Skip to main content
SpringerLink
Log in

Almost-sure enhanced dissipation and uniform-in-diffusivity exponential mixing for advection–diffusion by stochastic Navier–Stokes

  • Published:
Probability Theory and Related Fields Aims and scope Submit manuscript

Abstract

We study the mixing and dissipation properties of the advection–diffusion equation with diffusivity \(0 < \kappa \ll 1\) and advection by a class of random velocity fields on \(\mathbb {T}^d\), \(d=\{2,3\}\), including solutions of the 2D Navier–Stokes equations forced by sufficiently regular-in-space, non-degenerate white-in-time noise. We prove that the solution almost surely mixes exponentially fast uniformly in the diffusivity \(\kappa \). Namely, that there is a deterministic, exponential rate (independent of \(\kappa \)) such that all mean-zero \(H^1\) initial data decays exponentially fast in \(H^{-1}\) at this rate with probability one. This implies almost-sure enhanced dissipation in \(L^2\). Specifically that there is a deterministic, uniform-in-\(\kappa \), exponential decay in \(L^2\) after time \(t > rsim \left| \log \kappa \right| \). Both the \(\mathcal {O}(\left| \log \kappa \right| )\) time-scale and the uniform-in-\(\kappa \) exponential mixing are optimal for Lipschitz velocity fields. This work is also a major step in our program on scalar mixing and Lagrangian chaos necessary for a rigorous proof of the Batchelor power spectrum of passive scalar turbulence.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The case without diffusion follows almost immediately from the multiplicative ergodic theorem (see [14]), however, it requires an additional check to ensure that the random constant \(\overline{D}\) possesses good moment bounds (Lemma 7.3).

  2. Equivalently, we can think of \((u_t, x_t^\kappa , v_t^\kappa )\) as evolving on the sphere bundle \(\mathbf{H}\times S \mathbb {T}^d\), where \(S \mathbb {T}^d \cong \mathbb {T}^d \times \mathbb {S}^{d-1}\). In this parametrization, \(v_t^\kappa \) evolves according to the random ODE

    $$\begin{aligned} \dot{v}_t^\kappa = (1 - v_t^\kappa \otimes v_t^\kappa ) D u_t(x_t^\kappa ) v_t^\kappa \, . \end{aligned}$$

References

  1. Alberti, G., Crippa, G., Mazzucato, A.: Exponential self-similar mixing by incompressible flows. J. Am. Math. Soc. 32(2), 445–490 (2019)

    MathSciNet  MATH  Google Scholar 

  2. Amarouchene, Y., Kellay, H.: Batchelor scaling in fast-flowing soap films. Phys. Rev. Lett. 93(21), 214504 (2004)

    Google Scholar 

  3. Amon, C.H., Guzmán, A.M., Morel, B.: Lagrangian chaos, Eulerian chaos, and mixing enhancement in converging-diverging channel flows. Phys. Fluids 8(5), 1192–1206 (1996)

    MathSciNet  MATH  Google Scholar 

  4. Antonia, R.A., Orlandi, P.: Effect of schmidt number on small-scale passive scalar turbulence. Appl. Mech. Rev. 56(6), 615–632 (2003)

    Google Scholar 

  5. Antonsen Jr., T.M., Fan, Z., Ott, E., Garcia-Lopez, E.: The role of chaotic orbits in the determination of power spectra of passive scalars. Phys. Fluids 8(11), 3094–3104 (1996)

    MATH  Google Scholar 

  6. Aref, H., Blake, J.R., et al.: Frontiers of chaotic advection. Rev. Mod. Phys. 89(2), 025007 (2017)

    MathSciNet  Google Scholar 

  7. Arnold, L.: A formula connecting sample and moment stability of linear stochastic systems. SIAM J. Appl. Math. 44(4), 793–802 (1984)

    MathSciNet  MATH  Google Scholar 

  8. Bajer, K., Bassom, A.P., Gilbert, A.D.: Accelerated diffusion in the centre of a vortex. J. Fluid Mech. 437, 395–411 (2001)

