In the case of favorable pressure gradient, Oleinik obtained the global-in-x solutions to the steady Prandtl equations with low regularity (see Oleinik and Samokhin [9], P.21, Theorem 2.1.1). Due to the degeneracy of the equation near the boundary, the question of higher regularity of Oleinik's solutions remains open. See the local-in-x higher regularity established by Guo and Iyer [5]. In this paper, we prove that Oleinik's solutions are smooth up to the boundary $y=0$ for any $x>0$, using further maximum principle techniques. Moreover, since Oleinik only assumed low regularity on the data prescribed at $x=0$, our result implies instant smoothness (in the steady case, $x=0$ is often considered as initial time).

在有利的压力梯度的情况下，Oleinik 获得了具有低规律性的稳态 Prandtl 方程的global-in-x解（参见 Oleinik 和 Samokhin [9]，P.21，定理 2.1.1）。由于边界附近方程的简并性，Oleinik 解的更高正则性问题仍然悬而未决。参见由郭和艾尔 [5] 建立的local-in-x更高的规律。在本文中，我们证明了 Oleinik 的解在边界上是平滑的$是=0$ 对于任何 $X>0$，使用进一步的最大原理技术。此外，由于 Oleinik 仅假设在$X=0$，我们的结果意味着即时平滑（在稳定的情况下， $X=0$ 通常被认为是初始时间）。