Absence of shocks in an initially dilute collisionless plasma
type: Item
instance of: scholarly article
title: Absence of shocks in an initially dilute collisionless plasma
zbMATH Open document ID: 0646.35072
DOI: 10.1007/BF01223511
author: Robert T. Glassey
author: Walter A. Strauss
published in: Communications in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: The Cauchy problem for the relativistic Vlasov-Maxwell equations \[ \partial_ tf_{\alpha}+\hat v_{\alpha}\cdot \nabla_ xf_{\alpha}+e_{\alpha}+e_{\alpha}[E+c^{-1}\hat v_{\alpha}\wedge B]\cdot \nabla_ vf_{\alpha}=0 \] \[ E_ t=c Cur B-j,\quad \nabla \cdot E=\rho;\quad B_ t=-c Curl E,\quad \nabla \cdot B=0 \] is studied in three dimensions. The authors prove: If the initial data satisfy the constraints \((\nabla \cdot E_ 0=\rho_ 0\equiv 4\pi \int_{k^ 3}\sum_{\alpha}e_{\alpha}f_{\alpha_ 0}dv\), \(\nabla \cdot B_ 0=0)\) and have compact support and sufficiently small \(C^ 2\) norm, then there exists a unique global \(C^ 1\)-solution. This is proved using the iteration method. This class of problems have been studied by the authors, Bardos and Degond [the authors, Math. Methods Appl. Sci. 9, 46- 52 (1987); \textit{C. Bardos} and \textit{P. Degond}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2, 101-118 (1985; Zbl 0593.35076)].
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 82C70
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 4054200
zbMATH Keywords: Cauchy problem
zbMATH Keywords: relativistic Vlasov-Maxwell equations
zbMATH Keywords: initial data
zbMATH Keywords: unique global \(C^ 1\)-solution
zbMATH Keywords: iteration method
reviewed by: Boling Guo
MaRDI profile type: Publication
cites work: Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data
cites work: Q3732195
cites work: Singularity formation in a collisionless plasma could occur only at high velocities
cites work: High velocity particles in a collisionless plasma
cites work: Blow-up of solutions of nonlinear wave equations in three space dimensions
description: scientific article; zbMATH DE number 4054200