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

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 parabolic p-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.


Introduction
e general model describing the kinetic equations is about an evolution equation of unknown function f, 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. e 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 × R d , 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: (t, x, v)dxdv � 1, for all t ≥ 0: 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][2][3][4][5][6][7], where (1) is investigated in some particular cases. Indeed, in [3], the authors dealt with the heat equation: By fixing an probability density ρ 0 with R d |v| 2 ρ 0 (v)dv finite and a time step h > 0, they define the mass density ρ h k as a discrete solution of (4) at time t k � hk, which minimizes the functional on P 2 (R d ), where P 2 (R d ) is the set of all probability density on R d having finite second moments and W 2 is the 2-Wasserstein metric defined as By defining ρ h as follows: they tend h to 0 and then show that the sequence (ρ h ) h 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 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 obtained by choosing c * (x) � |x| 2 /2 and G(x) � x ln x, has been studied in [5] by using a discretization scheme basing on the "splitting method." is enables the authors to decompose a discrete solution f k of the kinetic equation (8) in the form f k � ρ h k F xh k , where ρ h k stands for a discrete solution of the free transport equation when v is fixed and F xh k a discrete solution of the diffusion equation in (8) when x is fixed.
Defining f h as they show that (f h ) h 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: 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 obtained when and the parabolic p− Laplacian equation in the case c * (x) � |x| p /p and G(x) � x c /c(c − 1) with c � (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 step h > 0 and define f k as a discrete solution of the kinetic equation (1) at time t k � hk, for k ∈ N (see Section 2.1). (2) Next, we prove that the solution T k of the Monge problem is defined by where F xh k and G xh k are as in Section 2.1. We use (16) to show that the sequence (f k ) k satisfies the time-discretization equation of the kinetic equation (1) weakly, for k ∈ N, where A h k tends to 0 and when h tends to 0.
(3) en, we define an approximate solution f h of the kinetic equation (1) (see (118)), and we prove that the sequence (f h ) h converges to a nonnegative function f which solves the kinetic equation (1) in a weak sense when h tends to 0.
e convergence result has been achieved as follows: ′ enable us to establish that f is a weak solution of the kinetic equation (1). e paper is structured as follows. In Section 2, 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.

Preliminaries
roughout this work, we will assume the following:

Remark 1.
Typical examples satisfying assumption (H G ) are the functions G(s) � s ln s, s > 0 and G(s) � s r , r > 1.
en, the functional is displacement convex.
From [1] and Proposition 3, we have that Replacing (21) in (20), we obtain Recalling again [1] and Proposition 3, we get that ∇T t is diagonalizable with positive eigenvalues. So, using the fact that the map A⟼(detA) 1/d is concave on the set of d × d diagonalizable matrices with positive eigenvalues, we get Since From (23) and (24) and the fact that

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Hence, we conclude that the functional E is displacement convex. □ Corollary 1. Since the functional E is displacement convex, we have that 2.1. e Flow and Descend Algorithm. Assume that the probability density f 0 (x, v) satisfies (H f 0 ) and fix h > 0 a time step, then we define the following: For each x fixed, F xh 1 denotes the unique minimizer of the variational problem: is the set of all probability density on R d having a finite q-moment, and W h c (F, G xh 1 ) stands for the Kantorovich work defined as We obtain the terms f h k , for k ≥ 2, by induction as follows: where and F xh k is the unique minimizer of the variational problem. with Existence of F xh 1 and F xh k will be proved farther in Sections 2 and 3, respectively.

