Low Mach number limit for non-isentropic magnetohydrodynamic equations with ill-prepared data and zero magnetic diffusivity in bounded domains

TL;DR

This paper rigorously justifies the low Mach number limit for non-isentropic magnetohydrodynamic equations with zero magnetic diffusivity in bounded domains, using weighted energy estimates and decomposition techniques.

Contribution

It introduces a new weighted energy functional and a decomposition method to handle ill-prepared data and boundary conditions in the low Mach number limit analysis.

Findings

Established uniform estimates with respect to Mach number.

Proved strong convergence of density and temperature.

Demonstrated the low Mach number limit in bounded domains.

Abstract

In this article, we verify the low Mach number limit of strong solutions to the non-isentropic compressible magnetohydrodynamic equations with zero magnetic diffusivity and ill-prepared initial data in three-dimensional bounded domains, when the density and the temperature vary around constant states. Invoking a new weighted energy functional, we establish the uniform estimates with respect to the Mach number, especially for the spatial derivatives of high order. Due to the vorticity-slip boundary condition of the velocity, we decompose the uniform estimates into the part for the fast variables and the other one for the slow variables. In particular, the weighted estimates of highest-order spatial derivatives of the fast variables are crucial for the uniform bounds. Finally, the low Mach number limit is justified by the strong convergence of the density and the temperature, the divergence-free component of the velocity, and the weak convergence of other variables. The methods in this paper can be applied to singular limits of general hydrodynamic equations of hyperbolic-parabolic type, including the full Navier-Stokes equations.