A proof of Onsager’s conjecture

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 [Isett], and in the class $L_t^1 C_x^\alpha $ for $\alpha < 1/3$ due to [Buckmaster, De Lellis, Székelyhidi], both based on the method of convex integration developed for the incompressible Euler equations by [De Lellis, Székelyhidi]. 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, Székelyhidi] and the framework of estimates developed in the author’s previous work.