Research Article | Open Access
Aboubacar Marcos, Ambroise Soglo, "Solutions of a Class of Degenerate Kinetic Equations Using Steepest Descent in Wasserstein Space", Journal of Mathematics, vol. 2020, Article ID 7489532, 30 pages, 2020. https://doi.org/10.1155/2020/7489532
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.
The general model describing the kinetic equations is about an evolution equation of unknown function , representing a time-depending density of probability distribution of a material in a given domain of the space. In the present work, may measure the density distribution of a system of identical particles of a bulk material. The density depends on the time and the position and the velocity of some particles at . Roughly speaking, the equation is considered as the evolution of the density function in the phase space , with as an open bounded domain with periodic boundary. As a probability density, remains positive in the court of time and satisfies the mass conservation principle: , for all :where the initial datum is a probability density on . Here, is an open, bounded, convex, and smooth domain of , with periodic, is the Legendre transform of a cost function , and 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
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 , the authors dealt with the heat equation:
By fixing an probability density with finite and a time step , they define the mass density as a discrete solution of (4) at time , which minimizes the functionalon , where is the set of all probability density on having finite second moments and is the 2-Wasserstein metric defined as
By defining as follows: , they tend to 0 and then show that the sequence converges to a nonnegative function , which solves (4) in a weak sense.
In , the existence of solutions for the spatially homogeneous equations associated with (1), that is, the equation for fixed has been proved by M. Agueh, using a similar variational scheme as in . Here, is an bounded and convex domain (see  for more details).
A particular case of (1), namely, the kinetic equationobtained by choosing and , has been studied in  by using a discretization scheme basing on the “splitting method.” This enables the authors to decompose a discrete solution of the kinetic equation (8) in the form , where stands for a discrete solution of the free transport equationwhen is fixed and a discrete solution of the diffusion equation in (8) when is fixed.
Defining as they show that converges to a nonnegative function 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  along with some techniques developed in  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 equationobtained whenand the parabolic Laplacian equationin the case and with .
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 and define as a discrete solution of the kinetic equation (1) at time , for (see Section 2.1).(2)Next, we prove that the solution of the Monge problem is defined by where and are as in Section 2.1. We use (16) to show that the sequence satisfies the time-discretization equation of the kinetic equation (1) weakly, for , where tends to 0 and when tends to 0.(3)Then, we define an approximate solution of the kinetic equation (1) (see (118)), and we prove that the sequence converges to a nonnegative function which solves the kinetic equation (1) in a weak sense when tends to 0.
The convergence result has been achieved as follows:(a)The weak convergence of to in for follows from the displacement convexity of the functional on the set of all probability density (see Proposition 3), and its strong convergence in is obtained, thanks to a diagonal method combined with a result obtained in .(b)The convexity of and the boundedness of in help to prove a weak convergence of the nonlinear term to in .(c)Finally, the strong convergence of to in and the weak convergence of the nonlinear term to in enable us to establish that is a weak solution of the kinetic equation (1).
The 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.
Throughout this work, we will assume the following: is a convex function of class such that is convex. is even and convex function such that and , for all , with . is a probability density on such that with and with .
Remark 1. Typical examples satisfying assumption are the functions and .
Proposition 1. Assume that satisfies . Let be a convex function such that is convex and decreasing.
Then, the functionalis displacement convex.
Proof. Let and be a -optimal map that pushes forward to . We define and , where .
Then, we haveFrom  and Proposition 3, we have thatReplacing (21) in (20), we obtainRecalling again  and Proposition 3, we get that is diagonalizable with positive eigenvalues. So, using the fact that the map is concave on the set of diagonalizable matrices with positive eigenvalues, we getSince is decreasing, thenFrom (23) and (24) and the fact that is convex, we obtainHence, we conclude that the functional is displacement convex.
Corollary 1. Since the functional is displacement convex, we have that
and is the -optimal map that pushes forward to .
