Communications in Mathematical Sciences

Volume 13 (2015)

Number 1

Cauchy problem of the magnetohydrodynamic Burgers system

Pages: 127 – 151

DOI: https://dx.doi.org/10.4310/CMS.2015.v13.n1.a7

Authors

Hai-Yang Jin (Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong)

Zhi-An Wang (Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong)

Linjie Xiong (School of Mathematics and Statistics, Wuhan University, Wuhan, China)

Abstract

In this paper, the asymptotic nonlinear stability of solutions to the Cauchy problem of a strongly coupled Burgers system arising in magnetohydrodynamic (MHD) turbulence [Fleischer and Diamond (2000), Yanase (1997)] is established. It is shown that, as time tends to infinity, the solutions of the Cauchy problem converge to constant states or rarefaction waves with large data, or viscous shock waves with arbitrarily large amplitude, where the precise asymptotic behavior depends on the relationship between the left and right end states of the initial value. Our results confirm the existence of shock waves (or turbulence) numerically found in [Fleischer and Diamond (2000), Yanase (1997)].

Keywords

MHD Burgers system, rarefaction waves, viscous shock waves, nonlinear stability, weighted energy estimates

2010 Mathematics Subject Classification

35A18, 35B35, 35C06, 35C07, 35K45

Published 16 July 2014