Global Strong Solutions to Magnetohydrodynamics with Density-Dependent Viscosity and Degenerate Heat-Conductivity

TL;DR

This paper proves the global existence and uniqueness of strong solutions for a magnetohydrodynamic flow with density-dependent viscosity and temperature-dependent heat conductivity, extending previous constant-property models.

Contribution

It generalizes Kazhikhov's theory to nonlinear viscosity and degenerate heat-conductivity in magnetohydrodynamics, ensuring no shock, vacuum, or concentration develop.

Findings

Global strong solutions exist and are unique.

No shock waves, vacuum, or concentration form in finite time.

Results extend classical models to nonlinear, degenerate cases.

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)], 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.