(s),𝒬_η_t(s)(ψ_t(s),ψ_t(s))⟩,

thus

    ℐ(t,g)
    
    =   -1+⟨η_t(t),ψ_t(t) ⟩+η_t(s),-D_sψ_t(s) -η_t(s),ψ_t(s)ϕ_t(s) +1/2η_t(s),𝒬_η_t(s)(ψ_t(s),ψ_t(s))
    
    =   -1+⟨η_t(t),ψ_t(t) ⟩+η_t(s),-D_sψ_t(s)-ψ_t(s)ϕ(s)+𝒬_η_t(s)(ψ_t(s),ψ_t(s))
       -1/2η_t(s),𝒬_η_t(s)(ψ_t(s),ψ_t(s)).

Since (ψ_t(s),η_t(s)) is the mild solution of the coupled Boltzmann equations (<ref>), we have

    ℐ(t,g)=-1+⟨η_t(t),ψ_t(t) ⟩_-1/2η_t(s),𝒬_η_t(s)(ψ_t(s),ψ_t(s))_.

Recall the perturbations ψ_t,p and η_t,p introduced in (<ref>). The estimate of term (I) is relatively immediate with L_β^∞ being a stronger topology than L^2 due to β>4

    ∫η_t(t)ψ_t(t)dvdx≤ C ‖η_t(t)‖_L_β^∞‖ψ_t(t)‖_L_β^∞   ≤ C (‖𝒢‖_L_β^∞+‖η_p,t(t)‖_L_β^∞)(‖𝒢‖_L_β^∞+‖ψ_p,t(t)‖_L_β^∞)
       ≤ C (‖𝒢‖_L_β^∞+a_*)(‖𝒢‖_L_β^∞+a_*).

To estimate the term (II), we first perform the perturbation decomposition for η_t

    =𝒢,𝒬_η_t(s)(ψ_t(s),ψ_t(s))_+η_t,p(s),𝒬_η_t(s)(ψ_t(s),ψ_t(s))_.

To estimate the (II.1) term, we first notice that Lemma <ref> implies 𝒬_η_t(s)(ψ_t(s),ψ_t(s))∈ L_β-1^∞. Consequently the integrand in (II.1) is absolutely integrable. The fact that (ψ_t,η_t) is the mild solution of the coupled Boltzmann equations (<ref>)

    ψ_t(t)=S_tψ_t(0)-∫_0^tS_t-sψ_t(s)ϕ_t(s)ds+∫_0^tS_t-s𝒬_η_t(s)(ψ_t(s),ψ_t(s))ds.

Using the fact that 𝒢 is a function independent of time and space, we have

    ∫_0^t(∫𝒢𝒬_η_t(s)(ψ_t(s),ψ_t(s))dvdx)ds   =∫_0^t(∫𝒢S_t-s𝒬_η_t(s)(ψ_t(s),ψ_t(s))dvdx)ds
       =∫𝒢(∫_0^tS_t-s𝒬_η_t(s)(ψ_t(s),ψ_t(s))ds)dvdx.

Since by assumption (<ref>) ϕ≡ 0, the equation above eventually yields the (II.1) term is equal to

    =∫𝒢(ψ_t(t)+∫_0^tS_t-sψ_t(s)ϕ_t(s)ds-S_tψ_t(0))dvdx=∫𝒢ψ_t(t)dvdx-∫𝒢ψ_t(0)dvdx.

For the (II.2) term, it has the upper bound as

    =   ∫_0^t[∫η_t,p(s)𝒬_η_t(s)(ψ_t(s),ψ_t(s))dvdx]ds
    ≤   C∫_0^t‖(1+|v|)η_p,t(s)‖_L_β-1‖(1+|v|)^-1𝒬_η_t(s)(ψ_t(s),ψ_t(s))‖_L_β^∞ds
    ≤   C∫_0^t ‖η_p,t(s)‖_L_β^∞‖η_t(s)‖_L_β^∞‖ψ_t(s)‖_L_β^∞‖ψ_t(s)‖_L_β^∞ds,

where in the last inequality we have used Lemma <ref>. This upper bound can be further written as

    ≤   C‖η_t,p^‖_P_β^σ∫_0^t (1+(t-s))^-σ‖η_t(s)‖_L_β^∞‖ψ_t(s)‖_L_β^∞‖ψ_t(s)‖_L_β^∞ds
    ≤    Ca_*(‖𝒢‖_L_β^∞+a_*)^3.

