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Onsager’s Conjecture Almost Everywhere in Time

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In recent works by Isett (Hölder continuous Euler flows in three dimensions with compact support in time, pp 1–173, 2012), and later by Buckmaster et al. (Ann Math 2015), iterative schemes were presented for constructing solutions belonging to the Hölder class C 1/5-ε of the 3D incompressible Euler equations which do not conserve the total kinetic energy. The cited work is partially motivated by a conjecture of Lars Onsager in 1949 relating to the existence of C 1/3-ε solutions to the Euler equations which dissipate energy. In this note we show how the later scheme can be adapted in order to prove the existence of non-trivial Hölder continuous solutions which for almost every time belong to the critical Onsager Hölder regularity C 1/3-ε and have compact temporal support.

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References

  1. Benzi R., Ciliberto S., Tripiccione R., Baudet C., Massaioli F., Succi S.: Extended self-similarity in turbulent flows. Phys. Rev. E 48 1, 29–32 (1993)

    Article  Google Scholar 

  2. Buckmaster, T., De Lellis, C., Isett, P., Székelyhidi, L. Jr.: Anomalous dissipation for 1/5-Hölder Euler flows. Ann. Math. (2015)

  3. Buckmaster, T., De Lellis, C., Székelyhidi, L. Jr.: Transporting microstructure and dissipative Euler flows, e-prints (Feb. 2013). arXiv:1302.2815

  4. Cheskidov A., Constantin P., Friedlander S., Shvydkoy R.: Energy conservation and Onsager’s conjecture for the Euler equations. Nonlinearity 21 6, 1233–1252 (2008)

    Article  MathSciNet  Google Scholar 

  5. Constantin P., Weinan E., Titi E.S.: Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Comm. Math. Phys. 165 1, 207–209 (1994)

    Article  ADS  Google Scholar 

  6. De Lellis C., Székelyhidi L. Jr: The Euler equations as a differential inclusion. Ann. Math. (2) 170(3), 1417–1436 (2009)

    Article  MATH  Google Scholar 

  7. De Lellis C., Székelyhidi L. Jr: On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195 1, 225–260 (2010)

    Article  Google Scholar 

  8. De Lellis C., Székelyhidi L. Jr.: Dissipative Euler flows and Onsager’s conjecture. J. Eur. Math. Soc. 16(7), 1467–1505 (2014). doi:10.4171/JEMS/466

  9. De Lellis C., Székelyhidi L. Jr: The h-principle and the equations of fluid dynamics. Bull. Am. Math. Soc. (N.S.) 49 3, 347–375 (2012)

    Article  Google Scholar 

  10. De Lellis C., Székelyhidi L. Jr: Dissipative continuous Euler flows. Invent. Math. 193, 377–407 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Duchon J., Robert R.: Inertial energy dissipation for weak solutions of incompressible Euler and Navier–Stokes equations. Nonlinearity 13 1, 249–255 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  12. Eyink G.L.: Energy dissipation without viscosity in ideal hydrodynamics I: Fourier analysis and local energy transfer. Phys. D 78 3–4, 222–240 (1994)

    Article  MathSciNet  Google Scholar 

  13. Eyink G.L., Sreenivasan K.R.: Onsager and the theory of hydrodynamic turbulence. Rev. Mod. Phys. 78 1, 87–135 (2006)

    Article  MathSciNet  Google Scholar 

  14. Frisch U.: Fully developed turbulence and intermittency. Ann. N. Y. Acad. Sci. 357 1, 359–367 (1980)

    Article  ADS  Google Scholar 

  15. Frisch, U.: Turbulence. In: The legacy of A. N. Kolmogorov, Cambridge University Press, Cambridge (1995)

  16. Isett, P.: Hölder continuous Euler flows in three dimensions with compact support in time, pp 1–173 (2012). arXiv:1211.4065

  17. Kolmogorov, A.N.: The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Proc. Roy. Soc. Lond. Ser. A 434, 1890. Translated from the Russian by V. Levin, Turbulence and stochastic processes: Kolmogorov’s ideas 50 years on, pp 9–13 (1991)

  18. Mandelbrot, B.: Turbulence and Navier-stokes equation. In: Proceedings of Lecture Notes in Mathematics, vol. 565, p 121 (1976)

  19. Onsager, L.: Statistical hydrodynamics. Nuovo Cimento (9) 6, Supplemento, 2 (Convegno Internazionale di Meccanica Statistica) 279–287 (1949)

  20. Scheffer V.: An inviscid flow with compact support in space-time. J. Geom. Anal. 3 4, 343–401 (1993)

    Article  MathSciNet  Google Scholar 

  21. Shnirelman A.: Weak solutions with decreasing energy of incompressible Euler equations. Comm. Math. Phys. 210 3, 541–603 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  22. Wiedemann E.: Existence of weak solutions for the incompressible Euler equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28 5, 727–730 (2011)

    Article  ADS  MathSciNet  Google Scholar 

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  1. Tristan Buckmaster

    Present address: Courant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York, NY, 10012, USA

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  1. Institut für Mathematik, Universität Leipzig, 04103, Leipzig, Germany

    Tristan Buckmaster

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  1. Tristan Buckmaster
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Correspondence to Tristan Buckmaster.

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Communicated by H.-T. Yau

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Buckmaster, T. Onsager’s Conjecture Almost Everywhere in Time. Commun. Math. Phys. 333, 1175–1198 (2015). https://doi.org/10.1007/s00220-014-2262-z

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  • DOI: https://doi.org/10.1007/s00220-014-2262-z

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