# Solutions of a Class of Degenerate Kinetic Equations Using Steepest Descent in Wasserstein Space

Authors: Aboubacar Marcos; Ambroise Soglo
Journal: Journal of Mathematics
 (2020)

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## Abstract

We use the steepest descent method in an Orlicz–Wasserstein space to study the existence of solutions for a very broad class of kinetic equations, which include the Boltzmann equation, the Vlasov–Poisson equation, the porous medium equation, and the parabolicp-Laplacian equation, among others. We combine a splitting technique along with an iterative variational scheme to build a discrete solution which converges to a weak solution of our problem.


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## Body

## 1. Introduction

The general model describing the kinetic equations is about an evolution equation of unknown functionf, representing a time-depending density of probability distribution of a material in a given domain of the space. In the present work, f may measure the density distribution of a system of identical particles of a bulk material. The density f depends on the time t and the position x and the velocity v of some particles at t. Roughly speaking, the equation is considered as the evolution of the density function in the phase space Ω×ℝd, with Ω as an open bounded domain with periodic boundary. As a probability density, f remains positive in the court of time and satisfies the mass conservation principle: ∫Ω×ℝdft,x,vdxdv=1, for all t≥0:(1)∂ft,x,v∂t+v⋅∇xft,x,v=divvft,x,v∇c∗∇vG′ft,x,v,in0,∞×Ω×ℝd,f0,x,v=f0x,v,inΩ×ℝd,where the initial datum f0x,v is a probability density on Ω×ℝd. Here, Ω is an open, bounded, convex, and smooth domain of ℝd, d≥1 with ∂Ω periodic, c∗ is the Legendre transform of a cost function c, and G:0,∞⟶ℝ is a convex function.Equation (1) can be viewed as a balance result of a streaming phenomenon with a general nonlinear interaction phenomenon between the particles described, respectively, as(2)∂ft,x,v∂t+v.∇xft,x,v=0,(3)∂ft,x,v∂t−divvft,x,v∇c∗∇vG′ft,x,v=0.Accordingly, the transport equation (2) can somewhat be interpreted as a relaxation of (1) at the absence of the interaction phenomena, whereas it reduced to (3) in absence of streaming.One of the interests in considering (1) under a general nonlinearity is that it covers a very broad range of problems which occurred in physics and it is a purely mathematical challenge. Of course, it has been motivated by some previous works in the literature, namely, the works in [1–7], where (1) is investigated in some particular cases. Indeed, in [3], the authors dealt with the heat equation:(4)∂ρt,v∂t=Δvρt,v,in0,∞×ℝd.By fixing an probability densityρ0 with ∫ℝdv2ρ0vdv finite and a time step h>0, they define the mass density ρkh as a discrete solution of (4) at time tk=hk, which minimizes the functional(5)Iρ≔∫ℝdρlnρdv+1hW22ρ,ρk−1h,on P2ℝd, where P2ℝd is the set of all probability density on ℝd having finite second moments and W2 is the 2-Wasserstein metric defined as(6)W2:P2ℝd×P2ℝd⟶ℝ,ρ1,ρ2⟼infT#ρ1=ρ2∫ℝdv−Tv2ρ1vdv1/2.By definingρh as follows: ρht,v=ρkhv,ift∈hk,hk+1,ρh0,v=ρ0v, they tend h to 0 and then show that the sequence ρhh converges to a nonnegative function ρ, which solves (4) in a weak sense.In [1], the existence of solutions for the spatially homogeneous equations associated with (1), that is, the equation for fixed x(7)∂f∂t=divvf∇c∗∇vG′f,in0,∞×Ω′,f=ft,v,has been proved by M. Agueh, using a similar variational scheme as in [6]. Here, Ω′ is an bounded and convex domain (see [6] for more details).A particular case of (1), namely, the kinetic equation(8)∂ft,x,v∂t+v.∇xft,x,v=Δvft,x,v,obtained by choosing c∗x=x2/2 and Gx=xlnx, has been studied in [5] by using a discretization scheme basing on the “splitting method.” This enables the authors to decompose a discrete solution fk of the kinetic equation (8) in the form fk=ρkhFkxh, where ρkh stands for a discrete solution of the free transport equation(9)∂ft,x,v∂t+v.∇xft,x,v=0,when v is fixed and Fkxh a discrete solution of the diffusion equation in (8) when x is fixed.Definingfh as (10)fht,x,v=fkx−t−tkv,v,ift∈kh,k+1h,fh0,x,v=f0x,v,withtk=kh,they show that fhh converges to a nonnegative function f which solves the kinetic equation (8) in a weak sense.Such a decomposition is not suitable in the case of problem (1) because of its nonlinear structure. To deal with the more general class of kinetic equation (1), we combine some ideas from the splitting method in [5] along with some techniques developed in [1] for the spatially homogeneous equations:(11)∂ρt,v∂t=divvρ∇c∗∇vG′ρ,in0,∞×ℝd.For the best of our knowledge, our technique is new and is stated in a more general setting. It is worth mentioning that the class of the kinetic equation (1) also includes the Vlasov–Poisson equation(12)∂ft,x,v∂t+v.∇xft,x,v=divvωft,x,v,obtained when(13)c∗x=∑i=1dxi,ω=1,1,…,1∈ℝd,and the parabolic p−Laplacian equation(14)∂ft,x,v∂t+v.∇xft,x,v=divv∇fp−2∇f,in the case c∗x=xp/p and Gx=xγ/γγ−1 with γ=2p−3/p−1.In order to facilitate the reading of the paper, we summarize below the main steps and technical schemes according to which ours results will be carried out:(1)
First of all, we fix a time steph>0 and define fk as a discrete solution of the kinetic equation (1) at time tk=hk, for k∈ℕ (see Section 2.1).(2)
Next, we prove that the solutionTk of the Monge problem(15)M:infS#Fkxh=Gkxh∫ℝdcv−SvhFkxhdv,is defined by(16)Tkv≔v+h∇c∗∇vG′ρkFkxh,whereFkxh and Gkxh are as in Section 2.1. We use (16) to show that the sequence fkk satisfies the time-discretization equation of the kinetic equation (1)(17)fk−fk−1h+v.∇xfk−1=divvfk∇c∗∇vG′fk+Akh,weakly, fork∈ℕ, where Akh tends to 0 and when h tends to 0.(3)
Then, we define an approximate solutionfh of the kinetic equation (1) (see (118)), and we prove that the sequence fhh converges to a nonnegative function f which solves the kinetic equation (1) in a weak sense when h tends to 0.The convergence result has been achieved as follows:(a)
The weak convergence offhh to f in Lp0,T×Ω×ℝd for 0<T<∞ follows from the displacement convexity of the functional HF=∫ℝdFpdv on the set of all probability density (see Proposition 3), and its strong convergence in Lp0,T×Ω×ℝd is obtained, thanks to a diagonal method combined with a result obtained in [1].(b)
The convexity ofc∗ and the boundedness of ∇vG′fhh in Lp0,T×Ω×ℝd help to prove a weak convergence of the nonlinear term divvf∇c∗∇vG′fhh to divvf∇c∗∇vG′f in Cc∞0,T×Ω×ℝd′.(c)
Finally, the strong convergence offhh to f in Lp0,T×Ω×ℝd and the weak convergence of the nonlinear term divvfh∇c∗∇vG′fhh to divvf∇c∗∇vG′f in Cc∞0,T×Ω×ℝd′ enable us to establish that f is a weak solution of the kinetic equation (1).The paper is structured as follows. In Section2, we state the required hypotheses and set some tools relevant for our problem. In Section 3, we set the variational formulation of the discrete problem related to our problem and construct the discrete solution. Section 4 concludes our main result by proving the convergence of the discrete problem to the considered problem. Section 5 ends the paper by giving an illustration example followed by an appendix on some regularity results.


## 2. Preliminaries

Throughout this work, we will assume the following:HGG:0,∞⟶ℝ is a convex function of class C20,∞ such that t⟶tdGt−d is convex.Hcc∈C1ℝd is even and convex function such that c0=0 and A1xq≤cx≤A2xq+1, for all x∈ℝd, with 1<q<∞.Hf0f0 is a probability density on Ω×ℝd such that(18)∫Ω×ℝdvqf0x,vdxdv<∞,∫Ω×ℝdGf0x,vdxdv<∞.f0∈LpΩ×ℝd with p=q/q−1 and N≤f0≤M with 0<N.Remark 1.
Typical examples satisfying assumptionHG are the functions Gs=slns,s>0 and Gs=sr,r>1.Proposition 1.
Assume thatc satisfies Hc. Let H:0,∞⟶ℝ be a convex function such that x⟼xdHx−d is convex and decreasing.Then, the functional(19)E:Pqℝd⟶ℝ,F⟼∫ℝdHFdv,is displacement convex.Proof.
LetF0,F1∈Pqℝd and T#F1=F0 be a c-optimal map that pushes F1 forward to F0. We define Tt=1−tid+tT and Ft=Tt#F1, where t∈0,1.
Then, we have(20)EFt=∫ℝdHFtdv=∫Ft>0HFtFtFtdv=∫ℝdHFt∘TtFt∘TtF1dv.
From [1] and Proposition 3, we have that(21)F1=det∇TtFt∘Tt.
Replacing (21) in (20), we obtain(22)EFt=∫ℝdHF1/det∇TtF1/det∇TtF1dv.
Recalling again [1] and Proposition 3, we get that ∇Tt is diagonalizable with positive eigenvalues. So, using the fact that the map A⟼detA1/d is concave on the set of d×d diagonalizable matrices with positive eigenvalues, we get(23)det∇Tt1/d≥1−t+tdet∇T1/d.
Sincex⟼xdHx−d is decreasing, then(24)HF1/det∇TtF1/det∇Tt≤1−tF1−1/d+tF1−1/ddet∇T1/ddH1−tF1−1/d+tF1−1/ddet∇T1d−d.
From (23) and (24) and the fact that x⟼xdHx−d is convex, we obtain(25)EFt≤1−tEF1+t∫ℝdHF1/det∇TF1/det∇TF1dv=1−tEF1+t∫ℝdHF0dv=1−tEF1+tEF0.
Hence, we conclude that the functionalE is displacement convex.Corollary 1.
Since the functionalE is displacement convex, we have that(26)EF0−EF1≥dEFtdtt=0=∫ℝd∇vH′F1,Tv−vF1dv.F1,F0∈Pqℝd and T#F1=F0 is the c-optimal map that pushes F1 forward to F0.Definition 1.
Letf0∈L1Ω×ℝd be nonnegative. We say that a nonnegative function ft,x,v on 0,∞×Ω×ℝd is a weak solution of (1) in the time’s interval 0,T for some T>0, if for every test function ψ∈Cc∞ℝ×Ω×ℝd with time support −T,T, we have(27)∫0T∫Ω×ℝd∂tψ+v.∇xψ−∇vψ.∇c∗∇vG′fft,x,vdtdxdv=−∫Ω×ℝdf0x,vψ0,x,vdxdv.


### 2.1. The Flow and Descend Algorithm

Assume that the probability densityf0x,v satisfies Hf0 and fix h>0 a time step, then we define the following:(1)
g1hx,v≔f0hx−hv,v=f0x−hv,v(2)
ρ1hx≔∫ℝdg1hx,vdv(3)
G1xhv≔g1hx,v/ρ1hx for x∈Ω fixed such that ρ1hx>0(4)
f1hx,v≔ρ1hxF1xhvFor eachx fixed, F1xh denotes the unique minimizer of the variational problem:(28)P1xh:infF∈PqℝdIF≔EF+hWchF,G1xh.Pqℝd is the set of all probability density on ℝd having a finite q-moment, and WchF,G1xh stands for the Kantorovich work defined as(29)WchF,G1xh=infγ∈ΠF,G1xh∫ℝd×ℝdcy−zhdγy,z,EF≔1ρ1hx∫ℝdGρ1hxFvdv.We obtain the termsfkh, for k≥2, by induction as follows:(i)
By fixingv, we define gkhx,v≔fk−1hx−hv,v.(ii)
By fixingx, we define(30)fkhx,v≔ρkhxFkxhv,where(31)ρkhx≔∫ℝdgkhx,vdv,and Fkxh is the unique minimizer of the variational problem.(32)Pkxh:infF∈PqℝdIF≔EF+hWchF,Gkxh,with(33)Gkxv≔gkhx,vρkhx,EF≔1ρkhx∫ℝdGρkhxFvdv.Existence ofF1xh and Fkxh will be proved farther in Sections 2 and 3, respectively.


### 2.2.c-Wasserstein Metric

In this section, we define a Wasserstein metric corresponding to a cost functionc, and we study its topology.Definition 2.
Assume thatc:ℝd⟶0,∞ satisfies Hc. Let ρ1,ρ2∈Pℝd two probability measures on ℝd. We define the c-Wasserstein metric between ρ1 and ρ2 by(34)Wcρ1,ρ2≔infλ>0,infγ∈Πρ1,ρ2∫ℝd×ℝdcx−yλdγx,y≤1.Theorem 1.
Assume thatc:ℝd⟶0,∞ satisfies Hc. Then, Wc is a distance on the probability space Pℝd. Furthermore, if ρnn is a sequence in Pℝd and ρ∈Pℝd, then ρnn converges to ρ in the metric space Pℝd,Wc if and only if ρnn converges narrowly to ρ in Pℝd.Proof.
Letρ1,ρ2∈Pℝd be two probability measures on ℝd such that Wcρ1,ρ2=0. Then, there exists a sequence λnn in 0,∞ which converges to 0 such that(35)infγ∈Πρ1,ρ2∫ℝd×ℝdcx−yλndγ≤1,for alln∈ℕ.
Denote byγn the solution of Kantorovich problem:(36)Kn:infγ∈Πρ1,ρ2∫ℝd×ℝdcx−yλndγ.
Then, we obtain(37)∫ℝd×ℝdcx−yλndγn≤1,for alln∈ℕ.
Sinceλn converges to 0, then x−y/λn tends to ∞ and when n goes to ∞, for all x,y∈ℝd such that x≠y. Then, using the fact that c is coercive, we deduce that there exists N0∈ℕ such that(38)cx−yλn≥x−yλn,for alln≥N0.
This with (37) implies that(39)infγ∈Πρ1,ρ2∫ℝd×ℝdx−ydγ≤λn,for alln≥N0.
Thus, we conclude that(40)infγ∈Πρ1,ρ2∫ℝd×ℝdx−ydγ=0.
Letγ0 be the solution of the Kantorovich problem:(41)K0:infγ∈Πρ1,ρ2∫ℝd×ℝdx−ydγ.
Then, using (40), we obtain(42)∫ℝd×ℝdx−ydγ0=0.
We deduce thatx=yγ0 a.e. So for all ϕ∈Cbℝd, we have(43)∫ℝdϕxρ1dx=∫ℝd×ℝdϕxdγ0x,y=∫ℝd×ℝdϕydγ0x,y=∫ℝdϕyρ2dy.
Consequently,ρ1=ρ2.
Let us fix two probability measuresρ1 and ρ2 on ℝd. Since c is even, then(44)infγ∈Πρ1,ρ2∫ℝd×ℝdcx−yλdγ=infγ¯∈Πρ2,ρ1∫ℝd×ℝdcx−yλdγ¯,for all λ>0. We deduce from (44) that(45)Wcρ1,ρ2=infλ>0,infγ∈Πρ1,ρ2∫ℝd×ℝdcx−yλdγx,y≤1=infλ>0,infγ¯∈Πρ2,ρ1∫ℝd×ℝdcx−yλdγ¯x,y≤1=Wcρ2,ρ1.
Letρ1,ρ2,ρ3 be three probability measures on ℝd. Define λ1=Wcρ1,ρ2 and λ2=Wcρ2,ρ3. Denote by γ1∈Πρ1,ρ2 the solution of Kantorovich problem(46)K1:infγ∈Πρ1,ρ2∫ℝd×ℝdcx−yλ1dγ,and denote by γ2∈Πρ2,ρ3 the solution of the Kantorovich problem:(47)K2:infγ∈Πρ2,ρ3∫ℝd×ℝdcx−yλ2dγ.
Using the Gluing lemma [8], there exists a probability measure σ on ℝd×ℝd×ℝd such that(48)σA×B×ℝd=γ1A×B,σℝd×A×B=γ2A×B,for some Borel subsets A and B of ℝd. Let γ3 be a probability measure on ℝd×ℝd defined by γ3A×B=σA×ℝd×B. Then, γ3∈Πρ1,ρ3, and we use the convexity of c to get that(49)∫ℝd×ℝdcx−zλ1+λ2dγ3=∫ℝd×ℝd×ℝdcx−zλ1+λ2dσ≤λ1λ1+λ2∫ℝd×ℝdcx−yλ1dγ1+λ2λ1+λ2∫ℝd×ℝdcy−zλ2dγ2≤λ1λ1+λ2+λ2λ1+λ2=1.
So,infγ∈Πρ1,ρ3∫ℝd×ℝdcx−z/λ1+λ2dγ≤1, and we conclude that(50)Wcρ1,ρ2≤λ1+λ2=Wcρ1,ρ2+Wcρ2,ρ3.
Hence,Wc is a distance on Pℝd.
Let us now study the topology ofWc.
Letρnn be a sequence on Pℝd and ρ∈Pℝd such that Wcρn,ρ converges to 0 when n tends to ∞. Define λn=Wcρn,ρ, since λnn converges to 0, then we use the fact that c is coercive to have(51)infγ∈Πρn,ρ∫ℝd×ℝdx−ydγ≤λn,when n⟶∞. We deduce that(52)limsupn⟶∞infγ∈Πρn,ρ∫ℝd×ℝdx−ydγ=0.
Note that the 1-Wasserstein metric betweenρn and ρ is(53)W1ρn,ρ=infγ∈Πρn,ρ∫ℝd×ℝdx−ydγ.
We deduce thatW1ρn,ρ converges to 0 when n tends to ∞. Since the 1-Wasserstein metric W1 induces the narrow topology of Pℝd, we conclude that the sequence ρnn converges narrowly to ρ in Pℝd.
Assume now that the sequenceρnn converges narrowly to ρ in Pℝd. Fix λ>0 and denote by γnλ the solution of Kantorovich problem:(54)K:infγ∈Πρn,ρ∫ℝd×ℝdcx−yλdγ.
Sinceρn converges narrowly to ρ, then γnλ converges narrowly to some γλ∈Πρ,ρ and(55)limn⟶∞∫ℝd×ℝdcx−yλdγnλ=∫ℝd×ℝdcx−yλdγλ=0.
So,(56)limn⟶∞infγ∈Πρn,ρ∫ℝd×ℝdcx−yλdγ=0,for all λ>0. Hence, for all λ>0, there exists Nλ>0 such that(57)infγ∈Πρn,ρ∫ℝd×ℝdcx−yλdγ≤1,for alln≥Nλ.
So,Wcρn,ρ≤λ for all n≥Nλ. Then,(58)limn⟶∞Wcρn,ρ=0.
Consequently,ρnn converges narrowly to ρ in the metric space PΩ,Wc.
We establish now the existence of solution for the variational problemP defined by(59)P:infρ∈Pqℝd∫ℝdGρdx+hWchρ0,ρ,in the metric space Pℝd,Wc, where(60)Wchρ0,ρ=infγ∈Πρ0,ρ∫ℝd×ℝdcx−yhdγ.
Pqℝd is the set of all probability measures on ℝd having q-finite moment, that is,(61)Pqℝd=ρ∈Pℝd,∫ℝdxqρdx<∞,and h>0 being a time step.Lemma 1.
Assume thatρ0, G, and c satisfy, respectively, Hf0, HG, and Hc. Then, the following is obtained:(i)
The mapρ⟶Wchρ0,ρ is lower semicontinuous in Pℝd,Wc.(ii)
The functionalρ⟶∫ℝdGρdx is lower semicontinuous in L1ℝd.(iii)
The setρ∈PMℝd,∫ℝdGρdx≤∫ℝdGρ0dx is a closed subset of L1ℝd,(62)wherePMℝd=ρ∈Pℝd,ρ≤M.Proof.
(i)
Letρ∈Pℝd. Let ρnn be a sequence in Pℝd such that ρnn converges to ρ in metric space Pℝd,Wc. Denote by γnh the solution of the Kantorovich problem:(63)Knh:infγ∈Πρn,ρ0∫ℝd×ℝdcx−yhdγ.We have(64)infγ∈Πρn,ρ0∫ℝd×ℝdcx−yhdγ=∫ℝd×ℝdcx−yhdγnh.Sinceρnn converges to ρ narrowly, then γnhn converges to γh∈Πρ0,ρ narrowly and(65)liminfn⟶∞∫ℝd×ℝdcx−yhdγnh≥∫ℝd×ℝdcx−yhdγh^≥Wchρ0,ρ.This implies that(66)liminfn⟶∞Wchρ0,ρn≥Wchρ0,ρ.Thus, we obtain the proof ofi.(ii)
SinceG is convex and G∈C10,∞,(67)∫ℝdGρndx≥∫ℝdGρdx+∫ℝdG′ρρn−ρdx.
Hence, ifρnn converges to ρ weakly in L1ℝd with ρ∈L∞ℝd, we have(68)liminfn⟶∞∫ℝdG′ρρn−ρdx=0,and then(69)liminfn⟶∞∫ℝdGρndx≥∫ℝdGρdx,which complete the proof of ii. The proof of iii is a consequence of ii.


