A vector field method for relativistic transport equations with applications

Fajman, David; Joudioux, Jérémie; Smulevici, Jacques

doi:10.2140/apde.2017.10.1539

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A vector field method for relativistic transport equations with applications

David Fajman, Jérémie Joudioux and Jacques Smulevici

Vol. 10 (2017), No. 7, 1539–1612

DOI: 10.2140/apde.2017.10.1539

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–Nordström 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.

Keywords

relativistic kinetic equations, wave equation, vector-field method, asymptotic behaviour, nonlinear stability, Vlasov–Nordström system

Mathematical Subject Classification 2010

Primary: 35B40, 35Q83, 83C30

Milestones

Received: 28 April 2016

Revised: 13 April 2017

Accepted: 9 May 2017

Published: 1 August 2017

Authors

David Fajman
Gravitational Physics Faculty of Physics University of Vienna Boltzmanngasse 5 1090 Vienna Austria

Jérémie Joudioux
Gravitational Physics Faculty of Physics University of Vienna Boltzmanngasse 5 1090 Vienna Austria

Jacques Smulevici
Laboratoire de Mathématiques Université Paris-Sud 11 Bât. 425 91405 Orsay Cedex France

Vol. 10, No. 7, 2017