Local-in-time well-posedness for Compressible MHD boundary layer

TL;DR

This paper establishes the local-in-time well-posedness of a Prandtl-type boundary layer system for the 2D non-isentropic viscous compressible MHD equations under specific boundary conditions, using a coordinate transformation and weighted Sobolev spaces.

Contribution

It extends the analysis of boundary layer systems to the compressible MHD context with new well-posedness results under non-degeneracy conditions.

Findings

Proves local-in-time well-posedness of the boundary layer system.

Uses a coordinate transformation based on stream functions.

Requires non-degeneracy of the tangential magnetic field.

Abstract

In this paper, we are concerned with the motion of electrically conducting fluid governed by the two-dimensional non-isentropic viscous compressible MHD system on the half plane, with no-slip condition for velocity field, perfect conducting condition for magnetic field and Dirichlet boundary condition for temperature on the boundary. When the viscosity, heat conductivity and magnetic diffusivity coefficients tend to zero in the same rate, there is a boundary layer that is described by a Prandtl-type system. By applying a coordinate transformation in terms of stream function as motivated by the recent work \cite{liu2016mhdboundarylayer} on the incompressible MHD system, under the non-degeneracy condition on the tangential magnetic field, we obtain the local-in-time well-posedness of the boundary layer system in weighted Sobolev spaces.