We consider two-dimensional (2D) pseudosteady flows around a sharp corner. This problem can be seen as a 2D Riemann initial and boundary value problem (IBVP) for the compressible Euler system. The initial state is a combination of a uniform flow in one quadrant and vacuum in the remaining domain. The boundary condition on the wall of the sharp corner is a slip boundary condition. By a self-similar transformation, the 2D Riemann IBVP is converted into a boundary value problem (BVP) for the 2D self-similar Euler system. Existence of global piecewise smooth (or Lipshitz-continuous) solutions to the BVP are obtained. One of the main difficulties for the global existence is that the type of the 2D self-similar Euler system is a priori unknown. In order to use the method of characteristic analysis, we establish some a priori estimates for the hyperboliciy of the system. The other main difficulty is that when the uniform flow is sonic or subsonic, the hyperbolic system becomes degenerate at the origin. Moreover, there is a multi-valued singularity at the origin. To solve this degenerate hyperbolic boundary value problem, we establish some uniform interior \(C^{0, 1}\) norm estimates for the solutions of a sequence of regularized hyperbolic boundary value problems, and then use the Arzela–Ascoli theorem and a standard diagonal procedure to construct a global Lipschitz continuous solution. The method used here may also be used to construct continuous solutions of some other degenerate hyperbolic boundary value problems and sonic-supersonic flow problems.

我们考虑围绕尖角的二维 (2D) 伪稳态流动。这个问题可以看作是可压缩欧拉系统的二维黎曼初边值问题 (IBVP)。初始状态是一个象限中的均匀流动和剩余域中的真空的组合。尖角壁上的边界条件为滑移边界条件。通过自相似变换，将二维黎曼 IBVP 转换为二维自相似欧拉系统的边界值问题 (BVP)。获得了 BVP 的全局分段平滑（或 Lipshitz 连续）解的存在性。全局存在的主要困难之一是二维自相似欧拉系统的类型是先验未知的。为了使用特征分析的方法，我们为系统的双曲线建立了一些先验估计。另一个主要困难是，当均匀流是音速或亚音速时，双曲系统在原点退化。此外，原点处存在多值奇点。为了解决这个退化的双曲边值问题，我们建立了一些统一的内部\(C^{0, 1}\)对一系列正则化双曲边值问题的解进行范数估计，然后使用 Arzela-Ascoli 定理和标准对角过程构建全局 Lipschitz 连续解。这里使用的方法也可以用来构造一些其他退化双曲边值问题和音速-超音速流动问题的连续解。