Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Inviscid limits for a stochastically forced shell model of turbulent flow

Susan Friedlander, Nathan Glatt-Holtz, and Vlad Vicol

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We establish the anomalous mean dissipation rate of energy in the inviscid limit for a stochastic shell model of turbulent fluid flow. The proof relies on viscosity independent bounds for stationary solutions and on establishing ergodic and mixing properties for the viscous model. The shell model is subject to a degenerate stochastic forcing in the sense that noise acts directly only through one wavenumber. We show that it is hypo-elliptic (in the sense of Hörmander) and use this property to prove a gradient bound on the Markov semigroup.


Nous étudions le taux anormal de la dissipation moyenne de l’énergie dans la limite non visqueuse d’un modèle en couche de fluide turbulent. La preuve se base sur des estimations indépendantes de la viscosité pour des solutions stationnaires, ainsi que sur des propriétés ergodiques et de mélange pour le modèle visqueux. Le modèle en couche subit un forçage aléatoire dégénéré, c’est à dire que le bruit n’agit seulement que sur un mode. Nous montrons que le système est hypoelliptique au sens d’Hörmander et utilisons cette propriété pour prouver une borne sur le gradient du semigroupe de Markov.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1217-1247.

Received: 4 April 2014
Revised: 1 December 2014
Accepted: 8 December 2014
First available in Project Euclid: 28 July 2016

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Zentralblatt MATH identifier

Primary: 76F05: Isotropic turbulence; homogeneous turbulence 35Q35: PDEs in connection with fluid mechanics 37L40: Invariant measures 37L55: Infinite-dimensional random dynamical systems; stochastic equations [See also 35R60, 60H10, 60H15]

Inviscid limits Invariant measures Dissipation anomaly Shell models Ergodicity


Friedlander, Susan; Glatt-Holtz, Nathan; Vicol, Vlad. Inviscid limits for a stochastically forced shell model of turbulent flow. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1217--1247. doi:10.1214/14-AIHP663.

