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

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 \begin{document}$(1+t)^{-3/4}$\end{document} . This time rate is faster than the optimal rate \begin{document}$(1+t)^{-1/4}$\end{document} 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.