Vanishing shear viscosity and boundary layers for plane magnetohydrodynamics flows

TL;DR

This paper investigates the behavior of plane magnetohydrodynamics flows, proving global existence of solutions and analyzing boundary layer thickness as shear viscosity approaches zero, under general heat conductivity conditions.

Contribution

It establishes the global existence of strong solutions for large initial data and justifies the vanishing shear viscosity limit in magnetohydrodynamics flows.

Findings

Proves global existence of strong solutions.

Justifies the limit as shear viscosity tends to zero.

Determines boundary layer thickness as a power of viscosity.

Abstract

In this paper, we consider an initial-boundary problem for plane magnetohydrodynamics flows under the general condition on the heat conductivity $\kappa$ that may depend on both the density $\rho$ and the temperature $\theta$ and satisfies $$ \kappa(\rho,\theta)\geq\kappa_1(1+\theta^{q}) \quad \hbox{\rm with constants}~ \kappa_1>0 ~\hbox{\rm and}~ q>0. $$ We prove the global existence of strong solutions for large initial data and justify the passage to the limit as the shear viscosity $\mu$ goes to zero. Furthermore, the value $\mu^\alpha$ with any $0<\alpha<1/2$ is established for the boundary layer thickness.