An Intermittent Onsager Theorem

Abstract

For any regularity exponent $\beta <\frac{1}{2}$, we construct non-conservative weak solutions to the 3D incompressible Euler equations in the class $C_t^0(H^{\beta }\cap L^{\frac{1}{(1-2\beta )}})$. By interpolation, such solutions belong to $C_t^0{B}_{3,\infty }^s$ for $s$ approaching $\frac{1}{3}$ as $\beta$ approaches $\frac{1}{2}$. Hence this result provides a new proof of the flexible side of the $L^3$-based Onsager conjecture. 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 $\frac{1}{3}$. Thus our result does not imply, and is not implied by, the work of Isett [A proof of Onsager's conjecture, Annals of Mathematics, 188(3):871, 2018], who gave a proof of the Hölder-based Onsager conjecture. Our proof builds on the authors' previous joint work with Buckmaster et al. (Intermittent convex integration for the 3D Euler equations: (AMS-217), Princeton University Press, 2023), 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.

54 pages, no figures, published version available at https://link.springer.com/article/10.1007/s00222-023-01185-6. arXiv admin note: text overlap with arXiv:2101.09278

References in corpus (1)

• Anomalous energy flux in critical $L^p$-based spaces

Cited by in corpus (5)

• Onsager's Conjecture for Subgrid Scale $\alpha$-Models of Turbulence
• A proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation
• Nonuniqueness of generalised weak solutions to the primitive and Prandtl equations
• An extension theorem for weak solutions of the 3d incompressible Euler equations and applications to singular flows
• Nontrivial absolutely continuous part of anomalous dissipation measures in time