A vector field method for massless relativistic transport equations and applications

Abstract

In this article, we present a vector field method for the study of solutions to massless relativistic transport equations. Compared to the methodologie developped by Fajman-Joudioux-Smulevici, we remove the Lorentz boosts of the commutation vector fields, which makes our method more suitable for the study of systems coupling a massless Vlasov equation with an equation which has a different speed of propagation. We also believe that our approach can be adapted more easily to the context of curved backgrounds. In a second time, we apply our method in order to derive sharp decay estimates on the spherically symmetric small data solutions of the relativistic massless Vlasov-Poisson system. No compact support assumption is required on the data and, in particular, the initial decay in the velocity variable is optimal. In order to compensate the slow decay rate of the particle density near the light cone, we take advantage of the null structure of the equations.