The proposal of this paper is to study the effect of magnetic field in stabilizing the hydrodynamic flow and preventing the occurrence of back flow in the Prandtl–Hartmann boundary layer. When both of the initial tangential velocity and upstream velocity are strictly increasing in the normal variable of the boundary, and the normal component of the magnetic field at the boundary is fairly strong, we obtain the global existence of a weak solution to the mixed Prandtl–Hartmann boundary layer problem, even for certain adverse pressure gradient, that may lead to the occurrence of a back-flow point for the classical Prandtl equation. This indicates that the magnetic field has certain stabilization effect on the hydrodynamic flow. On the other hand, under the monotonicity condition, we obtain that a first back-flow point, when the boundary layer evolves in time, should appear on the boundary if it occurs, notably the pressure gradient of the outer flow is not necessarily adverse. Moreover, when the adverse pressure gradient of the outer flow dominates the orthogonal magnetic field, and the initial velocity satisfies certain growth condition, we obtain the existence of a back-flow point of the Prandtl–Hartmann boundary layer.

本文的建议是研究磁场在稳定流体动力流和防止在Prandtl-Hartmann边界层中发生回流的作用。当边界的法向变量中的初始切线速度和上游速度都严格增加，并且边界处的磁场的法向分量相当强时，我们得到了混合普朗特尔–的弱解的整体存在性。即使对于某些不利的压力梯度，Hartmann边界层问题也可能导致经典Prandtl方程出现回流点。这表明磁场对流体动力流动具有一定的稳定作用。另一方面，在单调条件下，我们得到一个第一回流点，当边界层随时间变化时，如果边界层出现，应出现在边界上，特别是外流的压力梯度不一定不利。此外，当外部流的不利压力梯度主导正交磁场，并且初始速度满足某些增长条件时，我们获得了Prandtl–Hartmann边界层的回流点的存在。