Multi-dimensional Contact Discontinuities of the Ideal Compressible MHD

In this talk, we will present the local existence and uniqueness of multi-dimensional contact discontinuities for the ideal compressible magnetohydrodynamics (MHD) in Sobolev spaces, which are most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for hyperbolic systems of conservation laws. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversely, which lead to a two-phase free boundary problem where the pressure, velocity and magnetic field are continuous across the interface whereas the entropy and density may have jumps. To overcome the difficulties of possible nonlinear Rayleigh–Taylor instability and loss of derivatives, here we use crucially the Lagrangian formulation and the Cauchy formula (1882) for the magnetic field. This motivates us to define two special good unknowns; one enables us to capture the boundary regularizing effect of the transversal magnetic field on the flow map, and the other one allows us to get around the troublesome boundary integrals due to the transversality of the magnetic field. In particular, our result removes the additional assumption of the Rayleigh–Taylor sign condition required by Morando, Tarkhinin and Trebeschi (JDE 2015, ARMA 2018) and holds for both 2D and 3D and hence gives a complete answer to the two open questions raised therein. Moreover, there is no loss of derivatives in our well-posedness theory. The solution is constructed as the inviscid limit of solutions to well-chosen nonlinear approximate problems for the two-phase compressible viscous non-resistive MHD. This is a joint work with Professor Zhouping Xin (CUHK).