    MathSciNet  MATH  Google Scholar 

  9. Bakry, D., Cattiaux, P., Guillin, A.: Rate of convergence for ergodic continuous markov processes: Lyapunov versus poincaré. J. Funct. Anal. 254(3), 727–759 (2008)

    MathSciNet  MATH  Google Scholar 

  10. Baladi, V., Benedicks, M. Maume-Deschamps, V.: Almost sure rates of mixing for iid unimodal maps, Ann. Sci. de l’école normale supérieure 35(1), 77–126 (2002)

  11. Batchelor, G.K.: Small-scale variation of convected quantities like temperature in turbulent fluid Part 1. General discussion and the case of small conductivity. J. Fluid Mech. 5(1), 113–133 (1959)

    MathSciNet  MATH  Google Scholar 

  12. Baxendale, P.H., Stroock, D.W.: Large deviations and stochastic flows of diffeomorphisms. Probab. Theory Relat. Fields 80(2), 169–215 (1988)

    MathSciNet  MATH  Google Scholar 

  13. Bedrossian, J., Blumenthal, A., Punshon-Smith, S.: Almost-sure exponential mixing of passive scalars by the stochastic Navier–Stokes equations (2019). arXiv preprint arXiv:1905.03869

  14. Bedrossian, J., Blumenthal, A., Punshon-Smith, S.: Lagrangian chaos and scalar advection in stochastic fluid mechanics (2018). arXiv preprint arXiv:1809.06484

  15. Bedrossian, J., Blumenthal, A., Punshon-Smith, S.: Passive scalar turbulence in the batchelor regime. In preparation (2019)

  16. Bedrossian, J., Germain, P., Masmoudi, N.: Dynamics near the subcritical transition of the 3D Couette flow I: below threshold. Mem. AMS (2015a) (in press)

  17. Bedrossian, J., Germain, P., Masmoudi, N.: Dynamics near the subcritical transition of the 3D Couette flow II: above threshold. Mem. AMS (2015b) (in press)

  18. Bedrossian, J.: Suppression of plasma echoes and Landau damping in Sobolev spaces by weak collisions in a Vlasov–Fokker–Planck equation. Ann PDE 3(2), 19 (2017)

    MathSciNet  MATH  Google Scholar 

  19. Bedrossian, J., Coti Zelati, M.: Enhanced dissipation, hypoellipticity, and anomalous small noise inviscid limits in shear flows. Arch. Rat. Mech. Anal. 224(3), 1161–1204 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Bedrossian, J., Masmoudi, N., Vicol, V.: Enhanced dissipation and inviscid damping in the inviscid limit of the Navier–Stokes equations near the 2D Couette flow. Arch. Rat. Mech. Anal. 216(3), 1087–1159 (2016)

    MATH  Google Scholar 

  21. Bedrossian, J., Coti Zelati, M., Glatt-Holtz, N.: Invariant measures for passive scalars in the small noise inviscid limit. Commun. Math. Phys. 348(1), 101–127 (2016)

    MathSciNet  MATH  Google Scholar 

  22. Bernoff, A.J., Lingevitch, J.F.: Rapid relaxation of an axisymmetric vortex. Phys. Fluids 6, 3717 (1994)

    MATH  Google Scholar 

  23. Bohr, T., Jensen, M.H., Paladin, G., Vulpiani, A.: Dynamical Systems Approach to Turbulence. Cambridge University Press, Cambridge (2005)

    MATH  Google Scholar 

  24. Bressan, A.: A lemma and a conjecture on the cost of rearrangements. Rend. Sem. Mat. Univ. Padova 110, 97–102 (2003)

    MathSciNet  MATH  Google Scholar 

  25. Constantin, P., Kiselev, A., Ryzhik, L., Zlatoš, A.: Diffusion and mixing in fluid flow. Ann. Math. 2(168), 643–674 (2008)

    MathSciNet  MATH  Google Scholar 

  26. Coti Zelati, M., Delgadino, M.G., Elgindi, T.M.: On the relation between enhanced dissipation timescales and mixing rates. Commun. Pure Appl. Math. 73(6), 1205–1244 (2020)

