TY - JOUR

T1 - An intermittent Onsager theorem

AU - Novack, Matthew

AU - Vicol, Vlad

N1 - Funding Information:
MN thanks Hyunju Kwon and Vikram Giri for many stimulating discussions during the special year on the h -principle at the Institute for Advanced Study. MN was supported by the NSF under Grant DMS-1926686 while a member at the IAS. VV is grateful to Tristan Buckmaster for infinitely many (for all practical purposes) discussions about convex integration, and for teaching him everything he knows about this subject. VV was supported in part by the NSF CAREER Grant DMS-1911413. We thank Theodore Drivas for references and many discussions about structure function exponents in turbulent flows.
Funding Information:
MN thanks Hyunju Kwon and Vikram Giri for many stimulating discussions during the special year on the h-principle at the Institute for Advanced Study. MN was supported by the NSF under Grant DMS-1926686 while a member at the IAS. VV is grateful to Tristan Buckmaster for infinitely many (for all practical purposes) discussions about convex integration, and for teaching him everything he knows about this subject. VV was supported in part by the NSF CAREER Grant DMS-1911413. We thank Theodore Drivas for references and many discussions about structure function exponents in turbulent flows.
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 -