Asymptotic properties of small data solutions of the Vlasov-Maxwell system in high dimensions

Abstract

We prove almost sharp decay estimates for the small data solutions and their derivatives of the Vlasov-Maxwell system in dimension $n \geq 4$. The smallness assumption concerns only certains weighted $L^1$ or $L^2$ norms of the initial data. In particular, no compact support assumption is required on the Vlasov or the Maxwell fields. The main ingredients of the proof are vector field methods for both the kinetic and the wave equations, null properties of the Vlasov-Maxwell system to control high velocities and a new decay estimate for the velocity average of the solution of the relativistic massive transport equation. We also consider the massless Vlasov-Maxwell system under a lower bound on the velocity support of the Vlasov field. As we prove in this paper, the velocity support of the Vlasov field needs to be initially bounded away from $0$. We compensate the weaker decay estimate on the velocity averages of the massless Vlasov field near the light cone by an extra null decomposition of the velocity vector.