Relative entropy and compressible potential flow

TL;DR

This paper investigates the entropy structure of compressible potential flow, showing energy as the unique entropy, and discusses implications for solution uniqueness and stability in multiple dimensions.

Contribution

It demonstrates that energy is the only nontrivial entropy for compressible potential flow and explores the implications for solution uniqueness and stability.

Findings

Energy is the only nontrivial entropy in multiple dimensions.

Energy is strictly convex when flow speed is less than sound speed.

Smooth solutions influence the convergence of entropy-compatible perturbations.

Abstract

Compressible (full) potential flow is expressed as an equivalent first-order system of conservation laws for density $\rho$ and velocity $v$. Energy $E$ is shown to be the only nontrivial entropy for that system in multiple space dimensions, and it is strictly convex in $\rho,v$ if and only if $|v|<c$. For motivation some simple variations on the relative entropy theme of Dafermos/DiPerna are given, for example that smooth regions of weak entropy solutions shrink at finite speed, and that smooth solutions force solutions of singular entropy-compatible perturbations to converge to them. We conjecture that entropy weak solutions of compressible potential flow are unique, in contrast to the known counterexamples for the Euler equations.