An intermittent Onsager theorem

By Matthew Novack, Vlad Vicol. Inventiones Mathematicae, 2022

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For any regularity exponent $$\beta <\frac{1}{2}$$ β < 1 2 , we construct non-conservative weak solutions to the 3D incompressible Euler equations in the class $$C^0_t (H^{\beta } \cap L^{\frac{1}{(1-2\beta )}})$$ C t 0 ( H β ∩ L 1 ( 1 - 2 β ) ) . By interpolation, such solutions belong to $$C^0_t B^{s}_{3,\infty }$$ C t 0 B 3 , ∞ s for s approaching $$\frac{1}{3}$$ 1 3 as $$\beta $$ β approaches $$\frac{1}{2}$$ 1 2 . Hence this result provides a new proof of the flexible side of the $$L^3$$ 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 $$L^2$$ L 2 -based regularity index exceeding $$\frac{1}{3}$$ 1 3 . Thus our result does not imply, and is not implied by, the work of Isett (Ann Math 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.

Subfields: Applied Mathematics, Computational Mechanics

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