Incompressible limit of compressible ideal MHD flows inside a perfectly conducting wall

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 Hm(m≥2)...

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Bibliographic Details
Published in:Journal of Differential Equations 2025-04, Vol.425, p.846-894
Main Authors: Wang, Jiawei, Zhang, Junyan
Format: Article
Language:English
Subjects:
Compressible ideal MHD
Incompressible limit
Perfectly conducting wall
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Summary: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 Hm(m≥2), while the incompressible problem is still well-posed in Hm. 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 ε2. 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.
ISSN:0022-0396
DOI:10.1016/j.jde.2025.02.005