Now it has been proved that there is a uniform bound for all terms in the decomposition of ℐ. This yields the uniform boundedness of ℐ(t,g), and concludes the proof of the theorem.


With some additional effort, we can prove the proposition below for the mild solution ℐ(t,g) in Theorem <ref>.

 Fix g as a function in C_x,v^0 which is time independent. For arbitrary function g satisfying the assumptions (<ref>)-(<ref>) and such that g(t)=g, there is

    lim_t→ +∞ℐ(t,g)=ℐ_∞(g),

with

    ℐ_∞(g):=-1+⟨ e^g,M⟩.


In this framework the function g is fixed at terminal time g(t)=g. According to assumption (<ref>), there is D_sg≡ 0. Consequently the function g is determined on the whole time interval [0,t] by g(t)=g.

According to equations (<ref>), (<ref>), and (<ref>) in the proof of Theorem <ref>, the functional ℐ(t,g) can be rewritten as

    ℐ(t,g)=-1+⟨η_t(t),ψ_t(t) ⟩+1/2⟨𝒢, ψ_t(0)-ψ_t(t)⟩+1/2η_t,p(s),𝒬_η_t(s)(ψ_t(s),ψ_t(s))_.

We want to prove the (II.2) term converge to 0 as t→ +∞. This can be achieved by the decomposition of the collision operator

    =   2η_t,p(s),𝒬_η_t(s)(ψ_t,p(s),𝒢)+η_t,p(s),𝒬_η_t(s)(ψ_t,p(s),ψ_t,p(s))
    ≤    2∫_0^t ‖η_p,t(s)‖_L_β^∞‖η_t(s)‖_L_β^∞‖ψ_t,p(s)‖_L_β^∞‖ψ_t(s)‖_L_β^∞ds
       +∫_0^t ‖η_p,t(s)‖_L_β^∞‖η_t(s)‖_L_β^∞‖ψ_t,p(s)‖_L_β^∞‖ψ_t,p(s)‖_L_β^∞ds
    ≤   C∫_0^t (1+s)^-σ(1+(t-s))^-σds≤ C(1+t)^-σ→ 0,   t→ +∞,

where we have used Lemma <ref> and the fact according to Theorem <ref>

    ‖ψ_t,p‖_P_β^σ<a_*,  ‖η_t,p^‖_P_β^σ<a_*.

In Theorem <ref>, the bound ‖ψ_t,p‖_P_β^σ<a_* is uniform for all t>0. According to the definition (<ref>) of P_β^∞, there is

    (1+t)^σ‖ψ_t,p(t)‖_L_β^∞<a_*.


Consequently as t→ +∞, there is ψ_t(t)→𝒢 in the L_β^∞ norm. This implies

    lim_t→ +∞⟨η_t(t),ψ_t(t)⟩=lim_t→ +∞⟨ e^g(t)ℬ,𝒢⟩=lim_t→ +∞⟨ e^g(t),M⟩=⟨ e^g,M⟩,

where we have used the fact that η_t(t)=e^g(t) and g(t)=g.

These imply

    lim_t→ +∞ℐ(t,g)=-1+⟨ e^g,M⟩+1/2⟨𝒢,ℬ^-1f^0-𝒢⟩.

The orthogonality assumption (<ref>) means ⟨𝒢,ℬ^-1f^0-𝒢⟩=0, which concludes the proof of the lemma.

The relation between the result above and the Schrödinger problem has been discussed in Subsection <ref>. 

In addition, we can further prove the long-time limit of ℐ(t,g) given in Proposition <ref>, is also a non-trivial stationary solution of the Hamilton-Jacobi equation.

 The functional ℐ_∞(g) in (<ref>) is a stationary solution of the Hamilton-Jacobi equation.


It is sufficient to verify 

    ℋ(ℐ_∞(g)/g,g)=0.

By the definition (<ref>) of ℐ_∞(g), the derivative with respect to g is

    ℐ_∞(g)/g=⟨ e^g,M⟩/g=e^gM.

Then according to the definition (<ref>) of ℋ, we have

    ℋ(e^gM,g)
    
    =   1/2∫ M(v)M(v_*)e^g(v)+g(v_*)(e^g(v')+g(v_*')-g(v)-g(v_*)-1)((v_*-v)·ω)_+dω dv_* dvdx
    
    =   1/2∫ M(v)M(v_*)(e^g(v')+g(v_*')-e^g(v)+g(v_*))((v_*-v)·ω)_+dω dv_* dvdx=0,

where the last inequality is by the change of variables from (v,v_*,ω) to (v',v_*',-ω). This concludes the proof.


