Global strong solutions to magnetohydrodynamics with density-dependent viscosity and degenerate heat-conductivity**Partially supported by NNSFC 11671027 and 11471321.

Abstract

We deal with the equations of a planar magnetohydrodynamic compressible flow with the viscosity depending on the specific volume of the gas and the heat conductivity proportional to a positive power of the temperature. Under the same conditions on the initial data as those of the constant viscosity and heat conductivity case (Kazhikhov 1987 Boundary Value Problems for Equations of Mathematical Physics (Krasnoyarsk)), we obtain the global existence and uniqueness of strong solutions which means no shock wave, vacuum, or mass or heat concentration will be developed in finite time, although the motion of the flow has large oscillations and the interaction between the hydrodynamic and magnetodynamic effects is complex. Our result can be regarded as a natural generalization of the Kazhikhov’s theory for the constant viscosity and heat conductivity case to that of nonlinear viscosity and degenerate heat-conductivity.