### 2.3. Existence Results for the Discrete ProblemP1xh

First of all, we introduce here for unbounded domains analogous of the maximum principle stated for bounded domains in [1]. This maximum principle plays a central role in the searching of solution for the discrete problem P1xh. It is also used to further establish the convergence of our algorithm towards a weak solution of the kinetic equation (1).Proposition 2. (maximum principle).
Assume that the initial datumf0 satisfies N≤f0≤M with 0<N, c satisfies Hc, and G satisfies HG.
Then, any solutionF1xh of the variational problem(70)P1xh:infF∈PqℝdEF+hWchF,G1xh,satisfies ax≤F1xh≤bx, with ax=N/ρ1hx and bx=M/ρ1hx.Proof.
We defineEx≔y∈ℝd,F1xh>bx. Assume by contradiction that Ex has a positive Lebesgue measure. Then, γxExc×Ex>0, where γx is the minimizer of(71)K:infγ∈ΠG1xh,F1xh∫ℝd×ℝdcy−zhdγy,z.
Otherwise,(72)bxEx<∫ExF1xhdv=γxℝd×Ex=γxEx×Ex≤γxEx×ℝd=∫ExG1xhdy≤Exbx.
This yields a contradiction.
Defineνx=1Exc×Exγx, i.e., νxF=γxF∩Exc×Ex, for all F⊂ℝd, and denote by ν0,x and ν1,x the marginals of νx. We have νx≪γx with γx∈ΠG1xh,F1xh and then ν0,x≪G1xhdy and ν1,x≪F1xhdz.
Letv0,x and v1,x denote, respectively, the density functions of ν0,x and ν1,x. Because(73)∫Exv0,xdy=ν0,xEx=γxEx×ℝd∩Exc×Ex=0,∫Excv1,xdy=ν1,xExc=γxℝd×Exc∩Exc×Ex=0,we have v0,x=0 on Ex and v1,x=0 on Exc.
Forε>0 is small enough, define F1xh,ε=F1xh+εv0,x−v1,x and γxϵ by(74)∫ℝd×ℝdHx,ydγxε=∫ℝd×ℝdHx,ydγx+ε∫Exc×ExHx,x−Hx,ydγx,for all H∈C0ℝd×ℝd.
F1xh,ε∈Pqℝd, γxε∈ΠG1xh,F1xh,ε, and we get(75)IF1xh,ε−IF1xh=hWchF1xh,ε,G1xh−WchF1xh,G1xh+1ρ1hx∫ℝdGρ1hxF1xh,ε−Gρ1hxF1xh.
Taking into account some ideas from [1], we shall prove that IF1xh,ϵ−IF1xh<0, which leads to a contradiction.
Indeed,(76)WchF1xh,ε,G1xh−WchF1xh,G1xh=infγ∈ΠF1xh,ε,F1xh∫ℝd×ℝdcy−zhdγ−∫ℝd×ℝdcy−zhdγx≤∫ℝd×ℝdcy−zhdγxε−∫ℝd×ℝdcy−zhdγx=−ε∫Exc×Excy−zhdγx.
Sincey,z⟼cy−z/h is nonnegative on Exc×Ex and γxExc×Ex>0, we have(77)WchF1xh,ε,G1xh−WchF1xh,G1xh=−ε∫Exc×Excy−zhdγx<0.
Also, we have(78)1ρ1hx∫ℝdGρ1hxF1xh,ε−Gρ1hxF1xhdv=1ρ1hx∫ℝdGρ1hxF1xh+ερ1hxv0,x−v1,x−Gρ1hxF1xhdv=1ρ1hx∫ExGρ1hxF1xh−ερ1hv1,x−Gρ1hxF1xhdv+1ρ1hx∫ExcGρ1hxF1xh+ερ1hxv0,x−Gρ1hxF1xhdv.
G is being convex and of class C1, then(79)Gρ1hF1xh−Gρ1hF1xh−ερ1hv1,x≥G′ρ1hF1xh−ερ1hv1,xερ1hv1,x,on Ex and(80)Gρ1hxF1xh+ερ1hxv0,x−Gρ1hxF1xh≤ερ1hv0,xG′ρ1hF1xh+ερ1hv0,x,on Exc.
Hence, we have(81)1ρ1hx∫ℝdGρ1hxF1xh,ε−Gρ1hxF1xhdv≤ε∫ExcG′ρ1hF1xh+ερ1hv0,xv0,xdv−∫ExG′ρhF1xh−ερ1hv1,xv1,xdv=ε∫Exc×ExG′ρ1hF1xh+ερ1hv0,x−G′ρ1hF1xh−ερ1hv1,xdγx.
SinceG′ is continuous, then(82)limε⟶0G′ρ1hF1xh+ερ1hv0,x−G′ρ1hF1xh−ερ1hv1,x=0.
Consequently, there existsδ>0 such that, for 0<ε<δ, then(83)0≤G′ρ1hF1xh+ϵρ1hv0,x−G′ρ1hF1xh−ϵρ1hv1,x≤12γxExc×Ex∫Exc×Excy−zhdγx.
So, we fixε>0 small, such that 0<ε<δ, and we use (81) and (83) to obtain(84)1ρ1hx∫ℝdGρ1hxF1xh,ε−Gρ1hxF1xhdv≤ε2∫Exc×Excy−zhdγx.
Now, fixingϵ<δ and combining (77) and (84) yield(85)IF1xh,ε−IFxh<−ε∫Exc×Excy−zhdγx+ε2∫Exc×Excy−zhdγx=−ε2∫Exc×Excy−zhdγx<0.
It is a contradiction sinceF1xh is a minimizer of I on Pqℝd. Consequently, Ex is negligible and then F1xh≤bx.
The proof ofF1xh≥ax is analogous to that of F1x≤bx. Thus, we conclude that ax≤F1xh≤bx.Lemma 2.
Letf0∈PqΩ×ℝd be a probability density on Ω×ℝd such that 0<N≤f0≤M,∫Ω×ℝdGf0dxdv<∞ and ∫Ω×ℝdvqf0x,vdxdv<∞.The variational problemP1xh defined in (32) admits a unique minimizer F1xh in Pqℝd and N/ρ1hx≤F1xh≤M/ρ1hx.Proof.
Since0<N≤f0≤M and G1xhv=f0x,v/ρ1hx, then N/ρ1hx≤G1xh≤M/ρ1hx. Then, we use Proposition 2, and we obtain that any minimizer F1xh of the variational problem P1xh satisfies ax≤F1xh≤bx, with ax=N/ρ1hx and bx=M/ρ1hx.
By using Proposition2 and the fact that G is convex, we obtain that(86)EF≥1/ρ1hx∫ℝdGρ1hxG1xhvdv+G′N−G′M,for all probability density F∈Pqℝd such that N≤F≤M. We use now Lemma 1 to get that the functional F⟼1/ρhx∫ℝdGρhxFvdv is lower semicontinuous on the Wasserstein space Pqℝd,Wc. We conclude then that the problem P1xh admits a solution F1xh. The strict convexity of G and c implies the strict convexity of the map F⟼WchG1xh,F and that of the maps F⟼1/ρhx∫ℝdGρhxFvdv and accordingly the uniqueness of the minimizer F1xh of P1xh.


## 2.1. The Flow and Descend Algorithm

Assume that the probability densityf0x,v satisfies Hf0 and fix h>0 a time step, then we define the following:(1)
g1hx,v≔f0hx−hv,v=f0x−hv,v(2)
ρ1hx≔∫ℝdg1hx,vdv(3)
G1xhv≔g1hx,v/ρ1hx for x∈Ω fixed such that ρ1hx>0(4)
f1hx,v≔ρ1hxF1xhvFor eachx fixed, F1xh denotes the unique minimizer of the variational problem:(28)P1xh:infF∈PqℝdIF≔EF+hWchF,G1xh.Pqℝd is the set of all probability density on ℝd having a finite q-moment, and WchF,G1xh stands for the Kantorovich work defined as(29)WchF,G1xh=infγ∈ΠF,G1xh∫ℝd×ℝdcy−zhdγy,z,EF≔1ρ1hx∫ℝdGρ1hxFvdv.We obtain the termsfkh, for k≥2, by induction as follows:(i)
By fixingv, we define gkhx,v≔fk−1hx−hv,v.(ii)
By fixingx, we define(30)fkhx,v≔ρkhxFkxhv,where(31)ρkhx≔∫ℝdgkhx,vdv,and Fkxh is the unique minimizer of the variational problem.(32)Pkxh:infF∈PqℝdIF≔EF+hWchF,Gkxh,with(33)Gkxv≔gkhx,vρkhx,EF≔1ρkhx∫ℝdGρkhxFvdv.Existence ofF1xh and Fkxh will be proved farther in Sections 2 and 3, respectively.


## 2.2.c-Wasserstein Metric

In this section, we define a Wasserstein metric corresponding to a cost functionc, and we study its topology.Definition 2.
Assume thatc:ℝd⟶0,∞ satisfies Hc. Let ρ1,ρ2∈Pℝd two probability measures on ℝd. We define the c-Wasserstein metric between ρ1 and ρ2 by(34)Wcρ1,ρ2≔infλ>0,infγ∈Πρ1,ρ2∫ℝd×ℝdcx−yλdγx,y≤1.Theorem 1.
Assume thatc:ℝd⟶0,∞ satisfies Hc. Then, Wc is a distance on the probability space Pℝd. Furthermore, if ρnn is a sequence in Pℝd and ρ∈Pℝd, then ρnn converges to ρ in the metric space Pℝd,Wc if and only if ρnn converges narrowly to ρ in Pℝd.Proof.
Letρ1,ρ2∈Pℝd be two probability measures on ℝd such that Wcρ1,ρ2=0. Then, there exists a sequence λnn in 0,∞ which converges to 0 such that(35)infγ∈Πρ1,ρ2∫ℝd×ℝdcx−yλndγ≤1,for alln∈ℕ.
Denote byγn the solution of Kantorovich problem:(36)Kn:infγ∈Πρ1,ρ2∫ℝd×ℝdcx−yλndγ.
Then, we obtain(37)∫ℝd×ℝdcx−yλndγn≤1,for alln∈ℕ.
Sinceλn converges to 0, then x−y/λn tends to ∞ and when n goes to ∞, for all x,y∈ℝd such that x≠y. Then, using the fact that c is coercive, we deduce that there exists N0∈ℕ such that(38)cx−yλn≥x−yλn,for alln≥N0.
This with (37) implies that(39)infγ∈Πρ1,ρ2∫ℝd×ℝdx−ydγ≤λn,for alln≥N0.
Thus, we conclude that(40)infγ∈Πρ1,ρ2∫ℝd×ℝdx−ydγ=0.
Letγ0 be the solution of the Kantorovich problem:(41)K0:infγ∈Πρ1,ρ2∫ℝd×ℝdx−ydγ.
Then, using (40), we obtain(42)∫ℝd×ℝdx−ydγ0=0.
We deduce thatx=yγ0 a.e. So for all ϕ∈Cbℝd, we have(43)∫ℝdϕxρ1dx=∫ℝd×ℝdϕxdγ0x,y=∫ℝd×ℝdϕydγ0x,y=∫ℝdϕyρ2dy.
Consequently,ρ1=ρ2.
Let us fix two probability measuresρ1 and ρ2 on ℝd. Since c is even, then(44)infγ∈Πρ1,ρ2∫ℝd×ℝdcx−yλdγ=infγ¯∈Πρ2,ρ1∫ℝd×ℝdcx−yλdγ¯,for all λ>0. We deduce from (44) that(45)Wcρ1,ρ2=infλ>0,infγ∈Πρ1,ρ2∫ℝd×ℝdcx−yλdγx,y≤1=infλ>0,infγ¯∈Πρ2,ρ1∫ℝd×ℝdcx−yλdγ¯x,y≤1=Wcρ2,ρ1.
Letρ1,ρ2,ρ3 be three probability measures on ℝd. Define λ1=Wcρ1,ρ2 and λ2=Wcρ2,ρ3. Denote by γ1∈Πρ1,ρ2 the solution of Kantorovich problem(46)K1:infγ∈Πρ1,ρ2∫ℝd×ℝdcx−yλ1dγ,and denote by γ2∈Πρ2,ρ3 the solution of the Kantorovich problem:(47)K2:infγ∈Πρ2,ρ3∫ℝd×ℝdcx−yλ2dγ.
Using the Gluing lemma [8], there exists a probability measure σ on ℝd×ℝd×ℝd such that(48)σA×B×ℝd=γ1A×B,σℝd×A×B=γ2A×B,for some Borel subsets A and B of ℝd. Let γ3 be a probability measure on ℝd×ℝd defined by γ3A×B=σA×ℝd×B. Then, γ3∈Πρ1,ρ3, and we use the convexity of c to get that(49)∫ℝd×ℝdcx−zλ1+λ2dγ3=∫ℝd×ℝd×ℝdcx−zλ1+λ2dσ≤λ1λ1+λ2∫ℝd×ℝdcx−yλ1dγ1+λ2λ1+λ2∫ℝd×ℝdcy−zλ2dγ2≤λ1λ1+λ2+λ2λ1+λ2=1.
So,infγ∈Πρ1,ρ3∫ℝd×ℝdcx−z/λ1+λ2dγ≤1, and we conclude that(50)Wcρ1,ρ2≤λ1+λ2=Wcρ1,ρ2+Wcρ2,ρ3.
Hence,Wc is a distance on Pℝd.
Let us now study the topology ofWc.
Letρnn be a sequence on Pℝd and ρ∈Pℝd such that Wcρn,ρ converges to 0 when n tends to ∞. Define λn=Wcρn,ρ, since λnn converges to 0, then we use the fact that c is coercive to have(51)infγ∈Πρn,ρ∫ℝd×ℝdx−ydγ≤λn,when n⟶∞. We deduce that(52)limsupn⟶∞infγ∈Πρn,ρ∫ℝd×ℝdx−ydγ=0.
Note that the 1-Wasserstein metric betweenρn and ρ is(53)W1ρn,ρ=infγ∈Πρn,ρ∫ℝd×ℝdx−ydγ.
We deduce thatW1ρn,ρ converges to 0 when n tends to ∞. Since the 1-Wasserstein metric W1 induces the narrow topology of Pℝd, we conclude that the sequence ρnn converges narrowly to ρ in Pℝd.
Assume now that the sequenceρnn converges narrowly to ρ in Pℝd. Fix λ>0 and denote by γnλ the solution of Kantorovich problem:(54)K:infγ∈Πρn,ρ∫ℝd×ℝdcx−yλdγ.
Sinceρn converges narrowly to ρ, then γnλ converges narrowly to some γλ∈Πρ,ρ and(55)limn⟶∞∫ℝd×ℝdcx−yλdγnλ=∫ℝd×ℝdcx−yλdγλ=0.
So,(56)limn⟶∞infγ∈Πρn,ρ∫ℝd×ℝdcx−yλdγ=0,for all λ>0. Hence, for all λ>0, there exists Nλ>0 such that(57)infγ∈Πρn,ρ∫ℝd×ℝdcx−yλdγ≤1,for alln≥Nλ.
So,Wcρn,ρ≤λ for all n≥Nλ. Then,(58)limn⟶∞Wcρn,ρ=0.
Consequently,ρnn converges narrowly to ρ in the metric space PΩ,Wc.
We establish now the existence of solution for the variational problemP defined by(59)P:infρ∈Pqℝd∫ℝdGρdx+hWchρ0,ρ,in the metric space Pℝd,Wc, where(60)Wchρ0,ρ=infγ∈Πρ0,ρ∫ℝd×ℝdcx−yhdγ.
Pqℝd is the set of all probability measures on ℝd having q-finite moment, that is,(61)Pqℝd=ρ∈Pℝd,∫ℝdxqρdx<∞,and h>0 being a time step.Lemma 1.
Assume thatρ0, G, and c satisfy, respectively, Hf0, HG, and Hc. Then, the following is obtained:(i)
The mapρ⟶Wchρ0,ρ is lower semicontinuous in Pℝd,Wc.(ii)
The functionalρ⟶∫ℝdGρdx is lower semicontinuous in L1ℝd.(iii)
The setρ∈PMℝd,∫ℝdGρdx≤∫ℝdGρ0dx is a closed subset of L1ℝd,(62)wherePMℝd=ρ∈Pℝd,ρ≤M.Proof.
(i)
Letρ∈Pℝd. Let ρnn be a sequence in Pℝd such that ρnn converges to ρ in metric space Pℝd,Wc. Denote by γnh the solution of the Kantorovich problem:(63)Knh:infγ∈Πρn,ρ0∫ℝd×ℝdcx−yhdγ.We have(64)infγ∈Πρn,ρ0∫ℝd×ℝdcx−yhdγ=∫ℝd×ℝdcx−yhdγnh.Sinceρnn converges to ρ narrowly, then γnhn converges to γh∈Πρ0,ρ narrowly and(65)liminfn⟶∞∫ℝd×ℝdcx−yhdγnh≥∫ℝd×ℝdcx−yhdγh^≥Wchρ0,ρ.This implies that(66)liminfn⟶∞Wchρ0,ρn≥Wchρ0,ρ.Thus, we obtain the proof ofi.(ii)
SinceG is convex and G∈C10,∞,(67)∫ℝdGρndx≥∫ℝdGρdx+∫ℝdG′ρρn−ρdx.
Hence, ifρnn converges to ρ weakly in L1ℝd with ρ∈L∞ℝd, we have(68)liminfn⟶∞∫ℝdG′ρρn−ρdx=0,and then(69)liminfn⟶∞∫ℝdGρndx≥∫ℝdGρdx,which complete the proof of ii. The proof of iii is a consequence of ii.


