Onsager critical solutions of the forced Navier-Stokes equations

Elia Brué; Maria Colombo; Gianluca Crippa; Camillo De Lellis; Massimo Sorella

doi:10.3934/cpaa.2023071

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2024, Volume 23, Issue 10: 1350-1366. Doi: 10.3934/cpaa.2023071

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Onsager critical solutions of the forced Navier-Stokes equations

1.
Università Bocconi, via Sarfatti 25, 20136 Milano, Italy
2.
EPFL B, Station 8, CH-1015 Lausanne, Switzerland
3.
Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, CH-4051 Basel, Switzerland
4.
School of Mathematics, Institute for Advanced Study, 1 Einstein Dr., Princeton NJ 05840, U.S.A.
5.
EPFL B, Station 8, CH-1015 Lausanne, Switzerland

^* Corresponding author: Camillo De Lellis
^* Corresponding author: Camillo De Lellis

Dedicated to Vladimir Šverak on occasion of his 65th birthday.

Received: January 1, 2023

Revised: May 1, 2023

Early access: May 24, 2023

Published online: October 1, 2024

EB has been supported by the Giorgio and Elena Petronio Fellowship at the Institute for Advanced Study. CDL has been supported by the National Science Foundation under Grant No. DMS-1638352. MC and MS were supported by the SNSF Grant 182565 and by the Swiss State Secretariat for Education, Research and lnnovation (SERI) under contract number M822.00034. GC has been partially supported by the ERC Starting Grant 676675 FLIRT.

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Abstract

We answer positively to [3]*Question 2.4 by building new examples of solutions to the forced $ 3d $-Navier-Stokes equations with vanishing viscosity, which exhibit anomalous dissipation and which enjoy uniform bounds in the space $ L^3_t C^{^{1}\!\!\diagup\!\!_{3-\varepsilon}\;}_x $, for any fixed $ \varepsilon >0 $. Our construction combines ideas of [3] and [5].

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Mathematics Subject Classification: Primary: 35Q30, 35Q31; Secondary 35Q35, 35Q49, 76F05.

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References

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