The two propositions above together states that, the solution ℐ(t,g) converges to a stationary solution ℐ_∞(g) as t→ +∞. The stationary solution ℐ_∞(g)=⟨ e^g,M ⟩-1 is the cumulant generating functional of a random gas with Poisson-distributed total number, and i.i.d. distribution of the variables (x,v).






 §.§ Functional Solution for Theorem <ref>


Under the assumptions (<ref>)-(<ref>), the functional solution ℐ(t,g) constructed in Theorem <ref> of the Hamilton-Jacobi equation is uniformly bounded for arbitrary time t≥ 0 and function g satisfying Assumptions (<ref>)-(<ref>).


In the proof of Theorem <ref>, we have decomposed the functional ℐ into several terms: first it is decomposed as (<ref>), then the (II) term is decomposed into the summation of (II.1) term and (II.2) term in (<ref>). The same decomposition applies here. Except for the (II.2) term, the estimate of the others is exactly the same as in the proof of theorem <ref>. Thus we only detail the estimate of the (II.2) term here.

In equation (<ref>), it has been proved that

    =   ∫_0^t[∫η_t,p(τ)𝒬_η_t(τ)(ψ_t(τ),ψ_t(τ))dvdx]dτ
    ≤   C∫_0^t ‖η_t,p(τ)‖_L_β^∞‖η_t(τ)‖_L_β^∞‖ψ_t(τ)‖_L_β^∞‖ψ_t(τ)‖_L_β^∞dτ

According to the definition (<ref>) of the E_β^0 and the E_β^-σ norms, we get

    ≤ Ce^σ t‖η_t,p^‖_E_β^-σ‖η_t‖_E_β^0‖ψ_t‖_E_β^0‖ψ_t‖_E_β^0≤ Ce^σ ta_*e^-σ t(‖𝒢‖_L_β^∞+a_*)^3,

where we have used the condition ‖η_t,p^‖_E_β^-σ<a_*e^-σ t.

It shows the (II.2) term is also uniformly bounded. This concludes the proof of the uniform boundedness of ℐ(t,g).







§ DECOMPOSITION OF SEMIGROUP AND RELEVANT ESTIMATES

By Definition <ref>, the operator B^+ consists of a transport operator -v·∇_x, and a linearized collision operator 2𝒬_𝒢(·,𝒢). The operator 2𝒬_𝒢(·,𝒢) can be decomposed as the summation of a frequency multiplier -ν and a convolution operator K. This decomposition is initially due to <cit.> 

    2𝒬_𝒢(·,𝒢)=-ν+K,

with

    ν f(v)=∫_R^d∫_S^d-1((v_*-v)·ω)_+f(v)𝒢^2(v_*)dω dv_*,
       Kf(v)=∫_R^d∫_S^d-1((v_*-v)·ω)_+(f(v')𝒢(v_*')+f(v_*')𝒢(v')-f(v_*)𝒢(v))𝒢(v_*)dω dv_*.

The operator K can also be written using the related transition kernel, which is given explicitly on Page 19 of <cit.>. The following lemma about the convolution operator K is classical <cit.>.

The operator K is a self-adjoint compact operator on L^2. For any 2≤ p≤ r≤∞, it is also a bounded operator from L^p to L^r. If β≥ 0, then it is a bounded operator from L_β^∞ to L_β^∞.

For a detailed proof of the following lemma, the reader may see Section 2.2. of <cit.>.


The operator  (resp. ) in Definition <ref> generates a strongly continuous semigroup e^s (resp. e^s) on L^2(T_x^d×R_v^d). Both semigroups decay exponentially in the L_x,v^2-norm if the initial data is orthogonal to the kernel 𝒦: there exists constants ν_*>0 and C>0 such that if f∈𝒦^⊥, then

    ‖ e^sf‖_L_x,v^2≤ Ce^-ν_* s‖ f‖_L_x,v^2,  ‖ e^sf‖_L_x,v^2≤ Ce^-ν_* s‖ f‖_L_x,v^2.

If the function f belongs to the kernel 𝒦, then we have e^sf=e^sf=f.

