Local existence of solutions of the vlasov‐maxwell equations and convergence to the vlasov‐poisson equations for infinite light velocity

Abstract

We prove the local existence of smooth solutions for the Vlasov-Maxwell equations in three space variables. The existence time for such solutions is independent of the light velocity c. Then we derive regularity results for both the Vlasov-Poisson and the Vlasov-Maxwell equations. The last part of the paper is devoted to a proof of weak and strong convergence of the Vlasov-Maxwell equations towards the Vlasov-Poisson equations, when the light velocity c goes to infinity. 534P.Degond means of an abstract theorem of Kato [8]. In this paper, we want to prove a slightly different theorem (see 01) by a direct method using a fixed point theorem*). This leads to an explicit estimate for the life span T+, which is independent of the light velocity c.In the second part, we give some regularity results for the Vlasov-Poisson and the Vlasov-Maxwell equations. Essentially, we prove that the solutions have the same regularity properties, as the initial data, under the assumptions that these are compactly supported with respect to the velocity. It is probably easy to refine this result, but unuseful for our purpose, which is to provide tools for the third part.In the third part, we prove that, when c goes to infinity, the solution converges (strongly) in the regular time limit T* , towards the smooth solution of the Vlasov-Poisson equation. This may justify the electrostatic approximation, in all the physical cases where the characteristic speed of the particles is much smaller than the speed of light. In the same spirit, we should mention the work of A. Majda and S. Klainerman (91 about the convergence of the compressible Euler equation, to the incompressible one, or the one of S. and H. Added [lo] on the approximation of the Zakharov system, by the non linear Schrddinger equation.

Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in $1$ and $2$ space dimensions

“…We must differentiate equation (1) a certain number of times, and proceed by induction to give a linear Gronwall inequality for the derivatives. This type of proof is chieved in [5] for the Vlasov-Maxwell equation.…”

Boundary Value Problems for the Stationary Nordström-Vlasov System

“…It is well known that the Vlasov-Poisson model can be derived from the relativistic Vlasov-Maxwell model assuming that the particle velocities are small compared to the light speed [8,15,25,38].…”

Permanent regimes for the Vlasov–Maxwell equations with specular boundary conditions

“…This model is justified by studying the asymptotic behaviour of the relativistic Vlasov-Maxwell problem when the particle velocities are small with respect to the light speed [8], [24], [19], [5].…”