Discrete, Kinetic, and Hydrodynamic Descriptions of the Euler Alignment System with Adaptive Communication Strength
Roman Shvydkoy Trevor Teolis



Kinetic Model of Euler Alignment system with adaptive strength]Discrete, Kinetic, and Hydrodynamic descriptions of the Euler Alignment System with adaptive communication strength





851 S Morgan St, M/C 249, Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607

shvydkoy@uic.edu

 
 
 
851 S Morgan St, M/C 249, Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607

tteoli2@uic.edu
 
37A60, 92D50






Acknowledgment.  
	This work was  supported in part by NSF
	grant  DMS-2107956 and Simons Foundation.


§ ABSTRACT 
    The -model, introduced in <cit.>, is an alignment model with the property that the strength of the alignment force, , is transported along an averaged velocity field.  
    Inspired by the 1D threshold regularity criterion for the Cucker-Smale model in terms of the so-called e-quantity, e = ∂_x u + ρ∗ϕ, 
    the transport of the strength in the -model was explicitly designed so that it possesses its own e-quantity, e = ∂_x u +, which 
    yields an analogous threshold regularity criterion for global well-posedness in 1D. 
    Owing to the existence of this e-quantity, it also possesses similar long-time behavior to the Cucker-Smale model in 1D <cit.>.
    The utility of the -model is that it also has the versatility to behave qualitatively like the Motsch-Tadmor model, for which global regularity theory is not known.
    
    The goal of this paper is to put the -model on more firm physical grounds by formulating and justifying the microscopic and mesoscopic descriptions from which it arises. 
    A distinctive feature of the microscopic system is that it is a discrete-continuous system:
    the position and velocity of the particles are discrete objects, while the strength is an active continuum scalar function. 
    We establish a rigorous passage from the microscopic to the mesoscopic description via the Mean Field Limit and a passage from the mesoscopic to the macroscopic description 
    in the monokinetic and Maxwellian limiting regimes.  We present a survey of such results for the Cucker-Smale model and explain how to extend 
    these arguments to the -model, where the strength of the alignment force is transported. 
    We also address the long-time behavior of the kinetic Fokker-Planck-Alignment equation by establishing the relaxation to the Maxwellian in 1D when the velocity averaging is the Favre averaging. 
    As a supplement to the numerical results already presented in <cit.>, 
    we provide additional numerical evidence, via a particle simulation, that the -model behaves qualitatively like Motsch-Tadmor. 


[
    Trevor Teolis
    September 9, 2024
=====================







§ INTRODUCTION









Collective patterns and behavior can emerge from simply interacting agents.  Flocks of birds, schools of fish, and flashing of fireflies are examples of systems of simply interacting agents 
where there is a clear emergent behavior of the whole system.  This has spurred much interest in recent years in the mathematical modeling of this large class of phenomena.
The question is:  
how do we create a model that describes the behavior of a large class of communicating agents 
and is mathematically tractable in the sense of well-posedness analysis and provability of the emergent phenomena? 

Many classical alignment models possess some of these features. 
The 3-zone model was developed in 1987 by Craig Reynolds, a computer animator, who was interested in developing realistic models of birds for cinema, see <cit.>.  It is characterized 
by a repulsion force in the close range, alignment in the medium range, and attraction in the far range.  The repulsion force makes it challenging to describe   the long-time behavior due to the lack of a natural equilibrium state. 
The Viscek model  proposed in <cit.> incorporates stochastic forces that facilitate phase transitions as the strength of noise varies – an important feature of emergent dynamics. 
It evolves in discrete time steps whereby agents get assigned the average and normalized velocities of nearby agents.  
Phase transitions were observed numerically, however the model at the moment lacks a satisfactory analysis, see <cit.>. 
 In 2007 <cit.> Cucker and Smale proposed another alignment model with the following key features: it weights the alignment force with a radial interaction kernel ϕ inversely proportional to the distance between agents.
The model allows for an array of different dynamics and, most importantly, it was the first model that allowed a proof of alignment which depended only on the initial state and not on perpetual connectivity of the flock.  Moreover, the long-time behavior holds under the large crowd limit N→∞, which allows to carry such results over to the kinetic and macroscopic descriptions, <cit.>.  The universality of description and mathematical tractability of the model made it the subject of a series of studies  surveyed for example in <cit.>. Despite its success, however, the Cucker-Smale model does not respond well in certain modeling scenarios. 
Motsch and Tadmor argued in  <cit.> that in heterogeneous formations when, say, a small subflock separates itself from a distant large flock, its internal forces become annihilated by the latter if subjected to the Cucker-Smale protocol <cit.> creating unrealistic behavior. A proposed renormalization of the averaging operation restored the balance of forces and the alignment results under long range communication  were proved similar to those of Cucker and Smale  at the microscopic level and in the large crowd limit, <cit.>.  Such renormalization, however, destroys the symmetry of the alignment force and possible vacuum formation makes the system more singular, which is the reason for the lack of a coherent well-posedness theory at the moment. 

The -model has been designed to have an adaptive averaging protocol which allows for many similar regularity properties as those of the Cucker-Smale model, while maintaining the same level of universality as that of the Motsch-Tadmor model. 
One downside of the adaptive as opposed to prefixed protocol is the lack of control over kinematic properties of the alignment force, which makes the analysis of the long-time behavior a challenging but interesting problem, see <cit.>. 

The goal of this paper is to (1) introduce the microscopic and mesoscopic counterparts to the previously studied macroscopic description of the -model in <cit.>; (2) to 
establish a rigorous passage between the levels of description; 
and (3) to establish the relaxation to a thermodynamic state in one dimension for the mesoscopic description when the velocity averaging is the Favre averaging. 

Before we turn to the technical description of the results let us present a more detailed motivation of the -model by giving a brief survey of the classical Cucker-Smale and Motsch-Tadmor alignment models
as well as the more general environmental averaging models from which it originated. 



 §.§ Cucker-Smale model

The pressureless Euler alignment system based on the Cucker-Smale model on ^n or ^n is given by 

    ∂_t ρ + ∇· (ρ̆) = 0 
    ∂_t +̆·̆∇=̆  (ρ̆)_ϕ - ρ̆_ϕ

where ρ, $̆ are the density and velocity of the flock andϕ≥0is a smooth radially decreasing communication kernel. 
The notationf_ϕ = f ∗ϕdenotes a convolution. 
Provided the kernel has a fat tail (i.e. is non-integrable), 

    ∫_0^∞ϕ(r) dr = ∞

this model admits alignment and flocking:  there existsδ> 0such that 

    lim_t →∞(ρ) < ∞, lim_t →∞sup_x,y ∈(ρ) |(̆t,x) - (̆t,y)| ≤ C e^-δ t

It is a well-known result of Carillo, Choi, Tadmor, and Tan <cit.> that

    e = ∂_x u + ρ_ϕ, ∂_t e + ∂_x(ue) = 0

provides a threshold regularity criterion in 1D: the solution remains smooth iffe_0 ≥0. 
In fact, it is not only useful for global regularity; it is also instrumental in proving 1D strong flocking (i.e. convergence 
to a limiting distribution) and estimating the limiting distribution, <cit.>. 



 §.§ Motsch-Tadmor model

It was observed in <cit.> that the Cucker-Smale model displays unrealistic behavior in heterogeneous formations. 
In particular, when there is a small mass flock and a faraway, large mass flock, the internal dynamics of the small mass flock are hijacked by the large mass flock.  
The Motsch-Tadmor model was introduced in order to resolve this behavior. 
It was  proposed that the strength of the alignment force on a particle should be scaled by the total influence on that particle.  
With this modification, we get: 

    ∂_t ρ + ∇· (ρ̆) = 0 
    ∂_t +̆·̆∇=̆1/ρ_ϕ ((ρ̆)_ϕ - ρ̆_ϕ)

We will explain what this fix does to restore the balance of forces in Section <ref>.

It was shown in <cit.> that (<ref>) aligns under the same fat tail kernel condition (<ref>). 
The cost is that it no longer possesses thee-quantity and, as a result, there is no known 1D threshold regularity criterion. 
The-model aims to fix this lack of a threshold regularity criterion while also retaining the desired qualitative behavior of 
the Motsch-Tadmor model in heterogeneous formations.  Before we introduce it, we will mention the environmental averaging models 
from which it originated.  



 §.§ Environmental Averaging Models

Many  alignment models are characterized by an alignment force which pushes the velocity towards an averaged velocity field.  
The Vicsek, Cucker-Smale, and Motsch-Tadmor models are among many examples.  
For the Cucker-Smale model, we can rewrite the alignment force as 

    F = s_ρ ( - )̆, _ρ = ρ_ϕ,  = (ρ̆)_ϕ/ρ_ϕ .

The so-called strength_ρof the alignment force is a fixed function which depends on the density; andis an averaged velocity field.   
Written in this form, it is more clear that it is an alignment model– the velocity$̆ is being pushed towards the averaged velocity .
Alignment forces which take this more general form are called environmental averaging models and the macroscopic version is given by:

    ∂_t ρ + ∇· (ρ̆) = 0 
    ∂_t +̆·̆∇=̆_ρ ( - )̆ .

They generalize the Cucker-Smale model by treating _ρ and  as arbitrary modeling components in a proper functional framework.  
The general theory of environmental averaging models has been developed extensively in <cit.>. 










An important class of models are those for which the velocity averaging  has an integral representation against a smooth kernel Φ_ρ:  

    = ∫Φ_ρ(x,y) (̆y) ρ(y) , Φ_ρ is smooth.



  Models whose velocity averaging has the form (<ref>) will be an important sub-class for the as-yet defined -model 
  since the regularity of the model is tied to the regularity of the kernel Φ_ρ(x,y). 


The representation (<ref>) holds for many classical models: Cucker-Smale (M_CS), Motsch-Tadmor (M_MT), the overmollified version of (M_MT) introduced in <cit.>,

    = ( (ρ̆)_ϕ/ρ_ϕ)_ϕ,     _ρ = 1, M_ϕ

the segregation model based on a partition of unity ∑_l=1^L  g_l = 1, 

    = ∑_l=1^L g_l(x) ∫_Ø u g_l ρ/∫_Ø g_l ρ ,     _ρ = 1, M_seg

and other multi-flock, multi-species variants of the above, see <cit.>. 
All of these models possess a representation kernel as shown in table <ref>.





   
        Strength _ρ     Kernel Φ_ρ(x,y) 

    M_CS     ρ_ϕ     ϕ(x-y)/ρ_ϕ(x) 
 
    M_MT     1      ϕ(x-y)/ρ_ϕ(x) 
 
    M_ϕ     1     ∫_Øϕ(x-z) ϕ(y-z)/ρ_ϕ(z) 
 
    M_seg     1     ∑_l=1^L g_l(x) g_l(y)/∫_^n g_l ρ 
 
    
  






We can see that many of the models have the density-dependent renormalization at its core, _F: =  (ρ̆)_ϕ/ρ_ϕ. 
This is known as the Favre filtration, which was introduced in the context of non-homogeneous turbulence in <cit.>.  
In our context, we will refer to it as the "Favre averaging" since it represents the averaged velocity.  

  
  The Favre-averaged velocity is given by 
  
    _F :=  (ρ̆)_ϕ/ρ_ϕ









 §.§ The -model


The -model is a descendant of the environmental averaging models and was introduced in <cit.> at the macroscopic level.  
The only adendum to the environmental averaging models is that  is no longer a fixed function of the 
density, but instead evolves according to its own evolution equation along the average velocity field.  
This has a profound impact on the 1D regularity theory at the macroscopic level of description, and subsequently it lends to stronger long-time behavior results, 
all of which resemble the favorable behavior of the Cucker-Smale model in this more general setting.  






It was observed in <cit.> that the existence of the e-quantity in the Cucker-Smale case is owed to the transport of the strength function, _ρ = ρ_ϕ, 
along the Favre-averaged velocity field:

    ∂_t ρ_ϕ + ∂_x (ρ_ϕ_̆F) = 0  .

The new model therefore proposes that is more natural for the strength  to evolve according to its own transport equation along the averaged velocity field:

    ∂_t  + ∂_x () = 0 .

The -model is given by 

    SM∂_t ρ + ∇· (ρ̆) = 0 
    ∂_t  + ∇· () = 0,     ≥ 0 
    ∂_t +̆·̆∇=̆( - )̆

and, by design, it admits the e-quantity in 1D

    e = ∂_x u + , ∂_t e + ∂_x(ue) = 0.

The special case where the velocity averaging is the Favre averaging, i.e. = _̆F (see Definition <ref>), bears a similar alignment force to the classical Cucker-Smale and Motsch-Tadmor alignment models. 
Indeed, we can cast it into a similar form by introducing a new variable , which we call the "weight", defined by = ρ_ϕ. 
With this change of variables,  satisfies a pure transport along the Favre-averaged velocity field 

    ∂_t  + _̆F ·∇ = 0

and the system becomes 

    WM∂_t ρ + ∇· (ρ̆) = 0 
    ∂_t  + _̆F ·∇ = 0,     ≥ 0  
    ∂_t +̆·̆∇=̆ ((ρ̆)_ϕ - ρ̆_ϕ)

We obtain a model that looks like a hybrid of Cucker-Smale and Motsch-Tadmor. We refer to this particular variant of the -model as the -model. 
Setting _0 = 1, we recover the Cucker-Smale model; and setting _0 = 1/(ρ_0)_ϕ, we recover the Motsch-Tadmor model at the initial time 
(at later times, the transport of the weight  will cause it to deviate from the Motsch-Tadmor weight). 

