Incompressible limits at large Mach number for a reduced compressible MHD system

TL;DR

This paper investigates the incompressible limit of a reduced compressible MHD model at large Mach numbers, using compensated compactness to identify asymptotics and characterize the limiting dynamics.

Contribution

It introduces a novel analysis of the incompressible limit for a reduced MHD system, including the effects of a strong Coriolis term, with detailed asymptotic characterization.

Findings

Established the incompressible limit in the presence of ill-prepared data.

Identified the asymptotics of order O(ε) terms and their dynamics.

Extended analysis to include strong Coriolis effects.

Abstract

This paper studies a singular limit problem for a reduced model for compressible non-resistive MHD which was first introduced in \cite{Li-Sun_JDE, Li-Sun} in a two-dimensional setting. This system can also be related to a certain class of two-fluid models. By a suitable rescaling of the magnetic pressure in terms of some parameter $\varepsilon>0$, by letting $\varepsilon\to 0$ we perform the incompressible limit while keeping the Mach number of order $O(1)$. The study is conducted in the framework of global in time finite energy weak solutions and for ill-prepared initial data. We also consider a similar problem in presence of a strong Coriolis term. The key ingredient of the proof, based on a compensated compactness argument, is the use of the transport equation (well-known in the context of two-fluid models) underlying the dynamics. Thanks to it, and differently from previous studies about the incompressible limit, we are able to identify the asymptotics of the terms of order $O(\varepsilon)$ and to characterise their dynamics; such an information is in fact crucial to obtain a closed system in the limit.