An intermittent Onsager theorem

For any regularity exponent β<12, we construct non-conservative weak solutions to the 3D incompressible Euler equations in the class Ct0(Hβ∩L1(1-2β)). By interpolation, such solutions belong to Ct0B3,∞s for s approaching 13 as β approaches 12. Hence this result provides a new proof of the flexible side of theL³-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 L²-based regularity index exceeding 13. 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.