c-Wasserstein Metric.
In this section, we define a Wasserstein metric corresponding to a cost function c, and we study its topology.
Definition 2. Assume that c: R d ⟶ [0, ∞) satisfies H c . Let ρ 1 , ρ 2 ∈ P(R d ) two probability measures on R d . We define the c-Wasserstein metric between ρ 1 and ρ 2 by en, W c is a distance on the probability space P(R d ). Furthermore, if (ρ n ) n is a sequence in P(R d ) and ρ ∈ P(R d ), then (ρ n ) n converges to ρ in the metric space (P(R d ), W c ) if and only if (ρ n ) n converges narrowly to ρ in P(R d ).
Proof. Let ρ 1 , ρ 2 ∈ P(R d ) be two probability measures on R d such that W c (ρ 1 , ρ 2 ) � 0. en, there exists a sequence (λ n ) n in (0, ∞) which converges to 0 such that Denote by c n the solution of Kantorovich problem: 4 Journal of Mathematics en, we obtain Since λ n converges to 0, then |x − y/λ n | tends to ∞ and when n goes to ∞, for all x, y ∈ R d such that x ≠ y. en, using the fact that c is coercive, we deduce that there exists is with (37) implies that us, we conclude that Let c 0 be the solution of the Kantorovich problem: en, using (40), we obtain We deduce that x � yc 0 a.e. So for all ϕ ∈ C b (R d ), we have Consequently, ρ 1 � ρ 2 . Let us fix two probability measures ρ 1 and ρ 2 on R d . Since c is even, then for all λ > 0. We deduce from (44) that Let ρ 1 , ρ 2 , ρ 3 be three probability measures on R d . Define λ 1 � W c (ρ 1 , ρ 2 ) and λ 2 � W c (ρ 2 , ρ 3 ). Denote by c 1 ∈ Π(ρ 1 , ρ 2 ) the solution of Kantorovich problem and denote by c 2 ∈ Π(ρ 2 , ρ 3 ) the solution of the Kantorovich problem: Using the Gluing lemma [8], there exists a probability measure σ on for some Borel subsets A and B of R d . Let c 3 be a probability measure on en, c 3 ∈ Π(ρ 1 , ρ 3 ), and we use the convexity of c to get that So, inf c∈Π(ρ 1 ,ρ 3 ) R d ×R d c(x − z/λ 1 + λ 2 )dc ≤ 1, and we conclude that Hence, W c is a distance on P(R d ).
Let us now study the topology of W c . Let (ρ n ) n be a sequence on P(R d ) and ρ ∈ P(R d ) such that W c (ρ n , ρ) converges to 0 when n tends to ∞. Define λ n � W c (ρ n , ρ), since (λ n ) n converges to 0, then we use the fact that c is coercive to have when n ⟶ ∞. We deduce that Note that the 1-Wasserstein metric between ρ n and ρ is We deduce that W 1 (ρ n , ρ) converges to 0 when n tends to ∞. Since the 1-Wasserstein metric W 1 induces the narrow topology of P(R d ), we conclude that the sequence (ρ n ) n converges narrowly to ρ in P(R d ).

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Assume now that the sequence (ρ n ) n converges narrowly to ρ in P(R d ). Fix λ > 0 and denote by c λ n the solution of Kantorovich problem: Since ρ n converges narrowly to ρ, then c λ n converges narrowly to some c λ ∈ Π(ρ, ρ) and Consequently, (ρ n ) n converges narrowly to ρ in the metric space (P(Ω), W c ).
We establish now the existence of solution for the variational problem (P) defined by is the set of all probability measures on R d having q-finite moment, that is, and h > 0 being a time step.
□ Lemma 1. Assume that ρ 0 , G, and c satisfy, respectively, H f 0 , H G , and H c . en, the following is obtained: where Proof . Denote by c h n the solution of the Kantorovich problem: Since (ρ n ) n converges to ρ narrowly, then (c h n ) n converges to c h ∈ Π(ρ 0 , ρ) narrowly and us, we obtain the proof of (i).
and then lim inf which complete the proof of (ii). e proof of (iii) is a consequence of (ii). . First of all, we introduce here for unbounded domains analogous of the maximum principle stated for bounded domains in [1].
is maximum principle plays a central role in the searching of solution for the discrete problem (P xh 1 ). It is also used to further establish the convergence of our algorithm towards a weak solution of the kinetic equation (1).
. Assume by contradiction that E x has a positive Lebesgue measure. en, is yields a contradiction.
, and we get Taking into account some ideas from [1], we shall prove that I(F xh,ϵ 1 ) − I(F xh 1 ) < 0, which leads to a contradiction. Indeed, Also, we have G is being convex and of class C 1 , then on E x and Since G ′ is continuous, then Consequently, there exists δ > 0 such that, for 0 < ε < δ, then So, we fix ε > 0 small, such that 0 < ε < δ, and we use (81) and (83) to obtain Now, fixing ϵ < δ and combining (77) and (84) yield It is a contradiction since F xh 1 is a minimizer of I on P q (R d ). Consequently, E x is negligible and then F xh ). en, we use Proposition 2, and we obtain that any minimizer F xh 1 of the variational problem P xh . By using Proposition 2 and the fact that G is convex, we obtain that for all probability density F ∈ P q (R d ) such that N ≤ F ≤ M.
We use now Lemma 1 to get that the functional . We conclude then that the problem (P xh 1 ) admits a solution F xh 1 . e strict convexity of G and c implies the strict convexity of the map F⟼W h c (G xh 1 , F) and that of the maps )dv and accordingly the uniqueness of the minimizer F xh 1 of (P xh 1 ).

