Existence of solutions to gas expansion problem through a sharp corner for 2-D Euler equations with general equation of state

In this article, we study the gas expansion problem by turning a sharp corner into a vacuum for the two-dimensional (2-D) pseudosteady compressible Euler equations with a convex equation of state. This problem can be considered as the interaction of a centered simple wave with a planar rarefaction wave. To obtain the global existence of a solution up to the vacuum boundary of the corresponding 2-D Riemann problem, we consider several Goursat-type boundary value problems for 2-D self-similar Euler equations and use the ideas of characteristic decomposition and bootstrap method. Further, we formulate 2-D-modified shallow water equations newly and solve a dam-break-type problem for them as an application of this work. Moreover, we also recover the results from the available literature for certain equations of states that provide a check that the results obtained in this article are actually correct.