A proof of Onsager's conjecture

• 成果类型: Article
• 署名作者: Isett, Philip
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2018.188.3.4
• 发表日期: 2018
• 页码: 871-963
• 关键词: incompressible euler; WEAK SOLUTIONS; energy-conservation; DISSIPATION

摘要

For any alpha < 1/3, we construct weak solutions to the 3D incompressible Euler equations in the class CtCx alpha that have nonempty, compact support in time on R x 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 (LtCx alpha)-C-infinity. The previous best results were solutions in the class CtCx alpha for a < 1/5, due to [Isett], and in the class (LtCx alpha)-C-1 for a < 1/3 due to [Buckmaster, De Lellis, Szekelyhidi], both based on the method of convex integration developed for the incompressible Euler equations by [De Lellis, Szekelyhidi]. 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, Szekelyhidi] and the framework of estimates developed in the author's previous work.