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  • [1] S. Albeverio, F. Flandoli and Y. G. Sinai. SPDE in Hydrodynamic: Recent Progress and Prospects. Lecture Notes in Mathematics 1942. Springer, Berlin, 2008. Lectures given at the C.I.M.E. Summer School held in Cetraro, August 29–September 3, 2005. Edited by Giuseppe Da Prato and Michael Röckner.
  • [2] D. Barbato, F. Flandoli and F. Morandin. Uniqueness for a stochastic inviscid dyadic model. Proc. Amer. Math. Soc. 138 (7) (2010) 2607–2617.
  • [3] D. Barbato, F. Flandoli and F. Morandin. Anomalous dissipation in a stochastic inviscid dyadic model. Ann. Appl. Probab. 21 (6) (2011) 2424–2446.
  • [4] D. Barbato, F. Morandin and M. Romito. Smooth solutions for the dyadic model. Nonlinearity 24 (11) (2011) 3083–3097.
  • [5] A. Bensoussan. Stochastic Navier–Stokes equations. Acta Appl. Math. 38 (3) (1995) 267–304.
  • [6] A. Bensoussan and R. Temam. Équations stochastiques du type Navier–Stokes. J. Funct. Anal. 13 (1973) 195–222.
  • [7] H. Bessaih and B. Ferrario. Invariant Gibbs measures of the energy for shell models of turbulence: The inviscid and viscous cases. Nonlinearity 25 (4) (2012) 1075–1097.
  • [8] H. Bessaih and A. Millet. Large deviation principle and inviscid shell models. Electron. J. Probab. 14 (2009) 2551–2579.
  • [9] H. Bessaih, F. Flandoli and E. S. Titi. Stochastic attractors for shell phenomenological models of turbulence. J. Stat. Phys. 140 (4) (2010) 688–717.
  • [10] T. Buckmaster, C. De Lellis and L. Székelyhidi Jr. Transporting microstructure and dissipative Euler flows. Preprint, 2013. Available at arXiv:1302.2815.
  • [11] A. Cheskidov and S. Friedlander. The vanishing viscosity limit for a dyadic model. Phys. D 238 (8) (2009) 783–787.
  • [12] 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, 2011. Available at arXiv:1102.1944.
  • [13] A. Cheskidov and R. Shvydkoy. Euler equations and turbulence: Analytical approach to intermittency. SIAM J. Math. Anal. 46 (2014) 353–374.
  • [14] A. Cheskidov, S. Friedlander and N. Pavlović. Inviscid dyadic model of turbulence: The fixed point and Onsager’s conjecture. J. Math. Phys. 48 (6) (2007) 065503.
  • [15] A. Cheskidov, P. Constantin, S. Friedlander and R. Shvydkoy. Energy conservation and Onsager’s conjecture for the Euler equations. Nonlinearity 21 (6) (2008) 1233–1252.
  • [16] A. Cheskidov, S. Friedlander and N. Pavlović. An inviscid dyadic model of turbulence: The global attractor. Discrete Contin. Dyn. Syst. 26 (3) (2010) 781–794.
  • [17] P. Constantin, C. Foias and R. Temam. Attractors Representing Turbulent Flows. Mem. Amer. Math. Soc. 53. Amer. Math. Soc., Providence, RI, 1985.
  • [18] 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 (1) (1994) 207–209.
  • [19] P. Constantin, B. Levant and E. S. Titi. Regularity of inviscid shell models of turbulence. Phys. Rev. E (3) 75 (1) (2007) 016304.
  • [20] P. Constantin, N. Glatt-Holtz and V. Vicol. Unique ergodicity for fractionally dissipated, stochastically forced 2D Euler equations. Comm. Math. Phys. 330 (2014) 819–857.
  • [21] G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge, MA, 1992.
  • [22] C. De Lellis and L. Székelyhidi Jr. Dissipative continuous Euler flows. Invent. Math. 193 (2) (2013) 377–407.
  • [23] A. Debussche. Ergodicity results for the stochastic Navier–Stokes equations: An introduction. In Topics in Mathematical Fluid Mechanics 23–108. Lecture Notes in Mathematics 2073. Springer, Heidelberg, 2013.
  • [24] A. Debussche, N. Glatt-Holtz and R. Temam. Local martingale and pathwise solutions for an abstract fluids model. Phys. D 240 (2011) 1123–1144.
  • [25] V. N. Desnianskii and E. A. Novikov. Evolution of turbulence spectra toward a similarity regime. Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 10 (1974) 127–136.
  • [26] J. L. Doob. Asymptotic properties of Markoff transition prababilities. Trans. Amer. Math. Soc. 63 (1948) 393–421.
  • [27] W. E and J. C. Mattingly. Ergodicity for the Navier–Stokes equation with degenerate random forcing: Finite-dimensional approximation. Comm. Pure Appl. Math. 54 (11) (2001) 1386–1402.
  • [28] W. E, K. Khanin, A. Mazel and Y. Sinai. Invariant measures for Burgers equation with stochastic forcing. Ann. of Math. (2) 151 (3) (2000) 877–960.
  • [29] G. L. Eyink. Exact results on stationary turbulence in 2D: Consequences of vorticity conservation. Phys. D 91 (1) (1996) 97–142.