Due to the existence of the e-quantity and the similar structure of the alignment force to the Cucker-Smale model, 
many classical results of the Cucker-Smale case were extended to the -model in our joint work <cit.>.  The following is a list of the results 
which were obtained: 

  
  * Local well-posedness and global well-posedness for small data in multi-D;
  
  * Global well-posedness in 1D and for unidirectional flocks in multi-D under the threshold e ≥ 0;
  
  * L^∞-based alignment; conditional L^2-based alignment;
  
  * Strong flocking in 1D (i.e. convergence of the density to a limiting distribution);
  
  * Estimates on the limiting distribution of the flock in 1D.


The importance of the -model is that it retains the regularity and alignment characteristics of the Cucker-Smale 
system while also having the versatility to behave qualitatively like the Motsch-Tadmor model (when _0 = 1 / (ρ_0)_ϕ).  
In other words, the -model can be thought of as a more analytically tractable version of the Motsch-Tadmor model.  
Numerical evidence of the qualitative similarities between the -model and the Motsch-Tadmor model has been presented 
at the macroscopic level in 1D in <cit.>. 
Here, we present, perhaps more clear, numerical evidence of this qualitative behavior by showcasing simulations of the microscopic system in Section <ref>.
First, we introduce the microscopic and mesoscopic versions of the -model, and hence also the -model as a particular case. 





 §.§ Microscopic and mesoscopic levels of description

The -model was introduced in  <cit.> at the level of the hydrodynamic description, while the microscopic and mesoscopic levels have remained unattended.  
To fill this gap, we introduce  and justify them here.  
Due to its ancestral relation to the Cucker-Smale model, the microscopic and mesoscopic systems for the -model will have a similar structure
to the Cucker-Smale versions, albeit with important differences. As a reference point for the reader, we will first present the microscopic and mesoscopic descriptions for the Cucker-Smale model followed 
by those for the -model, highlighting the differences between the two.  


  We could have also juxtaposed the -model to the environmental averaging models, which also have a well developed theory at all levels of description. 









  §.§.§ Cucker-Smale: microscopic and mesoscopic descriptions


The classical discrete Cucker-Smale model, originally introduced by Cucker and Smale in <cit.>, is given by an ODE system of N interacting agents: 

    ẋ_̇i̇ = v_i 
    v̇_̇i̇ = λ∑_j=1^N m_j ϕ(x_i - x_j)(v_j - v_i) .

where x_i, v_i are the velocity and positions of the agents and λ > 0 is a scalar that affects the strength of the alignment force.  
Of course, it aligns and flocks under the same fat tail condition on the kernel, (<ref>). 
As the number of particles N →∞, it is more convenient to look at the evolution of the probability density, f(t,x,v), 
of finding a particle at position x and velocity v at time t instead of tracking individual particle trajectories. 
The evolution equation for the probability density, f(t,x,v), is given by a kinetic Vlasov equation derived
according to the BBGKY hierarchy <cit.>:

    ∂_t f + v ·∇_x f + λ∇_v · [f F(f)] = 0, 
       F(f)(t,x,v) = ∫_^2nϕ(|x-y|)(w - v)f(t,y,w)  .

The passage from the (<ref>) to (<ref>) in the scaling regime where the range of the communication between particles 
remains independent of N, i.e. the mean field limit, was established in <cit.>.

In practice, one cannot physically observe the probability density function.  
The physically observable (or macroscopic) variables are the density and the momentum, which are defined by 

    ρ(t,x) = ∫_^n f(t,x,v) ,  (ρ)̆(t,x) = ∫_^n v f(t,x,v)  .

To determine the evolution of macroscopic variables, one computes the v-moments of the kinetic Vlasov equation and obtains 

    ∂_t ρ + ∇· (ρ̆) = 0 
    ∂_t (ρ)̆ + ∇· (ρ⊗̆)̆ + ∇· = ρ ((ρ̆)_ϕ - ρ̆_ϕ),

where  is the Reynolds stress tensor 

    (t,x) = ∫_^n (v - (̆t,x)) ⊗ (v - (̆t,x)) f(t,x,v)  .

This results in the closure problem: the system (<ref>) depends not only on the macroscopic variables ρ and $̆,
but also on the kinetic distributionf.  
The system can be closed by adding a local alignment force which pushes the distributionfto be concentrated around the macroscopic 
velocity and thus removes the dependence onf.  
Two such alignment forces have been considered in the literature corresponding to the monokinetic and Maxwellian limiting regimes.  
For background, we will survey the monokinetic and Maxwellian limits for the Cucker-Smale model in Section <ref>. 


  The macroscopic pressureless Euler alignment system (<ref>) arises from the monokinetic limiting regime. 
  The as-yet defined macroscopic isentropic Euler alignment system (<ref>) (discussed in Section <ref>) 
  arises from the Maxwellian limit. 


  §.§.§ -model: microscopic and mesoscopic descriptions

The microscopic-model stands uniquely from the other microscopic alignment models as a discrete-continuous system. 
The strength satisfies its transport equation along the averaged velocity field and it is therefore not a discrete object. 
Instead, it is an active continuum scalar function that is transported along the continuous field(·, x). 
As far as the alignment force is concerned, it only keeps track ofandat the discrete pointsx_i.

The discrete densityρ^N ∈_M(^n)is given byρ^N = ∑_j=1^N m_j δ_x_jand the discrete velocity is given byu^N = ∑_j=1^N v_j 1_x_j.
The microscopic description of (<ref>) is then given by an ODE system describing the position and velocity of particles coupled 
with a PDE describing the transport of the strength function:

    ẋ_̇i̇ = v_i 
    v̇_̇i̇ = λ(x_i) ((x_i) - v_i ), ρ^N = ∑_j=1^N m_j δ_x_j,  u^N = ∑_j=1^N v_j 1_x_j, 
    ∂_t  + ∇_x · () = 0 .

Once again,λ> 0is a scalar that affects the strength of the alignment force.  
Althoughappears to be a discrete object, observe that when the velocity averaging satisfies the integral representation (<ref>), 
the velocity averaging is given by

    (t,x) = ∑_j=1^N m_j Φ_ρ^N(x, x_j) v_j(t) .

In this case,is a smooth object wheneverΦ_ρ^Nis smooth.  
The well-posedness of this discrete-continuous system for smooth kernelsΦ_ρ^Nis proved in Section <ref>.

The corresponding mesoscopic description is given by a Vlasov-type equation coupled with a transport of the strength function:

    f_t + v ·∇_x f + λ∇_v ( (v - ) f) = 0 
    ∂_t  + ∇_x · () = 0 .
 
For the mesoscopic case, when the velocity averaging satisfies the integral representation (<ref>),
the velocity averaging is given by 

    (x) = ∫_^nΦ_ρ(x,y) ∫_^n v f(t,y,v)   .

From table <ref>, we see that the discrete Cucker-Smale model (<ref>) can be 
recovered by setting= ρ^N_ϕandΦ_ρ(x,y) = ϕ(x-y) / ρ_ϕ(x).  
Sinceρ_ϕautomatically satisfies the transport along the averaged velocity, the strength equation drops out.  
For the kinetic system, (<ref>) is recovered similarly by setting= ρ_ϕandΦ_ρ(x,y) = ϕ(x-y) / ρ_ϕ(x).

For the-model, the mean field passage from the microscopic description (<ref>) to the mesoscopic description (<ref>) necessitates some uniform regularity onand. 
Let us make the following observations of the Cucker-Smale model, which will influence how we handle the mean field passage for the-model: 


  *  the regularity of the strength comes for free when the communication kernel ϕ is smooth, and
  
  *  the regularity of the velocity averaging is tied to the regularity of the kernel Φ_ρ .

For the-model, the regularity of the strength will not come for free, but we will see later that it is also tied to the regularity of the kernelΦ_ρ. 
Therefore, instead of imposing regularity conditions on the solution, we will subordinate the uniform regularity assumptions to the kernelΦ_ρ, (<ref>) and (<ref>), which are sufficient for 
achieving the desired regularity on the velocity averaging and the strength, provided that theL^1norm of the momentum is bounded.

The mean field limit under these uniform regularity assumptions on the kernelΦ_ρwill be established in Section <ref>.
This passage will simultaneously show existence and uniqueness of solutions to (<ref>) and that the solutions arise as a limit, in some sense, of solutions to (<ref>).


Taking thev-moments of the Vlasov equation (<ref>), we obtain 

    ∂_t ρ + ∇· (ρ̆) = 0 
    ∂_t  + ∇· () = 0 
    ∂_t (ρ)̆ + ∇· (ρ⊗̆)̆ + ∇· = ρ( - )̆,

where the densityρand momentumρ$̆ are defined as in (<ref>)
and  is the same Reynolds Stress tensor that appeared in the Cucker-Smale case, (<ref>). 
This results in the same closure problem and it will be addressed in the same way by adding a local alignment force. 
The monokinetic and Maxwellian limits for the -model will be proved in Section <ref>. 




  The pressureless macroscopic system (<ref>) arises from the monokinetic limiting regime. 
  The as-yet defined isentropic macroscopic system (<ref>) (introduced in Section <ref>) 
  arises from the Maxwellian limit. 













 §.§ Numerical evidence for similar qualitative behavior to Motsch-Tadmor


To make the case for physical relevance of the -model, 
we present numerical evidence, at the microscopic level, that the -model with _0 = 1/ρ_ϕ, and hence the -model,
displays similar qualitative behavior to that of Motsch-Tadmor in heterogeneous formations.  
We first clarify the qualitative behavior that we are seeking. 



  §.§.§ Qualitative behavior of Motsch-Tadmor


Here, we reproduce the motivation for the Motsch-Tadmor model given in <cit.>.  
Let m, M with M >> 1 ≈ m be the masses of the small and large flock, respectively.  
Let I = {i : agent  i  is in the small flock} and define J similarly for the large flock. 
Suppose that the flocks are far enough away so that M ∑_j ∈ Jϕ(|x_i - x_j|) << 1 for all i ∈ I and 
that the agents in the small flock are close enough so that ϕ(|x_i - x_i'|) ≈ 1 for all i, i' ∈ I. 
Then for any agent i, the alignment force of the Cucker-Smale system (<ref>) can be written as 

    v̇_̇i̇ 
        =    λ/m + M( ∑_j ∈ J m_j ϕ(|x_i - x_j|) (v_j - v_i) + ∑_i' ∈ I m_i'ϕ(|x_i - x_i'|) (v_i' - v_i) ) 
    
        :=    A_J + A_I.

Since the velocities are bounded and the flocks are far away, A_J << 1.  On the other hand, since M >> m, we also have A_I << 1. 
The result is that d/dt v_i << 1 for all i ∈ I so the small flock proceeds with essentially no force on it.  
The small flock is, in this sense, "hijacked" by the large flock. 

To see how the Motsch-Tadmor fixes this issue, let us introduce the discrete version of (<ref>): 

    ẋ_̇i̇ = v_i 
    v̇_̇i̇ = λ/∑_j=1^N m_j ϕ(|x_i - x_j|)∑_j=1^N m_j ϕ(x_i - x_j)(v_j - v_i) .

Under the same assumptions and notation as before, we estimate the alignment force.  Since the large and small flock are far away, 
the strength of the alignment force any agent in the small flock, i ∈ I, is approximately

    λ/∑_i' ∈ I^N m_i'ϕ(|x_i - x_i'|) + ∑_j ∈ J^N m_j ϕ(|x_i - x_j|)≈λ/∑_i' ∈ I^N m_i'ϕ(|x_i - x_i'|) .

The interactions with the agents from the large flock also drop out as before. 
So, the total alignment force on an agent i ∈ I is approximately 

    d/dt v_i 
           ≈λ/∑_i' ∈ I^N m_i'ϕ(|x_i - x_i'|)∑_i' ∈ I m_i'ϕ(|x_i - x_i'|) (v_i' - v_i) )

The small flock then behaves according to the Cucker-Smale model, but independently of the large flock. 
In our -model simulation, we are looking for this qualitative behavior.  
In particular, we aim to see:
 
  
  
  (Q_cs) For the Cucker-Smale model:  The small flock proceeds linearly as if there were no force on it. 
  
  (Q_) For the -model with _0 = 1/(ρ_0)_ϕ:  The small flock behaves according to Cucker-Smale, but independently of the large flock. 
  The velocities of the small flock will therefore align to the average velocity of the small flock. 




  §.§.§ Discrete -model

Let us state the discrete -model as a special case of (<ref>).  Setting Φ_ρ^N(x,y) = ϕ(x-y) / (ρ^N)_ϕ 
where ρ^N ∈_M(^n) is given by ∑_j=1^N m_j δ_x_j(t) in (<ref>), we get 

    = ∑_j=1^N ϕ(|x_i - x_j|) (v_j - v_i) .

Recalling that = ρ^N_ϕ, we obtain the discrete -model: 

    ẋ_̇i̇ = v_i 
    v̇_̇i̇ = λ(x_i) ∑_j=1^N m_j ϕ(|x_i - x_j|) (v_j - v_i) 
    ∂_t  + ·∇_x  = 0 .




  §.§.§ Numerical simulation

The solutions are computed on the 2D unit torus, ^2.  
We consider initial data consisting of a small mass and a faraway large mass flock.  
The aim is to illustrate that, in solutions to the Cucker-Smale system, the small flock proceeds linearly as if there were no force on it (Q_cs);
and that, in solutions to the -model (with _0 = 1/(ρ_0)_ϕ), the small flock aligns to the average velocity of the small flock, independently of the large flock (Q_). 

