Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces

Abstract

In this paper, we are concerned with the magnetic effect on the Sobolev solvability of boundary layer equations for the 2D incompressible MHD system without resistivity. The MHD boundary layer is described by the Prandtl type equations derived from the incompressible viscous MHD system without resistivity under the no-slip boundary condition on the velocity. Assuming that the initial tangential magnetic field does not degenerate, a local-in-time well-posedness in Sobolev spaces is proved without the monotonicity condition on the velocity field. Moreover, we show that if the tangential magnetic field of shear layer is degenerate at one point, then the linearized MHD boundary layer system around the shear layer profile is ill-posed in the Gevrey function space provided that the initial velocity shear flow is non-degenerately critical at the same point.