Prandtl-Meyer Reflection for Supersonic Flow past a Solid Ramp

TL;DR

This paper advances the mathematical understanding of steady supersonic weak shock solutions in potential flow past a solid ramp, establishing their stability as long-time limits of unsteady flows for all physical parameters.

Contribution

It removes previous assumptions and proves the stability of weak shock solutions as long-time asymptotics for all physical parameters in potential flow.

Findings

Established stability theorem for weak shock solutions.

Developed new mathematical techniques for potential flow analysis.

Extended applicability to all physical parameters.

Abstract

We present our recent results on the Prandtl-Meyer reflection for supersonic potential flow past a solid ramp. When a steady supersonic flow passes a solid ramp, there are two possible configurations: the weak shock solution and the strong shock solution. Elling-Liu's theorem (2008) indicates that the steady supersonic weak shock solution can be regarded as a long-time asymptotics of an unsteady flow for a class of physical parameters determined by certain assumptions for potential flow. In this paper we discuss our recent progress in removing these assumptions and establishing the stability theorem for steady supersonic weak shock solutions as the long-time asymptotics of unsteady flows for all the physical parameters for potential flow. We apply new mathematical techniques developed in our recent work to obtain monotonicity properties and uniform apriori estimates for weak solutions, which allow us to employ the Leray-Schauder degree argument to complete the theory for the general case.