Euler-Lagrange Equation for the Problem (P xh k )
In this section, we prove that the sequence (f h k ) k 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 that G satisfies H G and c satisfies H c . en, the Monge problem admits a unique solution T k such that where F x k is the unique minimizer of the variational problem.
Define the probability density G x ε � T ε# F x k on R d . Since G satisfies (H G ) and T ε is a diffeomorphism pushing F xh k forward to G x ε , we obtain the following Monge-Kantorovich-type energy inequality: Recalling the definition of T ε , we have Dividing (91) by ε > 0 and using (92) and the dominated convergence theorem, we have 8 Journal of Mathematics Furthermore, since T − 1 ε #G x ε � F xh k , then the Monge-Kantorovich-type energy inequality gives is implies that We combine now (93) and (95) to derive Since us, for ε > 0, we have We use the convexity of c and the fact that Now, from (92) and the dominated convergence theorem, we obtained Since F xh k minimizes the functional on the probability space, the Euler-Lagrange equation yields Consequently, by using (96) and (100), we have Next, by replacing ψ by − ψ in (96), we obtain the desired equation: us, we conclude that Note that ∇c is inversible and (∇c) − 1 � ∇c * . en, we obtain the explicit expression of the optimal map T k : e proof of this lemma is complete. We are now ready to show that the sequence (f h k ) k satisfies the time discretization (17) of the kinetic equation (1). Let Journal of Mathematics Next, we use the Taylor formula and the expression of the optimal T k to obtain Here If we show that A h k [ψ] tends to 0 as h goes to 0, then we are done.
Indeed, from the maximum principle, N ≤ f h k ≤ M, and then where K ψ is a compact subset of Ω × R d such that suppψ ⊂ K ψ . Since F xh k minimizes the general functional energy We use in (112) the expression of the optimal maps T k and the definition of the f h k , and then, we have (113) By using (114) and the fact that c(x) ≥ A 1 |x| q , then (113) becomes Recalling (115) and (111), we obtain We combine (116) and (117) to conclude that A h k (ψ) tends to 0, when h goes to 0. Accordingly, we conclude that the sequence (f h k ) k resulted from a time discretization of the kinetic equation (1).
Recalling Section 2.1, we define an approximate solution f h over [0, ∞[×Ω × R d of the kinetic equation (1) as follows:

⎧ ⎨ ⎩ (118)
In the next section, we establish the convergence of the sequence (f h ) h to a weak solution f of (1).

Weak
Convergence of (f h ) h . Let us consider the sequence (f h ) h as defined in (118). f 0 satisfies (H f 0 ). en, for all k ≥ 1, we have

Lemma 4. Assume that
Proof. Taking H(t) � t p , p > 1, in Corollary 1, we obtain where T k is c-optimal map that pushes F xh k forward to G xh k . We use expression of T k and we obtain From the convexity of c and c * , we have (123) Since c ≥ 0 and c(0) � 0, then c * ≥ 0; hence, Multiplying (125) by |ρ h k | p , we obtain after integration and then by an iteration process on k, we get the proof of Lemma 4.