Since B^+ is a bounded perturbation of A^+:=-v·∇_x-ν, according to Corollary 1.7 in Page 119 of <cit.>, the semigroup e^sB^+ can be written as

    e^sB^+=e^sA^++(e^sA^+K)∗ e^sB^+.

Here ∗ refers to the convolution over [0,s],

    (e^sA^+K)∗ e^sB^+ :=∫_0^s e^(s-τ)A^+Ke^τ B^+.

Iterate this and we will have

    e^sB^+   =e^sA^++(e^sA^+K)∗ e^sA^++(e^sA^+K)∗(e^sA^+K)∗ e^sA^++...+(e^sA^+K)^∗ Ne^sB^+
       =e^sA^++∑_j=1^N-1(e^sA^+K)^∗ j∗ e^sA^++(e^sA^+K)^∗ N∗ e^sB^+.

For the decomposition (<ref>) of e^sB^+=𝒟_1^+(s)+𝒟_2^+(s), we define 𝒟_1^+(s) and 𝒟_2^+(s) as

    𝒟_1^+(s):=e^sA^+,   D_2^+(s):=∑_j=1^N-1(e^sA^+K)^∗ j∗ e^sA^++(e^sA^+K)^∗ N∗ e^sB^+.

The operator 𝒟_1^+(s) has the explicit expression

    𝒟_1^+(s)f(x,v)=e^-ν(v)sf(x-sv,v).

The operator 𝒟_2^+ has a decay estimate as a map from L_β-1^∞ to L_β^∞ (see Lemma <ref>) due to the smoothing effect (Lemma <ref>) of K.

The analysis above is also true for the backward component, where for the decomposition (<ref>) of e^sB^-=𝒟_1^-(s)+𝒟_2^-(s), we define 𝒟_1^-(s) and 𝒟_2^-(s) as

    𝒟_1^-(s):=e^sA^-,  𝒟_2^-(s):=∑_j=1^N-1(e^sA^-K)^∗ j∗ e^sA^-+(e^sA^-K)^∗ N∗ e^sB^-.

The operator 𝒟_1^-(s) is explicitly written as

    𝒟_1^-(s)f(x,v)=e^-ν(v)sf(x+sv,v).

The decomposition has been well established in the literature <cit.>, whose modification gives the proof of Lemma <ref>.

We only give the proof for the forward component 𝒟_2^+(s). The proof for 𝒟_2^-(s) defined in (<ref>) is the same.

The case of f∈𝒦^⊥: Using Lemma <ref> with p=2 and r=∞, we can prove (e^sA^+K)∗ e^sB^+ is a bounded operator from L^2 to L_0^∞=L^∞, with Ce^-ν_*s as the upper bound for the operator norm

    ‖∫_0^s e^(s-τ)A^+Ke^τ B^+fdτ‖_L_0^∞≤ C∫_0^se^-ν_*(s-τ)e^-ν_*τ‖ f‖_L^2≤ Ce^-ν_*s‖ f‖_L^2.

Having an additional convolution with e^sA^+K, the operator (e^sA^+K)^∗ 2∗ e^sB^+ is a bounded operator from L^2 to L_1^∞. Iterating this bootstrap argument and choosing N=⌈β⌉, we have (e^sA^+K)^∗ N∗ e^sB^+ is a bounded operator from L^2 to L_β^∞ with β>4, also with Ce^-ν_*s as the upper bound for the operator norm. This implies

    ‖(e^sAK)^∗ N∗ e^sBf‖_L_β^∞≤ Ce^-ν_*s‖ f‖_L^2≤ Ce^-ν_*s‖ f‖_L_β-1^∞=Ce^-ν_*s‖ (1+|v|)^-1f‖_L_β^∞.

Using the explicit expression of e^sA^+ and the smoothing effect of K, we can prove for any j≥ 1 that (e^sA^+K)^∗ j∗ e^sA^+ has the decay estimate

    ‖(e^sA^+K)^∗ j∗ e^sA^+f‖_L_β^∞≤ Ce^-ν_* s‖ (1+|v|)^-1f‖_L_β^∞.

The case of f∈𝒦: Now the estimate (<ref>) is still true since it only depends on the explicit expression of e^sA^+ and the smoothing effect of K. For the other term in 𝒟_2^+, it becomes

    ‖(e^sAK)^∗ N∗ e^sBf‖_L_β^∞=‖(e^sAK)^∗ N∗f‖_L_β^∞≤ C‖ (1+|v|)^-1f‖_L_β^∞.

This concludes the proof of the lemma.

