Motivated by a recent paper of H. Narnhofer and G. Sewell, we investigate the problem of existence and uniqueness of solutions of the Vlasov hierarchy. It is shown that the unique solution of the Vlasow hierarchy is induced by the flow on the space of probability measures on R6 which is obtained from the solution of the Vlasov equation.
The existence of weak LL‐solutions of the initial value problem for Vlasov's equation is proved under rather general assumptions. In the three‐dimensional case the solutions exist on the entire time axis.
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.
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