The parameters of the experiment are as follows. 

  
  * The scalar strength of the alignment force is λ = 10.
  
  * ϕ(r) = 1/(1 + r^2)^80/2.
  
  * ρ_0^N is identical in the Cucker-Smale and the -model simulation.  It is shown in Figure <ref> (and Figure <ref>) as the leftmost picture. 

The kernel is periodized so that the distance r measures the distance on ^2. 
The large mass flock is indicated by red particles and 
the small mass flock is indicated by green particles. 
Each red particle has 100 times the mass of a green particle. 





We see that the small agents in the Cucker-Smale case proceed linearly while the small agents in -model case align to the average velocity of the small flock.  
This is the desired qualitative behavior.





 §.§ Relevant quantities and Notation

Let us introduce relevant quantities and notation. 
^n is the n-dimensional Torus.  
_M(Ω) denotes the set of measures on Ω with total mass M = ∫_Ωρ(x).  
R > 0 is a bound on the maximum velocity of the initial flock, i.e. μ_0 ⊂^n × B_R 
where B_R is the ball of radius R in ^n centered around zero. 


C^k is the space of k continuously differentiable functions with the usual norm f_C^k = ∑_i=0^k f_C^i.
As in the introduction, we will write f_ϕ := f ∗ϕ to denote convolutions. 
Subscripts - and + will be used as a shorthand for infima and suprema. For instance, f_- = inf_x ∈^n f(x), f_+ = sup_x ∈^n f(x). 
We will use (f_1, f_2) = ∫_^n f_1 f_2 to denote the L^2 inner product and (f_1, f_2)_h = ∫_^n f_1 f_2 h to denote the 
weighted L^2 inner product.  As per the notation in <cit.>, we will use the notation κ_ρ = ρ.
For instance, (·, ·)_κ_ρ denotes the L^2 inner product with respect to the measure κ_ρ. 
When a constant C depends on a parameter α, for instance, we will write C := C(α). 


Finally, let us define J, which will be a key quantity in the inheritance of regularity of the strength.  

  
  We will use J to denote the maximum L^1 norm of the velocity with respect to the density measure on the time interval [0,T]:
  
    J := sup_t ∈ [0,T](̆t, ·)_L^1(ρ) = sup_t ∈ [0,T]∫_Ω (|| ρ)(t,x)  .







 §.§ Outline


The remaining goal of this paper is to establish the passage between the microscopic, mesoscopic, and macroscopic levels of descriptions of the -model
and to address the relaxation to the Maxwellian in one dimension for the mesoscopic description.   

A key observation we make in this paper is that, when J < ∞ (given in Definition <ref>) and  satisfies the integral representation (<ref>),
the regularity conditions needed on  can be subordinated to regularity conditions on the kernel Φ_ρ(x,y), (<ref>) and (<ref>).  
That is,  and  inherit regularity from the kernel Φ_ρ.
This inherited regularity resembles the 'uniform regularity' conditions outlined in <cit.> for the environmental averaging models.








The only result considered in the paper for which we do not have inheritance of regularity is the Maxwellian limit. 
There, we instead resort to a set of continuity conditions on the strength and the velocity averaging, (<ref>)-(<ref>), which are stated in Section <ref>. 
Notably, these continuity conditions hold a priori for the -model when the communication kernel ϕ is bounded away from zero.  
The -model therefore serves as an important example where all of our results hold.   






The rest of the paper will be organized as follows. 
In Section <ref>, we will show that, when J < ∞, the velocity averaging and strength inherit the regularity of the kernel.  
To prepare our arguments for the passage from the microscopic to macroscopic system for the -model, we will survey the results related to the passage for the 
environmental averaging models, but catered to the Cucker-Smale model, in Sections <ref> and <ref>. 
Indeed, the mathematical tools and arguments used in the passage for the -model follow a similar outline to those used for the environmental averaging models, 
but adapted to accommodate for the transport of the strength.  
In preparation for the mean field limit, the well-posedness of the microscopic -model (<ref>) is established in Section <ref>. 
The mean field limit is proved in Section <ref>.  The hydrodynamic limits are proved in Section <ref>. 
Finally, in Section <ref>, we establish the relaxation to the Maxwellian in 1D for the mesoscopic -model (<ref>) provided the variation of the weight is small. 






 §.§ Assumptions


Our results will be stated for the torus ^n.  
Letting ρ, ρ', ρ”∈_M(^n), we assume throughout the paper, unless stated otherwise, that  has the integral representation (<ref>) 
and that its reproducing kernel Φ_ρ(x,y) ≥ 0 satisfies the following uniform regularity assumptions: 

    Reg1   ∂_x,y^k Φ_ρ_∞≤ C(k,M) 
    Reg2   ∂_x,y^k (Φ_ρ'- Φ_ρ”) _∞≤ C(k,M) W_1(ρ', ρ”),

where W_1 is the Wasserstein-1 distance.

In order to guarantee that  is an averaging operator, we also assume that Φ_ρ is right stochastic:

    ∫_^nΦ_ρ(x,y) ρ(y)  = 1.




  Table <ref> lists models for which  satisfies the integral representation (<ref>). 
  All of these models– namely, M_CS, M_MT, M_ϕ, and M_seg– have kernels Φ_ρ(x,y) which satisfy (<ref>) and (<ref>) when ϕ≥ c > 0 and is smooth.



  The Maxwellian limit is the only result for we which these assumptions do not guarentee that the strength inherits the regularity of the kernel. 
  There, we instead introduce a set of continuity assumptions, (<ref>)-(<ref>), which are verified for the -model when ϕ≥ c > 0. 









§ INHERITED REGULARITY FROM THE KERNEL


Our main assumptions (stated in Section <ref>) together with J < ∞ (given in Definition <ref>)  
imply that the velocity averaging and the strength inherit regularity from kernel. 
The inherited regularity is recorded in Proposition <ref> and Proposition <ref> 
below and will be the key ingredient to establishing the mean field limit and the monokinetic limit. 

 (Inherited regularity of the velocity averaging)
  
  Suppose that the velocity averaging has the integral representation (<ref>) and 
  that the kernel satisfies the uniform regularity assumptions (<ref>) and (<ref>). 
  Let ρ_0, ρ'_0, ρ”_0 ∈_M(^n).  
  If J < ∞, then for all k ≥ 0:
  
    $̆Reg1   ∂^k _∞≤ C_1 
    $̆Reg2   ∂^k ( - )_∞≤ C_1 W_1(ρ', ρ”) +  C_2  W_1('̆ρ', ”̆ρ”), 
      
      whereC_1 := C_1(k, M, J)andC_2 := C_2(k, M). 
    
      For (<ref>), place all of the derivatives on the kernel and use 
      
      For (<ref>), we write 
      
        -    = ∫_^nΦ_ρ'(x,y) '̆(y) ρ'(y)  - ∫_^nΦ_ρ”(x,y) ”̆(y) ρ”(y) 
           = ∫_^n (Φ_ρ'(x,y) - Φ_ρ”(x,y)) ('̆ρ')(y)   +  ∫_^nΦ_ρ”(x,y) ( ('̆ρ')(y) - (”̆ρ”)(y))  .
    
      Once again, placing the derivatives on the kernel and using J < ∞
      on the first term, we arrive at (<ref>). 
    J < ∞ holds a priori for the microscopic and mesoscopic -model, (<ref>) and (<ref>), 
      due to the maximum principle on the velocity.  
      The maximum principle also holds for the mesoscopic -model with the strong local alignment force for the monokinetic limiting regime, (<ref>)
      (the justification is provided in Section <ref>). 
      However, for the mesoscopic -model with strong Fokker-Planck penalization force, (<ref>), there is no control on J
      and, therefore, there is no inheritance of regularity from the kernel. 
    
    
    The strength subsequently inherits regularity from the velocity averaging. 
     (Inherited regularity of the strength)
        
        Suppose that the uniform regularity conditions on the velocity averaging (<ref>) and (<ref>) hold. 
        Let ρ_0', ρ_0”∈_M(^n). 
        If ', ”∈ C([0,T]; C^k+1(^n)) with _0' = _0” where ' solves ∂_t ' + ∇_x · (' ) = 0 and 
        ” solves ∂_t ” + ∇_x · (”) = 0, 
        then for all k ≥ 0:
        
        Reg1   ∂^k ' _∞≤ C 
        Reg2   ∂^k (' - ”) _∞≤ C ∂^k (s_0' - s_0”)_∞ +  C sup_t ∈ [0,T]( W_1(ρ', ρ”) +  W_1('̆ρ', ”̆ρ”) ) (t),
    
        where C := C(k, M, J, T). 
        
            We only prove (<ref>) since (<ref>) follows a similar computation.  
            Taking the k^th partial derivative of the strength equation, we get
            
        ∂_t ∂^k(' - ”) + ∇·∂^k (' - ”) = 0.
    
            This can be written as
            
        ∂_t ∂^k(' - ”) + ∇·∂^k ( ( - ) ' + (' - ”) ) = 0.
    
            Evaluating at a point of maximum of ∂^k (' - ”), the term ∂^k+1 (' - ”) drops out. 
            We obtain for some constant A depending on k: 
            
        ∂_t ∂^k (' - ”)_∞≤ A(k) (  - _C^k+1'_C^k+1 + _C^k+1' - ”_C^k).
    
            Summing over k, we obtain for some constant A' depending on k: 
            
        ∂_t ' - ”_C^k≤ A'(k) (  - _C^k+1'_C^k+1 + _C^k+1' - ”_C^k).
    
            Applying (<ref>), (<ref>), and (<ref>) along with Gronwall's inequality yields constants C_1'(k,M,J), C_2'(k,M,J,T) such that 
            
        ' - ”_C^k≤ e^C_1' Ts_0' - s_0”_C^k + C_2' e^C_1' Tsup_t ∈ [0,T]( W_1(ρ', ρ”) +  W_1('̆ρ', ”̆ρ”) ) (t,·)
    
            Setting C = max{ C_2' e^C_1' T,  e^C_1' T} gives (<ref>). 
        
    
    § PASSAGE FROM MICROSCOPIC TO MACROSCOPIC FOR CUCKER-SMALE
    
    It will be helpful to survey the mean field and hydrodynamic arguments for the Cucker-Smale model in order to understand the extension of our argument to the-model. 
    Indeed, these limiting arguments for the-model fit into the same framework as the arguments used in the Cucker-Smale case. 
    
    Each section will outline the argument for the Cucker-Smale case and conclude with the statement of the corresponding theorem for the-model.  
    The mean field and hyrdoynamic limits for the-model are stated in Theorems <ref>, <ref>, and <ref>. 
    The proofs are in Section <ref> and <ref>. 
    
    
    
    
    
    
    
     §.§ Mean field limit (Cucker-Smale survey)
    
    We will include details for pieces of the argument that do not depend (in a significant way) onand leave the pieces which depend onfor Section <ref>. 
    To rigorously pass from the discrete system (<ref>) to the kinetic one (<ref>), we would like to show that 
    solutions to (<ref>) are unique and that the solution arises as a limit, in some sense, of solutions to the discrete equation (<ref>).
    To make sense of discrete solutions converging to the kinetic one, the solutions to (<ref>) and (<ref>) must lie in the same space.  
    We therefore consider measure-valued solutions to both systems.  For (<ref>), a measure-valued solution is given by the empirical measure 
    
        μ^N_t(x,v) = ∑_i=1^N m_i δ_x_i(t)⊗δ_v_i(t)
    
    where(x_i(t), v_i(t))_i=1^Nsolve (<ref>). To make sense of measure valued solutions for (<ref>), we define a weak solution as follows. 
    
      Fix a time T > 0 and an integer k ≥ 0. We say 
      μ∈ C_w^*([0,T]; _M(^n ×^n)) is a weak solution to (<ref>) 
      if for all g(t,x,v) ∈ C_0^∞([0,T] ×^n ×^n) and for all 0 < t < T, 
      
        ∫_^n ×^n g(t,x,v) dμ_t(x,v)    = ∫_^n ×^n g(0,x,v) dμ_0(x,v) 
           + ∫_0^t ∫_^n ×^n (∂_τ g + v ·∇_x g + λ F(μ_τ) ·∇_v g  dμ_τ(x,v) .
    
    Observe that the empirical measure (<ref>) is a solution to (<ref>) if and only if(x_i(t), v_i(t))solve the discrete system (<ref>).
    We will always assume thatμ_0⊂^n × B_Rfor some fixedR>0. 
    We now aim to show the existence and uniqueness of weak solutions and that they arise as weak limits of these empirical measures (<ref>), where the 
    weak limit can be topologized by the Wasserstein-1 metric:
    
        W_1(μ, ν) = sup_Lip(g) ≤ 1| ∫_^n ×^n g(ø) (dμ(ø) - dν(ø)) | .
    
    The Wasserstein-1 distance metrizes the weak topology as long as the measures lie on some common compact set.
    Owing to the maximum principle on the velocity equation, this is the case here:μ_t⊂^n × B_Rfor allt ∈ [0,T).  
    
    The goal will be to establish a stability estimate in the Wasserstein-1 metric:  forμ_t', μ_t”∈_M(^n × B_R),
    there exists a constantC(M,R,T)such that 
    
        W_1 (μ_t', μ_t”) ≤ C(M,R,T) W_1(μ_0', μ_0”) .
    
