An Intermittent Onsager Theorem

TL;DR

This paper constructs non-conservative weak solutions to the 3D Euler equations with intermittent regularity properties, providing a new proof of the $L^3$-based Onsager conjecture and matching turbulence observations.

Contribution

It introduces an intermittent convex integration scheme with higher-order Reynolds stresses for the 3D Euler equations, advancing the understanding of Onsager's conjecture.

Findings

Constructed solutions in $C^0_t (H^{eta} igcap L^{rac{1}{(1-2eta)}})$ for $eta<rac{1}{2}$.

Solutions belong to $C^0_tB^{s}_{3, infty}$ with $s$ approaching $rac{1}{3}$ as $eta$ approaches $rac{1}{2}.

Matches the intermittency observed in turbulent flows, exceeding the $rac{1}{3}$ regularity threshold.

Abstract

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 $L^3$-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 $\frac 13$. 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 H\"older-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.