Next we prove the continuity of e^τ B^+ (resp. e^τ B^-) with respect to the forward initial perturbation f^0ℬ^-1-𝒢 (resp. the backward terminal perturbation e^g(t)ℬ-𝒢). This lemma is crucial to the proof of the continuity of (ψ_t(s),η_t(s)) (see Lemma <ref>).

 Under assumptions (<ref>)-(<ref>) or (<ref>)-(<ref>), for any parameter β>4 we have

    ‖ e^τ B^+(f^0ℬ^-1-𝒢)-(f^0ℬ^-1-𝒢)‖_L_β^∞≤ C(f_0)τ,
       ‖ e^τ B^-(e^g(t)ℬ-𝒢)-(e^g(t)ℬ-𝒢)‖_L_β^∞≤ C(g)τ,

with C(f_0) being a constant dependent on f_0, and C(g) being a constant dependent on g.


We only detail the proof of the second equation in (<ref>), since the other one is the same. For simplicity in notation and also in accordance with the choice of terminal data, we write G(x,v):=(e^g(t)ℬ-𝒢)(x,v). Using the bootstrap argument (<ref>), we have

    ‖ e^τ B^-G-G‖_L_β^∞
    ≤   ‖ e^τ A^-G-G‖_L_β^∞+∫_0^τ‖ e^u A^-Ke^(τ-u)B^-G‖_L_β^∞du
    
    =   ‖ G(x+τ v,v)e^-ν(v)τ-G(x,v)‖_L_β^∞+ ∫_0^τ‖ e^u A^-Ke^(τ-u)B^-G‖_L_β^∞du

The first term in the third line of (<ref>) is controlled as

    ‖ G(x+τ v,v)e^-ν(v)τ-G(x,v)‖_L_β^∞
    ≤   ‖ G(x+τ v,v)(1-e^-ν(v)τ)‖_L_β^∞+‖ G(x+τ v,v)-G(x,v)‖_L_β^∞
    ≤   ‖ (1+|v|)G(x+τ v,v)1-e^-ν(v)τ/(1+|v|)‖_L_β^∞+‖ e^g(t,x+τ v,v)ℬ(v)-e^g(t,x,v)ℬ(v)‖_L_β^∞
    ≤    C‖ G‖_L_β+1^∞τ+C‖ G‖_L_β+1^∞‖ g‖_C_t,x^1τ,

where the term C‖ G‖_L_β+1^∞τ is due to

    1-e^-ν(v)τ/(1+|v|)≤ Cτ  ,

and the term C‖ G‖_L_β+1^∞‖ g‖_C_t,x^1τ is because

    e^g(t,x+τ v,v)ℬ(v)-e^g(t,x,v)ℬ(v)
    =∫_0^τ[(v·∇_xg)e^gℬ](t,x+uv,v)du.

The norm ‖ g‖_C_t,x^1 is finite due to assumption (<ref>) or (<ref>), where we have assumed g has uniformly bounded derivatives in t and x. The control of the second term in the third line of (<ref>) is straightforward, since all the involved operators are bounded operators from L_β^∞ to L_β^∞

    ∫_0^τ‖ e^u A^-Ke^(τ-u)B^-G(x,v)‖_L_β^∞du≤ C‖ G‖_L_β^∞τ

This concludes the proof of the lemma.

In the end of this appendix, we give the proof of Lemma <ref> for completeness. The proof is elementary.

Given σ_1>1 and σ_2>1, we have the following inequality for a convolution,

    ∫_0≤ s≤ t(1+(t-s))^-σ_2(1+s)^-σ_1ds≤ C (1+t)^-min{σ_1,σ_2}



We split the integral

    ∫_0≤ s≤ t(1+(t-s))^-σ_2(1+s)^-σ_1ds
    
    =   ∫_0≤ s≤t/2(1+(t-s))^-σ_2(1+s)^-σ_1ds+∫_t/2≤ s≤ t(1+(t-s))^-σ_2(1+s)^-σ_1ds
    ≤      (1+t/2)^-σ_2∫_0≤ s≤t/2(1+s)^-σ_1ds+(1+t/2)^-σ_1∫_t/2≤ s≤ t(1+(t-s))^-σ_2ds

This is further less than

    ...   ≤ C(1+t/2)^-σ_2+C(1+t/2)^-σ_1
       ≤ C(1+t)^-min{σ_1,σ_2}

This concludes the proof.









plain