    For then, a Cauchy sequenceμ_0^NwithW_1(μ_0^N, μ_0) → 0yields a Cauchy sequenceμ_t^NwithW_1(μ_t^N, μ_t) → 0for someμ_t ∈ C_w^*([0,T]; (^n ×^n)). 
    Thatμis a solution to (<ref>) will follow from taking limits.
    
    Stability in the Wasserstein-1 metric is tied to the stability of the characteristic flow of (<ref>).
    
        X(t,s,x,v) = V(t,s,x,v),  X(s,s,x,v) = x, 
         V(t,s,x,v) = λ F(μ_t)(X,V),  V(s,s,x,v) = v .
     
    Indeed, letX(t,x,v) := X(t,0,x,v)andV(t,0,x,v) = V(t,x,v)and(x,v) = ø.  Givenh ∈ C_0^∞(^2n), 
    define the test functiong(s,ø) = h(X(t,s,ø), V(t,s,ø)). Then 
    
        ∂_s g + v ·∇_x g + λ F(μ_s) ·∇_v g = 0 .
    
    Pluggingginto (<ref>), we see thatμ_tis the push-forward measure ofμ_0along the characteristic flow(X,V): 
    
        ∫_^n ×^n h(X(t,ø), V(t,ø)) _0(ø) = ∫_^n ×^n h(ø) _t(ø) .
    
    LettingX' := X'(t,ø),V' := V'(t,ø)and similarly forX”,V”, we have for anyh ∈ Lip(^n)withLip(h) ≤ 1, 
    
        ∫_^n ×^n h(ø) (dμ_t' - dμ_t”) 
                 = ∫_^n ×^n h(X', V') dμ_0'  
                -∫_^n ×^n h(X”, V”) dμ_0”) 
           = ∫_^n ×^n h(X', V') (dμ_0' - dμ_0”) 
           +∫_^n ×^n (h(X', V') - h(X”, V”)) dμ_0”
           ≤ (∇ X'_∞ + ∇ V'_∞) W_1(μ_0', μ_0”)  
                + X' - X”_∞ + V' - V”_∞ .
    
    Therefore, 
    
        W_1(μ_t', μ_t”) 
               ≤ (∇ X'_∞ + ∇ V'_∞) W_1(μ_0', μ_0”)  
            + X' - X”_∞ + V' - V”_∞ .
    
    The stability estimate (<ref>) reduces to establishing the following stability estimates on the characteristic flow:  
    
        ∇ X _∞ + ∇ V _∞≤ C(M,R, T) 
           X' - X”_∞ + V' - V”_∞≤ C(M, R, T) W_1(μ_0', μ_0”) .
    
    This in turn implies the Wasserstein-1 stability (<ref>).  Since these estimates will depend on the 
    regularity of the strength, we will address these details for the-model in Lemmas <ref> and <ref>. 
    
    
    
    
    Wasserstein-1 stability implieslim_N →∞ W_1(μ_t^N, μ_t) = 0for eacht ∈ [0,T]and for someμ_t ∈(^n ×^n). 
    The last piece is to showμ∈ C_w^*([0,T]; (^n ×^n))is a weak solution to (<ref>). 
    The weak convergenceW_1(μ_t^N, μ_t) → 0immediately implies that the linear terms in (<ref>) converge.  
    For the non-linear term, the strength enters so we will address this for the-model later in Lemma <ref>. 
    
    We record the corresponding mean field limit theorem for themodel, which will be proven in Section <ref>. 
     (-model: Mean Field Limit)
      
      Suppose that the velocity averaging  has the integral representation (<ref>) with 
      the kernel satisfying (<ref>) and (<ref>). 
      Given T>0, R>0, k ≥ 0,  μ_0 ∈_M(^n × B_R), and _0 ∈ C^∞(^n), then there exists a unique 
      weak solution (μ, ) ∈ C_w^*([0,T]; _M(^n × B_R)) × C([0,T]; C^k(^n)) to (<ref>).  
      Moreover, the solution can be obtained as a limit of empirical measures (<ref>), μ_t^N, with corresponding 
      strength functions _t^N: If _0^N = _0 and the empirical measures μ_0^N are constructed from agents (x_i^0,v_i^0) ∈^n ×^n
      with total mass M = ∑_i=1^N m_i = ∫_^n ×^n dμ_0^N(x,v) in such a way that W_1(μ^N_0, μ_0) → 0, then 
      
      * sup_t ∈ [0,T] W_1(μ^N_t, μ_t) → 0, and 
        
      * sup_t ∈ [0,T]∂^k (_t^N - _t)_∞→ 0 for any k ≥ 0 where  solves the transport equation in (<ref>), ∂_t  + ∇· () = 0.
      
      The strength  enters in the stability estimates on the characteristic flow and in showing the limits (a) and (b).  
      The inherited regularity from the kernel will play a crucial role in each of these. 
    
    
     §.§ Hydrodynamic limits (Cucker-Smale survey)
    
    Once again, we will present the core arguments of the hydrodynamic limits for the Cucker-Smale case which do not depend (in a significant way) onand leave the pieces which depend onfor Section <ref>. 
    
    To resolve the closure problem for (<ref>), a strong local alignment force,F_la, is added to the kinetic Vlasov equation,
    
        ∂_t  + v ·∇_x  + ∇_v · [ F()] + 1/ϵ F_la() = 0 .
    
    Under the right local alignment forceF_la, the corresponding macroscopic system formed from taking thev-moments will lose its dependence
    on the kinetic solutionasϵ→ 0. 
    We will consider two such local alignment forces,F_la, corresponding to the monokinetic regime, where the distribution is forced to be concentrated around the macroscopic velocity$̆; 
    and maxwellian regime, where the distribution is forced to be a Gaussian distribution centered around the macroscopic velocity $̆.  
    We will denote the macroscopic density and momentum corresponding to the distributionbyand: 
    
        (t,x) = ∫_^n(t,x,v) ,  ()(t,x) = ∫_^n v (t,x,v)  .
    
    
      §.§.§ Monokinetic regime
    
    Under the monokinetic local alignment force,
    
        F_la = F_mono = ∇_v [  ( - v) ] = 0,
    
    the probability densityis forced to the monokinetic ansatz 
    
        f(t,x,v) = ρ(t,x) δ(v - (̆t,x))
    
    where(ρ, )̆solves:
    
    
        ∂_t ρ + ∇· (ρ̆) = 0 
        ∂_t (ρ)̆ + ∇· (ρ⊗̆)̆ = ρ ((ρ̆)_ϕ - ρ̆_ϕ) .
    
    If we know a priori that the solution remains non-vacuous, i.e.ρ > 0, then we can divide the momentum equation byρin order to rewrite it as an equation on the velocity$̆.  We arrive at the pressureless Euler alignment system from the introduction (<ref>).
    The system (<ref>) has the advantage that the velocity equation obeys the maximum principle (as does (<ref>)): _L^∞≤_̆0_L^∞, 
    which lends it to alignment analysis.  For this reason, it has received a lot of attention in the literature.
    
    This monokinetic limit was first proved by Figalli and Kang for non-vacuous solutions ρ > 0 on the torus ^n in <cit.>. 
    It was later improved to allow vacuous solutions on the open space ^n in <cit.> 
    and extended to environmental averaging models in <cit.>. 
    
    
    
    
    
    
    There, the local aligment is modified to force the system to a special averaged velocity field instead of the 
    rough velocity field .  
    
        F_la = F_mono reg = ∇_v [  (_δ - v) ] = 0
    
    where _δ is a special mollification given by 
    
        _̆δ = ( (ρ̆)_Ψ_δ/ρ_Ψ_δ)_Ψ_δ
    
    for some smooth mollifier Ψ_δ(x) = 1/δ^nΨ(x/δ).
    This special mollification has the key approximation property that _̆δ is close to $̆ for smallδwith a bound independent ofρ. 
    The following approximation lemma can be found in <cit.>.
    
        For any ∈̆Lip and for any 1 ≤ p < ∞, one has 
        
        _̆δ - _L^p(ρ)≤ C δ_Lip
    
        where C > 0 depends only on Ψ and p. 
    
    We will be interested in extending this argument with the local alignment forceF_mono regfrom the 
    environmental averaging models to the-model, see Theorem <ref>.  
    
    The convergence to the monokinetic ansatz is measured in the Wasserstein-2 metric, which is given by 
    
        W_2^2(f_1,f_2) = inf_γ∈∏(f_1,f_2)∫_^2n×^2n |w_1 - w_2|^2 γ(w_1, w_2) .
    
    The modulated kinetic energy,e( | )̆, which will be crucial in controllingW_2^2(_t, f_t), is defined by: 
    
        e( | )̆ = ∫_^n ×^n | v - (̆x)|^2 (x,v)  .
    
    Let us state the theorem for the Cucker-Smale model.
     (Cucker-Smale: Monokinetic Limit)
      Let (ρ, )̆ be a classical solution to (<ref>) on the time interval [0,T] and let f be the 
      monokinetic ansatz (<ref>). 
      Suppose that f_0^ϵ∈ C_0^k(^n ×^n) is a family of initial conditions satisfying: 
      
      (i)f_0^ϵ⊂^n × B_R for any fixed R>0, and 
          
      (ii)W_2(f_0^ϵ, f_0) < ϵ
      Then there exists a constant C(M,R,T) such that for all t ∈ [0,T] one has 
      
        W_2(f_t^ϵ, f_t) ≤ C √(ϵ + δ/ϵ) .
    
    
    In order to controlW_2(f_t^ϵ, f_t), it suffices to control  
    
        ∫_^2n×^2n |w_1 - w_2|^2 γ(w_1, w_2)
    
    for a particularγ_t ∈∏(f_t^ϵ, f_t).  The natural choice forγ_tis given by the flow 
    
        ∂_t γ + v_1 ·∇_x_1γ + v_2 ·∇_x_2γ 
              + ∇_v_1 [ γ (v_1 - ) + 1/ϵ (v_1 - _δ) ]
              + ∇_v_2 [ γ (v_2 - _F) ] = 0,
    
    where = ()_ϕ / _ϕ. 
    Sinceγ_t ∈∏(f_t^ϵ, f_t), 
    
        W := ∫_^2n×^2n |w_1 - w_2|^2 γ_t(w_1, w_2) ≥ W_2^2(_t, f_t)
    
    So, we aim to controlW.  We split it into the potential and kinetic components 
    
        W = ∫_^2n×^2n |v_1 - v_2|^2 γ_t + ∫_^2n×^2n |x_1 - x_2|^2 γ_t := W_v + W_x
     
    ForW_x, we have 
    
        d/dt W_x ≤ W_x + W_v .
    
    ForW_v, we have 
    
        W_v 
               ≤∫_^2n×^2n |v_1 - (̆x_1)|^2 γ_t + ∫_^2n×^2n |(̆x_1) - (̆x_2)|^2 γ_t + ∫_^2n×^2n |(̆x_2) - v_2|^2 γ_t 
           ≤∫_^n ×^n | v - (̆x)|^2 (x,v)  + C ∫_^2n×^2n |x_1 - x_2|^2 γ_t 
           = e( | )̆ + C W_x .
    
    We obtain 
    
        d/dt W_x ≲ e( | )̆ + W_x 
        
            W_v ≲ e( | )̆ + W_x .
    
    Expandinge( | )̆, we get
    
        e( | )̆ = _ϵ - ∫_^n·+̆1/2∫_^n ||̆^2 ,
    
    where_ϵis the kinetic energy: 
    
        _ϵ = 1/2∫ |v|^2 (x,v) .
    
    Under no assumptions on the strength, it is shown in <cit.> that 
      
        ≲ + 1/ϵ∫_^n (_δ - ) ·
           + ( - ,̆ - )_κ_ + ∫_^n(-̆) · ( - )̆ .
    
    The local alignment term is controlled with the help of Lemma <ref>, see <cit.>: 
    
        1/ϵ∫_^n (_δ - ) ·≲δ/ϵ .
    
    So, it remains to estimate the alignment terms 
    
        A := ( - ,̆ - )_κ_ + ∫_^n(-̆) · ( - )̆ .
    
    The estimate onAdoes depend on the regularity of the strength and it is addressed in Theorem <ref>, which is proved in Section <ref>.
    
     (-model: Monokinetic Limit)
      
      Suppose the velocity averaging has the integral representation (<ref>) 
      with a kernel Φ satisfying the regularity conditions (<ref>) and (<ref>). 
      Let (ρ, , )̆ be a smooth solution to (<ref>) on the time interval [0,T] with mass M. 
      Let f = ρ(t,x) δ(v - (̆t,x)). 
      Suppose _0 ∈ C^∞(^n), f_0^ϵ∈ C_0^k(^n ×^n) is a family of initial conditions satisfying:
      
      (i)f_0^ϵ⊂^n × B_R, for any fixed R>0. 
          
      (ii)W_2(f_0^ϵ, f_0) < ϵ
      (iii)_0 = _0 ∈ C^∞(^n)  . 
      
      Then there exists a constant C(M,R,T) such that for all t ∈ [0,T]: 
      
        W_2(f_t^ϵ, f_t) ≤ C √(ϵ + δ/ϵ) .
    
      As a consequence of this and Proposition <ref>, sup_t ∈ [0,T]( - )(t,·)_∞→ 0. 
    
