Momentum regularity and stability of the relativistic Vlasov-Maxwell-Boltzmann system
type: Item
instance of: scholarly article
title: Momentum regularity and stability of the relativistic Vlasov-Maxwell-Boltzmann system
zbMATH Open document ID: 1245.35130
author: Yan Guo
author: Robert M. Strain
published in: Communications in Mathematical Physics
publication date: 2012-04-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1012.1158
review text: The authors address the problem of the existence and uniqueness of classical solutions to the fully Lorentz invariant, relativistic Vlasov-Maxwell-Boltzmann system. The two-species relativistic equations studied are \[ \begin{cases} \partial_t F_+ + c\frac{p}{p_+^0} \cdot \nabla_x F_ + + e_+\Big (E + \frac{p}{p_+^0} \times B \Big ) \cdot \nabla_p F_+ &= {\mathcal Q}^+(F) + {\mathcal Q}^{\pm}(F), \\ \partial_t F_- + c\frac{p}{p_-^0} \cdot \nabla_x F_ - - e_-\Big (E + \frac{p}{p_-^0} \times B \Big ) \cdot \nabla_p F_- &= {\mathcal Q}^-(F) + {\mathcal Q}^{\mp}(F).\end{cases}\tag{1} \] Here \({\mathcal Q}\) represents the collision operator in which \({\mathcal Q}^+(F) = {\mathcal Q}(F_+,F_+),\; {\mathcal Q}^{\pm}(F) = {\mathcal Q}(F_+,F_-)\), and similarly with \({\mathcal Q}^-(F),\; {\mathcal Q}^{\mp}(F)\). The ``hard ball'' condition is assumed throughout for the collision kernel of \({\mathcal Q}\). Equations (1) are coupled with Maxwell's equations for the internally consistent electric field, \(E(t,x)\), and magnetic field, \(B(t,x)\). The functions \(F_{\pm}(t,x,p)\) are spatially periodic number density functions for ions (+) and electrons (\(-\)) in which position is \(x=(x_1,x_2,x_3) \in {\mathcal T}^3 = [-\pi,\pi]^3\), and momentum is \(p=(p_1,p_2,p_3) \in \mathbb R^3\). The energy of a particle is given by \(p_{\pm}^0 = \sqrt{(m_{\pm}c)^2 + |p|^2}\) where \(m_{\pm}\) are the particle masses and \(c\) the speed of light. The charge is \(e_{\pm}\). A steady state solution to (1) is the global relativistic Maxwellian (a.k.a the Juttner solution) represented by \(J_{\pm}(p)\) and with \(E(t,x) = 0\) and \(B(t,x) = \overline B\), constant. The goal of this paper is to prove the existence of solutions to (1) in the context of perturbations around the steady state. The standard perturbation \(f_{\pm}(t,x,p)\) to \(J_{\pm}\) is \(F_{\pm} = J_{\pm} + \sqrt{J_{\pm}} f_{\pm}\). The relativistic Vlasov-Maxwell-Boltzmann system is then written in terms of \(f_{\pm}\). Th…
Mathematics Subject Classification ID: 35Q83
Mathematics Subject Classification ID: 35Q20
Mathematics Subject Classification ID: 35Q61
Mathematics Subject Classification ID: 82D10
Mathematics Subject Classification ID: 83A05
zbMATH DE Number: 6021956
zbMATH Keywords: Vlasov-Maxwell-Boltzmann system
zbMATH Keywords: momentum reqularity
zbMATH Keywords: global in time existence of solutions
reviewed by: Stephen Wollman
MaRDI profile type: Publication
OpenAlex ID: W3105449148
arXiv ID: 1012.1158
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DOI: 10.1007/S00220-012-1417-Z
description: scientific article; zbMATH DE number 6021956