## Analysis & PDE

• Anal. PDE
• Volume 10, Number 7 (2017), 1539-1612.

### A vector field method for relativistic transport equations with applications

#### 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.

#### Article information

Source
Anal. PDE, Volume 10, Number 7 (2017), 1539-1612.

Dates
Revised: 13 April 2017
Accepted: 9 May 2017
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.apde/1510843556

Digital Object Identifier
doi:10.2140/apde.2017.10.1539

Mathematical Reviews number (MathSciNet)
MR3683922

Zentralblatt MATH identifier
1373.35046

#### Citation

Fajman, David; Joudioux, Jérémie; Smulevici, Jacques. A vector field method for relativistic transport equations with applications. Anal. PDE 10 (2017), no. 7, 1539--1612. doi:10.2140/apde.2017.10.1539. https://projecteuclid.org/euclid.apde/1510843556

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