Asymptotic Properties of the Solutions to the Vlasov–Maxwell System in the Exterior of a Light Cone

Abstract

Abstract This paper is concerned with the asymptotic behavior of small data solutions to the three-dimensional Vlasov–Maxwell system in the exterior of a light cone. The plasma does not have to be neutral, and no compact support assumptions are required on the data. In particular, the initial decay in the velocity variable of the particle density is optimal and we only require an $L^2$ bound on the electromagnetic field with no additional weight. We use vector field methods to derive improved decay estimates in null directions for the electromagnetic field, the particle density, and their derivatives. In contrast with [ 2], where we studied the behavior of the solutions in the whole spacetime, the initial data have less decay and we do not need to modify the commutation vector fields of the relativistic transport operator. To control the solutions under these assumptions, we crucially use the strong decay satisfied by the particle density in the exterior of the light cone, null properties of the Vlasov equation, and certain hierarchies in the energy norms.