Vol. 10, No. 7, 2017

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ISSN: 1948-206X (e-only)
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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
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 4 under some smallness assumptions. In the massless case, the system decouples and we prove optimal decay estimates for the solutions in dimensions n 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