  • [30] G. L. Eyink and K. R. Sreenivasan. Onsager and the theory of hydrodynamic turbulence. Rev. Modern Phys. 78 (1) (2006) 87–135.
  • [31] F. Flandoli and M. Romito. Markov selections for the 3d stochastic Navier–Stokes equations. Probab. Theory Related Fields 140 (3–4) (2008) 407–458.
  • [32] C. Foias, O. Manley, R. Rosa and R. Temam. Navier–Stokes Equations and Turbulence. Encyclopedia of Mathematics and Its Applications 83. Cambridge Univ. Press, Cambridge, MA, 2001.
  • [33] J. Foldes, N. Glatt-Holtz, G. Richards and E. Thomann. Ergodic and mixing properties of the Boussinesq equations with a degenerate random forcing. Preprint, 2013. Available at arXiv:1311.3620.
  • [34] S. Friedlander and N. Pavlović. Blowup in a three-dimensional vector model for the Euler equations. Comm. Pure Appl. Math. 57 (6) (2004) 705–725.
  • [35] U. Frisch. Turbulence: The Legacy A. N. Kolmogorov. Cambridge Univ. Press, Cambridge, MA, 1995.
  • [36] N. Glatt-Holtz, V. Sverak and V. Vicol. On inviscid limits for the stochastic Navier–Stokes equations and related models. Arch. Ration. Mech. Anal. 217 (2015) 619–649.
  • [37] M. Hairer and J. C. Mattingly. Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann. of Math. (2) 164 (3) (2006) 993–1032.
  • [38] M. Hairer and J. C. Mattingly. Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations. Ann. Probab. 36 (6) (2008) 2050–2091.
  • [39] M. Hairer and J. C. Mattingly. A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs. Electron. J. Probab. 16 (23) (2011) 658–738.
  • [40] R. Z. Has’minskiĭ. Ergodic properties of recurrent diffusion processes and stabilization of the solution to the Cauchy problem for parabolic equations. Theory Probab. Appl. 5 (2) (1960) 179–196.
  • [41] L. Hörmander. Hypoelliptic second order differential equations. Acta Math. 119 (1967) 147–171.
  • [42] P. Isett. Hölder continuous euler flows in three dimensions with compact support in time. Preprint, 2012. Available at arXiv:1211.4065.
  • [43] N. H. Katz and N. Pavlović. Finite time blow-up for a dyadic model of the Euler equations. Trans. Amer. Math. Soc. 357 (2) (2005) 695–708. (electronic).
  • [44] A. Kiselev and A. Zlatoš. On discrete models of the Euler equation. Int. Math. Res. Not. IMRN 2005 (38) (2005) 2315–2339.
  • [45] A. N. Kolmogorov. Local structure of turbulence in an incompressible fluid at very high Reynolds number. Dokl. Akad. Nauk SSSR 30 (4) (1941) 299–303. Translated from the Russian by V. Levin, Turbulence and stochastic processes: Kolmogorov’s ideas 50 years on.
  • [46] A. N. Kolmogorov. On degeneration of isotropic turbulence in an incompressible viscous liquid. Dokl. Akad. Nauk SSSR 31 (1941) 538–540.
  • [47] T. Komorowski and A. Walczuk. Central limit theorem for Markov processes with spectral gap in the Wasserstein metric. Stochastic Process. Appl. 122 (5) (2012) 2155–2184.
  • [48] S. Kuksin and A. Shirikyan. Mathematics of Two-Dimensional Turbulence. Cambridge Tracts in Mathematics 194. Cambridge Univ. Press, Cambridge, MA, 2012.
  • [49] J. C. Mattingly, T. Suidan and E. Vanden-Eijnden. Simple systems with anomalous dissipation and energy cascade. Comm. Math. Phys. 276 (1) (2007) 189–220.
  • [50] E. A. Novikov. Functionals and the random-force method in turbulence theory. Sov. Phys. JETP 20 (1965) 1290–1294.
  • [51] D. Nualart. The Malliavin Calculus and Related Topics, 2nd edition. Probability and Its Applications (New York). Springer, Berlin, 2006.
  • [52] L. Onsager. Statistical hydrodynamics. Nuovo Cimento (9) 6 (1949) 279–287.
  • [53] R. Robert. Statistical hydrodynamics (Onsager revisited). In Handbook of Mathematical Fluid Dynamics II 1–54. North-Holland, Amsterdam, 2003.
  • [54] M. Romito. Ergodicity of the finite dimensional approximation of the 3D Navier–Stokes equations forced by a degenerate noise. J. Stat. Phys. 114 (1–2) (2004) 155–177.
  • [55] M. Romito. Uniqueness and blow-up for the noisy viscous dyadic model. Preprint, 2011. Available at arXiv:1111.0536.
  • [56] R. Shvydkoy. On the energy of inviscid singular flows. J. Math. Anal. Appl. 349 (2) (2009) 583–595.
  • [57] T. Tao. Finite time blowup for an averaged three-dimensional Navier–Stokes equation. Preprint, 2014. Available at arXiv:1402.0290.
  • [58] R. Temam. Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition. Applied Mathematical Sciences 68. Springer, New York, 1997.
  • [59] M. I. Vishik, A. I. Komech and A. V. Fursikov. Some mathematical problems of statistical hydromechanics. Uspekhi Mat. Nauk 34 (5) (1979) 135–210, 256.