Magnetic effects on the solvability of 2D incompressible magneto-micropolar boundary layer equations without resistivity in Sobolev spaces

Abstract

In this paper, we consider the magnetic effect on the Sobolev solvability of the two-dimensional incompressible magneto-micropolar boundary layer system without resistivity. This gives a complement to the previous work of Lin et al. (2022), where the well-posedness and the convergence theory were established for the magneto-micropolar boundary layer system without monotonicity in Sobolev spaces. If the initial tangential magnetic field is not degenerate, a local-in-time well-posedness theory in Sobolev spaces is established without the monotonic condition on the velocity or the micro-rotational velocity. Moreover, when the tangential magnetic field of shear layer degenerates at the non-degenerate critical point of the initial velocity and the initial micro-rotational velocity, the linearized magneto-micropolar boundary layer system around a shear flow with general decay is ill-posed in Sobolev spaces.