□
Proof. Let 0 < T < ∞ and let h > 0 be a step such that T/h ∈ N * . We use Lemma 3 and the definition of f h in (118) to obtain  Moreover, Journal of Mathematics Proof.

see Proposition 2 and Lemma 4), then
(130) Since  First, we prove that the sequence that f 0 , c, and G satisfy, respectively,  (H f 0 ), (H c ), and (H G ). en, for all 0 < T < ∞ fixed,

Strong
Convergence of (f h ) h . In order to prove the strong convergence result, we need to establish the following compactness results for
Using the definition of f h in (118), we obtain where Using (152), we have Using S h,i k,N as a c-optimal map that pushes F xh On the contrary, Lemma 6), and from assumptions on G, G ′ (s)/s is continuous on R * , and then there exists a constant K such that .)) by C ∞ c (R d ) functions and using the dominated convergence theorem, we have

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So, by using Hölder's inequality, we have that Using the algorithm defined in Section 2.1 and Lemma 6, we have us, Applying the Hölder's inequality in the previous relation yields We notice that 14 Journal of Mathematics Next, we use Lemma 6, and we have We combine (160), (161), and (162) to get Writing Nh � τ, we then have On the contrary, Since (166) Next, we use the fact that us, where Let us process now to the following estimate.
Let B n be an open ball of radius n centered at origin, with n ∈ N * . en, B n ⊂ B n+1 and ∪ n∈N * B n � R d .
Using previous results, the sequence |B c n | tends to 0, when n goes to ∞; for ε > 0, there exists N ε > 0 such that for all n ≥ N ε . Since f h n converges strongly to f n in when h n goes to 0 and n ≥ N ε . We conclude that (f h n ) converges weakly to f n in L p ([0, T] × Ω × R d ), for n ≥ N ε . So f n � f for n ≥ N ε . Now, let prove that ‖f h n − f‖ L p ([0,T]×Ω×R d ) tends to 0 as h n ⟶ 0 and n ≥ N ε .
Note that Let us recall that Since |B c n | < (ε/2), for n ≥ N ε , as we can find n ε ∈ N such that for every n ≥ N ε , we have en, for all n ≥ N ε , Consequently, (f h n ) converges strongly to f in Using the previous lemma, (f h ) h converges strongly to On the contrary, We derive from the proof of eorem 3.10 in [1] that the sequence div v (f h ∇c * (∇ v G ′ (f h ))) converges weakly to div v (f∇c Proof. Let us recall the following: Since Also, we have Using (i), (ii), and the fact that To conclude with eorem 2, we need to establish the following limit, whose proof is derived from the three following lemmas: Proof. Since c * is convex and f h ≥ 0, u ≥ 0, then So, Passing to the limit, we obtain Taking into account (187)    Journal of Mathematics solution f. One can actually notice the convergence of our method in the L 2 and in L ∞ norms when h is decreasing to 0, which is explained as follows (Table 1).
Proof. Since supp f 0 (x, .) ⊂ B R , then f 0 (x, B c R ) � 0, where B c R is the complement of B R . In Section 2.1, we have  and F x 1 is the unique solution of the variational problem.
Since f 0 (x, B c R ) � 0, then G x 1 (B c R ) � 0. e c-optimal map T 1 that pushes F x 1 forward to G x 1 satisfies then We now use the explicit expression of T 1 (see Lemma 3): along with (A.5) and ∇c * (0) � 0 to get is implies that T − 1 1 (B c R ) � B c R , and then, F x 1 (B c R ) � 0. us, we deduce that supp F x 1 ⊂ B R . Since we obtain supp f 1 (x, .) ⊂ B R , for all x ∈ Ω. Finally, we obtain by induction supp f h k (x, .) ⊂ B R , for all x ∈ Ω. Since F xh k and G xh k have compact support and that c is strictly convex, then the c-optimal maps T k that pushes F xh k forward to G xh k is differentiable, and we have Hence, ∇ v T k (v) satisfies the Jacobian equation: Since N ≤ f 0 ≤ M, the maximum principle gives N ≤ f h k ≤ M. en, we deduce that Note that ∇ v T k is diagonalizable and has positive eigenvalues (see [1]). en, we deduce that ∇ v T k (v) ∈ L ∞ (Ω × B R ) and ‖∇ v T k (v)‖ L ∞ (Ω×B R ) ≤ (M/N). And by using (A.8), we obtain (A.11)

Data Availability
ere are no data underlying the findings in this paper to be shared.

Conflicts of Interest
e authors declare that they have no conflicts of interest.