Local-in-time well-posedness theory for the inhomogeneous MHD boundary layer equations without resistivity in lower regular Sobolev space

Abstract

In this paper, we consider the well-posedness theory in lower regular Sobolev space for the Prandtl-type equation arising from the inhomogeneous incompressible MHD equation in dimension two. Due to the degeneracy of horizontal dissipation, the well-posedness result of MHD boundary layer equation is established in higher regular Sobolev space rather than lower one generally. By using the anisotropic Sobolev inequality and characteristic line method, the well-posedness result of inhomogenous MHD Prandtl-type equation can be established under the H2−framework.