A proof of Onsager's conjecture
A proof of Onsager's conjecture
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DOI：
10.4007/annals.2018.188.3.4
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发表时间：
2018-11-01
影响因子：
4.9
通讯作者：
Isett, Philip
中科院分区：
文献类型：
--
作者：
Isett, Philip
对于任何 alpha < 1/3，我们构造 CtCx alpha 类中 3D 不可压缩欧拉方程的弱解，该方程在 R x T-3 上及时具有非空紧支持，因此无法守恒总动能。这一结果与 [Eyink] 和 [Constantin, E, Titi] 提出的 alpha > 1/3 的能量守恒证明一起，解决了 Onsager 的猜想，即指数 alpha = 1/3 标志着 (LtCx alpha)-C-无穷大类中弱解的能量守恒阈值。之前的最佳结果是 [Isett] 的 CtCx alpha 类中 a < 1/5 的解，以及 [Buckmaster, De Lellis, Szekelyhidi] 的 (LtCx alpha)-C-1 类中 a < 1/3 的解，两者都基于 [De Lellis, Szekelyhidi] 为不可压缩欧拉方程开发的凸积分方法。本证明结合了凸积分方法和新的“粘合近似”技术。证明的凸积分部分依赖于[Daneri，Szekelyhidi]引入的“Mikado 流”以及作者之前工作中开发的估计框架。
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.