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 C_tC_^xα 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 [Eyink] and [Constantin, E, Titi], 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_x^α. The previous best results were solutions in the classC_tC_x^α for α < 1/5, due to [Isett], and in the class L_t^1C_x^α for α < 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.

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