On the endpoint regularity in Onsager’s conjecture

Abstract

Onsager’s conjecture states that the conservation of energy may fail for three-dimensional incompressible Euler flows with Hölder regularity below $\frac{1}{3}$. This conjecture was recently solved by the author, yet the endpoint case remains an interesting open question with further connections to turbulence theory. In this work, we construct energy nonconserving solutions to the three-dimensional incompressible Euler equations with space-time Hölder regularity converging to the critical exponent at small spatial scales and containing the entire range of exponents $[0,\frac{1}{3})$.

Our construction improves the author’s previous result towards the endpoint case. To obtain this improvement, we introduce a new method for optimizing the regularity that can be achieved by a convex integration scheme. A crucial point is to avoid loss of powers in frequency in the estimates of the iteration. This goal is achieved using localization techniques of Isett and Oh (Arch. Ration. Mech. Anal. 221:2 (2016), 725–804) to modify the convex integration scheme.

We also prove results on general solutions at the critical regularity that may not conserve energy. These include a theorem on intermittency stating roughly that energy dissipating solutions cannot have absolute structure functions satisfying the Kolmogorov–Obukhov scaling for any $p>3$ if their singular supports have space-time Lebesgue measure zero.