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

TL;DR

This paper explores the mathematical property of weak-strong uniqueness in the Euler equations, its implications for high Reynolds number flows, and how extreme boundary layer events may challenge classical hydrodynamic theories.

Contribution

It provides a conditional proof of weak-strong uniqueness for the Euler equations and discusses physical scenarios where this property may be violated due to extreme boundary layer phenomena.

Findings

Weak-strong uniqueness may not hold during violent boundary layer eruptions.

Extreme events could invalidate the classical hydrodynamic description.

Observational signatures of boundary layer eruptions are discussed.

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.