Stability of the nonrelativistic  Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials

Renjun Duan; Shuangqian Liu; Tong Yang; Huijiang Zhao

doi:10.3934/krm.2013.6.159

Article Contents

2013, Volume 6, Issue 1: 159-204. Doi: 10.3934/krm.2013.6.159

This issue Previous Article Diffusion asymptotics of a kinetic model for gaseous mixtures Next Article On the Stokes approximation equations for two-dimensional compressible flows

Stability of the nonrelativistic Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials

1.
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
2.
Department of Mathematics, Jinan Unviersity, Guangdong
3.
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
4.
School of Mathematics and Statistics, Wuhan University

Received: July 31, 2012

Revised: September 30, 2012

Published: February 2013

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Abstract

Although there recently have been extensive studies on the perturbation theory of the angular non-cutoff Boltzmann equation (cf. [4] and [17]), it remains mathematically unknown when there is a self-consistent Lorentz force coupled with the Maxwell equations in the nonrelativistic approximation. In the paper, for perturbative initial data with suitable regularity and integrability, we establish the large time stability of solutions to the Cauchy problem of the Vlasov-Maxwell-Boltzmann system with physical angular non-cutoff intermolecular collisions including the inverse power law potentials, and also obtain as a byproduct the convergence rates of solutions. The proof is based on a new time-velocity weighted energy method with two key technical parts: one is to introduce the exponentially weighted estimates into the non-cutoff Boltzmann operator and the other to design a delicate temporal energy $X(t)$-norm to obtain its uniform bound. The result also extends the case of the hard sphere model considered by Guo [Invent. Math. 153(3): 593--630 (2003)] to the general collision potentials.

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Mathematics Subject Classification: Primary: 35Q20, 76P05; Secondary: 35B35, 35B40.

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