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Onsager critical solutions of the forced Navier-Stokes equations

  • * Corresponding author: Camillo De Lellis

    * Corresponding author: Camillo De Lellis 

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

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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  • 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].

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


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  • [1] D. AlbrittonE. Brué and M. Colombo, Non-uniqueness of Leray solutions of the forced Navier-Stokes equations, Ann. Math., 196 (2022), 415-455.  doi: 10.4007/annals.2022.196.1.3.
    [2] D. Albritton, E. Brué and M. Colombo et al., Instability and nonuniqueness for the 2d Euler equations in vorticity form, after M. Vishik, preprint, 2021, arXiv: 2112.04943.
    [3] E. Brué and C. De Lellis, Anomalous dissipation for the forced 3d Navier-Stokes equations, Comm. Math. Phys., (2023), 27 pp. doi: 10.1007/s00220-022-04626-0.
    [4] T. BuckmasterC. De Lellis and L. Székelyhidi, et al., Onsager's conjecture for admissible weak solutions, Commun. Pure Appl. Math., 72 (2019), 229-274.  doi: 10.1002/cpa.21781.
    [5] M. Colombo, G. Crippa and M. Sorella, Anomalous dissipation and lack of selection in the Obukhov-Corrsin theory of scalar turbulence, preprint, 2022, arXiv: 2207.06833
    [6] P. ConstantinW. E and E. Titi, Onsager's conjecture on the energy conservation for solutions of Euler's equation, Commun. Math. Phys., 165 (1994), 207-209. 
    [7] De LellisSzékelyhidi and Jr., The Euler equations as a differential inclusion, Ann. Math., 170 (2009), 1417-1436.  doi: 10.4007/annals.2009.170.1417.
    [8] De LellisSzékelyhidi and Jr., Dissipative continuous Euler flows, Invent. Math., 193 (2013), 377-407.  doi: 10.1007/s00222-012-0429-9.
    [9] L. De Rosa and P. Isett, Intermittency and lower dimensional dissipation in incompressible fluids: quantifying Landau, preprint, 2022, arXiv: 2212.08176
    [10] L. De Rosa, M. Inversi and G. Stefani, Weak-strong uniqueness and vanishing viscosity for incompressible Euler equations in exponential spaces, preprint, 2022, arXiv: 2204.12779.
    [11] G. L. Eying, Energy dissipation without viscosity in ideal hydrodynamics. Ⅰ. Fourier analysis and local energy transfer, Phys. D., 79 (1994), 222-240.  doi: 10.1016/0167-2789(94)90117-1.
    [12] P. Isett, A proof of Onsager's conjecture, Ann. Math., 188 (2018), 871-963.  doi: 10.4007/annals.2018.188.3.4.
    [13] C. J. P. Johansson and M. Sorella, Nontrivial absolutely continuous part of anomalous dissipation measures in time, preprint, 2023, arXiv: 2303.09486.
    [14] I. Jeong and T. Yoneda, Quasi-streamwise vortices and enhanced dissipation for the incompressible 3d Navier-Stokes equations, Proc. Amer. Math. Soc., 150 (2022), 1279-1286.  doi: 10.1090/proc/15754.
    [15] I. Jeong and T. Yoneda, Vortex stretching and anomalous dissipation for the incompressible 3d Navier-Stokes equations, Math. Ann., 380 (2021), 2041-2072.  doi: 10.1007/s00208-020-02019-z.
    [16] L. Onsager, Statistical hydrodynamics, Nuovo Cimento, Supplemento 2 (Convegno Internazionale di Meccanica Statistica), 6 (1949), 279-287.
    [17] M. Vishik, Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid, Part Ⅰ, preprint, 2018, arXiv: 1805.09426.
    [18] M. Vishik, Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid, Part Ⅱ, preprint, 2018, arXiv: 1805.09440.
    [19] E. Wiedemann, Weak-strong uniqueness in fluid dynamics, London Math. Soc. Lecture Note Ser., 452 (2018), 289-326. 
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