## 2.3. Existence Results for the Discrete ProblemP1xh

First of all, we introduce here for unbounded domains analogous of the maximum principle stated for bounded domains in [1]. This maximum principle plays a central role in the searching of solution for the discrete problem P1xh. It is also used to further establish the convergence of our algorithm towards a weak solution of the kinetic equation (1).Proposition 2. (maximum principle).
Assume that the initial datumf0 satisfies N≤f0≤M with 0<N, c satisfies Hc, and G satisfies HG.
Then, any solutionF1xh of the variational problem(70)P1xh:infF∈PqℝdEF+hWchF,G1xh,satisfies ax≤F1xh≤bx, with ax=N/ρ1hx and bx=M/ρ1hx.Proof.
We defineEx≔y∈ℝd,F1xh>bx. Assume by contradiction that Ex has a positive Lebesgue measure. Then, γxExc×Ex>0, where γx is the minimizer of(71)K:infγ∈ΠG1xh,F1xh∫ℝd×ℝdcy−zhdγy,z.
Otherwise,(72)bxEx<∫ExF1xhdv=γxℝd×Ex=γxEx×Ex≤γxEx×ℝd=∫ExG1xhdy≤Exbx.
This yields a contradiction.
Defineνx=1Exc×Exγx, i.e., νxF=γxF∩Exc×Ex, for all F⊂ℝd, and denote by ν0,x and ν1,x the marginals of νx. We have νx≪γx with γx∈ΠG1xh,F1xh and then ν0,x≪G1xhdy and ν1,x≪F1xhdz.
Letv0,x and v1,x denote, respectively, the density functions of ν0,x and ν1,x. Because(73)∫Exv0,xdy=ν0,xEx=γxEx×ℝd∩Exc×Ex=0,∫Excv1,xdy=ν1,xExc=γxℝd×Exc∩Exc×Ex=0,we have v0,x=0 on Ex and v1,x=0 on Exc.
Forε>0 is small enough, define F1xh,ε=F1xh+εv0,x−v1,x and γxϵ by(74)∫ℝd×ℝdHx,ydγxε=∫ℝd×ℝdHx,ydγx+ε∫Exc×ExHx,x−Hx,ydγx,for all H∈C0ℝd×ℝd.
F1xh,ε∈Pqℝd, γxε∈ΠG1xh,F1xh,ε, and we get(75)IF1xh,ε−IF1xh=hWchF1xh,ε,G1xh−WchF1xh,G1xh+1ρ1hx∫ℝdGρ1hxF1xh,ε−Gρ1hxF1xh.
Taking into account some ideas from [1], we shall prove that IF1xh,ϵ−IF1xh<0, which leads to a contradiction.
Indeed,(76)WchF1xh,ε,G1xh−WchF1xh,G1xh=infγ∈ΠF1xh,ε,F1xh∫ℝd×ℝdcy−zhdγ−∫ℝd×ℝdcy−zhdγx≤∫ℝd×ℝdcy−zhdγxε−∫ℝd×ℝdcy−zhdγx=−ε∫Exc×Excy−zhdγx.
Sincey,z⟼cy−z/h is nonnegative on Exc×Ex and γxExc×Ex>0, we have(77)WchF1xh,ε,G1xh−WchF1xh,G1xh=−ε∫Exc×Excy−zhdγx<0.
Also, we have(78)1ρ1hx∫ℝdGρ1hxF1xh,ε−Gρ1hxF1xhdv=1ρ1hx∫ℝdGρ1hxF1xh+ερ1hxv0,x−v1,x−Gρ1hxF1xhdv=1ρ1hx∫ExGρ1hxF1xh−ερ1hv1,x−Gρ1hxF1xhdv+1ρ1hx∫ExcGρ1hxF1xh+ερ1hxv0,x−Gρ1hxF1xhdv.
G is being convex and of class C1, then(79)Gρ1hF1xh−Gρ1hF1xh−ερ1hv1,x≥G′ρ1hF1xh−ερ1hv1,xερ1hv1,x,on Ex and(80)Gρ1hxF1xh+ερ1hxv0,x−Gρ1hxF1xh≤ερ1hv0,xG′ρ1hF1xh+ερ1hv0,x,on Exc.
Hence, we have(81)1ρ1hx∫ℝdGρ1hxF1xh,ε−Gρ1hxF1xhdv≤ε∫ExcG′ρ1hF1xh+ερ1hv0,xv0,xdv−∫ExG′ρhF1xh−ερ1hv1,xv1,xdv=ε∫Exc×ExG′ρ1hF1xh+ερ1hv0,x−G′ρ1hF1xh−ερ1hv1,xdγx.
SinceG′ is continuous, then(82)limε⟶0G′ρ1hF1xh+ερ1hv0,x−G′ρ1hF1xh−ερ1hv1,x=0.
Consequently, there existsδ>0 such that, for 0<ε<δ, then(83)0≤G′ρ1hF1xh+ϵρ1hv0,x−G′ρ1hF1xh−ϵρ1hv1,x≤12γxExc×Ex∫Exc×Excy−zhdγx.
So, we fixε>0 small, such that 0<ε<δ, and we use (81) and (83) to obtain(84)1ρ1hx∫ℝdGρ1hxF1xh,ε−Gρ1hxF1xhdv≤ε2∫Exc×Excy−zhdγx.
Now, fixingϵ<δ and combining (77) and (84) yield(85)IF1xh,ε−IFxh<−ε∫Exc×Excy−zhdγx+ε2∫Exc×Excy−zhdγx=−ε2∫Exc×Excy−zhdγx<0.
It is a contradiction sinceF1xh is a minimizer of I on Pqℝd. Consequently, Ex is negligible and then F1xh≤bx.
The proof ofF1xh≥ax is analogous to that of F1x≤bx. Thus, we conclude that ax≤F1xh≤bx.Lemma 2.
Letf0∈PqΩ×ℝd be a probability density on Ω×ℝd such that 0<N≤f0≤M,∫Ω×ℝdGf0dxdv<∞ and ∫Ω×ℝdvqf0x,vdxdv<∞.The variational problemP1xh defined in (32) admits a unique minimizer F1xh in Pqℝd and N/ρ1hx≤F1xh≤M/ρ1hx.Proof.
Since0<N≤f0≤M and G1xhv=f0x,v/ρ1hx, then N/ρ1hx≤G1xh≤M/ρ1hx. Then, we use Proposition 2, and we obtain that any minimizer F1xh of the variational problem P1xh satisfies ax≤F1xh≤bx, with ax=N/ρ1hx and bx=M/ρ1hx.
By using Proposition2 and the fact that G is convex, we obtain that(86)EF≥1/ρ1hx∫ℝdGρ1hxG1xhvdv+G′N−G′M,for all probability density F∈Pqℝd such that N≤F≤M. We use now Lemma 1 to get that the functional F⟼1/ρhx∫ℝdGρhxFvdv is lower semicontinuous on the Wasserstein space Pqℝd,Wc. We conclude then that the problem P1xh admits a solution F1xh. The strict convexity of G and c implies the strict convexity of the map F⟼WchG1xh,F and that of the maps F⟼1/ρhx∫ℝdGρhxFvdv and accordingly the uniqueness of the minimizer F1xh of P1xh.


## 3. Euler–Lagrange Equation for the ProblemPkxh

In this section, we prove that the sequencefkhk is a time discretization of the kinetic equation (1). In order to achieve it, we need the following lemma.Lemma 3. (explicit expression for optimal maps).
Assume thatG satisfies HG and c satisfies Hc. Then, the Monge problem(87)M:infT#Fkxh=Gkxh∫ℝdcv−TvhFkxhdv,admits a unique solution Tk such that(88)Tkv=v+h∇c∗∇vG′ρkhFkxh,where Fkx is the unique minimizer of the variational problem.(89)Pkxh:infF∈PqℝdIF≔1ρkh∫ℝdGρkhF+hWchGkxh,F.Proof.
Letψ∈Cc∞ℝd,ℝd be a test function and consider the diffeomorphism map Tεε∈ℝ in C∞ℝd,ℝd defined by(90)∂Tε∂ε=ψ∘Tε,T0=id.
Define the probability densityGεx=Tε#Fkx on ℝd. Since G satisfies HG and Tε is a diffeomorphism pushing Fkxh forward to Gεx, we obtain the following Monge–Kantorovich-type energy inequality:(91)1ρkhx∫ℝdGρkhGεxdv−∫ℝdGρkhFkxhdv≥∫ℝdTεv−v,∇vG′ρkhFkxhFkxhdv.
Recalling the definition ofTε, we have(92)Tεv−vε≤ψ∞,for allv∈ℝdandε>0.
Dividing (91) by ε>0 and using (92) and the dominated convergence theorem, we have(93)limε⟶0+1ερkhx∫ℝdGρkhGεxdv−∫ℝdGρkhFkxhdv≥limε⟶0+∫ℝdTεv−vε,∇vG′ρkhFkxhFkxhdv≥∫ℝdψ,∇vG′ρkhFkxhFkxhdv.
Furthermore, sinceTε−1#Gεx=Fkxh, then the Monge–Kantorovich-type energy inequality gives(94)limε⟶0+1ερkhx∫ℝdGρkhFkxhdv−∫ℝdGρkhGεxdv≥limε⟶0+∫ℝdTε−1v−vε,∇vG′ρkhGεxGεxdv=limε⟶0+∫ℝdv−Tεvε,∇vG′ρkhGεx∘TεFkxdv=−∫ℝdψ,∇vG′ρkhFkxhFkxhdv.
This implies that(95)limε⟶0+1ερkhx∫ℝdGρkhGεxdv−∫ℝdGρkhFkxhdv≤∫ℝdψ,∇vG′ρkhFkxhFkxhdv.
We combine now (93) and (95) to derive(96)limε⟶0+1ερkhx∫ℝdGρkhGεxdv−∫ℝdGρkhFkxhdv=∫ℝdψ,∇vG′ρkhFkxhFkxhdv.
SinceTk#Fkxh=Gkxh and Tε#Fkxh=Gεx, then Tε∘Tk−1#Gkxh=Gεx. So,(97)WchGkxh,Gεx=infS#Gkxh=Gεx∫ℝdcv−SvhGkxhdv≤∫ℝdcv−Tε∘Tk−1vhGkxhdv.
Thus, forε>0, we have(98)WchGkxh,Gεx−WchGkxh,Fkxhε≤1ε∫ℝdcv−Tε∘Tk−1vh−cv−TkvhGkxhdv.
We use the convexity ofc and the fact that Tk#Fkxh=Gkxh to obtain(99)hWchGkxh,Gεx−hWchGkxh,Fkxhε≤−∫ℝd∇cTkv−Tεvh,Tεv−vεFkxhdv.
Now, from (92) and the dominated convergence theorem, we obtained(100)limε⟶0+hWchGkxh,Gεx−hWchGkxh,Fkxhε≤−∫ℝd∇cTkv−vh,ψFkxhdv.
SinceFkxh minimizes the functional(101)IF=1ρkhx∫ℝdGρkhFdv+hWchGkxh,F,on the probability space, the Euler–Lagrange equation yields(102)limε⟶0+IGεx−IFkxhε=0.
Consequently, by using (96) and (100), we have(103)∫ℝdψ,∇vG′ρkhFkxhFkxhdv−∫ℝd∇cTkv−vh,ψFkxhdv≥0.
Next, by replacingψ by −ψ in (96), we obtain the desired equation:(104)∫ℝdψ,∇vG′ρkhFkxhFkxhdv−∫ℝd∇cTkv−vh,ψFkxhdv=0.
Thus, we conclude that(105)∇cTkv−vh=∇vG′ρkhFkxha.e−Fkxh.
Note that∇c is inversible and ∇c−1=∇c∗. Then, we obtain the explicit expression of the optimal map Tk:(106)Tkv=v+h∇c∗∇vG′ρkhFkxh−a.eFkxh.
The proof of this lemma is complete.
We are now ready to show that the sequencefkhk satisfies the time discretization (17) of the kinetic equation (1).
Letψ∈Cc∞Ω×ℝd, then(107)∫Ω×ℝdfkhx,v−fk−1hx,vψx,vdxdv=∫Ω×ℝdfkhx,v−gk−1x,vψx,vdxdv+∫Ω×ℝdgkhx,v−fk−1hx,vψx,vdxdv.
By usingTk#Fkxh=Gkxh, fkh=ρkhFkxh, and gkh=ρkhGkxh, we have(108)∫Ω×ℝdfkhx,v−gk−1x,vψx,vdxdv=∫Ω×ℝdψx,v−ψx,Tkxfkhx,vdv,∫Ω×ℝdgkhx,v−fk−1hx,vψx,vdxdv=∫Ω×ℝdψx+hv,v−ψx,vfk−1hdxdv.
Next, we use the Taylor formula and the expression of the optimalTk to obtain(109)∫Ω×ℝdfkhx,v−fk−1hx,vψx,vdxdv=h∫Ω×ℝdv,∇xψx,vfk−1hx,vdxdv−h∫Ω×ℝd∇c∗∇vG′fkhfkhdxdv+h2∫Ω×ℝdvτ,∇x2ψx+hθv,vvfk−1hdxdv+h2∫Ω×ℝdσkτ,∇v2ψx,v+αhσkσkfkhdxdv.
Here,σk=∇c∗∇vG′fkh and θ,α∈]0,1[.
Now define(110)Akhψ≔h22∫Ω×ℝdσkτ,∇2ψx,v+hθσk.σkfkhx,vdxdv+h22∫Ω×ℝdvτ,∇x2ψx+αhv,vvfk−1hdxdv.
If we show thatAkhψ tends to 0 as h goes to 0, then we are done.
Indeed, from the maximum principle,N≤fkh≤M, and then(111)Akhψ≤h22∇v2ψL∞Ω×ℝd∫Ω×ℝd∇c∗∇vG′fkh2fkhdxdv+h22∇x2ψL∞Ω×ℝdf0L∞Ω×ℝd∫Kψv2dxdv,where Kψ is a compact subset of Ω×ℝd such that suppψ⊂Kψ.
SinceFkxh minimizes the general functional energy IF=1/ρkh∫ℝdGρkhFdv+hWchF,Gkxh on the metric space Pqℝd,Wc, then(112)1ρkh∫ℝdGρkhFkxhdv+hWchFkxh,Gkxh≤1ρkh∫ℝdGρkhGkxhdv.
We use in (112) the expression of the optimal maps Tk and the definition of the fkh, and then, we have(113)∫Ω×ℝdGfk−1hdxdv−∫Ω×ℝdGfkhdxdv≥h∫Ω×ℝdc∇c∗∇vG′fkhfkhdxdv.
BecauseG∈C10,∞ is convex and N≤fkh≤M, we have(114)∫Ω×ℝdGfk−1hdxdv−∫Ω×ℝdGfkhdxdv≤G′M−G′N.
By using (114) and the fact that cx≥A1xq, then (113) becomes(115)h∫Ω×ℝd∇c∗∇vG′fkhqfkhdxdv≤G′M−G′NA1.
Recalling (115) and (111), we obtain(116)Akhψ≤h∇x2ψ∞Kψ2M+h‖∇v2ψ∞2q−1G′M−G′NA1+h2q−1,if,q≥2,(117)Akhψ≤h∇x2ψ∞Kψ2M+hq−1∇v2ψ∞G′M−G′NA1,if,q<2.
We combine (116) and (117) to conclude that Akhψ tends to 0, when h goes to 0. Accordingly, we conclude that the sequence fkhk resulted from a time discretization of the kinetic equation (1).
Recalling Section2.1, we define an approximate solution fh over 0,∞×Ω×ℝd of the kinetic equation (1) as follows:(118)fht,x,v=fkhx−t−tkv,v,ift∈tk,tk+1,fh0,x,v=f0x,v.
In the next section, we establish the convergence of the sequencefhh to a weak solution f of (1).


## 4. Convergence Results

### 4.1. Weak Convergence offhh

Let us consider the sequencefhh as defined in (118).Lemma 4.
Assume thatf0 satisfies Hf0. Then, for all k≥1, we have(119)∫Ω×ℝdfkhx,vpdxdv≤∫Ω×ℝdf0x,vpdxdv.Proof.
TakingHt=tp,p>1, in Corollary 1, we obtain(120)∫ℝdGkxhpdv−∫ℝdFkxhpdv≥∫ℝdTkv−v,∇vH′FkxhFkxhdv,where Tk is c-optimal map that pushes Fkxh forward to Gkxh. We use expression of Tk and we obtain(121)∫ℝdGkxhpdv−∫ℝdFkxhpdv≥∫ℝdTkv−v,∇vH′FkxhFkxhdv=pp−1h∫ℝd∇c∗∇vG′fkh,∇vFkxhFkxhp−1dv.
Since∇vG′fkh=G″fkh∇vfkh and fkh=ρkhFkxh, (121) becomes(122)∫ℝdGkxhpdv−∫ℝdFkxhpdv≥hpp−1ρkhx∫ℝdFkxhp−1G′′fkh∇c∗∇vG′fkh,∇vG′fkhdv.
From the convexity ofc and c∗, we have(123)a.b=ca+c∗b,ifa,b∈ℝd,witha=∇c∗b.
Thus, using (123) with a=∇c∗∇vG′fkh and b=∇vG′fkh, (122) becomes(124)∫ℝdGkxhpdv−∫ℝdFkxhpdv≥hpp−1ρkhx∫ℝdFkxhp−1G′′fkhc∇c∗∇vG′fkhdv+hpp−1ρkhx∫ℝdFkxhp−1G′′fkhc∗∇vG′fkhdv.
Sincec≥0 and c0=0, then c∗≥0; hence,(125)∫ℝdGkxhpdv−∫ℝdFkxhpdv≥0.
Multiplying (125) by ρkhp, we obtain after integration(126)∫Ω×ℝdfkhpdxdv≤∫Ω×ℝdgkhpdxdv=∫Ω×ℝdfk−1hpdxdv,and then by an iteration process on k, we get the proof of Lemma 4.Lemma 5.
Assume thatf0∈LpΩ×ℝd∩PqΩ×ℝd with ∫Ω×ℝdGf0dxdv<∞ and 0<N≤f0≤M. For 0<T<∞, we have(127)∫0,T×Ω×ℝdfht,x,vpdtdxdv≤T∫Ω×ℝdf0x,vpdxdv.Proof.
Let0<T<∞ and let h>0 be a step such that T/h∈ℕ∗.
We use Lemma3 and the definition of fh in (118) to obtain(128)∫0,T×Ω×ℝdfht,x,vpdtdxdv=∑k=1T/hh∫Ω×ℝdfkhpdxdv≤T∫Ω×ℝdf0x,vpdxdv<∞.


### 4.2. Weak Convergence of the Linear Term

Here, we study the weak convergence of the linear term.Proposition 3.
Assume thatf0 satisfy Hf0. Then, there exists a function f in Lp0,×Ω×ℝd∩L∞0,T×Ω×ℝd until a subsequence fhh converges to f weakly in Lp0,T×Ω×ℝd and weakly-∗ in L∞0,T×Ω×ℝd.Moreover,(129)limk⟶∞∫0,T×Ω×ℝd∂tψ+v.∇xψfhkdtdxdv=∫0,T×Ω×ℝd∂tψ+v.∇xψfdtdxdv,for every test function ψ∈Cc∞ℝ×Ω×ℝd, with suppψ.,x,v⊂−T,T.Proof.
Sincef0∈LpΩ×ℝd∩L∞Ω×ℝd (see Proposition 2 and Lemma 4), then(130)fhL∞≤f0L∞Ω×ℝd,fhLp≤T1/pf0LpΩ×ℝd.Hence, the sequence fhh is bounded in Lp0,T×Ω×ℝd∩L∞0,T×Ω×ℝd, and then, there is subsequence of fhh still denoted fhh that converges to a nonnegative function f weakly in Lp0,T×Ω×ℝd and weakly-∗ in L∞0,T×Ω×ℝd.
Since∂tψ+v.∇xψ∈Lq0,T×Ω×ℝd, for every ψ∈Cc∞ℝ×Ω×ℝd such that suppψ.,x,v⊂−T,T, we finally obtain Proposition 3.


### 4.3. Weak Convergence of the Nonlinear Term

In this section, we establish the weak convergence of the nonlinear term.First, we prove that the sequence∇vG′fh is bounded in Lp0,T×Ω×ℝd, and that ∇c∗∇vG′fh is bounded in Lq0,T×Ω×ℝd where 0<T<∞.Lemma 6.
Assume thatf0, c, and G satisfy, respectively, Hf0, Hc, and HG. Then, for all 0<T<∞ fixed,(131)∫0,T×Ω×ℝd∇c∗∇vG′fhqdtdxdv≤TA1NG′M−G′N,∫0,T×Ω×ℝd∇vG′fhpdtdxdv≤A2T+G′M−G′Npqp−1A2p−1N.Proof.
Using (115), we have(132)∫Ω×ℝd∇c∗∇vG′fkhqfkhdxdv≤G′M−G′NA1.
Butfh≥N, and then(133)∫0,T×Ω×ℝd∇c∗∇vG′fhqdtdxdv≤G′M−G′N′TNA1.
Recalling (123), with a=∇c∗∇vG′fh and b=∇vG′fh, we have(134)∇c∗∇vG′fh⋅∇vG′fh−c∇c∗∇vG′fh=c∗∇vG′fh,with c≥0.
Then,(135)∫0,T×Ω×ℝdc∗∇vG′fhfhdtdxdv≤∫0,T×Ω×ℝd∇c∗∇vG′fh∇vG′fhdtdxdv=∑k=1T/hh∫Ω×ℝd∇c∗∇vG′fkh∇vG′fkhdxdv≤∑k=1T/h∫Ω×ℝdGfk−1hdxdv−∫Ω×ℝdGfkhdxdv.
Moreover,(136)∑k=1T/h∫Ω×ℝdGfk−1hdxdv−∫Ω×ℝdGfkhdxdv=∫Ω×ℝdGf0dxdv−∫Ω×ℝdGfT/hdxdv≤G′M−G′N,and then,(137)∫0,T×Ω×ℝdc∗∇vG′fhfhdtdxdv≤G′M−G′N.
Becausecy≤A2yq+A2, we have x⋅y−cy≥x⋅y−A2yq−A2 for all x,y∈ℝd and then(138)c∗x≔supy∈ℝdx⋅y−cy≥−A2+xppqp−1A2p−1.
Consequently,(139)−A2T+1pqp−1A2p−1∫0,T×Ω×ℝd∇vG′fhpfhdtdxdv≤G′M−G′N.
Usingfh≥N, we obtain(140)∫0,T×Ω×ℝd∇vG′fhpdtdxdv≤TA2+G′M−G′Npqp−1A2p−1N.


