Communications in Mathematical Sciences
Volume 13 (2015)
Cauchy problem of the magnetohydrodynamic Burgers system
Pages: 127 – 151
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)].
MHD Burgers system, rarefaction waves, viscous shock waves, nonlinear stability, weighted energy estimates
2010 Mathematics Subject Classification
35A18, 35B35, 35C06, 35C07, 35K45