Weak-Strong Uniqueness and the d'Alembert Paradox

TL;DR

This paper establishes conditions under which weak and strong solutions of the potential Euler equations are unique in three dimensions, linking the vanishing of certain boundary terms to the classical d'Alembert paradox.

Contribution

It proves weak-strong uniqueness for potential Euler solutions under specific boundary conditions and discusses implications for the d'Alembert paradox in inviscid flows.

Findings

Weak-strong uniqueness holds if the streamwise skin friction vanishes in the inviscid limit.

The vanishing of the skin friction component is a weaker condition than Bardos-Titi's in bounded domains.

The Drivas-Nguyen condition on normal velocity continuity implies weak-strong uniqueness.

Abstract

We prove conditional weak-strong uniqueness of the potential Euler solution for external flow around a smooth body in three space dimensions, within the class of viscosity weak solutions with the same initial data. Our sufficient condition is the vanishing of the streamwise component of the skin friction in the inviscid limit, somewhat weaker than the condition of Bardos-Titi in bounded domains. Because global-in-time existence of the smooth potential solution leads back to the d'Alembert paradox, we argue that weak-strong uniqueness is not a valid criterion for "relevant" notions of generalized Euler solution and that our condition is likely to be violated in the inviscid limit. We prove also that the Drivas-Nguyen condition on uniform continuity at the wall of the normal velocity component implies weak-strong uniqueness within the general class of admissible weak Euler solutions in bounded domains.