Skip to main content
SpringerLink
Log in

Enhanced Dissipation, Hypoellipticity, and Anomalous Small Noise Inviscid Limits in Shear Flows

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We analyze the decay and instant regularization properties of the evolution semigroups generated by two-dimensional drift-diffusion equations in which the scalar is advected by a shear flow and dissipated by full or partial diffusion. We consider both the space-periodic \({\mathbb{T}^2}\) setting and the case of a bounded channel \({\mathbb{T} \times [0,1]}\) with no-flux boundary conditions. In the infinite Péclet number limit (diffusivity \({\nu\to 0}\)), our work quantifies the enhanced dissipation effect due to the shear. We also obtain hypoelliptic regularization, showing that solutions are instantly Gevrey regular even with only partial diffusion. The proofs rely on localized spectral gap inequalities and ideas from hypocoercivity with an augmented energy functional with weights replaced by pseudo-differential operators (of a rather simple form). As an application, we study small noise inviscid limits of invariant measures of stochastic perturbations of passive scalars, and show that the classical Freidlin scaling between noise and diffusion can be modified. In particular, although statistically stationary solutions blow up in \({H^1}\) in the limit \({\nu \to 0}\), we show that viscous invariant measures still converge to a unique inviscid measure.

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

References

  1. Alexakis A., Tzella A.: Bounding the scalar dissipation scale for mixing flows in the presence of sources. J. Fluid Mech. 688, 443–460 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Alexandre R., Morimoto Y., Ukai S., Xu C.-J., Yang T.: Regularizing effect and local existence for the non-cutoff Boltzmann equation. Arch. Ration. Mech. Anal. 198(1), 39–123 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Beauchard K., Zuazua E.: Some controllability results for the 2D Kolmogorov equation. Ann. Inst. Henri Poincaré Anal. Non Linéaire 26(5), 1793–1815 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Beauchard K.: Null controllability of kolmogorov-type equations. Math. Control Signals Syst. 26(1), 145–176 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Beck M., Wayne C.E.: Metastability and rapid convergence to quasi-stationary bar states for the two-dimensional Navier–Stokes equations. Proc. R. Soc. Edinb. Sect. A Math. 143(05), 905–927 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bedrossian, J., Germain, P., Masmoudi, N.: Dynamics near the subcritical transition of the 3D Couette flow I: Below threshold, arXiv:1506.03720, 2015

  8. Bedrossian, J., Germain, P., Masmoudi, N.: Dynamics near the subcritical transition of the 3D Couette flow II: Above threshold, arXiv:1506.03721, 2015

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Bedrossian J., Masmoudi N., Vicol V.: Enhanced dissipation and inviscid damping in the inviscid limit of the Navier–Stokes equations near the two dimensional Couette flow. Arch. Ration. Mech. Anal. 219(3), 1087–1159 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Berestycki H., Hamel F., Nadirashvili N.: Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena. Commun. Math. Phys. 253(2), 451–480 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  13. Chen, H., Li, W.-X., Xu, C.-J.: Gevrey hypoellipticity for linear and non-linear fokkerplanck equations, J. Differ. Equ. 246(1), 320–339, 2009

  14. Chen Y., Desvillettes L., He L.: Smoothing effects for classical solutions of the full Landau equation. Arch. Ration. Mech. Anal. 193(1), 21–55 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

  16. Da Prato, G., Zabczyk, J.: Ergodicity for infinite dimensional systems, Vol. 229, Cambridge University Press, Cambridge, 1996

  17. Desvillettes, L., Villani, C., et al., On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems. Part I: the linear Fokker–Planck equation, Commun. Pure Appl. Math. 54(1), 1–42, 2001

  18. Dolbeault J., Mouhot C., Schmeiser C.: Hypocoercivity for linear kinetic equations conserving mass. Trans. Am. Math. Soc. 367(6), 3807–3828 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dubrulle B., Nazarenko S.: On scaling laws for the transition to turbulence in uniform-shear flows. EPL (Europhys. Lett.) 27(2), 129 (1994)

    Article  ADS  Google Scholar 

  20. Freidlin M.: Reaction-diffusion in incompressible fluid: asymptotic problems. J. Differ. Equ. 179(1), 44–96 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Freidlin, M.I., Wentzell, A.D.: Random perturbations of Hamiltonian systems, Mem. Am. Math. Soc. 109(523), viii+82, 1994

  22. Gallagher, I., Gallay, T., Nier, F.: Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator, Int. Math. Res. Not. 2009, rnp013

