An intermittent Onsager theorem

Matthew Novack, Vlad Vicol

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)223-323
Number of pages101
JournalInventiones Mathematicae
Issue number1
StatePublished - Jul 2023

ASJC Scopus subject areas

  • General Mathematics


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