  27. Coti Zelati, M.: Stable mixing estimates in the infinite Péclet number limit (2019). arXiv preprint arXiv:1909.01310

  28. Crippa, G., Renato, L., Schulze, C.: Polynomial mixing under a certain stationary Euler flow. Physica D 394, 44–55 (2019)

    MathSciNet  MATH  Google Scholar 

  29. Crisanti, A., Falcioni, M., Vulpiani, A., Paladin, G.: Lagrangian chaos: transport, mixing and diffusion in fluids. In:L a Rivista del Nuovo Cimento (1978–1999), vol. 14, no. 12, pp. 1–80 (1991)

  30. Da Prato, G.: Introduction to Stochastic Analysis and Malliavin Calculus, vol. 13. Springer, New York (2014)

    MATH  Google Scholar 

  31. Da Prato, G., Zabczyk, J.: Ergodicity for Infinite-Dimensional Systems, London Mathematical Society Lecture Note Series, vol. 229. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  32. Dolgopyat, D., Kaloshin, V., Koralov, L., et al.: Sample path properties of the stochastic flows. Ann. Probab. 32(1A), 1–27 (2004)

    MathSciNet  MATH  Google Scholar 

  33. Donzis, D.A., Sreenivasan, K.R., Yeung, P.K.: The batchelor spectrum for mixing of passive scalars in isotropic turbulence. Flow Turbul. Combust. 85(3–4), 549–566 (2010)

    MATH  Google Scholar 

  34. Down, D., Meyn, S.P., Tweedie, R.L.: Exponential and uniform ergodicity of Markov processes. Ann. Prob. 23(4), 1671–1691 (1995)

    MathSciNet  MATH  Google Scholar 

  35. Eckmann, J.-P., Hairer, M.: Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise. Commun. Math. Phys. 219(3), 523–565 (2001)

    MathSciNet  MATH  Google Scholar 

  36. Elgindi, T.M., Zlatoš, A.: Universal mixers in all dimensions (2018). arXiv preprint arXiv:1809.09614

  37. Feng, Y., Iyer, G.: Dissipation enhancement by mixing. Nonlinearity 32(5), 1810 (2019)

    MathSciNet  MATH  Google Scholar 

  38. Flandoli, F., Maslowski, B.: Ergodicity of the 2-D Navier–Stokes equation under random perturbations. Commun. Math. Phys. 172(1), 119–141 (1995)

    MathSciNet  MATH  Google Scholar 

  39. Frisch, U.: Turbulence: The Legacy of A.N. Kolmogorov. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  40. Gallay, T.: Enhanced dissipation and axisymmetrization of two-dimensional viscous vortices. Arch. Ration. Mech. Anal. 230(3), 939–975 (2018)

    MathSciNet  MATH  Google Scholar 

  41. Galluccio, S., Vulpiani, A.: Stretching of material lines and surfaces in systems with lagrangian chaos. Phys. A 212(1–2), 75–98 (1994)

    Google Scholar 

  42. Gibson, C., Schwarz, W.: The universal equilibrium spectra of turbulent velocity and scalar fields. J. Fluid Mech. 16(3), 365–384 (1963)

    MATH  Google Scholar 

  43. Glatt-Holtz, N.E., Herzog, D.P., Mattingly, J.C.: Scaling and saturation in infinite-dimensional control problems with applications to stochastic partial differential equations. Ann. PDE 4(2), 16 (2018)

    MathSciNet  MATH  Google Scholar 

  44. Hairer, M., Mattingly, J.C.: Yet another look at Harris’s ergodic theorem for Markov chains. In: Seminar on Stochastic Analysis, Random Fields and Applications, vol. vi, pp. 109–117 (2011)

  45. Hairer, M.: On Malliavin’s proof of Hörmander’s theorem. Bull. Sci. Math. 135(6–7), 650–666 (2011)

    MathSciNet  MATH  Google Scholar 

  46. Hairer, M., Mattingly, J.C.: Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann. Math. 164(3), 993–1032 (2006)

    MathSciNet  MATH  Google Scholar 

  47. Hairer, M., Mattingly, J.C.: Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations. Ann. Probab. 36(6), 2050–2091 (2008)

