Global Solutions of the Two-Dimensional Riemann Problem with Four-Shock Interactions for the Euler Equations for Potential Flow

TL;DR

This paper rigorously constructs global solutions for the 2-D Riemann problem with four-shock interactions in potential flow Euler equations, introducing critical angles to classify configurations and reformulating the problem as a free boundary problem.

Contribution

It introduces a novel framework with critical angles and reformulates the problem as a free boundary problem, providing the first rigorous solution for four-shock interactions in 2-D Euler equations.

Findings

Identification of three critical angles determining shock configurations

Proof of strict monotonicity of detachment and sonic angles

First rigorous solution for 2-D Riemann problem with four-shock interactions

Abstract

We present a rigorous approach and related techniques to construct global solutions of the 2-D Riemann problem with four-shock interactions for the Euler equations for potential flow. With the introduction of three critical angles: the vacuum critical angle from the compatibility conditions, the detachment angle, and the sonic angle, we clarify all configurations of the Riemann solutions for the interactions of two-forward and two-backward shocks, including the subsonic-subsonic reflection configuration that has not emerged in previous results. To achieve this, we first identify the three critical angles that determine the configurations, whose existence and uniqueness follow from our rigorous proof of the strict monotonicity of the steady detachment and sonic angles for 2-D steady potential flow with respect to the Mach number of the upstream state. Then we reformulate the 2-D Riemann problem into the shock reflection-diffraction problem with respect to a symmetric line, along with two independent incident angles and two sonic boundaries varying with the choice of incident angles. With these, the problem can be further reformulated as a free boundary problem for a second-order quasilinear equation of mixed elliptic-hyperbolic type. The difficulties arise from the degenerate ellipticity of the nonlinear equation near the sonic boundaries, the nonlinearity of the free boundary condition, the singularity of the solution near the corners of the domain, and the geometric properties of the free boundary. To the best of our knowledge, this is the first rigorous result for the 2-D Riemann problem with four-shock interactions for the Euler equations. The approach and techniques developed for the Riemann problem for four-wave interactions should be useful for solving other 2-D Riemann problems for more general Euler equations and related nonlinear hyperbolic systems of conservation laws.