Spectrum structure and optimal decay rate of the relativistic Vlasov-Poisson-Landau system

Hai-Liang Li; Hongjun Yu; Mingying Zhong

doi:10.3934/krm.2017043

Article Contents

2017, Volume 10, Issue 4: 1089-1125. Doi: 10.3934/krm.2017043

This issue Previous Article The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks Next Article First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions

Spectrum structure and optimal decay rate of the relativistic Vlasov-Poisson-Landau system

1.
Department of Mathematics, Capital Normal University, China
2.
School of Mathematical Sciences, South China Normal University, China
3.
Department of Mathematics and Information Sciences, Guangxi University, China

* Corresponding author: Mingying Zhong
* Corresponding author: Mingying Zhong

Received: September 30, 2016

Revised: December 31, 2016

Published: November 2017

The first author is supported by the NNSFC grants No. 11225102,11231006,11461161007 and 11671384, and by the Key Project of Beijing Municipal Education Commission no. CIT and TCD2014 0323. The second author is supported by the NNSFC grants No. 11371151. The third author is supported by National Natural Science Foundation of China No. 11671100 and 11301094, Project supported by Guangxi Natural Science Foundation No. 2014GXNSFBA118020

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Abstract

It is interesting to analyze the mutual influence of relativistic effect and electrostatic potential force on the qualitative behaviors of charge particles simulated by the one-species relativistic Vlasov-Poisson-Landau (rVPL) system with the physical Coulombic interaction. In this paper, we first study the spectrum structure on the linearized rVPL system and obtain the optimal time decay rates of the solutions to the linearized system, and then we construct global strong solutions to the nonlinear system around a global relativistic Maxwellian. Finally we make use of time decay rates of the solutions to the linearized system and uniform energy estimates to establish the time decay of the global solution to the original Cauchy problem for the rVPL system to the absolute Maxwellian at the optimal convergence rate $(1+t)^{-3/4}$. This time rate is faster than the optimal rate $(1+t)^{-1/4}$ of classical Vlasov-Poisson-Boltzmann [2,10] and Vlasov-Poisson-Landau system [7,8,17] and this fast time decay rate is caused by the combined influence of relativistic effect and electrostatic potential force.

Keywords:

Relativistic Vlasov-Poisson-Landau system,
spectrum analysis,
optimal time decay rate.

Mathematics Subject Classification: 76P05, 82C40, 82D05.

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