Well-posedness in Sobolev spaces of the two-dimensional MHD Boundary layer equations without viscosity

Abstract

We consider the two-dimensional MHD Boundary layer system without hydrodynamic viscosity, and establish the existence and uniqueness of solutions in Sobolev spaces under the assumption that the tangential component of magnetic fields dominates. This gives a complement to the previous works of Liu-Xie-Yang [Comm. Pure Appl. Math. 72 (2019)] and Liu-Wang-Xie-Yang [J. Funct. Anal. 279 (2020)], where the well-posedness theory was established for the MHD boundary layer systems with both viscosity and resistivity and with viscosity only, respectively. We use the pseudo-differential calculation, to overcome a new difficulty arising from the treatment of boundary integrals due to the absence of the diffusion property for the velocity.