TY - JOUR

T1 - An intermittent Onsager theorem

AU - Novack, Matthew

AU - Vicol, Vlad

N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023/7

Y1 - 2023/7

N2 - 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.

AB - 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.

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U2 - 10.1007/s00222-023-01185-6

DO - 10.1007/s00222-023-01185-6

M3 - Article

AN - SCOPUS:85148865267

SN - 0020-9910

VL - 233

SP - 223

EP - 323

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 1

ER -