Energy Conservation for Rough Euler Flows: from Onsager's Threshold to Mixed Space-Time Criteria

Overview

The incompressible Euler equations formally conserve kinetic energy, but for weak solutions the nonlinear term may transfer energy to arbitrarily small scales and the classical cancellation is no longer justified. Onsager predicted that one third of a spatial derivative marks the threshold between rigidity and the possibility of anomalous dissipation. After reviewing the energy identity, Onsager’s conjecture, and the commutator argument of Eyink and Constantin–E–Titi, I will introduce Besov spaces through finite differences and explain why they provide the natural language for energy flux. I will then present a family of mixed space-time regularity criteria for energy conservation. I will also discuss gradient criteria, inviscid limits of Navier–Stokes solutions, and the relation between energy rigidity, anomalous dissipation, and wild solutions constructed by convex integration.