    
    
    
    
      §.§.§ Maxwellian regime
    
    Let us rewrite the equation for the macroscopic density and momentum (<ref>) as 
    
        ∂_t ρ + ∇· (ρ̆) = 0 
        ∂_t (ρ)̆ + ∇· (ρ⊗̆)̆ + ∇ρ + ∇· = ρ ( (ρ̆)_ϕ - ρ̆_ϕ )
    
    where 
    
        (t,x) = ∫_^n ((v-)̆⊗ (v - )̆ - 𝕀) f  .
    
    Under the local alignment force, 
    
        F_la = F_max reg =  Δ_v  +  ∇_v · ((v - _δ) ),
    
    where_δis given in (<ref>),  
    the probability densityis forced to the Maxwellianμ(t,x), 
    
        μ(t,x) = ρ(t,x)/(2π)^n/2 e^|v - (̆t,x)|^2/2,
    
    where(ρ, )solve the isentropic Euler alignment system: 
    
    
    
    
    
    
    
    
    
    
    
    
    
    
        ∂_t ρ + ∇· (ρ̆) = 0 
        ∂_t (ρ)̆ + ∇· (ρ⊗̆)̆ + ∇ρ = ρ ( (ρ̆)_ϕ - ρ̆_ϕ ) .
    
    As in the monokinetic limit, we are interested in extending this argument with the local alignment forceF_max regfrom the 
    environmental averaging models to the-model, see Theorem <ref>.  
    The distance betweenand the Maxwellianμis controlled by the relative entropy( | μ), which is defined as:
    
        ( | μ) = ∫_^n ×^nlog/μ .
    
    Due to the classical , for some constantc > 0, 
    
        c  - μ_L^1(^n ×^n)≤( | μ),
    
    it suffices to show that( | μ) → 0. We state the theorem for the Cucker-Smale model. 
    
     (Cucker-Smale: Maxwellian limit)
      
      Let (ρ, )̆ be a smooth, non-vacuous solution to (<ref>) on the torus ^n and on the time interval [0,T]. 
      Suppose that f_0^ϵ∈ C_0^k(^n ×^n) is a family of initial conditions satisfying 
      
        lim_ϵ→ 0(_0 | μ_0) = 0 .
    
      Then for δ = o(ϵ), 
      
        sup_t ∈ [0,T]( | μ) → 0 .
    
    Breaking the relative entropy into the kinetic and macroscopic parts, we have
    
        ( | μ) = _ϵ + _ϵ, 
           _ϵ = ∫_^n ×^n( log + 1/2 |v|^2 )  + n/2log(2π), 
           _ϵ = ∫_^n( 1/2 ||̆^2 - ·-̆logρ)   .
    
    We then seek to estimate_ϵand_ϵ.  
    Let us define the Fisher information,, which will be relevant for these estimates:
    
        = ∫_^n ×^n | ∇_v  + (1 + ϵ /2) (v - )  |^2 / .
    
    The following identities are from <cit.>: 
    
        = 
               -1/ϵ∫_^n ×^n[ |∇_v |^2/ + 2 ∇_v · (v - _δ) + |v - _δ|^2 ] 
           - 1/ϵ [(_δ, )_ - (_δ, _δ)] 
           -∫_^n ×^n [ ∇_v · (v - ) + v · (v - )  ]
    
    and 
    
        = 
               ∫_^n [ ∇:̆ -  ( - )̆·∇·̆( - )̆ ] 
            + 1/ϵ∫_^n (_δ - ) ·
           + _L^2()^2 - (, )_ + δ A  .
    
    whereδ Ais the same alignment term used in the proof of the monokinetic case.  
    From (<ref>), dropping the first two terms, we obtain 
    
        d/dt≤ -∫_^n ×^n [ ∇_v · (v - ) + v · (v - )  ] 
        
          = ∫_^n ×^n  -∫_^n ×^n |v|^2  + (, )_κ^ϵ .
    
    This estimate will be used to show thatis bounded which will in turn be used to control the Reynolds stress termin 
    the equation for.  All that is needed to complete this estimate is the boundedness of, which will indeed hold due to the 
    inherited regularity of the kernel.  We will nonetheless delay this detail until the proof of Theorem <ref> 
    (in order to isolate the pieces of the argument which depend on the regularity of the strength).  
    A stronger estimate onis achieved by retaining the first dissipative term in (<ref>). 
    From <cit.>, the first term and the last term in (<ref>) combine to control the Fisher information and we obtain
    
        d/dt≤   -1/ϵ + ϵ/4∫_^n ×^n |v - |^2  
              - _L^2(κ_ϵ)^2 + (, )_κ^ϵ .
    
    Turning to the terms in (<ref>) which do not depend on the strength, we have due to the assumed smoothness of$̆, 
    
        | ∫_^n ( - )̆·∇·̆( - )̆| 
            ≲∫_^n | - |̆^2 ≤( | )̆ .
    
    The local alignment term is the same as in the monokinetic case and is handled the same way with the help of Lemma <ref>, see <cit.>: 
    
        1/ϵ∫_^n (_δ - ) ·≲δ/ϵ .
    
    The remaining estimates depend on the regularity of the strength and will be addressed in the proof of Theorem <ref>.  
    The Fokker-Planck penalization force destroys any uniform control on the momentum uniformly in ϵ so the strength will not inherit the regularity of the kernel in this case.  
    Instead, we identify a set of continuity conditions (<ref>)-(<ref>) which are sufficient for proving the Maxwellian Limit.  
    Notably, these conditions hold for the -model, which we show (along with the proof of Theorem <ref>) in Section <ref>. 
    
     (-model: Maxwellian Limit)
      
      
      Suppose the continuity conditions (<ref>)-(<ref>) hold. 
      Let (ρ, , )̆ be a smooth, non-vacuous solution to (<ref>) on the torus ^n and on the time interval [0,T] with mass M. 
      Suppose also that _0 ∈ C^∞(^n) and f_0^ϵ∈ C_0^k(^n ×^n) is a family of initial conditions satisfying 
      
      (i)(_0 | μ_0) → 0 as ϵ→ 0,  
          
      (ii)_0 = _0 ∈ C^∞(^n).
      
      Then for δ = o(ϵ), 
      
        sup_t ∈ [0,T]( | μ) → 0.
    
      As a consequence of this and Proposition <ref>, sup_t ∈ [0,T]( - )(t,·)_∞→ 0. 
      
    
    
    § WELL-POSEDNESS OF THE DISCRETE-CONTINUOUS -MODEL
    
    The well-posedness of the discrete-continuous system must be established before addressing the mean field passage to the kinetic system. 
    This section is devoted to proving global existence and uniqueness of solutions  to (<ref>) in class 
    
        ({x_i(t)}_i=1^N, {v_i(t)}_i=1^N, (t)) ∈ C([0,T]; ^nN×^nN× H^k(^n)),     k > n/2 + 2.
    
    The choice of k > n/2 + 2 guarantees that ∂^2 _∞≲_H^k for an arbitrary second order partial derivative ∂^2 by the Sobolev Embedding Theorem. 
    
    Let us recall our main assumptions which will be relevant here: the velocity averaging satisfies the integral representation (<ref>) and the kernel Φ_ρ^N is smooth.  
    Written in the terms of the discrete empirical data ρ^N, ^̆N, the velocity averaging becomes: 
    
        (t,x) =  ∑_j=1^N m_j Φ_(ρ^N)(x, x_j) v_j(t),  (ρ^N)(t,x) = ∑_j=1^N m_j δ_x_j(t) .
    
    From Proposition <ref>,  inherits the regularity of the kernel and, in particular, satisfies (<ref>). 
    For the remainder of this section, we will use C := C(k,M) to denote a constant depending on k and M.  
    For instance, the inherited regularity of velocity averaging (<ref>) can be written:  ∂^k _∞≤ C.  The constant C may change line by line. 
    Consider the following regularized version of (<ref>):
    
        ẋ_̇i̇ = v_i 
        v̇_̇i̇ = λ(x_i) ((x_i) - v_i ) 
        ∂_t  + ∇_x · () = ϵΔ .
    
    For the moment, we avoid writing the explicit dependence of x_i, v_i, on ϵ for the sake of brevity.  They are not to be confused with solutions 
    to the unregularized system (<ref>).  
    Let X = C([0,T]; ^2nN× H^k(^n)) and define Z(t) := ({x_i(t)}_i=1^N, {v_i(t)}_i=1^N, (t)). 
    The norm ·_X is given by:
    
        Z_X = sup_t ∈ [0, T]max_i = 1 … Nx_i(t) + v_i(t) + (t,·) _H^k(^n).
    
    Define the map  by 
    
        ( Z(t) ) =   
            [       1;       1; e^ϵ t Δ ] Z_0 
          + ∫_0^t  
           [           1;           1; e^ϵ (t-τ) Δ ] 
          A(τ) dτ,
    
    where A represents all of the non-laplacian terms.  Existence and uniqueness of solutions to (<ref>) amounts to showing 
    that : B_1(Z_0) ↦ B_1(Z_0) and that it is a contraction mapping, where B_1(Z_0) is the ball of radius 1 centered at Z_0 in X.
    Contraction will follow similarly from invariance. 
    For invariance, we aim to show that: 
    
        ( Z(t) ) - Z_0 _X ≤ e^ϵ t Δ Z_0 - Z_0 _X + ∫_0^t e^ϵ (t-τ) Δ A( Z(τ) ) dτ_X ≤ 1 .
    
    The first term is small for small T due to the continuity of the heat semigroup.  
    For second term, we will treat each component individually. 
    The x_i-component gives 
    x_i(t) - x_i(0)≤ T max_i v_i(0) .
    For the v_i-component, we have 
    
        v_i(t) - v_i(0) ≤ T _∞ (_∞ + max_i v_i(0)) ≤ C T (Z_0 + 1) .
    
    For the -component, we will use the analyticity property of the heat semigroup: 
    
        ∇ e^ϵ t Δ f_L^2≤1/√(ϵ t)f_L^2
    
    along with the product estimate to get 
    
        ∫_0^t e^ϵ (t-τ) Δ∇_x · ()  dτ_H^k   ≤2 T^1/2/ϵ^1/2sup_t ∈ [0,T]_H^k
           ≤2 T^1/2/ϵ^1/2sup_t ∈ [0,T] ( _∞_H^k + _H^k_∞) 
           ≤2 T^1/2/ϵ^1/2 (Z_0_X + 1) .
    
    For small enough T, we have invariance.  Contractivity of  follows from similar estimates. 
    This time interval of existence T could depend on ϵ. 
    To establish that the there is a common time interval of existence independent of ϵ, we establish an ϵ-independent energy estimate.
    For the x_i and v_i components, the estimates follow easily from the invariance estimates.  We record them here. 
    x_i(t)≤x_i(0) + t max_i v_i(0),  
    and
    
        v_i(t)≤v_i(0) + t _∞ (_∞ + max_i v_i(0)) ≤ C( 1 + t _H^k ).
    
    For the strength, we multiply (<ref>) by ∂^2j and integrate by parts. We obtain for any 0 ≤ j ≤ k, 
    
        d/dt_Ḣ^j   = ∫∇_x · |∂^j |^2  - ∫ (∂^j (·∇_x ) - ·∇_x ∂^j ) ∂^j 
           - ∫∂^j ((∇_x ·)  ) ∂^j  - ϵ∫ |∂^j ∇|^2  .
    
    Dropping the ϵ term coming from the Laplacian, we obtain
    
        _Ḣ^j^2 ≤ C ( _Ḣ^j^2 + _Ḣ^j∇_∞ + _Ḣ^j_∞), for all  0 ≤ j ≤ k.
    
    Since k > n/2 + 2, 
    
        _H^k^2 ≤ C _H^k^2.
    
    We will now denote the explicit dependencies on ϵ and take ϵ→ 0. 
    From Grownwall, we conclude that Z^ϵ(t) exists on a common time interval independent of ϵ. 
    Writing the equation for ()^2, we have 
    
        _L^2^2 ≤ C _H^1 + ϵ_H^2≤ C _H^k.
    
    By local well-posedness Z^ϵ∈ C([0,T]; R^2nN× H^k(^n)) so that d/dt Z^ϵ∈ L^2([0,T]; ^2nN× L^2(^n)).  
    By the Aubin-Lions lemma, we obtain a subsequence, which we denote again by Z^ϵ such that Z^ϵ→ Z^0 in C([0,T]; R^2nN× H^k-1(^n)). 
    Since H^k-1(^n) is dense in H^k(^n), we have Z^0 ∈ C_w([0,T]; ^2nN× H^k(^n)).  Finally, since k > n/2 + 2, the terms A^ϵ converge 
    pointwise to A. Taking ϵ→ 0 in the Duhamel formula (<ref>), we get 
    
        Z^0(t) = Z^0_0 + ∫_0^t A(τ) dτ .
    
    That is, Z^0 ∈ C_w([0,T]; ^2nN× H^k(^n)) solves (<ref>). 
    Finally, we note that due to the ϵ-independent energy estimate (<ref>), Z^0_X remains bounded for any finite time and thus exists for all time. 
    This concludes existence and uniqueness of solutions to (<ref>) on the global time interval [0, ∞). 
    
    
    
    
    
    
    
    § MEAN FIELD LIMIT
    
    We establish the passage from from the discrete system (<ref>) to the kinetic system (<ref>) in the mean field limit for the -model, Theorem <ref>.
    The outline for this argument is discussed in Section <ref> for the Cucker-Smale model.  We will skip the pieces of the argument which are covered there 
    (in particular, we will skip the pieces of the argument in which  has an insignificant role). 
    
