Weak-Strong Uniqueness and the d’Alembert Paradox

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 integrated over the surface in the inviscid limit, slightly stronger than the condition of Kelliher and weakening that of Bardos-Titi, both for 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.