    MathSciNet  MATH  Google Scholar 

  48. Hairer, M., Mattingly, J.C.: A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs. Electron. J. Probab. 16(23), 658–738 (2011)

    MathSciNet  MATH  Google Scholar 

  49. Haynes, P.H., Vanneste, J.: What controls the decay of passive scalars in smooth flows? Phys. Fluids 17(9), 097103 (2005)

    MathSciNet  MATH  Google Scholar 

  50. Iyer, G., Kiselev, A., Xu, X.: Lower bounds on the mix norm of passive scalars advected by incompressible enstrophy-constrained flows. Nonlinearity 27(5), 973 (2014)

    MathSciNet  MATH  Google Scholar 

  51. Jaksic, V., Nersesyan, V., Pillet, C.-A., Shirikyan, A.: Large deviations and entropy production in viscous fluid flows (2019). arXiv preprint arXiv:1902.03278

  52. Kuksin, S., Nersesyan, V., Shirikyan, A.: Exponential mixing for a class of dissipative PDES with bounded degenerate noise (2018). arXiv preprint arXiv:1802.03250

  53. Kuksin, S., Nersesyan, V., Shirikyan, A.: Mixing via controllability for randomly forced nonlinear dissipative PDES (2019). arXiv preprint arXiv:1902.00494

  54. Kuksin, S., Shirikyan, A.: Mathematics of Two-Dimensional Turbulence, vol. 194. Cambridge University Press, Cambridge (2012)

    MATH  Google Scholar 

  55. Latini, M., Bernoff, A.J.: Transient anomalous diffusion in Poiseuille flow. J. Fluid Mech. 441, 399–411 (2001)

    MATH  Google Scholar 

  56. Lin, Z., Thiffeault, J.-L., Doering, C.R.: Optimal stirring strategies for passive scalar mixing. J. Fluid Mech. 675, 465–476 (2011)

    MathSciNet  MATH  Google Scholar 

  57. Lunasin, E., Lin, Z., Novikov, A., Mazzucato, A., Doering, C.R.: Optimal mixing and optimal stirring for fixed energy, fixed power, or fixed palenstrophy flows. J. Math. Phys. 53(11), 115611 (2012)

    MathSciNet  MATH  Google Scholar 

  58. Meyn, S.P., Tweedie, R.L., et al.: Computable bounds for geometric convergence rates of Markov Chains. Ann. Appl. Probab. 4(4), 981–1011 (1994)

    MathSciNet  MATH  Google Scholar 

  59. Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Springer, New York (2012)

    MATH  Google Scholar 

  60. Miles, C.J., Doering, C.R.: Diffusion-limited mixing by incompressible flows. Nonlinearity 31(5), 2346 (2018)

    MathSciNet  MATH  Google Scholar 

  61. Miller, P.L., Dimotakis, P.E.: Measurements of scalar power spectra in high schmidt number turbulent jets. J. Fluid Mech. 308, 129–146 (1996)

    Google Scholar 

  62. Ngan, K., Vanneste, J.: Scalar decay in a three-dimensional chaotic flow. Phys. Rev. E 83(5), 056306 (2011)

    Google Scholar 

  63. Nualart, D.: The Malliavin Calculus and Related Topics, vol. 1995. Springer, New York (2006)

    MATH  Google Scholar 

  64. Ottino, J.M.: Mixing, chaotic advection, and turbulence. Annu. Rev. Fluid Mech. 22(1), 207–254 (1990)

    MathSciNet  Google Scholar 

  65. Poon, C.C.: Unique continuation for parabolic equations. Commun. Partial Differ. Equ. 21(3–4), 521–539 (1996)

    MathSciNet  MATH  Google Scholar 

  66. Provenzale, A.: Transport by coherent barotropic vortices. Annu. Rev. Fluid Mech. 31(1), 55–93 (1999)

    MathSciNet  Google Scholar 

  67. Rhines, P.B., Young, W.R.: How rapidly is a passive scalar mixed within closed streamlines? J. Fluid Mech. 133, 133–145 (1983)

