Cauchy problem of the magnetohydrodynamic Burgers system (Q2257627)

In this paper, the asymptotic nonlinear stability of solutions to the Cauchy problem of a strongly coupled Burgers system arising in magnetohydrodynamic turbulence [\textit{J. Fleischer} and \textit{P. H. Diamond}, ``Burgers' turbulence with self-consistently evolved pressure'', Phy. Rev. E (3) 61, No. 4, 3912--3925 (2000; \url{doi:10.1103/PhysRevE.61.3912}); \textit{S. Yanase}, ``New one-dimensional model equations of magnetohydrodynamic turbulence'', Phys. Plasmas 4, 1010--1017 (1997; \url{doi:10.1063/1.872190})] 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. The results confirm the existence of shock waves (or turbulence) numerically found in Fleischer and Diamond [loc. cit.] and Yanase [loc. cit.].