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 of the flow is composed of a uniform incoming flow and a vacuum. The flow satisfies the slip condition on the wall of the sharp corner. By a self-similar transformation, the 2D Riemann IBVP can be changed into a boundary value problem (BVP) for the 2D self-similar Euler system. Existence of global piecewise smooth (or Lipshitz continuous) solution to the BVP is established. 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 characteristic method to construct a global solution, we establish an a priori estimate for the hyperboliciy of the system. The other main difficulty is that when the incoming 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 first 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 be also used to construct continuous solutions of some other degenerate hyperbolic boundary value problems and sonic-supersonic flow problems. This is a joint work with Prof. Wancheng Sheng.