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 shear layer is degenerate at one point, then the linearized MHD boundary layer system around the shear layer profile is ill-posed in the Sobolev settings provided that the initial velocity shear flow is non-degenerately critical at the same point.

中文翻译：

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磁效应对 Sobolev 空间中无电阻率的 2D MHD 边界层方程的可解性的影响

在本文中，我们关注的是磁效应对无电阻率的二维不可压缩 MHD 系统边界层方程的 Sobolev 可解性的影响。MHD 边界层由 Prandtl 型方程描述，该方程源自在速度上的无滑移边界条件下无电阻率的不可压缩粘性 MHD 系统。假设初始切向磁场不退化，则证明了 Sobolev 空间中的局部时间适定性，而没有速度场的单调性条件。此外，我们表明，如果切向磁场剪切层在某一点退化，那么在 Sobolev 设置中，剪切层轮廓周围的线性化 MHD 边界层系统是不适定的，前提是初始速度剪切流是非退化临界的在同一点。