Vanishing Shear Viscosity Limit and Boundary Layer Study on the Planar MHD system

TL;DR

This paper investigates the vanishing shear viscosity limit and boundary layer behavior in the planar MHD system with temperature-dependent heat conductivity, establishing global solutions and convergence rates.

Contribution

It proves the global existence of strong solutions for large initial data and justifies the vanishing shear viscosity limit with explicit convergence rates.

Findings

Global existence of strong solutions for large initial data

Justification of the vanishing shear viscosity limit

Estimation of boundary layer thickness

Abstract

We consider an initial boundary problem for the planar MHD system under the general condition on the heat conductivity $\kappa$ that may depend on both the density $\rho$ and the temperature $\theta$ satisfying $\kappa(\rho,\theta)\geq\kappa_1 \theta^{q}$ for some constants $\kappa_1>0$ and $q>0.$ Firstly, the global existence of strong solution for large initial data is obtained, and then the limit of the vanishing shear viscosity is justified. In addition, the $L^2$ convergence rate is obtained together with the estimation on the thickness of the boundary layer.