Abstract
We study the asymptotic properties of the small data solutions of the Vlasov–Maxwell system in dimension three. No neutral hypothesis nor compact support assumptions are made on the data. In particular, the initial decay in the velocity variable is optimal. We use vector field methods to obtain sharp pointwise decay estimates in null directions on the electromagnetic field and its derivatives. For the Vlasov field and its derivatives, we obtain, as in Fajman et al. (The Stability of the Minkowski space for the Einstein-Vlasov system, 2017. arXiv:1707.06141), optimal pointwise decay estimates by a vector field method where the commutators are modification of those of the free relativistic transport equation. In order to control high velocities and to deal with non integrable source terms, we make fundamental use of the null structure of the system and of several hierarchies in the commuted equations.
Highlights
This article is concerned with the asymptotic behavior of small data solutions to the three-dimensional Vlasov–Maxwell system
These equations, used to model collisionless plasma, describe, for one species of particles,1 a distribution function f and an electromagnetic field which will be reprensented by a two form Fμν The equations are given by2
In [3], using vector field methods, we proved optimal decay estimates on small data solutions and their derivatives of the Vlasov–Maxwell system in high dimensions d ≥ 4 without any compact support assumption on the initial data
Summary
This article is concerned with the asymptotic behavior of small data solutions to the three-dimensional Vlasov–Maxwell system. These informations will allow us to deduce pointwise decay estimates on the null components of F in both the exterior and the interior regions Another problem arises from the source terms of the commuted Maxwell equations, which need to be written with our modified vector fields. The estimate can be improved by removing the factor (1 + |t − r |)k ( one looses one power of r in the initial norm) This remarkable behavior reflects that the particles do not reach the speed of light so that v∈R3 |g|dv enjoys much better decay properties along null rays than along time-like directions and should be compared with solutions to the Klein-Gordon equation (see [16]).
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