Dissipative anomalies in singular Euler flows

We discuss Onsager’s conjecture that non-vanishing energy dissipation in high-Reynolds-number turbulence is associated to singular (distributional) solutions of the incompressible Euler equations. We carefully explain the physical and mathematical meaning of the conjecture and also review relevant theoretical, experimental and numerical work, emphasizing some of the dramatic successes of Onsager’s point of view. Finally, we present several new ideas and results on Lagrangian dynamics of circulations and vortex-lines that we believe will be important for future progress.