The Morawetz Problem for Supersonic Flow with Cavitation

TL;DR

This paper proves the existence and compactness of entropy solutions for the 2D steady compressible Euler equations around obstacles in supersonic flow, addressing cavitation and hyperbolicity loss.

Contribution

It provides the first complete existence theorem for the Morawetz problem using a novel entropy analysis combined with vanishing viscosity and compensated compactness.

Findings

Established existence of entropy solutions in supersonic cavitating flow.

Proved compactness and weak continuity of solutions.

Described entropy pairs via Loewner--Morawetz relations.

Abstract

We are concerned with the existence and compactness of entropy solutions of the compressible Euler system for two-dimensional steady potential flow around an obstacle for a polytropic gas with supersonic far-field velocity. The existence problem, initially posed by Morawetz \cite{morawetz85} in 1985, has remained open since then. In this paper, we establish the first complete existence theorem for the Morawetz problem by developing a new entropy analysis, coupled with a vanishing viscosity method and compensated compactness ideas. The main challenge arises when the flow approaches cavitation, leading to a loss of strict hyperbolicity of the system and a singularity of the entropy equation, particularly for the case of adiabatic exponent $\gamma=3$. Our analysis provides a complete description of the entropy and entropy-flux pairs via the Loewner--Morawetz relations, which, in turn, leads to the establishment of a compensated compactness framework. As direct applications of our entropy analysis and the compensated compactness framework, we obtain the compactness of entropy solutions and the weak continuity of the compressible Euler system in the supersonic regime.