Riemann boundary value problems for the Chaplygin gas outside a convex cornered wedge

TL;DR

This paper investigates Riemann boundary value problems for the Euler equations of Chaplygin gas outside a convex wedge, establishing non-existence of solutions in subsonic cases and unique solutions in supersonic cases, with implications for shock diffraction.

Contribution

It provides the first analysis of the existence and non-existence of solutions for these boundary value problems in different flow regimes, including the handling of degenerate elliptic equations in cornered domains.

Findings

No global Lipschitz solutions in subsonic flow cases.

Unique solutions exist when the flow is supersonic.

Results apply to shock diffraction problems with convex wedges.

Abstract

We consider two-dimensional Riemann boundary value problems of Euler equations for the Chaplygin gas with two piecewise constant initial data outside a convex cornered wedge. In self-similar coordinates, when the flow at the wedge corner is subsonic, this problem can be reformulated as a boundary value problem for nonlinear degenerate elliptic equations in concave domains containing a corner larger than $\pi$. It is shown that there does not exist a global Lipschitz solution for this case. We analyze the sign of the flow velocity along a certain direction, and then obtain this result by deriving a contradiction. Besides, the unique existence of the solution to the problem is established when the flow at the wedge corner is supersonic. The results obtained here are also valid for the problem of shock diffraction by a convex cornered wedge.