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

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 (f0, E0, B0) E L1 x H1 x H1, with f0 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 L1 x H1 x H1. 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.