Euler equations and turbulence: Analytical approach to intermittency

SIAM Journal on Mathematical Analysis

Volumn 46, Issue 1, 2014, Pages 353-374

  Cheskidov, A ^a   Shvydkoy, R ^a  

^a UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

Author keywords

Euler equations; Intermittency; Turbulence

Indexed keywords

ACTIVE VOLUMES; ANALYTICAL APPROACH; ENERGY ACCUMULATION; ENERGY CASCADE; ENERGY SPECTRA; INTERMITTENCY; PHYSICAL MODEL; SECOND ORDER STRUCTURE;

EULER EQUATIONS; MODELS;

TURBULENCE;

EID: 84897844602     PISSN: 00361410     EISSN: 10957111     Source Type: Journal    
DOI: 10.1137/120876447     Document Type: Article

References (27)

1. D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren Math. Wiss. 314, Springer-Verlag, Berlin, 1996.
2. F. Anselmet, Y. Gagne, E. J. Hopfinger, and R. A. Antonia, High-order velocity structure functions in turbulent shear flow, J. Fluid Mech., 140 (1984), pp. 63-89.
3. A. Cheskidov, P. Constantin, S. Friedlander, and R. Shvydkoy, Energy conservation and Onsager's conjecture for the Euler equations, Nonlinearity, 21 (2008), pp. 1233-1252.
4. A. Cheskidov and S. Friedlander, The vanishing viscosity limit for a dyadic model, Phys. D, 238 (2009), pp. 783-787.
5. A. Cheskidov, S. Friedlander, and R. Shvydkoy, A continuous model for turbulent energy cascade, in Mathematical Aspects of Fluid Mechanics, London Math. Soc. Lecture Note Ser. 402, Cambridge University Press, Cambridge, UK, 2012, pp. 52-69.
6. A. Cheskidov and R. Shvydkoy, A Unified Approach to Regularity Problems for the 3D Navier-Stokes and Euler equations: The Use of Kolmogorov's Dissipation Range, preprint, http://arXiv.com/abs/1102.1944, 2011.
7. P. Constantin, The Littlewood-Paley spectrum in 2D turbulence, Theoret. Comput. Fluid Dyn., 9 (1997), pp. 183-189.
8. P. Constantin, W. E, and E. S. Titi, Onsager's conjecture on the energy conservation for solutions of Euler's equation, Comm. Math. Phys., 165 (1994), pp. 207-209.
9. P. Constantin and C. Fefferman, Scaling exponents in fluid turbulence: Some analytic results, Nonlinearity, 7 (1994), pp. 41-57.
10. P. Constantin, Q. Nie, and S. Tanveer, Bounds for second order structure functions and energy spectrum in turbulence, Phys. Fluids, 11 (1999), pp. 2251-2256.
11. C. De Lellis and L. Székelyhidi, Dissipative continuous Euler flows, Invent. Math., 193 (2013), pp. 377-407.
12. J. Duchon and R. Robert, Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations, Nonlinearity, 13 (2000), pp. 249-255.
13. G. L. Eyink, Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer, Phys. D, 78 (1994), pp. 222-240.
14. G. L. Eyink, Besov spaces and the multifractal hypothesis, J. Statist. Phys., 78 (1995), pp. 353-375.
15. C. Foias, What do the Navier-Stokes equations tell us about turbulence?, in Harmonic Analysis and Nonlinear Differential Equations (Riverside, CA, 1995), Contemp. Math. 208, AMS, Providence, RI, 1997, pp. 151-180.
16. U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, Cambridge, UK, 1995.
17. U. Frisch, P.-L. Sulem, and M. Nelkin, A simple dynamical model of intermittent fully developed turbulence, J. Fluid Mech., 87 (1978), pp. 719-736.
18. P. Isett, Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time, preprint, 2012.
19. Y. Kaneda, T. Ishihara, M. Yokokawa, K. Itakura, and A. Uno, Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box, Phys. Fluids, 15 (2003), pp. 21-24.
20. B. Mandelbrot, Intermittent turbulence and fractal dimension: Kurtosis and the spectral exponent 5/3 + B, in Turbulence and Navier-Stokes Equations (Proc. Conf. Univ. Paris-Sud, Orsay, 1975), Lecture Notes in Math. 565, Springer, Berlin, 1976, pp. 121-145.
21. W. D. McComb, The Physics of Fluid Turbulence, Oxford Engrg. Sci. Ser. 25, The Clarendon Press, Oxford University Press, New York, 1991.
22. A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. II, Dover, Mineola, NY, 2007. Translated from the 1965 Russian original, edited and with a preface by John L. Lumley; English edition updated, augmented and revised by the authors; reprinted from the 1975 edition.
23. E. A. Novikov and R. W. Stewart, The intermittency of turbulence and the spectrum of energy dissipation, Izv. Akad. Nauk SSSR Ser. Geofiz., 1964, pp. 408-413.
24. L. Onsager, Statistical Hydrodynamics, Nuovo Cimento (9), 6 (Suppl. 2) (1949), pp. 279-287.
25. G. Parisi and U. Frisch, On the singularity structure of fully developed turbulence, in Turbulence and Predictability in Geophysical Fluid Dynamics, Proceed. Intern. School of Phys. "E. Fermi, " G. Parisi, M. Ghil, and R. Benzi, eds., North-Holland, Amsterdam, 1983, pp. 84-87.
26. R. Shvydkoy, On the energy of inviscid singular flows, J. Math. Anal. Appl., 349 (2009), pp. 583-595.
27. R. Shvydkoy, Lectures on the Onsager conjecture, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), pp. 473-496.