An Intermittent Onsager Theorem

Matthew Novack, Vlad Vicol

Published 2022-03-24Version 1

For any regularity exponent $\beta<\frac 12$, we construct non-conservative weak solutions to the 3D incompressible Euler equations in the class $C^0_t (H^{\beta} \cap L^{\frac{1}{(1-2\beta)}})$. By interpolation, such solutions belong to $C^0_tB^{s}_{3,\infty}$ for $s$ approaching $\frac 13$ as $\beta$ approaches $\frac 12$. Hence this result provides a new proof of the flexible side of the Onsager conjecture, which is independent from that of Isett [A proof of Onsager's conjecture, Annals of Mathematics, 188(3):871, 2018]. Of equal importance is that the intermittent nature of our solutions matches that of turbulent flows, which are observed to possess an $L^2$-based regularity index exceeding $\frac 13$. Our proof builds on the authors' previous joint work with Buckmaster and Masmoudi [Nonconservative $H^{\frac 12-}$ weak solutions of the incompressible 3D Euler equations, arXiv 2101.09278, 2021], in which an intermittent convex integration scheme is developed for the 3D incompressible Euler equations. We employ a scheme with higher-order Reynolds stresses, which are corrected via a combinatorial placement of intermittent pipe flows of optimal relative intermittency.