    The empirical measures are given by: 
    
        μ^N_t(x,v) = ∑_i=1^N m_i δ_x_i(t)⊗δ_v_i(t),  (x_i(t), v_i(t), (t,x))_i=1^N   solve (<ref>).
    
    To make sense of measure valued solutions, we consider a weak version of (<ref>). 
    
      Fix a time T > 0 and an integer k ≥ 0. We say the pair (μ, ) with 
      μ∈ C_w^*([0,T]; (^n ×^n)) and ∈ C([0,T]; C^k(^n)) is a weak solution to (<ref>) 
      if for all g(t,x,v) ∈ C_0^∞([0,T] ×^n ×^n) and for all 0 < t < T, 
      
      
        ∫_^n ×^n g(t,x,v) dμ_t(x,v)    = ∫_^n ×^n g(0,x,v) dμ_0(x,v) 
           + ∫_0^t ∫_^n ×^n (∂_τ g + v ·∇_x g +  ( - v) ·∇_v g) dμ_τ(x,v) 
        ∂_t  + ∇_x · () = 0.
    
      In other words, μ solves the first equation weakly and the strength, , solves the second equation strongly. 
    
    As in the Cucker-Smale case, the empirical measure (<ref>) is a solution to (<ref>) if and only if (x_i(t), v_i(t), (t,x)) solve the discrete system (<ref>).
    The well-posedness of (<ref>) for empirical measure-valued solutions is therefore equivalent to the well-posedness of the discrete-continuous system (<ref>) established in Section <ref>.
    To show existence of general weak solutions to (<ref>), we will show that the weak solution arises a weak limit of the empirical measures (<ref>).
    The goal is to establish the Wasserstein-1 stability estimate (<ref>). 
    
    Stability in the Wasserstein-1 metric amounts to showing the stability estimates (<ref>) and (<ref>) on the characteristic flow: 
    
        X(t,s,x,v) = V(t,s,x,v),  X(s,s,x,v) = x 
         V(t,s,x,v) = (X) ((X) - V),  V(s,s,x,v) = v,
     
    which will be established in the following Lemmas. 
    Let us first note that the total momentum ∫_^nρ = ∫_^n ×^n v dμ_t(x,v) in (<ref>) is conserved (by plugging in g = v). 
    In particular, J<∞ and we can apply Propositions <ref> and <ref> and use the inherited regularity from the kernel. 
    By a similar argument to the Cucker-Smale case, we also have that μ_t is the push-forward of μ_0 along the characteristic flow: 
    
        ∫_^n ×^n h(X(t,ø), V(t,ø)) _0(ø) = ∫_^n ×^n h(ø) _t(ø),     ø = (x,v)
     (Deformation Tensor Estimates)
       
      Suppose that the velocity averaging has the integral representation (<ref>) 
      and that the kernel satisfies the uniform regularity assumptions (<ref>) and (<ref>). 
      
      Let (μ, ) be a weak solution to (<ref>) on [0,T] with characterstics X,V given in (<ref>).  
      Then
      
        ∇ X _∞ + ∇ V _∞≤ C(M,R,J,T).
    
        Differentiating (<ref>),
        
        ∇ X = ∇ V 
        ∇ V = ∇ X^T ∇(X) ((X) - V) + (X) ∇ X^T ∇(X) - (X) ∇ V.
    
        By the maximum principle on the velocity and the inherited regularity (<ref>), (<ref>), and (<ref>)
        (∇ V_∞ + ∇ X_∞) ≤ C'(M, R, J, T) ( ∇ X_∞ + ∇ V_∞ ) .
    
        We conclude by Gronwall. 
       (Continuity Estimates)
      
      Suppose that the velocity averaging has the integral representation (<ref>) 
      and that the kernel satisfies the uniform regularity assumptions (<ref>) and (<ref>). 
      
      Let (μ', '), (μ”, ”) be weak solutions to (<ref>) on [0,T];
      and let X', X”, V', V” be the corresponding characteristics given by (<ref>). Then  
      
        X' - X”_∞ + V' - V”_∞≤ C(M, R, J, T) W_1(μ_0', μ_0”).
    
        We have 
        
        (X' - V') = V' - V”
         (V' - V”) = '(X') ((X') - V') - ”(X”) ((X”) - V”)  .
    
        By the maximum principle on the velocity and inherited regularity conditions (<ref>), (<ref>), (<ref>), (<ref>), 
        
        V' - V”_∞   ≤'(X') - '(X”)_∞(X') - V' _∞ + '(X”) - ”(X”)_∞(X') - V' _∞
           + ”(X”)_∞(X') - V' - (X”) + V”_∞
           ≤ C(M,R,J,T) ( V' - V”_∞ + sup_t ∈ [0,T] (W_1(ρ', ρ”) + W_1('̆ρ',  ”̆ρ”))(t) ).
    
        Combining with X' - X”_∞≤V' - V”_∞, we get
        
        ( X'    - X”_∞ + V' - V”_∞) 
           ≤ C(M,R,J,T) ( X' - X”_∞ + V' - V”_∞ + sup_t ∈ [0,T] (W_1(ρ', ρ”) + W_1('̆ρ',  ”̆ρ”))(t) ) .
    
        To estimate W_1(ρ',ρ”) and W_1('̆ρ', ”̆ρ”), we use the fact that μ_t is the push-forward of μ_0, (<ref>). Fix g_Lip≤ 1. Since M' = M,
        
        ∫_^n g(x) (dρ_t' - dρ_t”) 
                       = ∫_^n ×^n g(x) (dμ_t' - dμ_t”) = ∫_^n ×^n g(X') dμ_0' - ∫_^n ×^n g(X”) dμ_0”
           = ∫_^n ×^n (g(X') - g(X”)) dμ_0' + ∫_^n ×^n g(X”) (dμ_0' - dμ_0”) 
           ≤ M X' - X”_∞ + ∇ X”_∞ W_1(μ_0', μ_0”).
    
        For W_1('̆ρ', ”̆ρ”), we have
        
        ∫_^n g(x) (d('̆ρ_t') - d('̆ρ_t”)) 
                       = ∫_^n ×^n vg(x) (dμ_t' - dμ_t”) 
           = ∫_^n ×^n V'g(X') dμ_0' - ∫_^n ×^n V”g(X”) dμ_0”
           = ∫_^n ×^n (V'g(X') - V”g(X”)) dμ_0' + ∫_^n ×^n V”g(X”) (dμ_0' - dμ_0”) 
           ≤ M g_∞V' - V”_∞ + M X' - X”_∞ R 
           + ( g_∞∇ V”_∞ + R ∇ X”_∞) W_1(μ_0', μ_0”).
      
        These estimates hold uniformly in time. Plugging these into (<ref>) and using Lemma <ref>, we conclude by Gronwall. 
      
    An immediate Corollary is that the regularity conditions (<ref>) and (<ref>) can be restated in terms of the 
    distance W_1(μ_0',μ_0”). 
    
      
      Suppose that the velocity averaging has the integral representation (<ref>) 
      and that the kernel satisfies the uniform regularity assumptions (<ref>) and (<ref>). 
      Let (μ', '), (μ”, ”) be weak solutions to (<ref>) on [0,T].
      Then
      
        sup_t ∈ [0,T]( W_1(ρ', ρ”) + W_1('̆ρ', ”̆ρ”) )(t) ≤ C_1(M,R,J,T) W_1(μ_0', μ_0”),
    
      
      and 
      
        $̆Lip2'   sup_t ∈ [0,T]∂^k ( - )(t, ·)_∞≤ C_2(k,M,R,J,T) W_1(μ_0', μ_0”) , 
        Lip2'   sup_t ∈ [0,T]∂^k (' - ”)(t, ·) _∞≤ C_3(k,M,R,J,T) ( ∂^k(_0' - _0”)_∞ + W_1(μ_0', μ_0”) ) . 
          
        
        Lemmas <ref>, <ref> and inequality (<ref>) imply the desired Wasserstein-1 stability (<ref>).
        We conclude that the empirical measuresμ_t^Nconverge in the Wasserstein-1 metric to someμ∈ C_w^*([0,T]; (^n × B_R))uniformly on[0,T](the weak^*continuity ofμowes to the weak^*continuity of the empirical measures and the uniform convergence on[0,T]).
        In addition, by Corollary <ref>, for anyk≥ 0,_t^Nconverges inC^kto some_t ∈ C^k(^n)uniformly on[0,T]. 
        It remains to show that the limiting pair(μ, )is in fact a weak solution to (<ref>) in the sense of Definition <ref>. 
        
        
            Suppose a sequence of solutions μ^N ∈ C_w^*([0,T]; (^n × B_R)) converge weakly pointwise, i.e. μ_t^N →μ_t, uniformly 
            for all t ∈ [0,T]; and suppose that 
            for any k≥ 0, _t^N converges in C^k(^n) to _t, i.e. ∂^k(_t^N - _t)_∞→ 0, uniformly for all t ∈ [0,T]. 
            Then (μ, ) ∈ C_w^*([0,T]; (^n × B_R)) × C([0,T], C^k(^n)) is a weak solution to (<ref>). 
        
            In the following proof, C := C(M,R,J,T) and C' := C(k,M,R,J,T). 
            We have already observed that, due to Corollary (<ref>), for any k, _t^N converges in C^k to some _t ∈ C^k(^n) 
            uniformly on [0,T]. In addition ∂_t ^N = -∇· (^N ) is uniformly bounded on [0,T] due to (<ref>) and (<ref>). 
            So, ∈ C([0,T]; C^k(^n)). 
            To show that  solves the transport equation ∂_t  + ∇· () = 0, let X̃ be the characteristic flow, 
            
            X̃(t, α) = (t, X̃(t,α)), X̃(0, α) = α .
        
            Similarly, let X̃^N be the characteristic flow for the empirical strength _t^N which solves ∂_t ^N + ∇· (^N ) = 0,
            
            X̃^N(t, α) = (t, X̃^N(t,α)), X̃^N(0, α) = α .
        
            Abbreviating X̃(t,α), X̃^N(t,α) by X̃, X̃^N and using (<ref>) and (<ref>), we have for all k ≥ 0
            ∂^k    ( (τ, X̃^N) -  (τ, X̃)) _∞
               ≤∂^k ( (τ, X̃) -  (τ, X̃))_∞
                    + ∂^k ( (τ, X̃^N) -  (τ, X̃)) _∞
               ≤∂^k ( (τ, X̃) -  (τ, X̃)) _∞
                    + ∂^k+1_∞X̃^N - X̃_∞
               ≤ C' ( W_1(μ_0^N, μ_0) + X̃^N - X̃_∞).
        
            As a result, 
            
            (X̃^N - X̃)(t,·)_∞≤ C ( W_1(μ_0', μ_0”) +  (X̃^N - X̃) (t,·) _∞), for all  t ∈ [0,T) .
        
            By Gronwall and X̃(0,α) = X̃^N(0,α) = α, we obtain 
            
            (X̃^N- X̃)(t,·)_∞≤ C W_1(μ_0^N, μ_0),   for all  t ∈ [0,T) .
        
            Now solving along the characteristic X̃^N, 
            
            ^N (t, X̃^N(t, α)) = _0^N(α) exp{∫_0^t ∇· (τ, X̃^N(s, α)) dτ}.
        
            With (<ref>) and (<ref>) in hand, we obtain uniform in time convergence to 
            
            (t, X̃(t, α)) = _0(α) exp{∫_0^t ∇· (τ, X̃(s, α)) dτ}.
        
            So, _t is a solution to ∂_t  + ∇() = 0. 
            Turning to the convergence of the empirical measures, the weak convergence W_1(μ_t^N, μ_t) → 0 immediately implies that the linear terms in (<ref>) converge.  
            Let us address the nonlinear term.
            
            ∫_0^t    ∫_^2n∇_v g ^N ( - v)) dμ^N_τ(x,v) - ∫_0^t ∫_^2n∇_v g  ( - v)) dμ_τ(x,v) 
               ≤∇_v g_∞∫_0^T ∫_^2n^N ( - v)) -  ( - v))_∞ dμ^N_τ(x,v) 
               + ∇_v g_∞∫_0^T ∫_^2n ( - v))_∞ d(μ^N_τ(x,v) - μ_τ(x,v)).
        
            The second term goes to zero by weak convergence.  For the first term, we simply use the regularity conditions (<ref>), (<ref>), (<ref>), (<ref>) to get
            
            ^N ( - v)) -  ( - v))_∞   ≤^N - _∞_∞ + _∞ - _∞
               ≤ C W_1(μ_0^N, μ_0).
        
            This gives ^N ( - v)) -  ( - v))_∞→ 0 uniformly on [0,T]. 
        
        
        § HYDRODYNAMIC LIMITS
        
        The goal of this section is to establish a passage from the kinetic description (<ref>) to the 
        corresponding macroscopic description in the monokinetic and Maxwellian limiting regimes, Theorems <ref> and <ref>. 
        The arguments presented here contain similar estimates to the hydrodynamic limiting arguments presented for environmental averaging models. 
        We will therefore often refer to Section <ref> where relevant quantities and estimates 
        are presented in the context Cucker-Smale model (i.e. when the regularity of the strength is known). 
        
        From the introduction, thev-moments of the mesoscopic system (<ref>) yields (<ref>) with the Reynolds Stress tensorgiven by (<ref>). 
        The system is closed by adding a local alignment forceF()to the kinetic equation (<ref>):
        
            ∂_t  + v ·∂_x  = ∇_v · ( (v - ) ) + 1/ϵ F() 
            ∂_t  + ∇· () = 0  .
        
