On the Global Strong Solutions to Magnetohydrodynamics with Density-Dependent Viscosity and Degenerate Heat-Conductivity in Unbounded Domains

TL;DR

This paper proves the global existence of unique strong solutions for a one-dimensional planar magnetohydrodynamic flow with density-dependent viscosity and degenerate heat conductivity, extending classical results to more complex, nonlinear conditions.

Contribution

It generalizes the classical compressible Navier-Stokes results to MHD flows with nonlinear viscosity and degenerate heat conductivity, ensuring no shock, vacuum, or concentration forms in finite time.

Findings

Global existence of strong solutions established

No shock, vacuum, or concentration develops in finite time

Results apply to complex MHD interactions with large oscillations

Abstract

For the equations of a planar magnetohydrodynamic (MHD) compressible flow with the viscosity depending on the specific volume of the gas and the heat conductivity being proportional to a positive power of the temperature, we obtain global existence of the unique strong solutions to the Cauchy problem or the initial-boundary-value one under natural conditions on the initial data in one-dimensional unbounded domains. Our result generalizes the classical one of the compressible Navier-Stokes system with constant viscosity and heat conductivity ([Kazhikhov. Siberian Math. J. (1982)]) to the planar MHD compressible flow with nonlinear viscosity and degenerate heat-conductivity, which means no shock wave, vacuum, or mass or heat concentration will be developed in finite time, although the interaction between the magnetodynamic effects and hydrodynamic is complex and the motion of the flow has large oscillations.