  23. Gevrey M.: Sur la nature analytique des solutions des équations aux dérivées partielles. Premier mémoire. Ann. Sci. École Norm. Sup. 35(3), 129–190 (1918)

    MathSciNet  MATH  Google Scholar 

  24. Glatt-Holtz, N., Šverák, V., Vicol, V.: On inviscid limits for the stochastic navier-stokes equations and related models, Arch. Ration. Mech. Anal. 217, 619–649, 2015

  25. Hérau F.: Short and long time behavior of the Fokker-Planck equation in a confining potential and applications. J. Funct. Anal. 244(1), 95–118 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hérau F., Nier F.: Isotropic hypoellipticity and trend to equilibrium for the Fokker–Planck equation with a high-degree potential. Arch. Ration. Mech. Anal. 171(2), 151–218 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hörmander, L.: The analysis of linear partial differential operators III: Pseudo-differential operators, Vol. 274, Springer, Berlin, 1985

  28. Hörmander L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  29. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Kelvin L.: Stability of fluid motion-rectilinear motion of viscous fluid between two parallel plates. Phil. Mag. 24, 188 (1887)

    Article  Google Scholar 

  31. Kolmogoroff A.: Zufällige Bewegungen (zur Theorie der Brownschen Bewegung). Ann. Math. 35(1), 116–117 (1934)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kuksin S., Shirikyan A.: Randomly forced CGL equation: stationary measures and the inviscid limit. J. Phys. A 37, 3805–3822 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Kuksin S.: Damped-driven KdV and effective equations for long-time behaviour of its solutions. Geom. Funct. Anal. 20, 1431–1463 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kuksin, S., Shirikyan, A.: Mathematics of two-dimensional turbulence, Vol. 194, Cambridge University Press, Cambridge, 2012

  35. Kuksin, S.B.: The Eulerian limit for 2D statistical hydrodynamics, J. Stat. Phys. 115(1–2), 469–492, 2004

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

    Article  ADS  MATH  Google Scholar 

  37. Li, W., Wu, D., Xu, C.-J.: Gevrey class smoothing effect for the Prandtl equation, arXiv preprint, arXiv:1502.03569 2015

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Lin Z., Zeng C.: Inviscid dynamic structures near Couetteflow. Arch. Ration. Mech. Anal. 200, 1075–1097 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lundgren T.: Strained spiral vortex model for turbulent fine structure. Phys. Fluids 25, 2193 (1982)

    Article  ADS  MATH  Google Scholar 

  41. Mattingly J.C., Pardoux E.: Invariant measure selection by noise. An example. Discrete Contin. Dyn. Syst. 34(10), 4223–4257 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  42. Reed, M., Simon, B.: Methods of modern mathematical physics. I, Second, Academic Press, Inc. [Harcourt Brace Jo-vanovich, Publishers], New York, 1980. Functional analysis. MR751959 (85e:46002)

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

    Article  ADS  MATH  Google Scholar 

  44. Seis C.: Maximal mixing by incompressible fluid flows. Nonlinearity 26, 3279–3289 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, Vol. 43, Princeton University Press, Princeton, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. MR1232192 (95c:42002)

  46. Trefethen, L.N., Embree, M.: Spectra and pseudospectra: the behavior of nonnormal matrices and operators, Princeton University Press, Princeton, 2005

  47. Villani, C.: Hypocoercivity, American Mathematical Soc., 2009

  48. Vukadinovic J., Dedits E., Poje A.C., Schäfer T.: Averaging and spectral properties for the 2D advection–diffusion equation in the semi-classical limit for vanishing diffusivity. Phys. D 310, 1–18 (2015)

    Article  MathSciNet  Google Scholar 

  49. Zillinger C.: Linear inviscid damping for monotone shear flows in a finite periodic channel, boundary effects, blow-up and critical Sobolev regularity. Arch. Ration. Mech. Anal. 221(3), 1449–1509 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  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 & Michele Coti Zelati

Authors
  1. Jacob Bedrossian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michele Coti Zelati
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michele Coti Zelati.

Additional information

Communicated by P. Constantin

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bedrossian, J., Coti Zelati, M. Enhanced Dissipation, Hypoellipticity, and Anomalous Small Noise Inviscid Limits in Shear Flows. Arch Rational Mech Anal 224, 1161–1204 (2017). https://doi.org/10.1007/s00205-017-1099-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-017-1099-y

Search

Navigation