Vanishing viscosity limit for compressible magnetohydrodynamics equations with transverse background magnetic field

TL;DR

This paper establishes uniform regularity and justifies the vanishing viscosity limit for 2D viscous compressible MHD equations with a transverse magnetic field, showing the magnetic field prevents boundary layer formation.

Contribution

It provides the first uniform regularity estimates and vanishing viscosity limit results for compressible MHD with a transverse magnetic field under specific boundary conditions.

Findings

Uniform regularity estimates in conormal Sobolev spaces.

Vanishing viscosity limit justified in $L^ abla$ sense.

Transverse magnetic field prevents boundary layer formation.

Abstract

We are concerned with the uniform regularity estimates and vanishing viscosity limit of solution to two dimensional viscous compressible magnetohydrodynamics (MHD) equations with transverse background magnetic field. When the magnetic field is assumed to be transverse to the boundary and the tangential component of magnetic field satisfies zero Neumann boundary condition, even though the velocity is imposed the no-slip boundary condition, the uniform regularity estimates of solution and its derivatives still can be achieved in suitable conormal Sobolev spaces in the half plane $\mathbb{R}^2_+$, and then the vanishing viscosity limit is justified in $L^\infty$ sense based on these uniform regularity estimates and some compactness arguments. At the same time, together with \cite{CLX21}, our results show that the transverse background magnetic field can prevent the strong boundary layer from occurring for compressible magnetohydrodynamics whether there is magnetic diffusion or not.