A proof of Onsager's conjecture

Abstract

For any α < 1/3, we construct weak solutions to the 3D incompressible Euler equations in the class CtC αx that have nonempty, compact support in time on R × T 3 and therefore fail to conserve the total kinetic energy. This result, together with the proof of energy conservation for α > 1/3 due to [Eyi94, CET94], solves Onsager's conjecture that the exponent α = 1/3 marks the threshold for conservation of energy for weak solutions in the class L ∞ t C α x . The previous best results were solutions in the class CtC αx for α < 1/5, due to the author, and in the class L 1 t C α x for α < 1/3 due to Buckmaster, De Lellis and Székelyhidi, both based on the method of convex integration developed for the incompressible Euler equations in [DLS13b, DLS14]. 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 in [DS16] and the framework of estimates developed in the author's previous work. Contents I Introduction

On Turbulence and Geometry: from Nash to Onsager

“…where is an 1 function of time. The correct scale of spaces was finally achieved in [41], where P. Isett proved the existence of compactly supported nontrivial solutions in for every < 1 3 . The proof in [41] contains two new ideas.…”

“…The correct scale of spaces was finally achieved in [41], where P. Isett proved the existence of compactly supported nontrivial solutions in for every < 1 3 . The proof in [41] contains two new ideas. Firstly, in [22] S. Daneri and the second author introduced a new class of "master functions"…”

“…However, it is not possible to use such directly in the scheme of [27] (and its subsequent refinements), not even to produce continuous solutions. The second key idea in [41] is a gluing technique, which combines the convex integration technique with the free (unforced) Euler dynamics in an alternating fashion. A shortcoming of [41] is the insufficient control of the kinetic energy ( ).…”

“…The second key idea in [41] is a gluing technique, which combines the convex integration technique with the free (unforced) Euler dynamics in an alternating fashion. A shortcoming of [41] is the insufficient control of the kinetic energy ( ). The final proof of Theorem 1(b) was then given in [10].…”

“…Part (b) of Theorem 1 took another 25 years, with a series of partial results and gradual improvements [5,6,8,22, 25-28, 40, 54, 56, 57], finally culminating in [41] and the subsequent improvement [10].…”