Abstract
Adopting the setting for the study of existence and scale locality of the energy cascade in 3D viscous flows in physical space recently introduced by the authors to 3D inviscid flows, it is shown that the anomalous dissipation is – in the case of decaying turbulence – indeed capable of triggering the cascade which then continues ad infinitum, confirming Onsager’s predictions.
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Onsager, L.: Nuovo Cimento (9), 6 (Supplemento, 2 (Convegno Internazionale di Meccanica Statistica)), 279 (1949)
Scheffer V.: Hausdorff measure and the Navier-Stokes equations. Commun. Math. Phys 55, 97 (1977)
Caffarelli L., Kohn R., Nirenberg L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math 35, 771 (1982)
Scheffer V.: An inviscid flow with compact support in space-time. J. Geom. Anal 1993, 343 (1993)
Eyink G.: Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer. Phys. D 78, 222 (1994)
Constantin P., Weinan E., Titi E.: Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Commun. Math. Phys. 165, 207 (1994)
Frisch U.: Turbulence. Cambridge University Press, Cambridge (1995)
Lions P.-L.: Mathematical Topics in Fluid Mechanics Vol 1 Incompressible Models. Clarendon, Oxford (1996)
Shnirelman A: On the nonuniqueness of weak solution of the Euler equation. Comm. Pure Appl. Math 50, 1261 (1997)
Shnirelman A.: Weak solutions with decreasing energy of incompressible Euler equations. Comm. Math. Phys. 210, 541 (2000)
Duchon J., Robert R.: Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations. Nonlinearity 13, 249 (2000)
Eyink G., Sreenivasan K.: Onsager and the theory of hydrodynamic turbulence. Rev. Mod. Phy. 78, 87 (2006)
Cheskidov A., Constantin P., Friedlander S., Shvydkoy R: Energy conservation and Onsager’s conjecture for the Euler equations. Nonlinearity 21, 1233 (2008)
Cheskidov A., Friedlander S.: The vanishing viscosity limit for a dyadic model. Phys. D 238, 783 (2009)
Shvydkoy R.: On the energy of inviscid singular flows. J. Math. Anal. Appl. 349, 583 (2009)
De Lellis C., Szekelyhidi L. Jr.: The Euler equations as a differential inclusion. Ann. Math. 170(2), 1417 (2009)
De Lellis C., Szekelyhidi L. Jr.: On admissibility criteria for weak solutions of the Euler equations. Arch. Rat. Mech. Anal 195, 225 (2010)
Bardos C., Titi E.: Loss of smoothness and energy conserving rough weak solutions for the 3d Euler equations. Discrete Cont. Dynamical Sysytems S 3, 185 (2010)
Dascaliuc R., Grujić Z.: Energy cascades and flux locality in physical scales of the 3D NSE. Commun. Math. Phys. 305, 199 (2011)
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Communicated by P. Constantin
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Dascaliuc, R., Grujić, Z. Anomalous Dissipation and Energy Cascade in 3D Inviscid Flows. Commun. Math. Phys. 309, 757–770 (2012). https://doi.org/10.1007/s00220-011-1382-y
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DOI: https://doi.org/10.1007/s00220-011-1382-y