Relative entropy and compressible potential flow

2014/11/08 by Volker Elling, Elling, Volker
Engineering · Mathematics · #35L65 #35L67 #76L05 #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #math.AP #msc:35L65 #msc:35L67 #msc:76L05

paper · pdf · doi:10.48550/arxiv.1411.2063

openalex publication_date 2014/11/08 · arxiv created 2015/04/06 · arxiv updated 2015/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Compressible (full) potential flow is expressed as an equivalent first-order system of conservation laws for density ρ 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 ρ,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.

Citations

• Formation of singularities in one‐dimensional nonlinear wave propagation
• The second law of thermodynamics and stability
• Hyperbolic systems of conservation laws II

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