Weak-Strong Uniqueness and Extreme Wall Events at High Reynolds Number

Abstract

Singular or weak solutions of the incompressible Euler equations have been hypothesized to account for anomalous dissipation at very high Reynolds numbers and, in particular, to explain the d'Alembert paradox of non-vanishing drag. A possible objection to this explanation is the mathematical property called ``weak-strong uniqueness'', which requires that any admissable weak solution of the Euler equations must coincide with the smooth Euler solution for the same initial data. As an application of the Josephson-Anderson relation, we sketch a proof of conditional weak-strong uniqueness for the potential Euler solution of d'Alembert within the class of strong inviscid limits. We suggest that the mild conditions required for weak-strong uniqueness are, in fact, physically violated by violent eruption of very thin boundary layers. We discuss observational signatures of these extreme events and explain how the small length-scales involved could threaten the validity of a hydrodynamic description.