### 4.4. Strong Convergence offhh

In order to prove the strong convergence result, we need to establish the following compactness results forfhh in Lp0,T×Ω×ℝd.Lemma 7.(velocity-compactness).
Assume thatf0 satisfies Hf0. Fix 0<T<∞ and 0≠η∈ℝd such that η is small. Then,(141)∫0,T×Ω×ℝdfht,x,v+η−fht,x,vpdtdxdv≤ηpK,where K≔inft∈N,M1/G′′tpTA2+G′M−G′Npqp−1A2p−1/N is a constant.Proof.
We use∇vG′fh=G′′fh∇vfh, N≤fh≤M, and (140) to produce that(142)∫0,T×Ω×ℝd∇vfhpdtdxdv≤K.
Then,fh and ∇vfh∈Lp0,T×Ω×ℝd and approximating fht,x,. by Cc∞ℝd functions in W1,pℝd, we obtain(143)∫0,T×Ω×ℝdfht,x,v+η−fht,x,vpdtdxdv≤ηp∫0,T×Ω×ℝd∇vfhpdtdxdv.
This implies that(144)∫0,T×Ω×ℝdfht,x,v+η−fht,x,vpdtdxdv≤ηpK.Lemma 8. (position-compactness).
Assume thatf0 satisfies Hf0. We fix 0<T<∞ and 0≠η∈ℝ such that η is small. Then,(145)∫0,T×Ω×ℝdfht,x+hηv,v−fht,x,vpdtdxdv≤ηpTK1,where K1=2pf0LpΩ×ℝdp.Proof.
By fixingv, (17) becomes(146)fkh−fk−1h+hv⋅∇xfkh=oh,where oh tends to 0 when h goes to 0. We use (146), and the fact that fkhk is bounded in LpΩ×ℝd to deduce that hv⋅∇xfkhk is bounded in LpΩ×ℝd. Then, by using Lemma 4, we have(147)hv⋅∇xfkhLpΩ×ℝd≤2f0LpΩ×ℝd.
Thus,fht,.,v∈LpΩ, and approximating fht,.,v by Cc∞ℝd functions, we use the fact that hv⋅∇xfkhk is bounded in LpΩ×ℝd to deduce that(148)∫0,T×Ω×ℝdfht,x+hηv,v−fht,x,vpdtdxdv≤ηp∫0,T×Ω×ℝdhv⋅∇xfht,x,vpdtdxdv≤ηp2pT∫Ω×ℝdf0x,vpdxdv.Lemma 9.
Assume thatHf0 holds. Let τ>0 be quite small such that t,t+τ⊂0,T,∀t∈0,T, where 0<T<∞ is fixed. We have(149)∫0,T×Ω×ℝdG′fht+τ−G′fhtfht+τ−fhtdtdxdv≤τK2Ω,T,N,M,f0,G′,q,p,where K2Ω,T,N,M,f0,G′,q,p is a positive constant depending on Ω,T,N,M,f0,G′,q,p.Proof.
Chooseh>0 such that T/h∈ℕ∗ and τ/h=N∈ℕ∗. Define(150)A=∫0,T×Ω×ℝdG′fht+τ−G′fhtfht+τ−fhtdtdxdv.
Using the definition offh in (118), we obtain(151)A=∑k=1T/hh∫Ω×ℝdψh,kx,vfk−1+Nhx,v−fk−1hx,vdxdv,where ψh,kx,v=G′fk−1+Nhx,v−G′fk−1hx,v.
Note that(152)fk−1+Nhx,v−fk−1hx,v=∑i=0N−1fk−1+N−ihx,v−fk−2+N−ihx,v.
Using (152), we have(153)A=∑k=1T/h∑i=0N−1h∫Ω×ℝdψh,kx,vfk−1+N−ihx,v−fk−2+N−ihx,vdxdv=C+D,where(154)C=∑k=1T/h∑i=0N−1h∫Ω×ℝdψh,kx,vfk−1+N−ihx,v−gk−1+N−ihx,vdxdv,D=∑k=1T/h∑i=0N−1h∫Ω×ℝdψh,kx,vgk−1+N−ihx,v−fk−2+N−ihx,vdxdv.
UsingSk,Nh,i as a c-optimal map that pushes Fk−1+N−ixh forward to Gk−1+N−ixh, we have(155)C=∑k=1T/h∑i=0N−1h∫Ω×ℝdψh,kx,v−ψh,kx,Sk,Nh,ivfk−1+N−ihx,vdxdv.
On the contrary,∇vG′fk−1+N−ih∈LpΩ×ℝd (see Lemma 6), and from assumptions on G, G′s/s is continuous on ℝ∗, and then there exists a constant K such that G′s≤K⋅s for 0<N≤s≤M. Since 0<N≤fk−1+N−ih≤M (see Lemma 2) and fk−1+N−ih∈LpΩ×ℝd, then G′fk−1+N−ih∈LpΩ×ℝd. So G′fk−1+N−ihx,.∈W1,pℝd.
ApproximatingG′fk−1+N−ihx,. by Cc∞ℝd functions and using the dominated convergence theorem, we have(156)C=∑i=1N−1∑k=1T/hh∫Ω×ℝdfk−1+N−ihx,vv−Sk,Nh,ix,v,∇vψh,kx,v+θv−Sk,Nh,ix,vdxdv,forθ∈0,1.
Note thatv−Sk,Nh,i=−h∇c∗∇vG′fk−1+N−ih∈LqΩ×ℝd and ∇vψh,k=∇vG′fk−1+Nh−∇vG′fk−1h∈LpΩ×ℝd (see Lemma 6).
So, by using Hölder’s inequality, we have that(157)C≤∑i=1N−1∑k=1T/hh∫Ω×ℝdv−Sk,Nh,ivqfk−1+N−ihx,vdxdv1/q∫Ω×ℝd∇vψh,kp1/p≤∑i=1N−1∑k=1T/hhA11/q∫Ωρk−1+N−ihxWchFk−1+N−ixh,Gk−1+N−ixhdx1/q∫Ω×ℝd∇vψh,kp1/p.
Using the algorithm defined in Section2.1 and Lemma 6, we have(158)∫Ωρk−1+N−ihxWchFk−1+N−ixh,Gk−1+N−ixhdx≤hq−1∫Ω×ℝdGfk−2+N−ihx,vdxdv−∫Ω×ℝdGfk−1+N−ihx,vdxdv.
Thus,(159)C≤∑i=1N−1∑k=1T/hhA11/q∫Ω×ℝdGfk−2+N−ihx,vdxdv−∫Ω×ℝdGfk−1+N−ihx,vdxdv1/q×hq−1/q∫Ω×ℝd∇vψh,kp1/p.
Applying the Hölder’s inequality in the previous relation yields(160)C≤1A11/qh1−1/phq−1/q∑k=1T/h∫Ω×ℝdGfk−2+N−ihx,vdxdv−∫Ω×ℝdGfk−1+N−ihx,vdxdv1/q×∑i=1N−1∑k=1T/hh∫Ω×ℝd∇vψh,kp1/p.
We notice that(161)∑k=1T/h∫Ω×ℝdGfk−2+N−ihx,vdxdv−∫Ω×ℝdGfk−1+N−ihx,vdxdv=∫Ω×ℝdGf−1+N−ihx,vdxdv−∫Ω×ℝdGfT/h−1+N−ihx,vdxdv≤G′M−G′N,
Next, we use Lemma6, and we have(162)∑k=1T/hh∫Ω×ℝd∇vψh,kp1/p=∑k=1T/hh∫Ω×ℝd∇vG′fk−1+Nh−∇vG′fk−1hp1/p≤2p−1/p∑k=1T/hh∫Ω×ℝd∇vG′fk−1+Nhp+∑k=1T/hh∫Ω×ℝd∇vG′fk−1hp1/p≤2p−1/p2M0T,Ω1/p=2M0T,Ω1/p.
We combine (160), (161), and (162) to get(163)C≤∑i=1N−12h1−1/phq−1/qG′M−G′N1/qM0T,Ω1/p≤Nh2−1/p+1/qG′M−G′N1/qM0T,Ω1/p.
WritingNh=τ, we then have(164)C≤G′M−G′N1/qM0T,Ω1/pτ.
On the contrary,(165)D=∑k=1T/h∑i=0N−1h∫Ω×ℝdψh,kx,vfk−2+N−ihx,v−gk−1+N−ihx,vdxdv=∑k=1T/h∑i=0N−1h∫Ω×ℝdψh,kx,vfk−2+N−ihx,v−fk−2+N−ihx−hv,vdxdv.
SinceG′fk−1+N−ih.,v∈W1,pΩ and then ψh,k.,v∈W1,pΩ, we approximate ψh,k.,v by Cc∞Ω functions and we get(166)D=−∑k=1T/h∑i=0N−1h2∫Ω×ℝdv⋅∇xψh,kx,vfk−2+N−ihx,vdxdv.
Note thatv⋅∇xG′fkh=G′′fkhv⋅∇xfkh, and according to Lemma 8, v⋅∇xfkh∈LpΩ×ℝd. So, since G′′ is continuous and N≤fkh≤M, we have v⋅∇xG′fkhh,k∈LpΩ×ℝd.
Consequently,(167)D=−∑k=1T/h∑i=0N−1h2∫Ω×ℝdv⋅∇xψh,kx,vfk−2+N−ihx,vdxdv=−∑k=1T/h∑i=0N−1h2∫Ω×ℝdv⋅∇xG′fk−1+Nhx,vfk−2+N−ihdxdv+∑k=1T/h∑i=0N−1h2∫Ω×ℝdv⋅∇xG′fk−1hx,vfk−2+N−ihx,vdxdv≤∑k=1T/h∑i=0N−1h2∫Ω×ℝdv⋅∇xG′fk−1hx,vpdxdv1/pf0L∞+∑k=1T/h∑i=0N−1h2∫Ω×ℝdv⋅∇xG′fk−1+Nhx,vpdxdv1/pf0L∞.
Next, we use the fact thatv⋅∇xG′fkh=G′′fkhv⋅∇xfkh and the boundedness in LpΩ×ℝd, of v⋅∇xfkhh,k (see Lemma 8), to get(168)D≤2f0L∞Ω×ℝdNThsups∈N,MG′′sK1Ωh2=2f0L∞Ω×ℝdsups∈N,MG′′sK1ΩTτ.
Thus,(169)A=C+D≤K2A1,Ω,T,N,M,f0,G′,q,pτ,where K2A1,Ω,T,N,M,f0,G′,q,p≔2f0L∞Ω×ℝdsups∈N,MG′′sK1ΩT+G′M−G′N1/qM0T,Ω1/p.
Let us process now to the following estimate.Lemma 10.
Assume thatf0 satisfies Hf0. Then, for every 0<T<∞ fixed and for all τ>0 small, we have(170)∫0,T×Ω×ℝdfht+τ,x,v−fht,x,vpdtdxdv≤CA1,Ω,T,N,M,f0,G′,q,pτ,where CA1,Ω,T,N,M,f0,G′,q,p is positive constant.Proof.
SinceG∈C2, we use Taylor’s formula to have that(171)G′fht+τ,x,v−G′fht,x,v=G′′fθhfht+τ,x,v−fht,x,v,where θ∈]0,1[ and fθh≔θfht+1−θfht+τ.
Since0<N≤f0≤M, we have 0<N≤fh≤M (see Lemma 2). Thus, using (171) and the fact that G′′ is continuous and nonnegative, we have(172)∫0,T×Ω×ℝdG′fht+τ−G′fhfht+τ−fhtdtdxdv≥inft∈N,MG′′t∫0,T×Ω×ℝdfht+τ,x,v−fht,x,v2dtdxdv.
Defineβ=inft∈N,MG′′t>0. We use Lemma 9 and obtain(173)∫0,T×Ω×ℝdfht+τ,x,v−fht,x,v2dtdxdv≤1βK2A1,Ω,T,N,M,f0,G′,q,pτ.
UsingN≤fh≤M, we have(174)∫0,T×Ω×ℝdfht+τ,x,v−fht,x,vpdtdxdv≤1βCA1,Ω,T,N,M,f0,G′,q,pτ,where CΩ,T,N,M,f0,G′,q,p is positive constant.
We will use Lemmas7, 8, and 10, to prove that the sequence fhh is precompact in Lp0,T×Ω×ℝd.Lemma 11.
Assume thatf0, c, and G satisfy, respectively, Hf0, Hc, and HG. Then, fhh is precompact in Lp0,T×Ω×BR, where BR is the open ball centered at origin of radius R.Proof.
DefineK=0,T×Ω×BR. Then, K=∪δ>0Kδ, with Kδ=y∈K,disty,Kc>δ. Kδ⊂K, Kδ¯⊂K, and Kδ¯ is compact and distKδ,Kc>δ/2. So, there exists δ1,δ2,…,δl>0 such that Kδ¯⊂∪i=1lKδi.
Defineδ0=mini=1,2,…,dδi and we have Kδ0⋐K and distKδ0,Kc>δ0/2.
Letτ>0, η1,η2∈ℝd, such that 0<τ<δ0,ε, η1<δ0,ε, and η2<δ0,ε, where δ0,ε=minδ0/2,ε.
We use Lemmas7, 8, and 10 to have(175)∫Kδfht+τ,x,v−fht,x,vpdtdxdv<c1ε,∫Kδfht,x+η1,v−fht,x,vpdtdxdv<εpc2,∫Kδfht,x,v+η2−fht,x,vpdtdxdv<c3εp.
c1,c2, and c3 are defined in Lemmas 7, 8, and 10. So, by using the Riesz–Fréchet–Kolmogorov’s theorem, we deduce that the sequence fhh is precompact in LpKδ. Therefore, limδ⟶0K\Kδ=0, and then for all ε>0, there exists δε>0 such that K\Kδε<εf0L∞, and since fhh is bounded in L∞0,T×Ω×BR, we have fhLpK\Kδεp≤f0L∞K\Kδε<ε.
We use corollaryIV.26 of [9], to conclude that the sequence fhh is precompact in Lp0,T×Ω×BR, which implies that fhh converges strongly to some function fR in Lp0,T×Ω×BR.
Now, let us prove that the sequencefhh is precompact in Lp0,T×Ω×ℝd.Lemma 12.
Assume thatf0, c, and G satisfy, respectively, Hf0, Hc, and HG. Then, the sequence fhh is precompact in Lp0,T×Ω×ℝd.Proof.
Sincef0∈Lp0,T×Ω×ℝd, Lemma 3 implies that fhh is bounded in Lp0,T×Ω×ℝd. Then, fhh converges weakly to some function f in Lp0,T×Ω×ℝd (up to a subsequence).
LetBn be an open ball of radius n centered at origin, with n∈ℕ∗. Then, Bn⊂Bn+1 and ∪n∈ℕ∗Bn=ℝd.
Using previous results, the sequencefhh is precompact in Lp0,T×Ω×Bn. A subsequence fhh converges strongly to some function fn in Lp0,T×Ω×Bn. Let us denote by fhn the subsequence of fhh which converges strongly to fn in Lp0,T×Ω×Bn. Fix ϕ in Lq0,T×Ω×ℝd. We have(176)∫0,T×Ω×ℝdfhn−fnϕdtdxdv=∫0,T×Ω×Bnfhn−fnϕdtdxdv+∫0,T×Ω×Bncfhn−fnϕdtdxdv,with(177)∫0,T×Ω×Bncfhn−fnϕdtdxdv≤ϕLq0,T×Ω×ℝd2MBnc1/p.
Bnc tends to 0, when n goes to ∞; for ε>0, there exists Nε>0 such that(178)∫0,T×Ω×Bncfhn−fnϕdtdxdv≤ε2,for all n≥Nε. Since fhn converges strongly to fn in Lp0,T×Ω×Bn, we deduce that(179)∫0,T×Ω×ℝdfhn−fnϕdtdxdv≤ε,when hn goes to 0 and n≥Nε. We conclude that fhn converges weakly to fn in Lp0,T×Ω×ℝd, for n≥Nε. So fn=f for n≥Nε.
Now, let prove thatfhn−fLp0,T×Ω×ℝd tends to 0 as hn⟶0 and n≥Nε.
Note that(180)fhn−fLp0,T×Ω×ℝdp=∫0,T×Ω×Bnfhn−fpdtdxdv+∫0,T×Ω×Bncfhn−fpdtdxdv≤∫0,T×Ω×Bnfhn−fpdtdxdv+2pf0L∞pTΩBnc.
Let us recall that(181)limhn⟶0∫0,T×Ω×Bnfhn−fpdtdxdv=0,∀n≥Nε.
SinceBnc<ε/2, for n≥Nε, as(182)limhn⟶0∫0,T×Ω×BNεfhn−fpdtdxdv=0,we can find nε∈ℕ such that for every n≥Nε, we have(183)∫0,T×Ω×BNεfhn−fpdtdxdv<ε2.
Then, for alln≥Nε,(184)∫0,T×Ω×ℝdfhn−fpdtdxdv<ε.
Consequently,fhn converges strongly to f in Lp0,T×Ω×ℝd.
Using the previous lemma,fhh converges strongly to f in Lp0,T×Ω×ℝd (up to a subsequence). G′ is being continuous, then G′fhh converges strongly to G′f in Lp0,T×Ω×ℝd. We conclude that the sequence ∇vG′fhh converges to ∇vG′f in Cc∞ℝ×Ω×ℝd′. Let us recall that ∇vG′fhh is bounded in Lp0,T×Ω×ℝd, so ∇vG′fhh converges weakly to ∇vG′f in Lp0,T×Ω×ℝd.
On the contrary,∇c∗∇vG′fhh is bounded in Lq0,T×Ω×ℝd. Then, a subsequence ∇c∗∇vG′fhh converges weakly to σ in Lq0,T×Ω×ℝd.
We derive from the proof of Theorem 3.10 in [1] that the sequence divvfh∇c∗∇vG′fh converges weakly to divvf∇c∗∇vG′f in Cc∞ℝ×Ω×ℝd′.Theorem 2.
Assume thatf0, c, and G satisfy, respectively, Hf0, Hc, and HG. Let 0<T<∞ and u∈Cc∞ℝ be such that u≥0 and suppu⊂−T,T. Then,(185)limh⟶0∫0,T×Ω×ℝdfh∇c∗∇vG′fh,∇vG′fhutdtdxdv=∫0,T×Ω×ℝdfσ,∇vG′futdtdxdv.Moreover,divvfh∇c∗∇vG′fhh converges weakly to divvfσ in Cc∞ℝ×Ω×ℝd′ and divvfσ=divvf∇c∗∇vG′f in the weak sense.Proof.
Let us recall the following:(i)fhh converges strongly to f in Lp0,T×Ω×ℝd(ii)∇vG′fhh converges weakly to ∇vG′f in Lp0,T×Ω×ℝd(iii)σh=∇c∗∇vG′fhh converges weakly to σ in Lq0,T×Ω×ℝd
Then, we have(186)∫0,T×Ω×ℝdfh∇c∗∇vG′fh,∇vG′futdtdxdv=∫0,T×Ω×ℝdfh−f∇c∗∇G′fh,∇vG′futdtdxdv+∫0,T×Ω×ℝdf∇c∗∇vG′fh,∇vG′futdtdxdv.
Since∇vG′f∈Lp0,T×Ω×ℝd and ∇c∗∇vG′fhh is bounded in Lq0,T×Ω×ℝd, we use i and iii to have(187)limh⟶0∫0,T×Ω×ℝdfhσh,∇vG′futdtdxdv=∫0,T×Ω×ℝdfσ,∇vG′futdtdxdv.
Also, we have(188)∫0,T×Ω×ℝdfh∇c∗∇vG′f,∇vG′fh−∇vG′futdtdxdv=∫0,T×Ω×ℝdfh−f∇c∗∇vG′f,∇vG′fh−∇vG′futdtdxdv+∫0,T×Ω×ℝdf∇c∗∇vG′f,∇vG′fh−∇vG′futdtdxdv.
Usingi, ii, and the fact that ∇c∗∇vG′f∈Lq0,T×Ω×ℝd and that ∇vG′fhh is bounded in Lp0,T×Ω×ℝd, we get(189)limh⟶0∫0,T×Ω×ℝdfh∇c∗∇vG′f,∇vG′fh−∇vG′futdtdxdv=0.
To conclude with Theorem2, we need to establish the following limit, whose proof is derived from the three following lemmas:(190)limh⟶0∫0,∞×Ω×ℝdfh∇c∗∇vG′fh,∇vG′fhutdtdxdv=∫0,∞×Ω×ℝdfσ,∇vG′futdtdxdv.Lemma 13.
(191)liminfh⟶0∫0,T×Ω×ℝdfhσh,∇vG′fhutdtdxdv≥∫0,T×Ω×ℝdfσ,∇vG′futdtdxdv.Proof.
Sincec∗ is convex and fh≥0,u≥0, then(192)fh∇c∗∇vG′fh−∇c∗∇vG′f,∇vG′fh−∇vG′fut≥0.
So,(193)∫0,T×Ω×ℝdfh∇c∗∇vG′fh,∇vG′fhutdtdxdv≥∫0,T×Ω×ℝdfh∇c∗∇vG′fh,∇vG′futdtdxdv+∫0,T×Ω×ℝdfh∇c∗∇vG′f,∇vG′fh−∇vG′futdtdxdv.
Passing to the limit, we obtain(194)liminfh⟶0∫0,T×Ω×ℝdfh∇c∗∇vG′fh,∇vG′fhutdtdxdv≥liminfh⟶0∫0,T×Ω×ℝdfh∇c∗∇vG′fh,∇vG′futdtdxdv+liminfh⟶0∫0,T×Ω×ℝdfh∇c∗∇vG′f,∇vG′fh−∇vG′futdtdxdv.
Taking into account (187) and (189) in (194) enables us to conclude the proof of Lemma 13.Lemma 14.
(195)limsuph⟶0∫0,T×Ω×ℝdfhσh,∇vG′fhutdtdxdv≤∫Ω×ℝdf0G′f0−G∗G′f0u0dxdv+∫0,T×Ω×ℝdfG′f−G∗G′fu′tdtdxdv.Proof.
Using the proof of Lemma 6, we have(196)∫Ω×ℝdGfk−1hdxdv−∫Ω×ℝdGfkhdxdv≥∫Ω×ℝdTkv−v,∇vG′fkhfkhdxdv,where(197)Tkv=v+h∇c∗∇vG′fkh,is a c-optimal map that pushes Fkxh forward Gkxh. Thus,(198)∫Ω×ℝdGfkh−Gfk−1hhdxdv≤−∫Ω×ℝd∇c∗∇vG′fkh,∇vG′fkhfkhdxdv.
Sinceu≥0, then(199)∑k=1T/h∫tk−1tk∫Ω×ℝdGfkh−Gfk−1hhutdxdv≤−∫0,T×Ω×ℝd∇c∗∇vG′fh,∇vG′fhfhutdtdxdv.
Also,(200)∑k=1T/h∫tk−1tk∫Ω×ℝdGfkh−Gfk−1hhutdxdv=−∫0,T×Ω×ℝdGfh∂thutdtdxdv−1h∫0h∫Ω×ℝdGf0utdtdxdv,where ∂thut=ut+h−ut/h.
Let us prove that(201)limh⟶0∫0,T×Ω×ℝdGfh∂thutdtdxdv=∫0,T×Ω×ℝdGfu′tdtdxdv.
Indeed,(202)∫0,T×Ω×ℝdGfh∂thut−Gfu′tdtdxdv≤∫0,T×Ω×ℝdGfh−Gfu′tdtdxdv+∫0,T×Ω×ℝd∂thut−u′tGfhdtdxdv.
Usingi, the continuity of G, the boundedness of fhh in L∞0,∞×Ω×ℝd, and the fact that ∂thuh converges uniformly to u′ on the compact 0,T, we obtain(203)limh⟶0∫0,T×Ω×ℝdGfh∂thutdtdxdv=∫0,T×Ω×ℝdGfu′tdtdxdv.
Next, we combine (199), (200), and (203) to get(204)∫0,T×Ω×ℝdGfu′tdtdxdv+∫Ω×ℝdGf0u0dxdv≥limsuph⟶0∫0,T×Ω×ℝdfh∇c∗∇vG′fh,∇vG′fhutdtdxdv.
SinceG∈C1 is strictly convex, G∗G′s=sG′s−Gs,∀s>0. Consequently, replacing Gf=fG′f−G∗G′f and Gf0=f0G′f0−G∗G′f0 in (5), we obtain Lemma 14.Lemma 15.
(205)∫Ω×ℝdf0G′f0−G∗G′f0u0dxdv+∫0,T×Ω×ℝdfG′f−G∗G′fu′tdtdxdv≤∫0,T×Ω×ℝdfσ,∇vG′futdtdxdv.Proof.
Let us writeψt,x,v=G′fut for t,,x,v∈0,T×Ω×ℝd. we have ψt,.,.∈W1,pΩ×BR.
Approximatingψt,.,. by Cc∞Ω×ℝd functions and using (17), we have(206)Akhψh=∫Ω×BRfkh−gkhhψt,x+t−tkv,vdxdv+∫Ω×BRfkh∇c∗∇G′fkh,∇vψt,x+t−tkv,vdxdv,where Akhψ is defined in (17) and tends to 0 as h⟶0. By integrating (206) over 0,T, one obtains(207)∑i=1T/h∫titi+1Aihψh=∑i=1T/h∫titi+1∫Ω×BRfih−gihhψt,x+t−tiv,vdtdxdv+∑i=1T/h∫titi+1∫Ω×BRfih∇c∗∇vG′fih,∇vψt,x+t−tiv,vdtdxdv.
Next, from the definition offh, we derive that(208)∑k=1T/h∫tktk+1∫Ω×BRfkh−gkhhψt,x+t−tkv,vdtdxdv=−∫0,T×Ω×BRfhψt+h,x,v−ψt,x,vhdtdxdv−1h∫0h∫Ω×BRfhψt,x,vdtdxdv,(209)∑i=1T/h∫titi+1∫Ω×BRfih∇c∗∇vG′fih,∇vψt,x+t−tiv,vdtdxdv=∫0,T×Ω×BRfh∇c∗∇vG′fh,∇vψt,x,vdtdxdv−∫0h∫Ω×BRf0∇c∗∇vG′f0,∇vψt,x,vdtdxdv.
We combine (208) and (209) to obtain(210)Bhψ=−∫0,T×Ω×BRfhG′ft+hut+h−G′ftuthdtdxdv−1h∫0h∫Ω×BRf0G′futdtdxdv+∫0,T×Ω×BRfh<∇c∗∇vG′fh,∇vψdtdxdv−∫0h∫Ω×BRf0∇c∗∇vG′f0,∇vψdtdxdv,where(211)Bhψ=∑i=1T/h∫titi+1Aihψhdt.
Let us show thatBhψ tends to 0 as h⟶0.
Assume thatψ∈Cc2ℝ×Ω×ℝd.(i) Ifq≥2, we use (116) and we have(212)Akhψ≤h2∇v2ψL∞h∫Ω×ℝd∇vG′fkhpfkhdxdv+h+h22∇x2ψL∞Ω×ℝdf0L∞Ω×ℝd∫Ω×Kψv2dxdv.Recalling the fact thatcx≥A1xq, we have(213)h∫Ω×ℝd∇vG′fkhpfkhdxdv≥hA1∫ΩρkhxWchFkxh,Gkxhdx.From the statements in Section2.1, we get(214)∫ΩρkhxWchFkxh,Gkxhdx≤1A1∫Ω×ℝdGfk−1h−∫Ω×ℝdGfkhdxdv,and accordingly,(215)Akhψ≤h2A1∇v2ψL∞∫Ω×ℝdGfk−1hdxdv−∫Ω×ℝdGfkhdxdv+h+h22∇x2ψL∞Ω×ℝdf0L∞Ω×ℝd∫Ω×Kψv2dxdv.So(216)Bhψ≤12A1∇v2ψL∞∑k=1T/h∫tktk+1∫Ω×ℝdGfk−1hdxdv−∫Ω×ℝdGfkhdxdv+hT+h2T∇x2ψL∞Ω×ℝdf0L∞Ω×ℝd∫Ω×Kψv2dxdv,and then,(217)Bhψ≤12∇v2ψL∞h∫Ω×ℝdGf0dxdv−∫Ω×ℝdGfT/hhdxdv+hT+h2T∇x2ψL∞Ω×ℝdf0L∞Ω×ℝd∫Ω×Kψv2dxdv.From the proof of Lemma 6, we have(218)∫Ω×ℝdGf0dxdv−∫Ω×ℝdGfT/hhdxdv≤G′M−G′N.Then, we obtain(219)Bhψ≤12∇v2ψL∞hG′M−G′N+hT+h2T∇x2ψL∞Ω×ℝdf0L∞Ω×ℝd∫Ω×Kψv2dxdv.So,Bhψ tends to 0 as h⟶0.(ii) Ifq<2, we have(220)∫Ω×ℝdh∇vG′fkhp2/qfkhdxdv≤∫Ω×ℝdh∇vG′fkhpfkhdxdv,and then,(221)Akhψ≤2h2−2/q2∇v2ψL∞∫Ω×ℝdh∇vG′fkhpfkhdxdv+h22∇x2ψL∞f0L∞∫Ω×Kψv2dxdv.
From the statements in Section2.1, we obtain(222)Akhψ≤h2−2/q2∇v2ψL∞∫Ω×ℝdGfk−1hdxdv−∫Ω×ℝdGfkhdxdv+h22∇x2ψL∞f0L∞∫Ω×Kψv2dxdv,and then,(223)Bhψ≤h2−2/q2∇v2ψL∞G′M−G′N+h2T∇x2ψL∞f0L∞∫Ω×Kψv2dxdv.
Thus,Bhψ tends to 0 as h⟶0.
By the change of variablet=t+h, we have(224)∫0,T×Ω×BRfhG′ft+hut+h−G′ftuthdtdxdv=X1h+X2h+X3h,where(225)X1h=∫0,T×Ω×BRfht−hG′ftut−G′ft−hut−hhdtdxdv,X2h=∫−h,0×Ω×BRfht−hG′ftut−G′ft−hut−hhdtdxdv,X3h=−∫T−h,T×Ω×BRfht−hG′ftut−G′ft−hut−hhdtdxdv.
Recalling the definition offh, we get(226)X2h=∫−h,0×Ω×BRf0G′f0ut−ut−hhdtdxdv.
Thus,(227)limh⟶0X2h=0.
Besides,(228)X3h≤MG′M1h∫0,h×Ω×BRut+T−h+1h∫0,h×Ω×BRut+T−2h,and since suppu⊂−T,T,(229)limh⟶0X3h≤MG′MuT+uT=0.
We also have(230)X1h=∫0,T×Ω×BRfht−h−fG′ftut−G′ft−hut−hhdtdxdv+∫0,T×Ω×BRfG′ftut−Gu′ft−ht−hhdtdxdv.
But(231)∫0,T×Ω×BRfht−h−fG′ftut−G′ft−hut−hhdtdxdv≤fht−h−fL∞∫0,T×Ω×BRG′ftut−G′ft−hut−hhpdtdxdv,(232)limh⟶0fht−h−fL∞=0,and then by the dominated convergence theorem, we have(233)limh⟶0∫0,T×Ω×BRG′ftut−G′ft−hut−hhpdtdxdv=∫0,T×Ω×BR∂tG′ftutp<∞,and using (232) and (233), we obtain that(234)limh⟶0∫0,T×Ω×BRfht−h−fG′ftut−G′ft−hut−hhdtdxdv=0.
Next, the equality(235)limh⟶0∫0,T×Ω×BRfhG′ft+hut+h−G′ftuthdtdxdv=limh⟶0∫0,T×Ω×BRfG′ftut−G′ft−hut−hhdtdxdv,follows by recalling (229), (227), and (234), and accordingly,(236)limh⟶0∫0T∫Ω×BRfG′fut−G′ft−hut−hhdtdxdv+∫Ω×BRf0G′f0u0dxdv=∫0T∫Ω×BRfσ,∇vG′futdtdxdv.
We notice that(237)fG′fut−G′ft−hut−hh=fG′f∂t−hut+1hfut−hG′f−G′ft−h,where ∂t−hut=ut−ut−h/h, and since G∈C1 is convex, then(238)G∗G′f−G∗G′ft−h≤fG′f−G′ft−h.
Accordingly,(239)fG′fut−G′ft−hut−hh≥fG′f∂t−hut+1hut−hG∗G′f−G∗G′ft−h,and after integration, we get(240)∫0T∫Ω×BRfG′fut−G′ft−hut−hhdtdxdv≥∫0T∫Ω×BRfG′f∂t−hutdtdxdv+1h∫0T∫Ω×BRut−hG∗G′f−G∗G′ft−hdtdxdv.
Sincesuppu⊂−T,T, ut=0 for t−h∈T−h,T, and then, we obtain(241)∫0T∫Ω×BRfG′fut−G′ft−hut−hhdtdxdv≥∫0T∫Ω×BRfG′f−G∗G′f∂t−hutdtdxdv+−1h∫0T∫Ω×BRG∗G′ft−hut−hdtdxdv.
Tendingh to 0 in the relation above enables us to have(242)limh⟶0∫0T∫Ω×BRfG′fut−G′ft−hut−hhdtdxdv≥∫0T∫Ω×BRfG′f−G∗G′fu′tdtdxdv−∫Ω×BRG∗G′f0u0dxdv.
Moreover, expressions (236) and (242) give(243)∫Ω×BRf0G′f0−G∗G′f0u0dxdv+∫0,T×Ω×BRfG′f−G∗G′fu′tdtdxdv≤∫0,T×Ω×BRfσ,∇vG′futdtdxdv.
So, tendingR to ∞ in (243), we obtain Lemma 15.
Now, we derive (185) by combining Lemmas 13–15.
So, let us show that the sequencedivvfhσhh converges weakly to divvfσ in Cc∞ℝ×Ω×ℝd′ and divvfσ=divvf∇c∗∇vG′f in the weak sense.
So, fixT>0 and let ψ∈Cc∞ℝ×Ω×ℝd such that suppψ.,x,v⊂−T,T.
We have(244)limh⟶0∫0T∫Ω×ℝdfh∇c∗∇G′fh−fσ,∇vψdtdxdv=0.
Indeed,(245)∫0T∫Ω×ℝdfh∇c∗∇vG′fh−fσ,∇vψdtdxdv=∫0T∫Ω×ℝdf∇c∗∇G′fh−σ,∇vψdtdxdv+∫0T∫Ω×ℝdfh−f∇c∗∇vG′fh,∇vψdtdxdv.
We usei, iii, and the fact that ∇vψ∈Lp0,T×Ω×ℝd and ∇c∗∇vG′fhh is bounded in Lq0,T×Ω×ℝd, and we tend h to 0 in (245) to obtain (244).
Let us show thatdivvfσ=divvf∇c∗∇vG′f in the weak sense. So, let ε>0 and ψ∈Cc∞Ω×ℝd be a test function and u∈Cc∞ℝ,u≥0, with suppu⊂−T,T. Define ωεt,x,v=G′f−εψ.
Sincec∗ is convex and fh≥0, we have(246)∫0,T×Ω×ℝdfh∇c∗∇vG′fh−∇c∗∇ωε,∇vG′fh−∇ωεut≥0.
Splitting (246) as follows,(247)∫0,T×Ω×ℝdfh∇c∗∇vG′fh−∇c∗∇ωε,∇G′fh−∇ωεut=∫0,T×Ω×ℝdfh∇c∗∇vG′fh−∇c∗∇ωε,∇vG′fh−∇vG′fut+∫0,T×Ω×ℝdfh∇c∗∇vG′fh−∇c∗∇ωε,∇vG′f−∇ωεut,we derive from (185) that(248)limh⟶0∫0,T×Ω×ℝdfh∇c∗∇vG′fh−∇c∗∇ωε,∇vG′fh−∇G′fut=0.
Consequently,(249)limh⟶0∫0,T×Ω×ℝdfh∇c∗∇vG′fh−∇c∗∇ωε,∇vG′fh−∇ωεut=∫0,T×Ω×ℝdfσ−f∇c∗∇ωε,∇vG′f−∇ωεut≥0.
We divide (249) by ε, and we tend ε to 0 and have(250)∫0,T×Ω×ℝdfσ−f∇c∗∇vG′f,∇vψut≤0.
Replacingψ by −ψ in the previous relationship, we obtain(251)∫0,T×Ω×ℝdfσ−f∇c∗∇vG′f,∇vψut=0.
We conclude thatdivvfh∇c∗∇vG′fh converges weakly to divvf∇c∗∇vG′f in Cc∞ℝ×Ω×ℝd′, and then, the proof of Theorem 2 is complete.


