The Incompressible Limit of the Equations of Compressible Ideal Magneto-Hydrodynamics with perfectly conducting boundary

TL;DR

This paper proves that solutions of compressible ideal magneto-hydrodynamics equations converge to incompressible solutions as Mach number approaches zero, accounting for boundary effects with anisotropic Sobolev spaces.

Contribution

It establishes the convergence in the presence of characteristic boundaries, addressing regularity loss and boundary singularities in ideal MHD.

Findings

Solutions converge to incompressible MHD as Mach number tends to zero.

Convergence is proven in anisotropic Sobolev spaces.

Boundary effects are rigorously handled in the analysis.

Abstract

We consider the initial-boundary value problem in the halfspace for the system of equations of ideal Magneto-Hydrodynamics with a perfectly conducting wall boundary condition. We show the convergence of solutions to the solution of the equations of incompressible MHD as the Mach number goes to zero. Because of the characteristic boundary, where a loss of regularity in the normal direction to the boundary may occur, the convergence is shown in suitable anisotropic Sobolev spaces which take account of the singular behavior at the boundary.