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

TL;DR

This paper investigates the behavior of solutions to one-dimensional compressible MHD equations as shear viscosity vanishes, establishing convergence, boundary layer characteristics, and global well-posedness for large initial data.

Contribution

It provides the first rigorous justification of the vanishing shear viscosity limit and analyzes boundary layer behavior for large data in 1D compressible MHD.

Findings

Convergence rates for shear viscosity limit are obtained.

Boundary-layer thickness and solutions are characterized.

Global well-posedness for 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. More important, to capture the behavior of the solutions at vanishing shear viscosity, both the boundary-layer thickness and the boundary-layer solution are discussed. 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.