A note on the ill-posedness of shear flow for the MHD boundary layer equations

For the two-dimensional Magnetohydrodynamics (MHD) boundary layer system, it has been shown that the non-degenerate tangential magneticfield leads to the well-posedness in Sobolev spaces and high Reynolds number limits without any monotonicity condition on the velocityfield in our previous works. This paper aims to show that suffcient degeneracy in the tangential magneticfield at a non-degenerate critical point of the tangential velocityfield of shear flow indeed yields instability as for the classical Prandtl equations without magneticfield studied by Gérard-Varet and Dormy (2010). This partially shows the necessity of the non-degeneracy in the tangential magneticfield for the stability of the boundary layer of MHD in 2D at least in Sobolev spaces.