A vector field method for relativistic transport equations with applications

TL;DR

This paper develops a vector field method to analyze relativistic transport equations, establishing decay estimates and applying them to prove global existence and decay for Vlasov-Nordström systems in higher dimensions.

Contribution

It adapts Klainerman's vector field method to relativistic transport equations and applies it to obtain decay and existence results for Vlasov-Nordström systems.

Findings

Robust decay estimates for velocity averages without compact support.

Global existence and decay in dimensions n ≥ 4 for massive case.

Optimal decay estimates for massless case in n ≥ 4 and dimension 3.

Abstract

We adapt the vector field method of Klainerman to the study of relativistic transport equations. First, we prove robust decay estimates for velocity averages of solutions to the relativistic massive and massless transport equations, without any compact support requirements (in $x$ or $v$) for the distribution functions. In the second part of this article, we apply our method to the study of the massive and massless Vlasov-Nordstr\"om systems. In the massive case, we prove global existence and (almost) optimal decay estimates for solutions in dimensions $n \geq 4$ under some smallness assumptions. In the massless case, the system decouples and we prove optimal decay estimates for the solutions in dimensions $n \geq 4$ for arbitrarily large data, and in dimension $3$ under some smallness assumptions, exploiting a certain form of the null condition satisfied by the equations. The $3$-dimensional massive case requires an extension of our method and will be treated in future work.