A proof of Onsager's conjecture

作者：Isett, Philip^*

来源：Annals of Mathematics, 2018, 188(3): 871-963.

DOI：10.4007/annals.2018.188.3.4

• Euler equations
• incompressible
• weak solutions
• Onsager conjecture
• Onsager's conjecture
• energy conservation
• conservation of energy
• dissipation
• energy dissipation
• regularity
• anomalous dissipation
• turbulence
• fluid dynamics
• convex integration
• nonuniqueness

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.

• 出版日期2018-11