### 4.5. Existence of Solutions

Now, we combine the weak convergence of the sequencefhh and that of the nonlinear term divvfh∇c∗∇vG′fh to prove that the kinetic equation (1) admits a weak solution f in the sense of Definition 1.Theorem 3.
Letf0 be a density of probability on Ω×ℝd such that f0+1/f0∈L∞Ω×ℝd, ∫Ω×ℝdvqf0dxdv<∞, and ∫Ω×ℝdGf0dxdv<∞ with G and c satisfying, respectively, HG and Hc.Then, for any test functionψ∈Cc∞ℝ×Ω×ℝd such that suppψ.,x,v⊂−T,T, we have(252)∫0,T×Ω×ℝdf∂tψ+v⋅∇xψ−∇vψ⋅∇c∗∇vG′fdtdxdv=−∫Ω×ℝdf0ψ0dxdv,where f is the limit function obtained in Proposition 3.Proof.
FixT>0 and let ψ∈Cc∞ℝ×Ω×ℝd be a test function such that suppψ.,x,v⊂−T,T. Integrating (17) over 0,T×Ω×ℝd, we obtain(253)∑k=1T/h∫tk−1tk∫Ω×ℝdfkh−fk−1hhψt,x+t−tk−1v,vdtdxdv+∑k=1T/h∫tk−1tk∫Ω×ℝdfk−1hv⋅∇xψt,x+t−tk−1v,vdtdxdv+∑k=1T/h∫tk−1tk∫Ω×ℝdfkh∇vψt,x+t−tk−1v,v⋅∇c∗∇vG′fkh=Bhψ,where Bhψ tends to 0 as h⟶0.
We observe that(254)∑k=1T/h∫tk−1tk∫Ω×ℝdfkh−fk−1hhψt,x+t−tk−1v,vdtdxdv=∫0,T×Ω×ℝdfhψt−h,x,v−ψt,x,vhdtdxdv−1h∫0h∫Ω×ℝdf0x−tv,vψt−h,x,vdtdxdv,(255)∑k=1T/h∫tk−1tk∫Ω×ℝdfk−1hv⋅∇xψt,x+t−tk−1v,vdtdxdv=∫0,T×Ω×ℝdfhv⋅∇xψdtdxdv.
Hence,(256)∑k=1T/h∫tk−1tk∫Ω×ℝdfkh∇vψt,x+t−tk−1v,v⋅∇c∗∇vG′fkhdtdxdv=∫0,T×Ω×ℝdfh∇v.ψ⋅∇c∗∇vG′fhdtdxdv−∫0,h×Ω×ℝdf0x−tv,v∇vψt,x,v⋅∇c∗∇G′f0x−tv,vdtdxdv.
We combine (254), (255), and (256) to obtain(257)Bhψ=−∫0,T×Ω×ℝdfh∂t−hψt,x,vdtdxdv−1h∫0h∫Ω×ℝdf0x−tv,vψt−h,x,vdtdxdv+∫0,T×Ω×ℝdfh∇vψ⋅∇c∗∇vG′fhdtdxdv−∫0,T×Ω×ℝdfhv⋅∇xψdtdxdv−∫0,h×Ω×ℝdf0x−tv,v∇vψt,x,v⋅∇c∗∇vG′f0x−tv,vdtdxdv.
Here,(258)∂t−hψt,x,v=ψt,x,v−ψt−h,x,vh.
Finally, we use the weak convergence offhh to f, the weak convergence of fh∇c∗∇vG′fh to f∇c∗∇vG′f, and the uniform convergence of ∂thψ.,x,v to ∂tψ.,x,v on the compact set 0,T when h tend to 0, to derive the following equality:(259)−∫0,T×Ω×ℝdf∂tψdtdxdv−∫Ω×ℝdf0ψ0dxdv−∫0,T×Ω×ℝdfv⋅∇xψdtdxdv+∫0,T×Ω×ℝd∇vψ⋅∇c∗∇vG′ffdtdxdv=0,i.e.,(260)∫0,T×Ω×ℝd∂tψ+v⋅∇xψ−∇vψ⋅∇c∗∇vG′ffdtdxdv=−∫Ω×ℝdf0ψ0dxdv.
In conclusion, we prove thatf solves (1) in the weak sense.


## 4.1. Weak Convergence offhh

Let us consider the sequencefhh as defined in (118).Lemma 4.
Assume thatf0 satisfies Hf0. Then, for all k≥1, we have(119)∫Ω×ℝdfkhx,vpdxdv≤∫Ω×ℝdf0x,vpdxdv.Proof.
TakingHt=tp,p>1, in Corollary 1, we obtain(120)∫ℝdGkxhpdv−∫ℝdFkxhpdv≥∫ℝdTkv−v,∇vH′FkxhFkxhdv,where Tk is c-optimal map that pushes Fkxh forward to Gkxh. We use expression of Tk and we obtain(121)∫ℝdGkxhpdv−∫ℝdFkxhpdv≥∫ℝdTkv−v,∇vH′FkxhFkxhdv=pp−1h∫ℝd∇c∗∇vG′fkh,∇vFkxhFkxhp−1dv.
Since∇vG′fkh=G″fkh∇vfkh and fkh=ρkhFkxh, (121) becomes(122)∫ℝdGkxhpdv−∫ℝdFkxhpdv≥hpp−1ρkhx∫ℝdFkxhp−1G′′fkh∇c∗∇vG′fkh,∇vG′fkhdv.
From the convexity ofc and c∗, we have(123)a.b=ca+c∗b,ifa,b∈ℝd,witha=∇c∗b.
Thus, using (123) with a=∇c∗∇vG′fkh and b=∇vG′fkh, (122) becomes(124)∫ℝdGkxhpdv−∫ℝdFkxhpdv≥hpp−1ρkhx∫ℝdFkxhp−1G′′fkhc∇c∗∇vG′fkhdv+hpp−1ρkhx∫ℝdFkxhp−1G′′fkhc∗∇vG′fkhdv.
Sincec≥0 and c0=0, then c∗≥0; hence,(125)∫ℝdGkxhpdv−∫ℝdFkxhpdv≥0.
Multiplying (125) by ρkhp, we obtain after integration(126)∫Ω×ℝdfkhpdxdv≤∫Ω×ℝdgkhpdxdv=∫Ω×ℝdfk−1hpdxdv,and then by an iteration process on k, we get the proof of Lemma 4.Lemma 5.
Assume thatf0∈LpΩ×ℝd∩PqΩ×ℝd with ∫Ω×ℝdGf0dxdv<∞ and 0<N≤f0≤M. For 0<T<∞, we have(127)∫0,T×Ω×ℝdfht,x,vpdtdxdv≤T∫Ω×ℝdf0x,vpdxdv.Proof.
Let0<T<∞ and let h>0 be a step such that T/h∈ℕ∗.
We use Lemma3 and the definition of fh in (118) to obtain(128)∫0,T×Ω×ℝdfht,x,vpdtdxdv=∑k=1T/hh∫Ω×ℝdfkhpdxdv≤T∫Ω×ℝdf0x,vpdxdv<∞.


