Incompressible Limit of Compressible Ideal MHD Flows Inside a Perfectly Conducting Wall

We prove the incompressible limit of compressible idealmagnetohydrodynamic(MHD) flows in a reference domain where the magnetic fieldis tangential to the boundary. Unlike the case of transversal magnetic fields,the linearized problem of our case is not well-posed in standard Sobolev spaceH^m (m≥ 2), while the incompressible problem is still well-posed in H^m.The key observation to overcome the difficulty is a hidden structurecontributed by Lorentz force in the vorticity analysis, which reveals that oneshould trade one normal derivative for two tangential derivatives together witha gain of Mach number weight ε^2. Thus, the energy functionalshould be defined by using suitable anisotropic Sobolev spaces. The weights ofMach number should be carefully chosen according to the number of tangentialderivatives, 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 offree-boundary problems in ideal MHD.