Regularity of Solutions to Regular Shock Reflection for Potential Flow

TL;DR

This paper investigates the regularity of solutions to regular shock reflection in potential flow, establishing optimal regularity results and analyzing the behavior near the pseudo-sonic circle and reflected shock.

Contribution

It proves that the $C^{1,1}$ regularity is optimal across the pseudo-sonic circle and at the intersection with the reflected shock, and shows $C^{2,eta}$ regularity up to the pseudo-sonic circle.

Findings

$C^{1,1}$ regularity is optimal at key intersection points

Solutions are $C^{2,eta}$ up to the pseudo-sonic circle in the pseudo-subsonic region

Analysis of transonic flow transitions across the pseudo-sonic circle and shock

Abstract

The shock reflection problem is one of the most important problems in mathematical fluid dynamics, since this problem not only arises in many important physical situations but also is fundamental for the theory of multidimensional conservation laws. However, most of the fundamental issues for shock reflection have not been understood. Therefore, it is important to establish the regularity of solutions to shock reflection in order to understand fully the phenomena of shock reflection. On the other hand, for a regular reflection configuration, the potential flow governs the exact behavior of the solution in $C^{1,1}$ across the pseudo-sonic circle even starting from the full Euler flow, that is, both of the nonlinear systems are actually the same in an physically significant region near the pseudo-sonic circle; thus, it becomes essential to understand the optimal regularity of solutions for the potential flow across the pseudo-sonic circle and at the point where the pseudo-sonic circle meets the reflected shock. In this paper, we study the regularity of solutions to regular shock reflection for potential flow. In particular, we prove that the $C^{1,1}$-regularity is optimal for the solution across the pseudo-sonic circle and at the point where the pseudo-sonic circle meets the reflected shock. We also obtain the $C^{2,\alpha}$ regularity of the solution up to the pseudo-sonic circle in the pseudo-subsonic region. The problem involves two types of transonic flow: one is a continuous transition through the pseudo-sonic circle from the pseudo-supersonic region to the pseudo-subsonic region; the other a jump transition through the transonic shock as a free boundary from another pseudo-supersonic region to the pseudo-subsonic region.