## 4.2. Weak Convergence of the Linear Term

Here, we study the weak convergence of the linear term.Proposition 3.
Assume thatf0 satisfy Hf0. Then, there exists a function f in Lp0,×Ω×ℝd∩L∞0,T×Ω×ℝd until a subsequence fhh converges to f weakly in Lp0,T×Ω×ℝd and weakly-∗ in L∞0,T×Ω×ℝd.Moreover,(129)limk⟶∞∫0,T×Ω×ℝd∂tψ+v.∇xψfhkdtdxdv=∫0,T×Ω×ℝd∂tψ+v.∇xψfdtdxdv,for every test function ψ∈Cc∞ℝ×Ω×ℝd, with suppψ.,x,v⊂−T,T.Proof.
Sincef0∈LpΩ×ℝd∩L∞Ω×ℝd (see Proposition 2 and Lemma 4), then(130)fhL∞≤f0L∞Ω×ℝd,fhLp≤T1/pf0LpΩ×ℝd.Hence, the sequence fhh is bounded in Lp0,T×Ω×ℝd∩L∞0,T×Ω×ℝd, and then, there is subsequence of fhh still denoted fhh that converges to a nonnegative function f weakly in Lp0,T×Ω×ℝd and weakly-∗ in L∞0,T×Ω×ℝd.
Since∂tψ+v.∇xψ∈Lq0,T×Ω×ℝd, for every ψ∈Cc∞ℝ×Ω×ℝd such that suppψ.,x,v⊂−T,T, we finally obtain Proposition 3.


## 4.3. Weak Convergence of the Nonlinear Term

In this section, we establish the weak convergence of the nonlinear term.First, we prove that the sequence∇vG′fh is bounded in Lp0,T×Ω×ℝd, and that ∇c∗∇vG′fh is bounded in Lq0,T×Ω×ℝd where 0<T<∞.Lemma 6.
Assume thatf0, c, and G satisfy, respectively, Hf0, Hc, and HG. Then, for all 0<T<∞ fixed,(131)∫0,T×Ω×ℝd∇c∗∇vG′fhqdtdxdv≤TA1NG′M−G′N,∫0,T×Ω×ℝd∇vG′fhpdtdxdv≤A2T+G′M−G′Npqp−1A2p−1N.Proof.
Using (115), we have(132)∫Ω×ℝd∇c∗∇vG′fkhqfkhdxdv≤G′M−G′NA1.
Butfh≥N, and then(133)∫0,T×Ω×ℝd∇c∗∇vG′fhqdtdxdv≤G′M−G′N′TNA1.
Recalling (123), with a=∇c∗∇vG′fh and b=∇vG′fh, we have(134)∇c∗∇vG′fh⋅∇vG′fh−c∇c∗∇vG′fh=c∗∇vG′fh,with c≥0.
Then,(135)∫0,T×Ω×ℝdc∗∇vG′fhfhdtdxdv≤∫0,T×Ω×ℝd∇c∗∇vG′fh∇vG′fhdtdxdv=∑k=1T/hh∫Ω×ℝd∇c∗∇vG′fkh∇vG′fkhdxdv≤∑k=1T/h∫Ω×ℝdGfk−1hdxdv−∫Ω×ℝdGfkhdxdv.
Moreover,(136)∑k=1T/h∫Ω×ℝdGfk−1hdxdv−∫Ω×ℝdGfkhdxdv=∫Ω×ℝdGf0dxdv−∫Ω×ℝdGfT/hdxdv≤G′M−G′N,and then,(137)∫0,T×Ω×ℝdc∗∇vG′fhfhdtdxdv≤G′M−G′N.
Becausecy≤A2yq+A2, we have x⋅y−cy≥x⋅y−A2yq−A2 for all x,y∈ℝd and then(138)c∗x≔supy∈ℝdx⋅y−cy≥−A2+xppqp−1A2p−1.
Consequently,(139)−A2T+1pqp−1A2p−1∫0,T×Ω×ℝd∇vG′fhpfhdtdxdv≤G′M−G′N.
Usingfh≥N, we obtain(140)∫0,T×Ω×ℝd∇vG′fhpdtdxdv≤TA2+G′M−G′Npqp−1A2p−1N.


