Global Solutions of a Two-Dimensional Riemann Problem for the Pressure Gradient System

TL;DR

This paper solves a complex 2D Riemann problem for the pressure gradient system by reformulating it as a free boundary problem and establishing global existence and regularity of the solution, including the diffracted shock.

Contribution

It introduces a novel approach to solving a 2D Riemann problem via free boundary formulation and proves global existence and regularity of the solution.

Findings

Global existence of the free boundary solution.

Optimal regularity of the diffracted shock and solution.

Solution contains two vortex sheets and a global 2D shock.

Abstract

We are concerned with a two-dimensional ($2$-D) Riemann problem for compressible flows modeled by the pressure gradient system that is a $2$-D hyperbolic system of conservation laws. The Riemann initial data consist of four constant states in four sectorial regions such that two shock waves and two vortex sheets are generated between the adjacent states. This Riemann problem can be reduced to a boundary value problem in the self-similar coordinates with the Riemann initial data as its asymptotic boundary data, along with two sonic circles determined by the Riemann initial data, for a nonlinear system of mixed-composite type. The solutions keep the four constant states and four planar waves outside the outer sonic circle. The two shocks keep planar until they meet the outer sonic circle at two different points and then generate a diffracted shock to be expected to connect these two points, whose exact location is {\it apriori} unknown which is regarded as a free boundary. Then the $2$-D Riemann problem can be reformulated as a free boundary problem, in which the diffracted transonic shock is the one-phase free boundary to connect the two points, while the other part of the outer sonic circle forms the part of the fixed boundary of the problem. We establish the global existence of a solution of the free boundary problem, as well as the $C^{0,1}$--regularity of both the diffracted shock across the two points and the solution across the outer sonic boundary which is optimal. One of the key observations here is that the diffracted transonic shock can not intersect with the inner sonic circle in the self-similar coordinates. As a result, this $2$-D Riemann problem is solved globally, whose solution contains two vortex sheets and one global $2$-D shock connecting the two original shocks generated by the Riemann data.