2018/10/19 by Philip Isett · 6 citations
Mathematics · Engineering · #Navier-Stokes equation solutions #Fluid Dynamics and Turbulent Flows #Geometric Analysis and Curvature Flows #Regular polygon #Conjecture #Exponent #Mathematics #Class (philosophy) #Alpha (finance) #Energy (signal processing) #Combinatorics #Pure mathematics #Mathematical analysis #Discrete mathematics #Geometry #Computer science #Statistics
paper · doi:10.4007/annals.2018.188.3.4
openalex publication_date 2018/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
For any α \lt 1/3, we construct weak solutions to the 3D incompressible Euler equations in the class CtCxα that have nonempty, compact support in time on ℝ× \mathbbT3 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 Lt^∞ Cxα. The previous best results were solutions in the class CtCxα for α \lt 1/5, due to [Isett], and in the class Lt1 Cxα for α \lt 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.