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 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.
Highlights
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 zf(t, x, v) zt + v.∇xf(t, x, v) 0
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 fk as a discrete solution of the kinetic equation (1) at time tk hk, for k ∈ N
(2) we prove that the solution Tk of the Monge problem (M): v−
Summary
(3) en, we define an approximate solution fh of the kinetic equation (1) (see (118)), and we prove that the sequence (fh)h converges to a nonnegative function f which solves the kinetic equation (1) in a weak sense when h tends to 0. Assume that the probability density f0(x, v) satisfies (Hf0) and fix h > 0 a time step, we define the following:. We establish the existence of solution for the variational problem (P) defined by (P): inf ρ∈Pq(Rd) Rd. Pq(Rd) is the set of all probability measures on Rd having q-finite moment, that is, PqRd. Define the probability density Gxε Tε#Fxk on Rd. Since G satisfies (HG) and Tε is a diffeomorphism pushing Fxkh forward to Gxε , we obtain the following Monge–Kantorovich-type energy inequality:.
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