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2017 A vector field method for relativistic transport equations with applications
David Fajman, Jérémie Joudioux, Jacques Smulevici
Anal. PDE 10(7): 1539-1612 (2017). DOI: 10.2140/apde.2017.10.1539


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.


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David Fajman. Jérémie Joudioux. Jacques Smulevici. "A vector field method for relativistic transport equations with applications." Anal. PDE 10 (7) 1539 - 1612, 2017.


Received: 28 April 2016; Revised: 13 April 2017; Accepted: 9 May 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 1373.35046
MathSciNet: MR3683922
Digital Object Identifier: 10.2140/apde.2017.10.1539

Primary: 35B40 , 35Q83 , 83C30

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

Rights: Copyright © 2017 Mathematical Sciences Publishers


Vol.10 • No. 7 • 2017
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