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We consider the two-dimensional MHD Boundary layer system withouthydrodynamic viscosity, and establish the existence and uniqueness of solutionsin Sobolev spaces under the assumption that the tangential component ofmagnetic fields dominates. This gives a complement to the previous works ofLiu-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 forthe MHD boundary layer systems with both viscosity and resistivity and withviscosity only, respectively. We use the pseudo-differential calculation, toovercome a new difficulty arising from the treatment of boundary integrals dueto the absence of the diffusion property for the velocity.

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