## 4.4. Strong Convergence offhh

In order to prove the strong convergence result, we need to establish the following compactness results forfhh in Lp0,T×Ω×ℝd.Lemma 7.(velocity-compactness).
Assume thatf0 satisfies Hf0. Fix 0<T<∞ and 0≠η∈ℝd such that η is small. Then,(141)∫0,T×Ω×ℝdfht,x,v+η−fht,x,vpdtdxdv≤ηpK,where K≔inft∈N,M1/G′′tpTA2+G′M−G′Npqp−1A2p−1/N is a constant.Proof.
We use∇vG′fh=G′′fh∇vfh, N≤fh≤M, and (140) to produce that(142)∫0,T×Ω×ℝd∇vfhpdtdxdv≤K.
Then,fh and ∇vfh∈Lp0,T×Ω×ℝd and approximating fht,x,. by Cc∞ℝd functions in W1,pℝd, we obtain(143)∫0,T×Ω×ℝdfht,x,v+η−fht,x,vpdtdxdv≤ηp∫0,T×Ω×ℝd∇vfhpdtdxdv.
This implies that(144)∫0,T×Ω×ℝdfht,x,v+η−fht,x,vpdtdxdv≤ηpK.Lemma 8. (position-compactness).
Assume thatf0 satisfies Hf0. We fix 0<T<∞ and 0≠η∈ℝ such that η is small. Then,(145)∫0,T×Ω×ℝdfht,x+hηv,v−fht,x,vpdtdxdv≤ηpTK1,where K1=2pf0LpΩ×ℝdp.Proof.
By fixingv, (17) becomes(146)fkh−fk−1h+hv⋅∇xfkh=oh,where oh tends to 0 when h goes to 0. We use (146), and the fact that fkhk is bounded in LpΩ×ℝd to deduce that hv⋅∇xfkhk is bounded in LpΩ×ℝd. Then, by using Lemma 4, we have(147)hv⋅∇xfkhLpΩ×ℝd≤2f0LpΩ×ℝd.
Thus,fht,.,v∈LpΩ, and approximating fht,.,v by Cc∞ℝd functions, we use the fact that hv⋅∇xfkhk is bounded in LpΩ×ℝd to deduce that(148)∫0,T×Ω×ℝdfht,x+hηv,v−fht,x,vpdtdxdv≤ηp∫0,T×Ω×ℝdhv⋅∇xfht,x,vpdtdxdv≤ηp2pT∫Ω×ℝdf0x,vpdxdv.Lemma 9.
Assume thatHf0 holds. Let τ>0 be quite small such that t,t+τ⊂0,T,∀t∈0,T, where 0<T<∞ is fixed. We have(149)∫0,T×Ω×ℝdG′fht+τ−G′fhtfht+τ−fhtdtdxdv≤τK2Ω,T,N,M,f0,G′,q,p,where K2Ω,T,N,M,f0,G′,q,p is a positive constant depending on Ω,T,N,M,f0,G′,q,p.Proof.
Chooseh>0 such that T/h∈ℕ∗ and τ/h=N∈ℕ∗. Define(150)A=∫0,T×Ω×ℝdG′fht+τ−G′fhtfht+τ−fhtdtdxdv.
Using the definition offh in (118), we obtain(151)A=∑k=1T/hh∫Ω×ℝdψh,kx,vfk−1+Nhx,v−fk−1hx,vdxdv,where ψh,kx,v=G′fk−1+Nhx,v−G′fk−1hx,v.
Note that(152)fk−1+Nhx,v−fk−1hx,v=∑i=0N−1fk−1+N−ihx,v−fk−2+N−ihx,v.
Using (152), we have(153)A=∑k=1T/h∑i=0N−1h∫Ω×ℝdψh,kx,vfk−1+N−ihx,v−fk−2+N−ihx,vdxdv=C+D,where(154)C=∑k=1T/h∑i=0N−1h∫Ω×ℝdψh,kx,vfk−1+N−ihx,v−gk−1+N−ihx,vdxdv,D=∑k=1T/h∑i=0N−1h∫Ω×ℝdψh,kx,vgk−1+N−ihx,v−fk−2+N−ihx,vdxdv.
UsingSk,Nh,i as a c-optimal map that pushes Fk−1+N−ixh forward to Gk−1+N−ixh, we have(155)C=∑k=1T/h∑i=0N−1h∫Ω×ℝdψh,kx,v−ψh,kx,Sk,Nh,ivfk−1+N−ihx,vdxdv.
On the contrary,∇vG′fk−1+N−ih∈LpΩ×ℝd (see Lemma 6), and from assumptions on G, G′s/s is continuous on ℝ∗, and then there exists a constant K such that G′s≤K⋅s for 0<N≤s≤M. Since 0<N≤fk−1+N−ih≤M (see Lemma 2) and fk−1+N−ih∈LpΩ×ℝd, then G′fk−1+N−ih∈LpΩ×ℝd. So G′fk−1+N−ihx,.∈W1,pℝd.
ApproximatingG′fk−1+N−ihx,. by Cc∞ℝd functions and using the dominated convergence theorem, we have(156)C=∑i=1N−1∑k=1T/hh∫Ω×ℝdfk−1+N−ihx,vv−Sk,Nh,ix,v,∇vψh,kx,v+θv−Sk,Nh,ix,vdxdv,forθ∈0,1.
Note thatv−Sk,Nh,i=−h∇c∗∇vG′fk−1+N−ih∈LqΩ×ℝd and ∇vψh,k=∇vG′fk−1+Nh−∇vG′fk−1h∈LpΩ×ℝd (see Lemma 6).
So, by using Hölder’s inequality, we have that(157)C≤∑i=1N−1∑k=1T/hh∫Ω×ℝdv−Sk,Nh,ivqfk−1+N−ihx,vdxdv1/q∫Ω×ℝd∇vψh,kp1/p≤∑i=1N−1∑k=1T/hhA11/q∫Ωρk−1+N−ihxWchFk−1+N−ixh,Gk−1+N−ixhdx1/q∫Ω×ℝd∇vψh,kp1/p.
Using the algorithm defined in Section2.1 and Lemma 6, we have(158)∫Ωρk−1+N−ihxWchFk−1+N−ixh,Gk−1+N−ixhdx≤hq−1∫Ω×ℝdGfk−2+N−ihx,vdxdv−∫Ω×ℝdGfk−1+N−ihx,vdxdv.
Thus,(159)C≤∑i=1N−1∑k=1T/hhA11/q∫Ω×ℝdGfk−2+N−ihx,vdxdv−∫Ω×ℝdGfk−1+N−ihx,vdxdv1/q×hq−1/q∫Ω×ℝd∇vψh,kp1/p.
Applying the Hölder’s inequality in the previous relation yields(160)C≤1A11/qh1−1/phq−1/q∑k=1T/h∫Ω×ℝdGfk−2+N−ihx,vdxdv−∫Ω×ℝdGfk−1+N−ihx,vdxdv1/q×∑i=1N−1∑k=1T/hh∫Ω×ℝd∇vψh,kp1/p.
We notice that(161)∑k=1T/h∫Ω×ℝdGfk−2+N−ihx,vdxdv−∫Ω×ℝdGfk−1+N−ihx,vdxdv=∫Ω×ℝdGf−1+N−ihx,vdxdv−∫Ω×ℝdGfT/h−1+N−ihx,vdxdv≤G′M−G′N,
Next, we use Lemma6, and we have(162)∑k=1T/hh∫Ω×ℝd∇vψh,kp1/p=∑k=1T/hh∫Ω×ℝd∇vG′fk−1+Nh−∇vG′fk−1hp1/p≤2p−1/p∑k=1T/hh∫Ω×ℝd∇vG′fk−1+Nhp+∑k=1T/hh∫Ω×ℝd∇vG′fk−1hp1/p≤2p−1/p2M0T,Ω1/p=2M0T,Ω1/p.
We combine (160), (161), and (162) to get(163)C≤∑i=1N−12h1−1/phq−1/qG′M−G′N1/qM0T,Ω1/p≤Nh2−1/p+1/qG′M−G′N1/qM0T,Ω1/p.
WritingNh=τ, we then have(164)C≤G′M−G′N1/qM0T,Ω1/pτ.
On the contrary,(165)D=∑k=1T/h∑i=0N−1h∫Ω×ℝdψh,kx,vfk−2+N−ihx,v−gk−1+N−ihx,vdxdv=∑k=1T/h∑i=0N−1h∫Ω×ℝdψh,kx,vfk−2+N−ihx,v−fk−2+N−ihx−hv,vdxdv.
SinceG′fk−1+N−ih.,v∈W1,pΩ and then ψh,k.,v∈W1,pΩ, we approximate ψh,k.,v by Cc∞Ω functions and we get(166)D=−∑k=1T/h∑i=0N−1h2∫Ω×ℝdv⋅∇xψh,kx,vfk−2+N−ihx,vdxdv.
Note thatv⋅∇xG′fkh=G′′fkhv⋅∇xfkh, and according to Lemma 8, v⋅∇xfkh∈LpΩ×ℝd. So, since G′′ is continuous and N≤fkh≤M, we have v⋅∇xG′fkhh,k∈LpΩ×ℝd.
Consequently,(167)D=−∑k=1T/h∑i=0N−1h2∫Ω×ℝdv⋅∇xψh,kx,vfk−2+N−ihx,vdxdv=−∑k=1T/h∑i=0N−1h2∫Ω×ℝdv⋅∇xG′fk−1+Nhx,vfk−2+N−ihdxdv+∑k=1T/h∑i=0N−1h2∫Ω×ℝdv⋅∇xG′fk−1hx,vfk−2+N−ihx,vdxdv≤∑k=1T/h∑i=0N−1h2∫Ω×ℝdv⋅∇xG′fk−1hx,vpdxdv1/pf0L∞+∑k=1T/h∑i=0N−1h2∫Ω×ℝdv⋅∇xG′fk−1+Nhx,vpdxdv1/pf0L∞.
Next, we use the fact thatv⋅∇xG′fkh=G′′fkhv⋅∇xfkh and the boundedness in LpΩ×ℝd, of v⋅∇xfkhh,k (see Lemma 8), to get(168)D≤2f0L∞Ω×ℝdNThsups∈N,MG′′sK1Ωh2=2f0L∞Ω×ℝdsups∈N,MG′′sK1ΩTτ.
Thus,(169)A=C+D≤K2A1,Ω,T,N,M,f0,G′,q,pτ,where K2A1,Ω,T,N,M,f0,G′,q,p≔2f0L∞Ω×ℝdsups∈N,MG′′sK1ΩT+G′M−G′N1/qM0T,Ω1/p.
Let us process now to the following estimate.Lemma 10.
Assume thatf0 satisfies Hf0. Then, for every 0<T<∞ fixed and for all τ>0 small, we have(170)∫0,T×Ω×ℝdfht+τ,x,v−fht,x,vpdtdxdv≤CA1,Ω,T,N,M,f0,G′,q,pτ,where CA1,Ω,T,N,M,f0,G′,q,p is positive constant.Proof.
SinceG∈C2, we use Taylor’s formula to have that(171)G′fht+τ,x,v−G′fht,x,v=G′′fθhfht+τ,x,v−fht,x,v,where θ∈]0,1[ and fθh≔θfht+1−θfht+τ.
Since0<N≤f0≤M, we have 0<N≤fh≤M (see Lemma 2). Thus, using (171) and the fact that G′′ is continuous and nonnegative, we have(172)∫0,T×Ω×ℝdG′fht+τ−G′fhfht+τ−fhtdtdxdv≥inft∈N,MG′′t∫0,T×Ω×ℝdfht+τ,x,v−fht,x,v2dtdxdv.
Defineβ=inft∈N,MG′′t>0. We use Lemma 9 and obtain(173)∫0,T×Ω×ℝdfht+τ,x,v−fht,x,v2dtdxdv≤1βK2A1,Ω,T,N,M,f0,G′,q,pτ.
UsingN≤fh≤M, we have(174)∫0,T×Ω×ℝdfht+τ,x,v−fht,x,vpdtdxdv≤1βCA1,Ω,T,N,M,f0,G′,q,pτ,where CΩ,T,N,M,f0,G′,q,p is positive constant.
We will use Lemmas7, 8, and 10, to prove that the sequence fhh is precompact in Lp0,T×Ω×ℝd.Lemma 11.
Assume thatf0, c, and G satisfy, respectively, Hf0, Hc, and HG. Then, fhh is precompact in Lp0,T×Ω×BR, where BR is the open ball centered at origin of radius R.Proof.
DefineK=0,T×Ω×BR. Then, K=∪δ>0Kδ, with Kδ=y∈K,disty,Kc>δ. Kδ⊂K, Kδ¯⊂K, and Kδ¯ is compact and distKδ,Kc>δ/2. So, there exists δ1,δ2,…,δl>0 such that Kδ¯⊂∪i=1lKδi.
Defineδ0=mini=1,2,…,dδi and we have Kδ0⋐K and distKδ0,Kc>δ0/2.
Letτ>0, η1,η2∈ℝd, such that 0<τ<δ0,ε, η1<δ0,ε, and η2<δ0,ε, where δ0,ε=minδ0/2,ε.
We use Lemmas7, 8, and 10 to have(175)∫Kδfht+τ,x,v−fht,x,vpdtdxdv<c1ε,∫Kδfht,x+η1,v−fht,x,vpdtdxdv<εpc2,∫Kδfht,x,v+η2−fht,x,vpdtdxdv<c3εp.
c1,c2, and c3 are defined in Lemmas 7, 8, and 10. So, by using the Riesz–Fréchet–Kolmogorov’s theorem, we deduce that the sequence fhh is precompact in LpKδ. Therefore, limδ⟶0K\Kδ=0, and then for all ε>0, there exists δε>0 such that K\Kδε<εf0L∞, and since fhh is bounded in L∞0,T×Ω×BR, we have fhLpK\Kδεp≤f0L∞K\Kδε<ε.
We use corollaryIV.26 of [9], to conclude that the sequence fhh is precompact in Lp0,T×Ω×BR, which implies that fhh converges strongly to some function fR in Lp0,T×Ω×BR.
Now, let us prove that the sequencefhh is precompact in Lp0,T×Ω×ℝd.Lemma 12.
Assume thatf0, c, and G satisfy, respectively, Hf0, Hc, and HG. Then, the sequence fhh is precompact in Lp0,T×Ω×ℝd.Proof.
Sincef0∈Lp0,T×Ω×ℝd, Lemma 3 implies that fhh is bounded in Lp0,T×Ω×ℝd. Then, fhh converges weakly to some function f in Lp0,T×Ω×ℝd (up to a subsequence).
LetBn be an open ball of radius n centered at origin, with n∈ℕ∗. Then, Bn⊂Bn+1 and ∪n∈ℕ∗Bn=ℝd.
Using previous results, the sequencefhh is precompact in Lp0,T×Ω×Bn. A subsequence fhh converges strongly to some function fn in Lp0,T×Ω×Bn. Let us denote by fhn the subsequence of fhh which converges strongly to fn in Lp0,T×Ω×Bn. Fix ϕ in Lq0,T×Ω×ℝd. We have(176)∫0,T×Ω×ℝdfhn−fnϕdtdxdv=∫0,T×Ω×Bnfhn−fnϕdtdxdv+∫0,T×Ω×Bncfhn−fnϕdtdxdv,with(177)∫0,T×Ω×Bncfhn−fnϕdtdxdv≤ϕLq0,T×Ω×ℝd2MBnc1/p.
Bnc tends to 0, when n goes to ∞; for ε>0, there exists Nε>0 such that(178)∫0,T×Ω×Bncfhn−fnϕdtdxdv≤ε2,for all n≥Nε. Since fhn converges strongly to fn in Lp0,T×Ω×Bn, we deduce that(179)∫0,T×Ω×ℝdfhn−fnϕdtdxdv≤ε,when hn goes to 0 and n≥Nε. We conclude that fhn converges weakly to fn in Lp0,T×Ω×ℝd, for n≥Nε. So fn=f for n≥Nε.
Now, let prove thatfhn−fLp0,T×Ω×ℝd tends to 0 as hn⟶0 and n≥Nε.
Note that(180)fhn−fLp0,T×Ω×ℝdp=∫0,T×Ω×Bnfhn−fpdtdxdv+∫0,T×Ω×Bncfhn−fpdtdxdv≤∫0,T×Ω×Bnfhn−fpdtdxdv+2pf0L∞pTΩBnc.
Let us recall that(181)limhn⟶0∫0,T×Ω×Bnfhn−fpdtdxdv=0,∀n≥Nε.
SinceBnc<ε/2, for n≥Nε, as(182)limhn⟶0∫0,T×Ω×BNεfhn−fpdtdxdv=0,we can find nε∈ℕ such that for every n≥Nε, we have(183)∫0,T×Ω×BNεfhn−fpdtdxdv<ε2.
Then, for alln≥Nε,(184)∫0,T×Ω×ℝdfhn−fpdtdxdv<ε.
Consequently,fhn converges strongly to f in Lp0,T×Ω×ℝd.
Using the previous lemma,fhh converges strongly to f in Lp0,T×Ω×ℝd (up to a subsequence). G′ is being continuous, then G′fhh converges strongly to G′f in Lp0,T×Ω×ℝd. We conclude that the sequence ∇vG′fhh converges to ∇vG′f in Cc∞ℝ×Ω×ℝd′. Let us recall that ∇vG′fhh is bounded in Lp0,T×Ω×ℝd, so ∇vG′fhh converges weakly to ∇vG′f in Lp0,T×Ω×ℝd.
On the contrary,∇c∗∇vG′fhh is bounded in Lq0,T×Ω×ℝd. Then, a subsequence ∇c∗∇vG′fhh converges weakly to σ in Lq0,T×Ω×ℝd.
We derive from the proof of Theorem 3.10 in [1] that the sequence divvfh∇c∗∇vG′fh converges weakly to divvf∇c∗∇vG′f in Cc∞ℝ×Ω×ℝd′.Theorem 2.
Assume thatf0, c, and G satisfy, respectively, Hf0, Hc, and HG. Let 0<T<∞ and u∈Cc∞ℝ be such that u≥0 and suppu⊂−T,T. Then,(185)limh⟶0∫0,T×Ω×ℝdfh∇c∗∇vG′fh,∇vG′fhutdtdxdv=∫0,T×Ω×ℝdfσ,∇vG′futdtdxdv.Moreover,divvfh∇c∗∇vG′fhh converges weakly to divvfσ in Cc∞ℝ×Ω×ℝd′ and divvfσ=divvf∇c∗∇vG′f in the weak sense.Proof.
Let us recall the following:(i)fhh converges strongly to f in Lp0,T×Ω×ℝd(ii)∇vG′fhh converges weakly to ∇vG′f in Lp0,T×Ω×ℝd(iii)σh=∇c∗∇vG′fhh converges weakly to σ in Lq0,T×Ω×ℝd
Then, we have(186)∫0,T×Ω×ℝdfh∇c∗∇vG′fh,∇vG′futdtdxdv=∫0,T×Ω×ℝdfh−f∇c∗∇G′fh,∇vG′futdtdxdv+∫0,T×Ω×ℝdf∇c∗∇vG′fh,∇vG′futdtdxdv.
Since∇vG′f∈Lp0,T×Ω×ℝd and ∇c∗∇vG′fhh is bounded in Lq0,T×Ω×ℝd, we use i and iii to have(187)limh⟶0∫0,T×Ω×ℝdfhσh,∇vG′futdtdxdv=∫0,T×Ω×ℝdfσ,∇vG′futdtdxdv.
Also, we have(188)∫0,T×Ω×ℝdfh∇c∗∇vG′f,∇vG′fh−∇vG′futdtdxdv=∫0,T×Ω×ℝdfh−f∇c∗∇vG′f,∇vG′fh−∇vG′futdtdxdv+∫0,T×Ω×ℝdf∇c∗∇vG′f,∇vG′fh−∇vG′futdtdxdv.
Usingi, ii, and the fact that ∇c∗∇vG′f∈Lq0,T×Ω×ℝd and that ∇vG′fhh is bounded in Lp0,T×Ω×ℝd, we get(189)limh⟶0∫0,T×Ω×ℝdfh∇c∗∇vG′f,∇vG′fh−∇vG′futdtdxdv=0.
To conclude with Theorem2, we need to establish the following limit, whose proof is derived from the three following lemmas:(190)limh⟶0∫0,∞×Ω×ℝdfh∇c∗∇vG′fh,∇vG′fhutdtdxdv=∫0,∞×Ω×ℝdfσ,∇vG′futdtdxdv.Lemma 13.
(191)liminfh⟶0∫0,T×Ω×ℝdfhσh,∇vG′fhutdtdxdv≥∫0,T×Ω×ℝdfσ,∇vG′futdtdxdv.Proof.
Sincec∗ is convex and fh≥0,u≥0, then(192)fh∇c∗∇vG′fh−∇c∗∇vG′f,∇vG′fh−∇vG′fut≥0.
So,(193)∫0,T×Ω×ℝdfh∇c∗∇vG′fh,∇vG′fhutdtdxdv≥∫0,T×Ω×ℝdfh∇c∗∇vG′fh,∇vG′futdtdxdv+∫0,T×Ω×ℝdfh∇c∗∇vG′f,∇vG′fh−∇vG′futdtdxdv.
Passing to the limit, we obtain(194)liminfh⟶0∫0,T×Ω×ℝdfh∇c∗∇vG′fh,∇vG′fhutdtdxdv≥liminfh⟶0∫0,T×Ω×ℝdfh∇c∗∇vG′fh,∇vG′futdtdxdv+liminfh⟶0∫0,T×Ω×ℝdfh∇c∗∇vG′f,∇vG′fh−∇vG′futdtdxdv.
Taking into account (187) and (189) in (194) enables us to conclude the proof of Lemma 13.Lemma 14.
(195)limsuph⟶0∫0,T×Ω×ℝdfhσh,∇vG′fhutdtdxdv≤∫Ω×ℝdf0G′f0−G∗G′f0u0dxdv+∫0,T×Ω×ℝdfG′f−G∗G′fu′tdtdxdv.Proof.
Using the proof of Lemma 6, we have(196)∫Ω×ℝdGfk−1hdxdv−∫Ω×ℝdGfkhdxdv≥∫Ω×ℝdTkv−v,∇vG′fkhfkhdxdv,where(197)Tkv=v+h∇c∗∇vG′fkh,is a c-optimal map that pushes Fkxh forward Gkxh. Thus,(198)∫Ω×ℝdGfkh−Gfk−1hhdxdv≤−∫Ω×ℝd∇c∗∇vG′fkh,∇vG′fkhfkhdxdv.
Sinceu≥0, then(199)∑k=1T/h∫tk−1tk∫Ω×ℝdGfkh−Gfk−1hhutdxdv≤−∫0,T×Ω×ℝd∇c∗∇vG′fh,∇vG′fhfhutdtdxdv.
Also,(200)∑k=1T/h∫tk−1tk∫Ω×ℝdGfkh−Gfk−1hhutdxdv=−∫0,T×Ω×ℝdGfh∂thutdtdxdv−1h∫0h∫Ω×ℝdGf0utdtdxdv,where ∂thut=ut+h−ut/h.
Let us prove that(201)limh⟶0∫0,T×Ω×ℝdGfh∂thutdtdxdv=∫0,T×Ω×ℝdGfu′tdtdxdv.
Indeed,(202)∫0,T×Ω×ℝdGfh∂thut−Gfu′tdtdxdv≤∫0,T×Ω×ℝdGfh−Gfu′tdtdxdv+∫0,T×Ω×ℝd∂thut−u′tGfhdtdxdv.
Usingi, the continuity of G, the boundedness of fhh in L∞0,∞×Ω×ℝd, and the fact that ∂thuh converges uniformly to u′ on the compact 0,T, we obtain(203)limh⟶0∫0,T×Ω×ℝdGfh∂thutdtdxdv=∫0,T×Ω×ℝdGfu′tdtdxdv.
Next, we combine (199), (200), and (203) to get(204)∫0,T×Ω×ℝdGfu′tdtdxdv+∫Ω×ℝdGf0u0dxdv≥limsuph⟶0∫0,T×Ω×ℝdfh∇c∗∇vG′fh,∇vG′fhutdtdxdv.
SinceG∈C1 is strictly convex, G∗G′s=sG′s−Gs,∀s>0. Consequently, replacing Gf=fG′f−G∗G′f and Gf0=f0G′f0−G∗G′f0 in (5), we obtain Lemma 14.Lemma 15.
(205)∫Ω×ℝdf0G′f0−G∗G′f0u0dxdv+∫0,T×Ω×ℝdfG′f−G∗G′fu′tdtdxdv≤∫0,T×Ω×ℝdfσ,∇vG′futdtdxdv.Proof.
Let us writeψt,x,v=G′fut for t,,x,v∈0,T×Ω×ℝd. we have ψt,.,.∈W1,pΩ×BR.
Approximatingψt,.,. by Cc∞Ω×ℝd functions and using (17), we have(206)Akhψh=∫Ω×BRfkh−gkhhψt,x+t−tkv,vdxdv+∫Ω×BRfkh∇c∗∇G′fkh,∇vψt,x+t−tkv,vdxdv,where Akhψ is defined in (17) and tends to 0 as h⟶0. By integrating (206) over 0,T, one obtains(207)∑i=1T/h∫titi+1Aihψh=∑i=1T/h∫titi+1∫Ω×BRfih−gihhψt,x+t−tiv,vdtdxdv+∑i=1T/h∫titi+1∫Ω×BRfih∇c∗∇vG′fih,∇vψt,x+t−tiv,vdtdxdv.
Next, from the definition offh, we derive that(208)∑k=1T/h∫tktk+1∫Ω×BRfkh−gkhhψt,x+t−tkv,vdtdxdv=−∫0,T×Ω×BRfhψt+h,x,v−ψt,x,vhdtdxdv−1h∫0h∫Ω×BRfhψt,x,vdtdxdv,(209)∑i=1T/h∫titi+1∫Ω×BRfih∇c∗∇vG′fih,∇vψt,x+t−tiv,vdtdxdv=∫0,T×Ω×BRfh∇c∗∇vG′fh,∇vψt,x,vdtdxdv−∫0h∫Ω×BRf0∇c∗∇vG′f0,∇vψt,x,vdtdxdv.
We combine (208) and (209) to obtain(210)Bhψ=−∫0,T×Ω×BRfhG′ft+hut+h−G′ftuthdtdxdv−1h∫0h∫Ω×BRf0G′futdtdxdv+∫0,T×Ω×BRfh<∇c∗∇vG′fh,∇vψdtdxdv−∫0h∫Ω×BRf0∇c∗∇vG′f0,∇vψdtdxdv,where(211)Bhψ=∑i=1T/h∫titi+1Aihψhdt.
Let us show thatBhψ tends to 0 as h⟶0.
Assume thatψ∈Cc2ℝ×Ω×ℝd.(i) Ifq≥2, we use (116) and we have(212)Akhψ≤h2∇v2ψL∞h∫Ω×ℝd∇vG′fkhpfkhdxdv+h+h22∇x2ψL∞Ω×ℝdf0L∞Ω×ℝd∫Ω×Kψv2dxdv.Recalling the fact thatcx≥A1xq, we have(213)h∫Ω×ℝd∇vG′fkhpfkhdxdv≥hA1∫ΩρkhxWchFkxh,Gkxhdx.From the statements in Section2.1, we get(214)∫ΩρkhxWchFkxh,Gkxhdx≤1A1∫Ω×ℝdGfk−1h−∫Ω×ℝdGfkhdxdv,and accordingly,(215)Akhψ≤h2A1∇v2ψL∞∫Ω×ℝdGfk−1hdxdv−∫Ω×ℝdGfkhdxdv+h+h22∇x2ψL∞Ω×ℝdf0L∞Ω×ℝd∫Ω×Kψv2dxdv.So(216)Bhψ≤12A1∇v2ψL∞∑k=1T/h∫tktk+1∫Ω×ℝdGfk−1hdxdv−∫Ω×ℝdGfkhdxdv+hT+h2T∇x2ψL∞Ω×ℝdf0L∞Ω×ℝd∫Ω×Kψv2dxdv,and then,(217)Bhψ≤12∇v2ψL∞h∫Ω×ℝdGf0dxdv−∫Ω×ℝdGfT/hhdxdv+hT+h2T∇x2ψL∞Ω×ℝdf0L∞Ω×ℝd∫Ω×Kψv2dxdv.From the proof of Lemma 6, we have(218)∫Ω×ℝdGf0dxdv−∫Ω×ℝdGfT/hhdxdv≤G′M−G′N.Then, we obtain(219)Bhψ≤12∇v2ψL∞hG′M−G′N+hT+h2T∇x2ψL∞Ω×ℝdf0L∞Ω×ℝd∫Ω×Kψv2dxdv.So,Bhψ tends to 0 as h⟶0.(ii) Ifq<2, we have(220)∫Ω×ℝdh∇vG′fkhp2/qfkhdxdv≤∫Ω×ℝdh∇vG′fkhpfkhdxdv,and then,(221)Akhψ≤2h2−2/q2∇v2ψL∞∫Ω×ℝdh∇vG′fkhpfkhdxdv+h22∇x2ψL∞f0L∞∫Ω×Kψv2dxdv.
From the statements in Section2.1, we obtain(222)Akhψ≤h2−2/q2∇v2ψL∞∫Ω×ℝdGfk−1hdxdv−∫Ω×ℝdGfkhdxdv+h22∇x2ψL∞f0L∞∫Ω×Kψv2dxdv,and then,(223)Bhψ≤h2−2/q2∇v2ψL∞G′M−G′N+h2T∇x2ψL∞f0L∞∫Ω×Kψv2dxdv.
Thus,Bhψ tends to 0 as h⟶0.
By the change of variablet=t+h, we have(224)∫0,T×Ω×BRfhG′ft+hut+h−G′ftuthdtdxdv=X1h+X2h+X3h,where(225)X1h=∫0,T×Ω×BRfht−hG′ftut−G′ft−hut−hhdtdxdv,X2h=∫−h,0×Ω×BRfht−hG′ftut−G′ft−hut−hhdtdxdv,X3h=−∫T−h,T×Ω×BRfht−hG′ftut−G′ft−hut−hhdtdxdv.
Recalling the definition offh, we get(226)X2h=∫−h,0×Ω×BRf0G′f0ut−ut−hhdtdxdv.
Thus,(227)limh⟶0X2h=0.
Besides,(228)X3h≤MG′M1h∫0,h×Ω×BRut+T−h+1h∫0,h×Ω×BRut+T−2h,and since suppu⊂−T,T,(229)limh⟶0X3h≤MG′MuT+uT=0.
We also have(230)X1h=∫0,T×Ω×BRfht−h−fG′ftut−G′ft−hut−hhdtdxdv+∫0,T×Ω×BRfG′ftut−Gu′ft−ht−hhdtdxdv.
But(231)∫0,T×Ω×BRfht−h−fG′ftut−G′ft−hut−hhdtdxdv≤fht−h−fL∞∫0,T×Ω×BRG′ftut−G′ft−hut−hhpdtdxdv,(232)limh⟶0fht−h−fL∞=0,and then by the dominated convergence theorem, we have(233)limh⟶0∫0,T×Ω×BRG′ftut−G′ft−hut−hhpdtdxdv=∫0,T×Ω×BR∂tG′ftutp<∞,and using (232) and (233), we obtain that(234)limh⟶0∫0,T×Ω×BRfht−h−fG′ftut−G′ft−hut−hhdtdxdv=0.
Next, the equality(235)limh⟶0∫0,T×Ω×BRfhG′ft+hut+h−G′ftuthdtdxdv=limh⟶0∫0,T×Ω×BRfG′ftut−G′ft−hut−hhdtdxdv,follows by recalling (229), (227), and (234), and accordingly,(236)limh⟶0∫0T∫Ω×BRfG′fut−G′ft−hut−hhdtdxdv+∫Ω×BRf0G′f0u0dxdv=∫0T∫Ω×BRfσ,∇vG′futdtdxdv.
We notice that(237)fG′fut−G′ft−hut−hh=fG′f∂t−hut+1hfut−hG′f−G′ft−h,where ∂t−hut=ut−ut−h/h, and since G∈C1 is convex, then(238)G∗G′f−G∗G′ft−h≤fG′f−G′ft−h.
Accordingly,(239)fG′fut−G′ft−hut−hh≥fG′f∂t−hut+1hut−hG∗G′f−G∗G′ft−h,and after integration, we get(240)∫0T∫Ω×BRfG′fut−G′ft−hut−hhdtdxdv≥∫0T∫Ω×BRfG′f∂t−hutdtdxdv+1h∫0T∫Ω×BRut−hG∗G′f−G∗G′ft−hdtdxdv.
Sincesuppu⊂−T,T, ut=0 for t−h∈T−h,T, and then, we obtain(241)∫0T∫Ω×BRfG′fut−G′ft−hut−hhdtdxdv≥∫0T∫Ω×BRfG′f−G∗G′f∂t−hutdtdxdv+−1h∫0T∫Ω×BRG∗G′ft−hut−hdtdxdv.
Tendingh to 0 in the relation above enables us to have(242)limh⟶0∫0T∫Ω×BRfG′fut−G′ft−hut−hhdtdxdv≥∫0T∫Ω×BRfG′f−G∗G′fu′tdtdxdv−∫Ω×BRG∗G′f0u0dxdv.
Moreover, expressions (236) and (242) give(243)∫Ω×BRf0G′f0−G∗G′f0u0dxdv+∫0,T×Ω×BRfG′f−G∗G′fu′tdtdxdv≤∫0,T×Ω×BRfσ,∇vG′futdtdxdv.
So, tendingR to ∞ in (243), we obtain Lemma 15.
Now, we derive (185) by combining Lemmas 13–15.
So, let us show that the sequencedivvfhσhh converges weakly to divvfσ in Cc∞ℝ×Ω×ℝd′ and divvfσ=divvf∇c∗∇vG′f in the weak sense.
So, fixT>0 and let ψ∈Cc∞ℝ×Ω×ℝd such that suppψ.,x,v⊂−T,T.
We have(244)limh⟶0∫0T∫Ω×ℝdfh∇c∗∇G′fh−fσ,∇vψdtdxdv=0.
Indeed,(245)∫0T∫Ω×ℝdfh∇c∗∇vG′fh−fσ,∇vψdtdxdv=∫0T∫Ω×ℝdf∇c∗∇G′fh−σ,∇vψdtdxdv+∫0T∫Ω×ℝdfh−f∇c∗∇vG′fh,∇vψdtdxdv.
We usei, iii, and the fact that ∇vψ∈Lp0,T×Ω×ℝd and ∇c∗∇vG′fhh is bounded in Lq0,T×Ω×ℝd, and we tend h to 0 in (245) to obtain (244).
Let us show thatdivvfσ=divvf∇c∗∇vG′f in the weak sense. So, let ε>0 and ψ∈Cc∞Ω×ℝd be a test function and u∈Cc∞ℝ,u≥0, with suppu⊂−T,T. Define ωεt,x,v=G′f−εψ.
Sincec∗ is convex and fh≥0, we have(246)∫0,T×Ω×ℝdfh∇c∗∇vG′fh−∇c∗∇ωε,∇vG′fh−∇ωεut≥0.
Splitting (246) as follows,(247)∫0,T×Ω×ℝdfh∇c∗∇vG′fh−∇c∗∇ωε,∇G′fh−∇ωεut=∫0,T×Ω×ℝdfh∇c∗∇vG′fh−∇c∗∇ωε,∇vG′fh−∇vG′fut+∫0,T×Ω×ℝdfh∇c∗∇vG′fh−∇c∗∇ωε,∇vG′f−∇ωεut,we derive from (185) that(248)limh⟶0∫0,T×Ω×ℝdfh∇c∗∇vG′fh−∇c∗∇ωε,∇vG′fh−∇G′fut=0.
Consequently,(249)limh⟶0∫0,T×Ω×ℝdfh∇c∗∇vG′fh−∇c∗∇ωε,∇vG′fh−∇ωεut=∫0,T×Ω×ℝdfσ−f∇c∗∇ωε,∇vG′f−∇ωεut≥0.
We divide (249) by ε, and we tend ε to 0 and have(250)∫0,T×Ω×ℝdfσ−f∇c∗∇vG′f,∇vψut≤0.
Replacingψ by −ψ in the previous relationship, we obtain(251)∫0,T×Ω×ℝdfσ−f∇c∗∇vG′f,∇vψut=0.
We conclude thatdivvfh∇c∗∇vG′fh converges weakly to divvf∇c∗∇vG′f in Cc∞ℝ×Ω×ℝd′, and then, the proof of Theorem 2 is complete.