        We will consider two such local alignment forces corresponding to the monokinetic and Maxwellian regimes. 
        In the Monokinetic regime,F()is given by (<ref>). 
        This monokinetic local alignment force pushes the kinetic solution towards the monokinetic ansatz 
        
            f(t,x,v) = ρ(t,x) δ(v - (̆t,x)),
        
        where(ρ, , )̆solve (<ref>) (Theorem <ref>).
        The Reynolds stress tensor becomes zero and the macroscopic description becomes 
        
            ∂_t ρ + ∇· (ρ̆) = 0 
            ∂_t  + ∇· () = 0 
            ∂_t (ρ)̆ + ∇· (ρ⊗̆)̆ = ρ( - )̆ .
        
        In the non-vacuous case, we can divide the$̆-equation by ρ and this becomes (<ref>). 
        The inherited regularity of the strength from the kernel will once again play a crucial role in the proof of the monokinetic limit. 
        
        For the Maxwellian regime, we will rewrite the v-moments system (<ref>) as: 
        
            ∂_t ρ + ∇· (ρ) = 0 
            ∂_t  + ∇· () = 0 
            ∂_t (ρ) + ∇· (ρ⊗) + ∇ρ
                    + ∇· = ρ ( - ) ,
        
        where _δ is given in (<ref>) and  
        
            (t,x) = ∫_^n ((v-) ⊗ (v - ) - 𝕀) .
        
        To close this system, we consider a strong Fokker-Planck force F() given by (<ref>), which pushes the kinetic solution towards the Maxwellian,
        
            μ(t,x) = ρ(t,x)/(2π)^n/2 e^-|v - (̆t,x)|^2/2,
        
        where (ρ, , )̆ solve (<ref>) (Theorem <ref>).
        The Reynolds stress term vanishes and the system becomes 
        
            ∂_t ρ + ∇· (ρ̆) = 0 
            ∂_t  + ∇· () = 0  
            ∂_t (ρ)̆ + ∇· (ρ⊗̆)̆ + ∇ρ = ρ ( - )̆ .
        
        Unfortunately, the total momentum of solutions to (<ref>) with the strong Fokker-Planck force F()(<ref>) is not necessarily bounded uniformly in ϵ, 
        so Propsitions <ref> and <ref> do not apply and  does not inherit regularity from the kernel.
        We've instead identified a set of sufficient continuity conditions for the solution (<ref>)-(<ref>) given in Section <ref>. 
        An important class of models for which these conditions can be verified is the w-model with the interaction kernel ϕ bounded from below away from zero.
        
        
        
        
         §.§ Monokinetic Regime
         
        We consider (<ref>) under the local alignment force (<ref>):
        
            ∂_t  + v ·∂_x  = ∇_v · ( (v - ) ) + 1/ϵ∇_v · ((v - _δ )) 
            ∂_t  + ∇· () = 0  ,
        
        where _̆δ is the special mollification given in (<ref>). 
        
        
        We will first verify the maximum principle on the velocity so that we can use the inherited regularity of the strength. 
        The characteristic equations of (<ref>) are given by: 
        
            X(t,s,x,v) = V(t,s,x,v),  X(s,s,x,v) = x 
             V(t,s,x,v) = (X)((X) - V) + 1/ϵ (_δ - V),  V(s,s,x,v) = v.
         
        As in the Cucker-Smale and -model, the measure f_t(x,v) is the push-forward of f_0(x,v) along the characteristic flow. 
        Letting ø := (x,v), X' := X(t, ø'), V' := V(t, ø'), and using the right stochasticity of Φ_ρ, (<ref>), we compute: 
        
            (X)((X) - V) 
                   = ∫_^n ×^n(X)  Φ_(X, x) v _t(x,v)  - V 
               = ∫_^n ×^n(X)  Φ_(X, x) (v - V) _t(x,v) 
               = ∫_^n ×^n(X)  Φ_(X, X') (V' - V) _0(ø') ' .
        
        Considering compactly supported initial data, f_0 ⊂^n × B_R, and evaluating at a point of maximum of V, we have 
        
            ∫_^n ×^n(X) Φ_(X, X') (V' - V_+) _0(x,v) ' ≤ 0,      V_+(t) = max_(x,v) ∈^n × B_R |V(t,0,x,v)| .
        
        For the local alignment term, recall that _δ is the averaging for the M_ϕ model with kernel given in Table (<ref>) with ϕ = Ψ_δ for a smooth mollifier Ψ_δ. 
        Therefore, it is handled similarly. Denoting the kernel by  Φ_, δ, we have: 
        
            1/ϵ∫_^n ×^nΦ_, δ(X, X') (V' - V_+) f_0(x,v) ' ≤ 0 .
        
        Using the classical Rademacher Lemma, we obtain 
        
            V_∞≤ 0 .
        
        Of course, this implies boundedness of the macroscopic velocity:
        
            |  | = | ∫_B_R v  dv | /∫_B_R dv≤ R ∫_B_R dv/∫_B_R dv = R .
        
        With the maximum principle in hand, we can readily apply the inheritance of regularity.  
        Furthermore, since ,  f ⊂^n × B_R, 
        we have W_1(, ρ̆) ≤ C W_1(f^ϵ, f) ≤ CW_2(f^ϵ, f).
        Similarly, W_1(, ρ) ≤ CW_2(f^ϵ, f).  We can then modify (<ref>) and (<ref>) to get 
        
            $̆Reg2' - _L^∞≤ C(M,J) W_2(, f) 
               Reg2' - _L^∞≤ C(M,J,T) sup_t ∈ [0,T] W_2(, f) (t)  provided _0 = _0.
            
            Since we are working on^n, theL^∞norm can be interchanged withL^2norm (with the only difference being the constantC, which is inconsequential for our arguments). 
            
            
             (proof of Theorem <ref>)
                
                
                
                When an identity or inequality does not depend on the regularity of the strength and can be found in the proof for the Cucker-Smale case, we refer the reader to Section <ref>.
                In order to control W_2(_t, f_t), we consider the flow γ_t whose marginals are _t and f_t (i.e. γ_t ∈∏(_t, f_t))
                
                ∂_t γ + v_1 ·∇_x_1γ + v_2 ·∇_x_2γ 
                            + ∇_v_1 [ γ (v_1 - ) + 1/ϵ (v_1 - _δ) ]
                            + ∇_v_2 [ γ (v_2 - ) ] = 0.
            
                Since γ_t ∈∏(_t, f_t),  
                
                W := ∫_^2n×^2n |w_1 - w_2|^2 μ̣_t(w_1, w_2) ≥ W_2^2(_t, f_t).
            
                So, we aim to control W.  Splitting it into the potential and kinetic components, we get: 
                
                W = ∫_^2n×^2n |v_1 - v_2|^2 γ_t + ∫_^2n×^2n |x_1 - x_2|^2 γ_t := W_v + W_x .
             
                We will from here on use the notation A ≲ B as a shorthand for A ≤ C(M,J,T) B. 
                As in the Cucker-Smale case, we get 
                
                W_x ≲ +  W_x 
                
                            W_v ≲ +  W_x ,
            
                where the modulated kinetic energy  is given in (<ref>).
                The modulated kinetic energy can also be estimated as in the Cucker-Smale case: 
                
                ≲ + δ/ϵ + ( - ,̆ - )_κ_ + ∫_^n(-̆) · ( - )̆ .
            
                The alignment terms can be written as  
                
                δ A    = ( - ,̆ - )_κ_ + ∫_^n(-̆) · ( - )̆
                   = ∫_^n (-̆) ·(  -  + -̆) 
                   = ( - ,̆ [ - ]̆_)_κ_ + ∫_^nρ^ϵ (-̆^̆ϵ) · ( []̆_ - ) 
                   - ∫_^n |-̆|^2  + ∫_^n (-̆) ·(̆ - )  := I + II + III + IV .
            
                By (<ref>) and (<ref>), 
                
                I ≲∫_^n | - |̆^2 .
            
                From (<ref>) and (<ref>) and the assumed smoothness of , we get 
                
                II ≲ + sup_t ∈ [0,T] W^2_2(, f) .
            
                The term III is negative so we drop it.  Finally, from (<ref>) and the assumed smoothness of $̆,
                
                IV ≲ + sup_t ∈ [0,T] W^2_2(, f) .
            
                Using (<ref>), the estimate for the modulated kinetic energy becomes 
                
                ≲ + δ/ϵ + sup_t ∈ [0,T] W_2^2(, f) 
                   ≤ + δ/ϵ + sup_t ∈ [0,T] W_x + sup_t ∈ [0,T] W_v 
                   ≲δ/ϵ + sup_t ∈ [0,T] (W_x + ).
            
                All together, 
                
                W_x ≲ + W_x 
                ≲δ/ϵ + sup_t ∈ [0,T] (W_x + ).
            
                Since the initial value of + W_xis smaller thanϵ, Gronwall implies 
                
                + W_x ≲ϵ + δ/ϵ,
            
             and by (<ref>),W_v ≲ϵ + δ/ϵ.
            
            
             §.§ Maxwellian Limit
            
            
            We consider (<ref>) under a strong Fokker-Planck penalization force (<ref>): 
            
                ∂_t  + v ·∂_x  = ∇_v · ( (v - ) ) + 1/ϵ[ Δ_v  + ∇_v · (v - _δ) ] 
                ∂_t  + ∇· () = 0
            
            To obtain the equation for the macroscopic quantitiesand, we take thev-moments to get: 
            
                ∂_t  + ∇· () = 0 
                ∂_t  + ∇· () = 0 
                ∂_t (ρ) + ∇· (ρ⊗) + ∇
                        + ∇· =  ( - ) + 1/ϵ (_δ - ) ,
            
            where_δis given in (<ref>) and  
            
                (t,x) = ∫_^n ((v-) ⊗ (v - ) - 𝕀) .
            
            To justify the Maxwellian limit, we will assume the following continuity conditions on solutions to (<ref>) and on the averaging[·]_ρ:
            
                R1 is bounded uniformly in ϵ: sup_t ∈ [0,T](t, ·)_∞≤ C 
                   R2 - _L^2≲sup_t ∈ [0,T] (W_1(, ρ) + W_1(, ρ̆))(t, ·) 
                   R3 - _L^2≲sup_t ∈ [0,T] (W_1(, ρ) + W_1(, ρ̆))(t, ·)   provided _0 = _0 
                   R4 [·]_ρ: L^2(ρ) ↦ L^2(ρ)  is bounded uniformly over ρ∈_M(Ω)  with equal mass
            
            Note that (<ref>) clearly holds with our usual assumption on the velocity averaging and kernel.
            
            These conditions hold for the-model, which we will show at the end of this section. 
            
            Define the Maxwellians associated to solutions(ρ, , )̆to (<ref>) and solutions(, , )to (<ref>) by
            
                μ(t,x) = ρ(t,x)/(2π)^n/2 e^|v - (̆t,x)|^2/2, (t,x) = (t,x)/(2π)^n/2 e^|v - (t,x)|^2/2 .
            
            and the relative entropy by 
            
                ( | μ) = ( | ) + ( | μ) 
                   ( | μ) = 1/2∫_^n | - |̆^2  + ∫_^nlog( / ρ)  .
            
            Due to the ,( | μ) → 0implies 
            
                →ρ
                   →ρ̆
            
            inL^1(^n). In particular,( | μ) → 0implies that 
            
                W_1(, ρ) → 0,  
                   W_1(, ρ̆) → 0.
             (Proof of Theorem <ref>)
              When an identity or inequality does not depend on the regularity of the strength and can be found in the proof for the Cucker-Smale case, we refer the reader to Section <ref>.
              As in the Cucker-Smale case, we break the relative entropy into the kinetic and macroscopic parts, ( | μ) = _ϵ + _ϵ with _ϵ, _ϵ given (<ref>) and (<ref>). 
              By (<ref>), 
              
                d/dt_ϵ 
                    =    - 1/ϵ∫_^n ×^n[ |∇_v |^2/ + 2 ∇_v · (v - _δ + |v - _δ|^2 ) ] 
                   - 1/ϵ [ (_δ, )_ - (_δ, _δ)_ ] 
                   - ∫_^n ×^n [ ∇_v · (v - ) + v · (v - )  ] 
                   := A_1 + A_2 + A_3.
            A_1 is strictly negative so it can be dropped; according to <cit.> (due to ball-positivity of the overmollified M_ϕ model), 
              A_2 is also strictly negative. 
              And 
              
                A_3 =  n ∫_^n ×^n  -∫_^n ×^n |v|^2  + (, )_κ^ϵ .
            
              Due to boundedness of (<ref>), 
              
                A_3 ≤ c_1 + c_1 _L^2()^2 + (, )_κ^ϵ  .
            
              Recalling that _ϵ = ∫ |v|^2 (x,v)  = _L^2()^2 ≤ and
              using L^2(ρ) boundedness of [·]_ρ, we have 
              
                A_3 ≤ c_1 + c_2 _ϵ≤ c_3 + c_4 ,
             
              where the last inequality can be found in <cit.>.  Thus, 
              
                ≤ c_3 + c_4 ,
            
              and by Gronwall  is bounded uniformly in ϵ.  
              Uniform boundedness of  in ϵ will be used to control the Reynolds stress term. 
              
              
              
              
              
              By (<ref>) and boundedness of (<ref>), 
              
              
              
              
              
              
              
              
              
              
                d/dt_ϵ≤ -1/ϵ + c ϵ e( | ) - _L^2(κ_ϵ)^2 + (, )_κ^ϵ.
            
