Vanishing shear viscosity limit and boundary layer for the one-dimensional full compressible MHD equations with large data

TL;DR

This paper investigates the limit of vanishing shear viscosity in one-dimensional compressible MHD equations with large data, establishing convergence rates, boundary layer behavior, and global well-posedness of strong solutions.

Contribution

It provides the first rigorous justification of the vanishing shear viscosity limit for large data in 1D compressible MHD equations, including boundary layer analysis and new density bounds.

Findings

Convergence rates for shear viscosity limit are obtained.

Boundary-layer thickness and solutions are characterized.

Global well-posedness of strong solutions with large data is established.

Abstract

This paper is concerned with an initial and boundary value problem of the one-dimensional planar MHD equations for viscous, heat-conducting, compressible, ideal polytropic fluids with constant transport coefficients and large data. The vanishing shear viscosity limit is justified and the convergence rates are obtained. To capture the behavior of the solutions at small shear viscosity, we also discuss the boundary-layer thickness and the boundary-layer solution. As by-products, the global well-posedness of strong solutions with large data is established. The proofs are based on the global (uniform) estimates which are achieved by making a full use of the "effective viscous flux", the material derivatives and the structure of the one-dimensional equations. Moreover, the lower positive bound of the density is obtained by using some new ideas, which are rather different from those in the existing literature.