Definition 1. Let be nonnegative. We say that a nonnegative function on is a weak solution of (1) in the time’s interval for some , if for every test function with time support , we have
2.1. The Flow and Descend Algorithm
Assume that the probability density satisfies and fix a time step, then we define the following:(1)(2)(3) for fixed such that (4)
For each fixed, denotes the unique minimizer of the variational problem:
is the set of all probability density on having a finite -moment, and stands for the Kantorovich work defined as
We obtain the terms , for , by induction as follows:(i)By fixing , we define .(ii)By fixing , we definewhereand is the unique minimizer of the variational problem.with
2.2. c-Wasserstein Metric
In this section, we define a Wasserstein metric corresponding to a cost function , and we study its topology.
Definition 2. Assume that satisfies . Let two probability measures on . We define the -Wasserstein metric between and by
Theorem 1. Assume that satisfies . Then, is a distance on the probability space . Furthermore, if is a sequence in and , then converges to in the metric space if and only if converges narrowly to in .
Proof. Let be two probability measures on such that . Then, there exists a sequence in which converges to 0 such thatDenote by the solution of Kantorovich problem:Then, we obtainSince converges to 0, then tends to and when goes to , for all such that . Then, using the fact that is coercive, we deduce that there exists such thatThis with (37) implies thatThus, we conclude thatLet be the solution of the Kantorovich problem:Then, using (40), we obtainWe deduce that a.e. So for all , we haveConsequently, .
Let us fix two probability measures and on . Since is even, thenfor all . We deduce from (44) thatLet be three probability measures on . Define and . Denote by the solution of Kantorovich problemand denote by the solution of the Kantorovich problem:Using the Gluing lemma , there exists a probability measure on such thatfor some Borel subsets and of . Let be a probability measure on defined by . Then, , and we use the convexity of to get thatSo, , and we conclude thatHence, is a distance on .
Let us now study the topology of .
Let be a sequence on and such that converges to 0 when tends to . Define , since converges to 0, then we use the fact that is coercive to havewhen . We deduce thatNote that the 1-Wasserstein metric between and isWe deduce that converges to 0 when tends to . Since the 1-Wasserstein metric induces the narrow topology of , we conclude that the sequence converges narrowly to in .
Assume now that the sequence converges narrowly to in . Fix and denote by the solution of Kantorovich problem:Since converges narrowly to , then converges narrowly to some andSo,for all . Hence, for all , there exists such thatSo, for all . Then,Consequently, converges narrowly to in the metric space .
We establish now the existence of solution for the variational problem defined byin the metric space , where is the set of all probability measures on having -finite moment, that is,and being a time step.
Lemma 1. Assume that , , and satisfy, respectively, , , and . Then, the following is obtained:(i)The map is lower semicontinuous in .(ii)The functional is lower semicontinuous in .(iii)The set is a closed subset of ,
Proof. (i)Let . Let be a sequence in such that converges to in metric space . Denote by the solution of the Kantorovich problem: We have Since converges to narrowly, then converges to narrowly and This implies that Thus, we obtain the proof of .(ii)Since is convex and ,Hence, if converges to weakly in with , we haveand thenwhich complete the proof of . The proof of is a consequence of .
2.3. Existence Results for the Discrete Problem
First of all, we introduce here for unbounded domains analogous of the maximum principle stated for bounded domains in . This maximum principle plays a central role in the searching of solution for the discrete problem . 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 datum satisfies with , satisfies , and satisfies .
Then, any solution of the variational problemsatisfies , with and .
Proof. We define . Assume by contradiction that has a positive Lebesgue measure. Then, , where is the minimizer ofOtherwise,This yields a contradiction.
Define , i.e., , for all , and denote by and the marginals of . We have with and then and .
Let and denote, respectively, the density functions of and . Becausewe have on and on .
For is small enough, define and byfor all .
, , and we getTaking into account some ideas from , we shall prove that , which leads to a contradiction.
Indeed,Since is nonnegative on and , we haveAlso, we have is being convex and of class , thenon andon .
Hence, we haveSince is continuous, then