Loss of Regularity of Solutions to Shock Reflection Problems by a Non-symmetric Convex Wedge with Potential Flow Equations

TL;DR

This paper investigates shock reflection by a non-symmetric wedge using potential flow equations, demonstrating the non-existence of ideal Lipschitz solutions and indicating the presence of singularities in non-symmetric cases.

Contribution

It extends existing symmetric shock reflection theory to non-symmetric cases, developing an integral method to handle free boundary and degeneracy issues.

Findings

Ideal Lipschitz solutions do not exist in non-symmetric cases.

Potential flow solutions exhibit singularities in non-symmetric shock reflection.

Developed new estimates near wedge corners for non-symmetric configurations.

Abstract

In this paper, we study the problem of shock reflection by a wedge, with the potential flow equation, which is a simplification of the Euler System. In the work of M. Feldman and G. Chen, the existence theory of shock reflection problems with the potential flow equation was established, when the wedge is symmetric w.r.t. the direction of the upstream flow. As a natural extension, we study non-symmetric cases, i.e. when the direction of the upstream flow forms a nonzero angle with the symmetry axis of the wedge. The main idea of investigating the existence of solutions to non-symmetric problems is to study the symmetry of the solution. Then difficulties arise such as free boundaries and degenerate ellipticity, especially when ellipticity degenerates on the free boundary. We developed an integral method to overcome these difficulties. Some estimates near the corner of wedge is also established, as an extension of G.Lieberman's work. We proved that in non-symmetric cases, the ideal Lipschitz solution to the potential flow equation, which we call regular solution, does not exist. This suggests that the potential flow solutions to the non-symmetric shock reflection problem, should have some singularity which is not encountered in symmetric case.