Stabilizing effect of magnetic field on the compressible magnetohydrodynamic flow with vanishing shear viscosity

Abstract As noted by Landau and Lifshitz in their work Course of Theoretical Physics (Vol. 6, Pergamon Press, Oxford, 2nd edition, 1987), “if the relaxation time of these processes is long, a considerable dissipation of energy occurs when the fluid is compressed or expanded,” then the volume viscosity coefficient becomes relatively large, making the shear viscosity coefficient much smaller in comparison. In this article, we take this phenomenon into consideration and investigate isentropic compressible magnetohydrodynamic flow with zero shear viscosity in a periodic domain. We prove the global existence of smooth solutions when the initial data are close to a background magnetic field. In addition, stability and large-time decay rates are also obtained. Mathematically, the zero shear viscosity makes the elliptic operator “ $\mu \Delta +\nu \nabla \text{div}\, $ ” lose its uniform ellipticity, rendering the classical dissipation mechanisms for velocity inapplicable. The stabilizing effect of the background magnetic field plays a crucial role in our analysis.