Uniform Anisotropic Regularity and Low Mach Number Limit of Non-isentropic Ideal MHD Equations with a Perfectly Conducting Boundary

TL;DR

This paper establishes the uniform anisotropic regularity and low Mach number limit for non-isentropic ideal MHD equations with perfectly conducting boundary, overcoming key analytical challenges posed by entropy and magnetic field interactions.

Contribution

It introduces novel anisotropic Sobolev space estimates and leverages Alinhac good unknowns to handle the complex structure of non-isentropic MHD equations.

Findings

Proved uniform estimates in anisotropic Sobolev spaces.

Established the low Mach number limit for non-isentropic ideal MHD.

Identified enhanced regularity of entropy along magnetic fields.

Abstract

We prove the low Mach number limit of non-isentropic ideal magnetohydrodynamic (MHD) equations with general initial data in the half-space whose boundary satisfies the perfectly conducting wall condition. By observing a special structure contributed by Lorentz force in vorticity analysis, we establish uniform estimates in suitable anisotropic Sobolev spaces with weights of Mach number determined by the number of material derivatives. We also observe that the entropy has the enhanced regularity in the direction of the magnetic field. These two observations help us get rid of the loss of derivatives and weights of Mach number in vorticity analysis caused by the simultaneous appearance of entropy, general initial data and the magnetic field, which is one of the major difficulties that do not appear in Euler equations or the isentropic problems. By utilizing the technique of Alinhac good unknowns, the anti-symmetric structure is preserved in the tangential estimates for the system differenetiated by high-order material derivatives.