Incompressible Limit of Compressible Ideal MHD Flows inside a Perfectly Conducting Wall

TL;DR

This paper establishes the incompressible limit of compressible ideal MHD flows with tangential magnetic fields, overcoming well-posedness issues via anisotropic Sobolev spaces and revealing a hidden Lorentz force structure.

Contribution

It introduces a novel approach using anisotropic Sobolev spaces and uncovers a hidden Lorentz force structure to handle boundary conditions in the incompressible limit of MHD flows.

Findings

Proves uniform energy estimates in Mach number for MHD flows

Identifies a hidden structure in Lorentz force aiding analysis

Lays groundwork for free-boundary MHD problems

Abstract

We prove the incompressible limit of compressible ideal magnetohydrodynamic(MHD) flows in a reference domain where the magnetic field is tangential to the boundary. Unlike the case of transversal magnetic fields, the linearized problem of our case is not well-posed in standard Sobolev space $H^m~(m\geq 2)$, while the incompressible problem is still well-posed in $H^m$. The key observation to overcome the difficulty is a hidden structure contributed by Lorentz force in the vorticity analysis, which reveals that one should trade one normal derivative for two tangential derivatives together with a gain of Mach number weight $\varepsilon^2$. Thus, the energy functional should be defined by using suitable anisotropic Sobolev spaces. The weights of Mach number should be carefully chosen according to the number of tangential derivatives, such that the energy estimates are uniform in Mach number. Besides, part of the proof is similar to the study of compressible water waves, so our result opens the possibility to study the incompressible limit of free-boundary problems in ideal MHD.