Abstract
This paper is concerned with the asymptotic properties of the small data solutions to the massless Vlasov–Maxwell system in 3d. We use vector field methods to derive almost optimal decay estimates in null directions for the electromagnetic field, the particle density and their derivatives. No compact support assumption in x or v is required on the initial data, and the decay in v is in particular initially optimal. Consistently with Proposition 8.1 of Bigorgne (Asymptotic properties of small data solutions of the Vlasov–Maxwell system in high dimensions. arXiv:1712.09698, 2017), the Vlasov field is supposed to vanish initially for small velocities. In order to deal with the slow decay rate of the solutions near the light cone and to prove that the velocity support of the particle density remains bounded away from 0, we make crucial use of the null properties of the system.
Highlights
This paper is concerned with the asymptotic properties of the small data solutions to the massless Vlasov–Maxwell system in 3d
In [2], we studied the massless Vlasov–Maxwell system in high dimensions (n ≥ 4) and we proved that if the particle densities initially vanish for small velocities and if certain weighted L1 and L2 norms of the initial data are small enough, the unique classical solution to the system exists globally in time
One of the goals of this article is to describe in full details the null structure of the system, which appears to be fundamental for proving integrability and controlling the velocity support of the particle density
Summary
This article is part of a series of works concerning the asymptotic behavior of small data solutions to the Vlasov–Maxwell equations. One of the goals of this article is to describe in full details the null structure of the system, which appears to be fundamental for proving integrability and controlling the velocity support of the particle density. In view of their physical meaning, the functions fk are usually supposed to be nonnegative. Wang proved in [15] a similar result for the 3d case Using both vector field method and Fourier analysis, he replaced the compact support assumption by strong polynomial decay hypotheses in (x, v) on f and obtained optimal pointwise decay estimates on v f dv and its derivatives
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