The relativistic Vlasov–Maxwell system: Local smooth solvability for a weak topology

Abstract

This article is devoted to the relativistic Vlasov–Maxwell system in space dimension three. We prove the local well-posedness (existence and uniqueness) for initial data $({\mathrm{f}}_0,{\mathrm{E}}_0,{\mathrm{B}}_0)\in L^{\infty }\times H^1\times H^1$, with ${\mathrm{f}}_0$ compactly supported in momentum. As a byproduct, we obtain the uniqueness of weak solutions to the 3D relativistic Vlasov–Maxwell system. This result is at the interface of the classical solutions in the sense of Glassey–Strauss, and the weak solutions in the sense of DiPerna–Lions. It is the consequence of the local smooth solvability for the weak topology associated with $L^{\infty }\times H^1\times H^1$. We derive our result from a representation formula decoding how the momentum spreads and revealing that the domain of influence in momentum is controlled by mild information. We do so by developing a Radon Fourier analysis on the RVM system, leading to the study of a class of singular weighted integrals. In parallel, we implement our method to construct smooth solutions to the RVM system in the regime of dense, hot and strongly magnetized plasmas.