In this paper, we establish the mathematical validity of the Prandtl boundary layer theory for a class of nonlinear plane parallel flows of viscous incompressible magnetohydrodynamic flow with the no-slip boundary condition of velocity and perfectly conducting walls for magnetic fields. The convergence is shown under various Sobolev norms, including the physically important space–time uniform norm L^∞(H¹). In addition, similar convergence results are also obtained under the case with uniform magnetic fields. This implies the stabilizing effects of magnetic fields. Besides, the higher-order expansion is also considered.

中文翻译：

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3D平面平行MHD流的边界层扩展的稳定性

在本文中，我们建立了Prandtl边界层理论的数学有效性，该理论对于一类具有速度的无滑移边界条件和磁场的理想传导壁的粘性不可压缩磁流体动力流的非线性平面平行流。的收敛性在各种的Sobolev规范中所示，包括物理上重要的空间-时间均匀规范大号^∞（ħ ¹）。此外，在磁场均匀的情况下，也可以获得类似的收敛结果。这暗示了磁场的稳定作用。此外，还考虑了高阶展开。