    MATH  Google Scholar 

  68. Romito, M., Xu, L.: Ergodicity of the 3D stochastic Navier–Stokes equations driven by mildly degenerate noise. Stoch. Processes Appl. 121(4), 673–700 (2011)

    MathSciNet  MATH  Google Scholar 

  69. Seidler, J.: A note on the strong Feller property, Unpublished lecture notes (2001)

  70. Seis, C.: Maximal mixing by incompressible fluid flows. Nonlinearity 26(12), 3279 (2013)

    MathSciNet  MATH  Google Scholar 

  71. Shraiman, B.I., Siggia, E.D.: Scalar turbulence. Nature 405(6787), 639 (2000)

    Google Scholar 

  72. Thiffeault, J.-L.: Using multiscale norms to quantify mixing and transport. Nonlinearity 25(2), R1–R44 (2012)

    MathSciNet  MATH  Google Scholar 

  73. Tristani, I.: Landau damping for the linearized Vlasov Poisson equation in a weakly collisional regime. J. Stat. Phys. 169(1), 107–125 (2017)

    MathSciNet  MATH  Google Scholar 

  74. Tsang, Y.-K., Antonsen Jr., T.M., Ott, E.: Exponential decay of chaotically advected passive scalars in the zero diffusivity limit. Phys. Rev. E 71(6), 066301 (2005)

    MathSciNet  Google Scholar 

  75. Varadhan, S.S.: Large Deviations and Applications, vol. 46. Siam, Philadelphia (1984)

    MATH  Google Scholar 

  76. Warhaft, Z.: Passive scalars in turbulent flows. Annu. Rev. Fluid Mech. 32(1), 203–240 (2000)

    MathSciNet  MATH  Google Scholar 

  77. Wei, D., Zhang, Z., Zhao, W.: Linear inviscid damping and enhanced dissipation for the kolmogorov flow (2017). arXiv preprint arXiv:1711.01822

  78. Weinan, E., Mattingly, J.C.: Ergodicity for the Navier–Stokes equation with degenerate random forcing: finite-dimensional approximation. Commun. Pure Appl. Math. 54(11), 1386–1402 (2001)

    MathSciNet  MATH  Google Scholar 

  79. Wunsch, C., Ferrari, R.: Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281–314 (2004)

    MathSciNet  MATH  Google Scholar 

  80. Yao, Y., Zlatoš, A.: Mixing and un-mixing by incompressible flows. J. Eur. Math. Soc. 19(7), 1911–1948 (2017)

    MathSciNet  MATH  Google Scholar 

  81. Yuan, G.-C., Nam, K., Antonsen Jr., T.M., Ott, E., Guzdar, P.N.: Power spectrum of passive scalars in two dimensional chaotic flows. Chaos: Interdiscip. J. Nonlinear Sci. 10(1), 39–49 (2000)

    MathSciNet  MATH  Google Scholar 

  82. Zillinger, C.: On geometric and analytic mixing scales: comparability and convergence rates for transport problems (2018). arXiv preprint arXiv:1804.11299

  83. Zlatoš, A.: Diffusion in fluid flow: dissipation enhancement by flows in 2D. Commun. Partial Differ. Equ. 35(3), 496–534 (2010)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Jacob Bedrossian

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Alex Blumenthal

  3. Division of Applied Mathematics, Brown University, Providence, RI, 02906, USA

    Sam Punshon-Smith

Authors
  1. Jacob Bedrossian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alex Blumenthal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sam Punshon-Smith
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jacob Bedrossian.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This material is based upon work supported by the National Science Foundation under Award No’s DMS-1552826, RNMS-1107444, DMS-1604805 and DMS-1803481.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bedrossian, J., Blumenthal, A. & Punshon-Smith, S. Almost-sure enhanced dissipation and uniform-in-diffusivity exponential mixing for advection–diffusion by stochastic Navier–Stokes. Probab. Theory Relat. Fields 179, 777–834 (2021). https://doi.org/10.1007/s00440-020-01010-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00440-020-01010-8

Keywords

Mathematics Subject Classification

Search

Navigation