Incompressible limits at large Mach number for a reduced compressible MHD system
Francesco Fanelli, Young-Sam Kwon, Aneta Wróblewska-Kamińska
TL;DR
The paper establishes a rigorous incompressible limit for a reduced compressible MHD system with a large magnetic pressure term, preserving a fixed Mach number. A novel scaling penalises magnetic pressure while leveraging a transport structure for the ratio Z=b/ρ to control density fluctuations and identify O(ε) dynamics via an α-β framework. Using compensated compactness and a detailed wave analysis (including acoustic and acoustic–Poincaré waves under rotation), the authors prove that, without rotation (κ=0), the limit is the 3D incompressible Navier–Stokes system, whereas with strong Coriolis force (κ=1) the limit is a coupled quasi-geostrophic-type system for α and β with horizontal velocity linked to β. The results advance the understanding of singular limits in two-fluid–like MHD models, provide explicit limiting dynamics, and illuminate how fast rotation interacts with large magnetic forces in complex fluid systems.
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.