A Proof of Onsager's Conjecture

2016/08/30 by Isett, Philip · 12 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1608.08301

Abstract

For any α< 1/3, we construct weak solutions to the 3D incompressible Euler equations in the class C_tC_x^α that have nonempty, compact support in time on \mathbb R × \mathbb T³ 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 class C_tC_x^α for α< 1/5, due to the author, and in the class L_t¹ C_x^α 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 by De Lellis and 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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Related

• Onsager's conjecture almost everywhere in time
• Onsager's conjecture for admissible weak solutions
• The L³-based strong Onsager theorem
• Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time
• Energy conservation and Onsager's conjecture for the Euler equations