## 4.5. Existence of Solutions

Now, we combine the weak convergence of the sequencefhh and that of the nonlinear term divvfh∇c∗∇vG′fh to prove that the kinetic equation (1) admits a weak solution f in the sense of Definition 1.Theorem 3.
Letf0 be a density of probability on Ω×ℝd such that f0+1/f0∈L∞Ω×ℝd, ∫Ω×ℝdvqf0dxdv<∞, and ∫Ω×ℝdGf0dxdv<∞ with G and c satisfying, respectively, HG and Hc.Then, for any test functionψ∈Cc∞ℝ×Ω×ℝd such that suppψ.,x,v⊂−T,T, we have(252)∫0,T×Ω×ℝdf∂tψ+v⋅∇xψ−∇vψ⋅∇c∗∇vG′fdtdxdv=−∫Ω×ℝdf0ψ0dxdv,where f is the limit function obtained in Proposition 3.Proof.
FixT>0 and let ψ∈Cc∞ℝ×Ω×ℝd be a test function such that suppψ.,x,v⊂−T,T. Integrating (17) over 0,T×Ω×ℝd, we obtain(253)∑k=1T/h∫tk−1tk∫Ω×ℝdfkh−fk−1hhψt,x+t−tk−1v,vdtdxdv+∑k=1T/h∫tk−1tk∫Ω×ℝdfk−1hv⋅∇xψt,x+t−tk−1v,vdtdxdv+∑k=1T/h∫tk−1tk∫Ω×ℝdfkh∇vψt,x+t−tk−1v,v⋅∇c∗∇vG′fkh=Bhψ,where Bhψ tends to 0 as h⟶0.
We observe that(254)∑k=1T/h∫tk−1tk∫Ω×ℝdfkh−fk−1hhψt,x+t−tk−1v,vdtdxdv=∫0,T×Ω×ℝdfhψt−h,x,v−ψt,x,vhdtdxdv−1h∫0h∫Ω×ℝdf0x−tv,vψt−h,x,vdtdxdv,(255)∑k=1T/h∫tk−1tk∫Ω×ℝdfk−1hv⋅∇xψt,x+t−tk−1v,vdtdxdv=∫0,T×Ω×ℝdfhv⋅∇xψdtdxdv.
Hence,(256)∑k=1T/h∫tk−1tk∫Ω×ℝdfkh∇vψt,x+t−tk−1v,v⋅∇c∗∇vG′fkhdtdxdv=∫0,T×Ω×ℝdfh∇v.ψ⋅∇c∗∇vG′fhdtdxdv−∫0,h×Ω×ℝdf0x−tv,v∇vψt,x,v⋅∇c∗∇G′f0x−tv,vdtdxdv.
We combine (254), (255), and (256) to obtain(257)Bhψ=−∫0,T×Ω×ℝdfh∂t−hψt,x,vdtdxdv−1h∫0h∫Ω×ℝdf0x−tv,vψt−h,x,vdtdxdv+∫0,T×Ω×ℝdfh∇vψ⋅∇c∗∇vG′fhdtdxdv−∫0,T×Ω×ℝdfhv⋅∇xψdtdxdv−∫0,h×Ω×ℝdf0x−tv,v∇vψt,x,v⋅∇c∗∇vG′f0x−tv,vdtdxdv.
Here,(258)∂t−hψt,x,v=ψt,x,v−ψt−h,x,vh.
Finally, we use the weak convergence offhh to f, the weak convergence of fh∇c∗∇vG′fh to f∇c∗∇vG′f, and the uniform convergence of ∂thψ.,x,v to ∂tψ.,x,v on the compact set 0,T when h tend to 0, to derive the following equality:(259)−∫0,T×Ω×ℝdf∂tψdtdxdv−∫Ω×ℝdf0ψ0dxdv−∫0,T×Ω×ℝdfv⋅∇xψdtdxdv+∫0,T×Ω×ℝd∇vψ⋅∇c∗∇vG′ffdtdxdv=0,i.e.,(260)∫0,T×Ω×ℝd∂tψ+v⋅∇xψ−∇vψ⋅∇c∗∇vG′ffdtdxdv=−∫Ω×ℝdf0ψ0dxdv.
In conclusion, we prove thatf solves (1) in the weak sense.


## 5. Numerical Implementation

In this section, we use the iterative operator-splitting methods as in [10] to construct an approximate weak solution of the kinetic equation (1) as stated in Section 2.1, and we establish the rate of convergence of our algorithm to the exact solution of the kinetic equation. The scheme consisting after fixing a time step h>0 and denoting tk=hk, with k∈ℕ∗, in defining the approximate solution ui=uit,x,v, t,x,v∈tk,tk+1×Ω×ℝd of (1) is as follows:(261)∂ui∂t+v⋅∇xui=divvui−1∇c∗∇vG′ui−1,intk,tk+1×Ω×ℝd,∂ui+1∂t+v⋅∇xui=divvui+1∇c∗∇vG′ui+1,intk,tk+1×Ω×ℝd,where i=1;3;5;…;2j+1, with j∈ℕ∗ and uitk,x,v=fkhx,v, ui+1tk,x,v=fkhx,v; fkh denotes the approximate solution of (1) at t=tk defined in Section 2.1.Remark 2.
By fixingui−1, we obtain(262)uit,x,v=fkhx−t−tkv,v+∫tktdivvui−1s,x−s−tkv,v∇c∗∇vG′ui−1s,x−s−tkv,vds.
By fixingui, we obtain ui+1 as follows:(263)ui+1t,x,v=χi+1t,xUx,i+1t,v,where(264)χi+1t,x=∫ℝduit−h,x−hv,vdv,Ux,i+1∈Argmin1χi+1t,x∫ℝdGχi+1t,xUdv+hWchU,Vx,i+1,with(265)Vx,i+1t,v=uit−h,x−hv,vχi+1t,x.
Note that, fort=tk and for i=1;3;…;2j+1, j∈ℕ∗, we obtain uitk,x,v=fkhx,v, ui+1tk,x,v=fkhx,v, χi+1tk,x=ρkhx, Vx,i+1tk,v=Gkxhv, and Ux,i+1tk,v=Fkxhv, where ρkh, Gkxh, Fkxh, and fkh are defined as in the descent algorithm (see Section 2.1).


### 5.1. Error Analysis

An estimate of the rate of convergence ofui to the exact solution of (1) is given as follows.Theorem 4.
Assume thatf0, c, and G satisfy, respectively, Hf0, Hc, and HG, and suppf0x,.⊂BR, for all x∈Ω; ∇c∗0=0. Then, we obtain the following error estimation:(266)ejt∞,LqΩ×BR≤Khj−1e0∞,LqΩ×BR,where ejt,x,v=ft,x,v−ujt,x,v, j∈ℕ∗ is the error between the exact solution f and the approximate solution uj and q=p/p−1≤p and e0=f−fkh, and K is a constant depending on N, M, R, and G′.Proof.
We use the error functionei=f−ui and the linearized equation of (1) via iterative splitting methods. Then, we have for i=1;3;…;2j+1:(267)∂ei∂t+v⋅∇xei=divvui−1∇c∗∇vG′ui−1ei−1+divvui−1D2c∗∇vG′ui−1∘Dv2G′ui−1ei−1+oei−1.
Sincesuppf0x,.⊂BR and N≤f0≤M, then Lemma 16 gives that suppuit,x,.⊂BR and D2c∗∇vG′ui−1°Dv2G′ui−1∈L∞tk,tk+1×Ω×BR and(268)D2c∗∇vG′ui−1∘Dv2G′ui−1L∞tk,tk+1×Ω×BR≤1+MN.
We conclude thatdivvui−1∇c∗∇vG′ui−1∈LqΩ×BR and divvui−1D2c∗∇vG′ui−1∘Dv2G′ui−1∈LqΩ×BR.
Thus,(269)divxvui−1∞,LqΩ×BR≤K1,(270)divvui−1∇c∗∇vG′ui−1∞,LqΩ×BR≤K2,(271)divvui−1D2c∗∇vG′ui−1∘Dv2G′ui−1∞,LqΩ×BR≤K3,where K1, K2, and K3 are constant depending of N, M, R, and G′. By using (267), we have(272)eit,x+tv,v=∫tktdivvui−1∇c∗∇vG′ui−1ei−1ds+∫tktdivvui−1D2c∗∇vG′ui−1∘Dv2G′ui−1ei−1ds+∫tktoei−1ds.
We now use (270) and (271) to obtain(273)ei∞,LqΩ×BR≤K4hei−1∞,LqΩ×BR,with K4=K2+K3+1. As in (272), we have(274)ei+1t,x,v=−∫tktv⋅∇xuieids+∫tktdivvui+1∇c∗∇vG′ui+1ei+1ds+∫tktdivvui+1D2c∗∇vG′ui+1∘Dv2G′ui+1ei+1ds+∫tktoei+1ds.
Next, from (269), (270), and (271), we derive(275)ei+1∞,LqΩ×BR≤h2K1K4ei−1∞,LqΩ×BR+K4hei+1∞,LqΩ×BR.
Accordingly,(276)ei+1∞,LqΩ×BR≤Khei−1∞,LqΩ×BR,where K depends on K4 and K1. Finally, we use (273) and (276) to conclude that(277)ej∞,LqΩ×BR≤Khj−1e0∞,LqΩ×BR,for j∈ℕ∗


### 5.2. Numerical Example

In this section, we solves (1), when d=1, Gt=tlnt, and c∗t=t2/2, i.e., the following Boltzmann equation:(278)∂f∂t+v⋅∂xf=∂v2f,f=ft,x,v,t,x,v∈0,∞×Ω×0,∞.We supposeΩ≔0,1/ℤ.=x=x¯+n,x¯∈0,1,nbrowsingℤ.The initial datumf0,x,v=f0x,v is a probability density on Ω×0,∞.Let us show that the functionf defined by(279)ft,x,v=x¯+v36+1112π4t+1e−v2/4t+1,ift,x¯,v∈0,+∞×0,1×0,1,ft,x,v=1112π4t+1e−v2/4t+1,ift,x¯,v∈0,+∞×0,1×1,+∞,is one solution of the kinetic equation (278). Indeed,(280)∂ft,x,v∂t+v⋅∂xf=∂v2f,in0,+∞×Ω×0,+∞,with the initial datum(281)f0x,v=x¯+v36+1112πe−v2,ifx¯,v∈0,1×0,1,f0x,v=1112πe−v2,ifx,v∈0,1×1,+∞.Moreover,(282)∂f∂t=11v234t+12π4t+1−1164t+1π4t+1e−v2/4t+1,v.∂xf=v,ift,x¯,v∈0,+∞×0,1×0,1,0,ift,x¯,v∈0,+∞×0,1×1,+∞,∂v2f=v+11v234t+12π4t+1−1164t+1π4t+1e−v2/4t+1,ift,x¯,v∈0,+∞×0,1×0,1,11v234t+12π4t+1−1164t+1π4t+1e−v2/4t+1,ift,x¯,v∈0,+∞×0,1×1,+∞.Hence,f satisfies the kinetic equation (280).Supposet∈0,+∞ fixed, one has(283)∫Ω×0,+∞ft,x,vdxdv=∫0,1×0,1x¯+v36+1112π4t+1e−v2/4t+1dx¯dv+∫0,1×1,+∞1112π4t+1e−v2/4t+1dx¯dv=∫0,1x¯dx¯+∫0,1v36dv+∫0,+∞1112π4t+1e−v2/4t+1dv,with(284)∫0,1x¯dx¯=12,∫0,1v36dv=124,∫0,+∞1112π4t+1e−v2/4t+1dv=1124.Consequently,∫Ω×0,+∞ft,x,vdxdv=1 for all t≥0.


### 5.3. The Figures

Here, we perform the first-order approximationf1 of the solution of (278) using the algorithm in (261). We limit our self to the first-order approximation just for the sake of simplicity and to avoid very complex and cumbersome calculations. Numerical computation carried out for h∈1/10,1/50,1/100 and the graphical representations of both the analytical solution f and the approximate solution f1 with Scilab software shows a coherence between the approximate solution and the analytical solution.Moreover, an error tables obtained for some values of the timet in the interval h,2h for different values of the step h are presented below.By a simple computation, the expression off1h appears as follows:(285)f1hx,v=x¯+v36+11hv23πe−v2+111−2h12πe−v2,isx¯,v∈0,1×0,1,f1hx,v=11hv23πe−v2+111−2h12πe−v2,isx¯,v∈0,1×1,+∞.For some values ofh, we draw the figures of both the analytical solution and the numerical solution (approximate solution). We notice that Figure 1 represents an approximation of Figure 2 when h=1/10, Figure 3 represents an approximation of Figure 4 when h=1/50, and Figure 5 is an approximation of Figure 6 when h=1/100.Figure 1
Numerical solutionf11/10x,v for h=1/10.Figure 2
Analytical solutionf3/20,x,v for h=1/10 and t=3/20∈1/10,2/10.Figure 3
Numerical solutionf11/50x,v for h=1/50.Figure 4
Analytical solutionf3/100,x,v for h=1/50 and t=3/100∈1/50,1/25.Figure 5
Numerical solutionf11/100x,v for h=1/100.Figure 6
Analytical solutionf3/200,x,v for h=1/100 and t=3/200∈1/100,1/50.Errors progression is in terms of theL∞ and the L2 norms between the numerical solution f1h and the analytical solution f. One can actually notice the convergence of our method in the L2 and in L∞ norms when h is decreasing to 0, which is explained as follows (Table 1).Table 1
Errors' progression.
h1/101/501/1001/5001/1000δt3/203/1003/2003/10003/2000f−f1hL20.034227910.034227910.0024780.0014201910.00027920f−f1hmax0.01456190.0780350.00450670.00100670.0008798


## 5.1. Error Analysis

An estimate of the rate of convergence ofui to the exact solution of (1) is given as follows.Theorem 4.
Assume thatf0, c, and G satisfy, respectively, Hf0, Hc, and HG, and suppf0x,.⊂BR, for all x∈Ω; ∇c∗0=0. Then, we obtain the following error estimation:(266)ejt∞,LqΩ×BR≤Khj−1e0∞,LqΩ×BR,where ejt,x,v=ft,x,v−ujt,x,v, j∈ℕ∗ is the error between the exact solution f and the approximate solution uj and q=p/p−1≤p and e0=f−fkh, and K is a constant depending on N, M, R, and G′.Proof.
We use the error functionei=f−ui and the linearized equation of (1) via iterative splitting methods. Then, we have for i=1;3;…;2j+1:(267)∂ei∂t+v⋅∇xei=divvui−1∇c∗∇vG′ui−1ei−1+divvui−1D2c∗∇vG′ui−1∘Dv2G′ui−1ei−1+oei−1.
Sincesuppf0x,.⊂BR and N≤f0≤M, then Lemma 16 gives that suppuit,x,.⊂BR and D2c∗∇vG′ui−1°Dv2G′ui−1∈L∞tk,tk+1×Ω×BR and(268)D2c∗∇vG′ui−1∘Dv2G′ui−1L∞tk,tk+1×Ω×BR≤1+MN.
We conclude thatdivvui−1∇c∗∇vG′ui−1∈LqΩ×BR and divvui−1D2c∗∇vG′ui−1∘Dv2G′ui−1∈LqΩ×BR.
Thus,(269)divxvui−1∞,LqΩ×BR≤K1,(270)divvui−1∇c∗∇vG′ui−1∞,LqΩ×BR≤K2,(271)divvui−1D2c∗∇vG′ui−1∘Dv2G′ui−1∞,LqΩ×BR≤K3,where K1, K2, and K3 are constant depending of N, M, R, and G′. By using (267), we have(272)eit,x+tv,v=∫tktdivvui−1∇c∗∇vG′ui−1ei−1ds+∫tktdivvui−1D2c∗∇vG′ui−1∘Dv2G′ui−1ei−1ds+∫tktoei−1ds.
We now use (270) and (271) to obtain(273)ei∞,LqΩ×BR≤K4hei−1∞,LqΩ×BR,with K4=K2+K3+1. As in (272), we have(274)ei+1t,x,v=−∫tktv⋅∇xuieids+∫tktdivvui+1∇c∗∇vG′ui+1ei+1ds+∫tktdivvui+1D2c∗∇vG′ui+1∘Dv2G′ui+1ei+1ds+∫tktoei+1ds.
Next, from (269), (270), and (271), we derive(275)ei+1∞,LqΩ×BR≤h2K1K4ei−1∞,LqΩ×BR+K4hei+1∞,LqΩ×BR.
Accordingly,(276)ei+1∞,LqΩ×BR≤Khei−1∞,LqΩ×BR,where K depends on K4 and K1. Finally, we use (273) and (276) to conclude that(277)ej∞,LqΩ×BR≤Khj−1e0∞,LqΩ×BR,for j∈ℕ∗


## 5.2. Numerical Example

In this section, we solves (1), when d=1, Gt=tlnt, and c∗t=t2/2, i.e., the following Boltzmann equation:(278)∂f∂t+v⋅∂xf=∂v2f,f=ft,x,v,t,x,v∈0,∞×Ω×0,∞.We supposeΩ≔0,1/ℤ.=x=x¯+n,x¯∈0,1,nbrowsingℤ.The initial datumf0,x,v=f0x,v is a probability density on Ω×0,∞.Let us show that the functionf defined by(279)ft,x,v=x¯+v36+1112π4t+1e−v2/4t+1,ift,x¯,v∈0,+∞×0,1×0,1,ft,x,v=1112π4t+1e−v2/4t+1,ift,x¯,v∈0,+∞×0,1×1,+∞,is one solution of the kinetic equation (278). Indeed,(280)∂ft,x,v∂t+v⋅∂xf=∂v2f,in0,+∞×Ω×0,+∞,with the initial datum(281)f0x,v=x¯+v36+1112πe−v2,ifx¯,v∈0,1×0,1,f0x,v=1112πe−v2,ifx,v∈0,1×1,+∞.Moreover,(282)∂f∂t=11v234t+12π4t+1−1164t+1π4t+1e−v2/4t+1,v.∂xf=v,ift,x¯,v∈0,+∞×0,1×0,1,0,ift,x¯,v∈0,+∞×0,1×1,+∞,∂v2f=v+11v234t+12π4t+1−1164t+1π4t+1e−v2/4t+1,ift,x¯,v∈0,+∞×0,1×0,1,11v234t+12π4t+1−1164t+1π4t+1e−v2/4t+1,ift,x¯,v∈0,+∞×0,1×1,+∞.Hence,f satisfies the kinetic equation (280).Supposet∈0,+∞ fixed, one has(283)∫Ω×0,+∞ft,x,vdxdv=∫0,1×0,1x¯+v36+1112π4t+1e−v2/4t+1dx¯dv+∫0,1×1,+∞1112π4t+1e−v2/4t+1dx¯dv=∫0,1x¯dx¯+∫0,1v36dv+∫0,+∞1112π4t+1e−v2/4t+1dv,with(284)∫0,1x¯dx¯=12,∫0,1v36dv=124,∫0,+∞1112π4t+1e−v2/4t+1dv=1124.Consequently,∫Ω×0,+∞ft,x,vdxdv=1 for all t≥0.


## 5.3. The Figures

Here, we perform the first-order approximationf1 of the solution of (278) using the algorithm in (261). We limit our self to the first-order approximation just for the sake of simplicity and to avoid very complex and cumbersome calculations. Numerical computation carried out for h∈1/10,1/50,1/100 and the graphical representations of both the analytical solution f and the approximate solution f1 with Scilab software shows a coherence between the approximate solution and the analytical solution.Moreover, an error tables obtained for some values of the timet in the interval h,2h for different values of the step h are presented below.By a simple computation, the expression off1h appears as follows:(285)f1hx,v=x¯+v36+11hv23πe−v2+111−2h12πe−v2,isx¯,v∈0,1×0,1,f1hx,v=11hv23πe−v2+111−2h12πe−v2,isx¯,v∈0,1×1,+∞.For some values ofh, we draw the figures of both the analytical solution and the numerical solution (approximate solution). We notice that Figure 1 represents an approximation of Figure 2 when h=1/10, Figure 3 represents an approximation of Figure 4 when h=1/50, and Figure 5 is an approximation of Figure 6 when h=1/100.Figure 1
Numerical solutionf11/10x,v for h=1/10.Figure 2
Analytical solutionf3/20,x,v for h=1/10 and t=3/20∈1/10,2/10.Figure 3
Numerical solutionf11/50x,v for h=1/50.Figure 4
Analytical solutionf3/100,x,v for h=1/50 and t=3/100∈1/50,1/25.Figure 5
Numerical solutionf11/100x,v for h=1/100.Figure 6
Analytical solutionf3/200,x,v for h=1/100 and t=3/200∈1/100,1/50.Errors progression is in terms of theL∞ and the L2 norms between the numerical solution f1h and the analytical solution f. One can actually notice the convergence of our method in the L2 and in L∞ norms when h is decreasing to 0, which is explained as follows (Table 1).Table 1
Errors' progression.
h1/101/501/1001/5001/1000δt3/203/1003/2003/10003/2000f−f1hL20.034227910.034227910.0024780.0014201910.00027920f−f1hmax0.01456190.0780350.00450670.00100670.0008798



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*Source: 7489532-2020-06-09.xml*