Vanishing shear viscosity limit for the compressible planar MHD system with boundary layer

Abstract

This paper is devoted to the study of the vanishing shear viscosity limit and strong boundary layer problem for the compressible, viscous, and heat-conducting planar MHD equations. The main aim is to obtain a sharp convergence rate which is usually connected to the boundary layer thickness. However, The convergence rate would be possibly slowed down due to the presence of the strong boundary layer effect and the interactions among the magnetic field, temperature, and fluids through not only the velocity equations but also the strongly nonlinear terms in the temperature equation. Our main strategy is to construct some new functions via asymptotic matching method which can cancel some quantities decaying in a lower speed. It leads to a sharp $L^{\infty }$ convergence rate as the shear viscosity vanishes for global-in-time solution with arbitrarily large initial data.