Vol. 10, No. 7, 2017

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
Editors' interests
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author index
To appear
Other MSP journals
A vector field method for relativistic transport equations with applications

David Fajman, Jérémie Joudioux and Jacques Smulevici

Vol. 10 (2017), No. 7, 1539–1612

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.

relativistic kinetic equations, wave equation, vector-field method, asymptotic behaviour, nonlinear stability, Vlasov–Nordström system
Mathematical Subject Classification 2010
Primary: 35B40, 35Q83, 83C30
Received: 28 April 2016
Revised: 13 April 2017
Accepted: 9 May 2017
Published: 1 August 2017
David Fajman
Gravitational Physics
Faculty of Physics
University of Vienna
Boltzmanngasse 5
1090 Vienna
Jérémie Joudioux
Gravitational Physics
Faculty of Physics
University of Vienna
Boltzmanngasse 5
1090 Vienna
Jacques Smulevici
Laboratoire de Mathématiques
Université Paris-Sud 11
Bât. 425
91405 Orsay Cedex