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

TL;DR

This paper proves the global existence of solutions to a gas expansion problem involving a sharp corner turning into vacuum in 2D Euler equations with a convex equation of state, using boundary value problems and characteristic methods.

Contribution

It introduces a new approach to establish global solutions for the 2D gas expansion problem with vacuum boundary, including applications to shallow water equations.

Findings

Global existence of solutions up to vacuum boundary

Formulation and solution of a dam-break problem for shallow water equations

Validation of results against existing literature

Abstract

In this article, we study the gas expansion problem by turning a sharp corner into vacuum for the two-dimensional pseudo-steady compressible Euler equations with a convex equation of state. This problem can be considered as interaction of a centered simple wave with a planar rarefaction wave. In order to obtain the global existence of solution up to vacuum boundary of the corresponding two-dimensional 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 two-dimensional 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 equation of states which provide a check that the results obtained in this article are actually correct.