              Recall that  is the Fisher information and is given by (<ref>). 
              Turning to , the evolution equation is given by (<ref>). 
              The term δ A is the same alignment term used in the proof of the monokinetic case.  
              As in the monokinetic case, the inherited regularity is precisely what is needed to control this term. 
              The difference here from the monokinetic case is that we use (<ref>) and (<ref>) instead of (<ref>), (<ref>); and 
              we retain the term ∫_^n | - |̆^2 and bound it by  instead of the modulated kinetic energy.  We get 
              
                δ A ≲∫_^n | - |̆^2  + sup_t ∈ [0,T] (W_1^2(, ρ) + W_1^2(, ρ̆))(t,·)
             
              and 
              
                W_1^2(, ρ̆) 
                         ≤ - ρ̆_L^1^2  
                   ≤ ( - )̆_L^1^2 + _∞^2  - ρ_L^1^2 .
            
              By , 
              
                ( - )̆_L^1^2 ≲ - _L^2(ρ_ϵ)^2 ≤( | μ) ≤( | μ) .
            
              By 
                - ρ_L^1^2 ≲( | ρ) ≤( | μ) ≤( | μ) .
            
              So W_1^2(, ρ̆)  ≲( | μ) and therefore by (<ref>)
                δ A ≲( | μ) .
            
              The Reynolds stress term can be written
              
                = ∫_^n [2 ∇_v √() + (1 + ϵ /2) (v - ) √()]
                      ⊗ [(v - ) √()]  - ϵ/2 ∫_^n (v - ) ⊗ (v - )   .
            
              Note that 
              
                e( | ) = _ϵ - 1/2∫_^n ||^2 ≤_ϵ≤ .
            
              Due to boundednes of (<ref>) and the boundedness of , we get 
              
                ≲√(e( | ) ) + ϵ e( | ) ≲√() + ϵ.
            
              The remaining terms do not depend on  are estimated in Section <ref>. In total, we get 
              
                d/dt≲( | μ) + √() + ϵ + δ/ϵ + _L^2() - (, )_.
            
              Combining the estimates on  and  and using √()≤1/2ϵ + 2 ϵ we arrive at 
              
                d/dt( | μ) ≲( | μ) - 1/ϵ + ϵ + δ/ϵ + √()≲( | μ) + ϵ + δ/ϵ.
            
              Gronwall concludes the proof. 
            
            
            We conclude this section by verifying (<ref>)-(<ref>) for the-model forϕ≥ c > 0. 
            Recall that = _̆F = (ρ̆)_ϕ / ρ_ϕand = ρ_ϕwheresolves the pure transport along_Fgiven in (<ref>). 
            Let(ρ, , )̆be a smooth solution to (<ref>) and(, , )be a solution to (<ref>);
            and let(ρ, , )and(, , )be the corresponding change of variables. 
            
            For (<ref>),is transported sois clearly bounded.  For (<ref>), we have 
            
                - _L^2   =  ()_ϕ -  (ρ̆)_ϕ_L^2
                   ≤ - _L^2 (ρ̆)_ϕ_∞ + ()_ϕ - (ρ̆)_ϕ_L^2_L^∞.
            
            To control - _L^2, we turn to control the velocities_F - _̆F_∞. 
            
            Sinceϕ≥ c > 0, 
            
                _F - _̆F_∞ = ρ_ϕ ()_ϕ - _ϕ (ρ̆_ϕ) /ρ_ϕ_ϕ_∞≲( W_1(, ρ) + W_1(, ρ̆) ) .
            
            Integrating∂_t ( - )along characteristics, we get: 
            
                - _L^2≲sup_t ∈ [0,T] (W_1(, ρ) + W_1(, ρ̆))(t, ·) .
            
            Moreover, 
            
                ()_ϕ -  (ρ̆)_ϕ_L^2≲sup_t ∈ [0,T] (W_1(, ρ) + W_1(, ρ̆))(t, ·) .
            (<ref>) follows similarly.  For (<ref>), the Favre averaging is clearly a bounded operator onL^2when theϕis bounded from below away from zero.  
            
            
            
            
            
            
            
            
            § RELAXATION TO GLOBAL MAXWELLIAN IN 1D
            
            
            Adding noise with the strength coefficient to the velocity equation leads to the Fokker-Planck-Alignment system given by
            
                ∂_t f + v ·∇_x f + ∇_v · ( (v - ) f) = σ_v f 
                ∂_t  + ∇_x · () = 0.
            
            We study the case where the velocity averaging is given by the Favre averaging:_̆F := (ρ̆)_ϕ / ρ_ϕ(i.e. the-model). 
            
            Recall that the weightis defined by = ρ_ϕand the system becomes 
            
                ∂_t f + v ·∇_x f = ρ_ϕ_v ·(  σ∇_v f - v f) +  (ρ̆)_ϕ·∇_v f 
                ∂_t  + _̆F ·∇_x   = 0.
            
            
            Let us discuss well-posedness of (<ref>).  General well-posedness theory for kinetic alignment equations based on a predefined strength_ρhas been developed in <cit.>.  
            The new system, under the uniform regularity assumptions (<ref>) and (<ref>), which for the Favre filtration (withΦ_ρ(x,y) = ϕ(x-y) / ρ_ϕ(x)) means all-to-all interactions 
            
                inf_Øϕ = c_0 >0,
            
            falls directly into the same framework of <cit.>. In fact, in this case every flock is automatically "thick", meaning that 
            
                inf_Øρ_ϕ = c_0 M >0.
            
            We also have the a priori bounded energy 
            
                ≤ c_1  + c_2, (t) = 1/2∫_Ω×^n |v|^2 f_t(x,v)
            
            on any finite time interval. 
            Indeed, multiplying (<ref>) by1/2|v|^2and integrating by parts, we have: 
            
                1/2∫_Ω×^n |v|^2 f    = ∫_Ω×^nρ_ϕ v · (vf - σ∇_v f)  - ∫_Ω×^n (ρ)_ϕ· v f 
                   = ∫_Ω×^nρ_ϕ· |v|^2 f  + n σ∫_Ω×^nρ_ϕ f  - ∫_Ω×^n (ρ)_ϕ· v f 
                   = ∫_Ωρ_ϕ ||^2 ρ + n σ∫_Ωρ_ϕρ - ∫_Ω (ρ)_ϕ· (ρ)   .
            
            Sinceremains bounded, the first term bounds the energy and the second term is bounded. 
            For the last term, we estimate, 
            
                ∫_Ω (ρ)_ϕ· (ρ) ≤_+ ϕ_+ _L^1(ρ)^2 ≲_+ ϕ_+ _L^2(ρ)^2 = _+ ϕ_+ ,
            
            which yields the desired energy inequality. 
            Since_L^1(ρ)≲_L^2(ρ) = √(), we have the inheritance of regularity: 
            
                _x^k _̆F _∞ < C(k,M,J,T),
            
            for anyk∈.  Consequently, according to (a trivial adaptation of) <cit.> for any data in the classical weighted Sobolev spacesf_0∈ H^m_q()and_0 ∈ C^m,inf_Ø_0 >0, where
            
                H^m_q() =  { f :   ∑_ k + l ≤ m∑_|| = k, || = l∫_ | v^q - k - l^_x^_v f |^2  <∞},    q > m,
            
            there exists a unique global in time classical solution in the same data space.  Let us note that the transport ofpreserves the lower bound onuniformly in time, and by the automatic thickness condition we have the ellipticity coefficientρ_ϕuniformly bounded away from zero as well.  
            
            Let us now recall a set of conditions on a given global solutionfthat are sufficient to imply global relaxation offto the Maxwellian
            
                μ_σ,  = 1/|^n| (2πσ)^n/2 e^|v-|^2/2σ,
            
            where = 1/M∫_Øρ. We note that this total momentumis not  preserved in time generally, but rather satisfies the equation
            
                ∂_t  = ∫_Ø (_̆F - )̆_ρ,     _ρ = ρ.
            
            We can't determine whethereventually settles to a fixed vector, as it does for all the classical alignment models including those that do not preserve the momentum. 
            
            So, according to <cit.> a given solutionfconverges to the Maxwellian in the sense defined in (<ref>) provided there exists a set of fixed constantsc_0,... >0such that 
            
              * c_0 ≤≤ c_1 and ∇_∞≤ c_2 for all ρ. 
              
              * 
                sup{ (,̆)_κ_ρ : ∈̆L^2(κ_ρ),  = 0, _L^2(κ_ρ) = 1 }≤ 1 - ϵ_0
            
              * [·]_ρ_L^2(ρ) → L^2(ρ) + ∇_x ( [·]_ρ)_L^2(ρ) → L^2(ρ)≤ c_3.  
            
            
            We will be able to show (i)-(iii) in one dimensional case. 
            Let us discuss these conditions starting from (i).  The key condition here is∂_x _∞≤ c_2.  We can establish such uniform control in 1D only. Indeed, since
            
                _t ∂_x  + ∂_x (u_F ∂_x ) = 0,
            
            we can see that∂_x satisfies the same continuity equation asρ_ϕ(this only holds in 1D). Thus, 
            
                _t ∂_x /ρ_ϕ + u_F _x ∂_x /ρ_ϕ = 0,
            
            Since initially|∂_x |/ρ_ϕ≤ Cfor some largeC>0due to the all-to-all interaction assumption, this bound will persist in time.  Hence,_∞≤ C ρ_ϕ_∞≤ c_2.
            
            Condition (iii) follows from (i). Indeed, since, remain uniformly bounded, it reduces to 
            
                ∫_Ø | (uρ)_ϕ |^2  ρ + ∫_Ø | (uρ)_ϕ |^2  ρ≤ c_3 u^2_L^2(ρ).
            
            This follows by the .
            
            
            Finally, let us get to (ii). We reduce the computation of the spectral gap to the low-energy method and the classical Cucker-Smale model.  To this end, we assume thatϕ = ψ∗ψwhereψ>0is another positive kernel. In other words,ϕis Bochner-positive.    In order to establish (<ref>) it suffices to show that 
            
                (u, u)__̨ρ - (u, u_ρ)__̨ρ≥ϵ_0 (u, u)__̨ρ.
            
            Let us start  by symmetrizing and using cancellation  in the second obtained integral:
            
                (u, u)__̨ρ - (u, u_ρ)__̨ρ    = ∫_Ø×Ø u(x) · (u(x) - u(y)) ρ(x) ρ(y) ϕ(x-y) (x) 
                   = 1/2∫_Ø×Ø |u(x) - u(y)|^2 ρ(x) ρ(y) ϕ(x-y) (x) 
                   - 1/2∫_Ø×Ø  | u(y)|^2 ρ(x) ρ(y)((x) - (y)) ϕ(x-y)  = I + II.
            
            Notice that
            
                I ≥_-  ∫_Ø×Ø u(x) · (u(x) - u(y)) ρ(x) ρ(y) ϕ(x-y)  = _- [ (u, u)_ρ_ϕρ - (u, u_ρ)_ρ_ϕρ ].
            
            The difference of the inner productions inside the bracket represents exactly the spectral gap of the Cucker-Smale model computed in <cit.>. From that computation it follows that 
            
                (u, u)_ρ_ϕρ - (u, u_ρ)_ρ_ϕρ≥ c M^3 (u, u)_ρ_ϕρ≥ c _- M^3 (u, u)__̨ρ,
            
            wherecdepends only on the kernelψ. 
            
            Let us now estimateII:
            
                II ≤ (_+ - _-)  (u, u)_ρ_ϕρ≤_+ - _-/_- (u, u)__̨ρ.
            
            We can see that provided
            
                _+ - _-/_-≤1/2 c _- M^3,
            
            we obtain
            
                (u, u)__̨ρ - (u, u_ρ)__̨ρ≥ϵ_0 (u, u)__̨ρ.
            
            which is the needed result. 
            
            Let us collect now all the assumptions we have made and state the main result.
            
            
              Suppose n=1 and the kernel is Bochner-positive, ϕ = ψ∗ψ, with infψ >0.  Then any initial distribution f_0∈ H^m_q() and strength _0 ∈ C^m satisfying the following small variation assumption
              
                sup (_0) - inf (_0)/inf (_0)^2≤ c M^3,
            
            for some absolute c>0, gives rise to a global classical solution f, which relaxes to the Maxwellian exponentially fast 
            
                f(t) - μ_σ, u̅_L^1(^n ×^n)≤ c_1 σ^-1/2 e^-c_2 σ t.
            
            
            We note once again that the solution relaxes to a moving Maxwellian centered around a time-dependent momentumu̅. There two conceivable mechanisms of stabilizingu̅. One is sufficiently fast alignment:
            
                ∫_0^∞sup_x,y |u(x,t)- u(y,t)|  <∞,
            
            which our relaxation seems to be not strong enough to imply. And another is stabilization of the density to a uniform distributionρ→1/|Ø|, which we do have from (<ref>)  and in the case when≡it does imply exponential stabilization of the momentum for Favre-based models, see <cit.>. 
            Ifvaries, even ifρ = 1/|Ø|, from (<ref>) we have,
            
                ∂_t u̅ = ∫_Ø (u_F - u) _ρ = 1/|Ø|∫_Ø ( u_ϕ - u ϕ_1) .
            
            We can see that unless (<ref>) holds, the persistent variations ofmay keep this term large leaving a possibility forever moving momentumu̅.
            
            
            
            
            
            
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