A Proof of Onsager's Conjecture

TL;DR

This paper constructs weak solutions to the 3D Euler equations that demonstrate energy non-conservation below a certain regularity threshold, thereby proving Onsager's conjecture that the critical Hölder exponent is 1/3.

Contribution

The paper proves Onsager's conjecture by constructing solutions with Hölder regularity below 1/3 that do not conserve energy, using a novel combination of convex integration and a new gluing technique.

Findings

Solutions with regularity $oldsymbol{eta < 1/3}$ fail to conserve energy.

Solutions with regularity $oldsymbol{eta > 1/3}$ conserve energy.

The proof introduces a new gluing approximation method combined with convex integration.

Abstract

For any $\alpha < 1/3$, we construct weak solutions to the $3D$ incompressible Euler equations in the class $C_tC_x^\alpha$ that have nonempty, compact support in time on ${\mathbb R} \times {\mathbb T}^3$ and therefore fail to conserve the total kinetic energy. This result, together with the proof of energy conservation for $\alpha > 1/3$ due to [Eyink] and [Constantin, E, Titi], solves Onsager's conjecture that the exponent $\alpha = 1/3$ marks the threshold for conservation of energy for weak solutions in the class $L_t^\infty C_x^\alpha$. The previous best results were solutions in the class $C_tC_x^\alpha$ for $\alpha < 1/5$, due to the author, and in the class $L_t^1 C_x^\alpha$ for $\alpha < 1/3$ due to Buckmaster, De Lellis and Sz\'{e}kelyhidi, both based on the method of convex integration developed for the incompressible Euler equations by De Lellis and Sz\'ekelyhidi. The present proof combines the method of convex integration and a new "gluing approximation" technique. The convex integration part of the proof relies on the "Mikado flows" introduced by [Daneri, Sz\'ekelyhidi] and the framework of estimates developed in the author's previous work.