Abstract
We consider a class of invariant measures for a passive scalar f driven by an incompressible velocity field \({\varvec{u}}\), on a d-dimensional periodic domain, satisfying
The measures are obtained as limits of stochastic viscous perturbations. We prove that the span of the H 1 eigenfunctions of the operator \({\varvec{u} \cdot \nabla}\) contains the support of these measures. We also analyze several explicit examples: when \({\varvec{u}}\) is a shear flow or a relaxation enhancing flow (a generalization of weakly mixing), we can characterize the limiting measure uniquely and compute its covariance structure. We also consider the case of two-dimensional cellular flows, for which further regularity properties of the functions in the support of the measure can be deduced. The main results are proved with the use of spectral theory results, in particular the RAGE theorem, which are used to characterize large classes of orbits of the inviscid problem that are growing in H 1.
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Alexakis A., Tzella A.: Bounding the scalar dissipation scale for mixing flows in the presence of sources. J. Fluid Mech. 688, 443–460 (2011)
Bajer K., Bassom A.P., Gilbert A.D.: Accelerated diffusion in the centre of a vortex. J. Fluid Mech. 437, 395–411 (2001)
Beck, M., Wayne, C.E.: Metastability and rapid convergence to quasi-stationary bar states for the two-dimensional Navier–Stokes equations. Proc. R. Soc. Edinburgh Sect. A 143, 905–927 (2013)
Bedrossian, J., Masmoudi, N.: Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations, Publ. Math. de l’IHES (2015) (in print)
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. Ration. Mech. Anal. 219, 1087 (2016)
Berestycki H., Hamel F., Nadirashvili N.: Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena. Commun. Math. Phys. 253, 451–480 (2005)
Constantin P., Kiselev A., Ryzhik L., Zlatoš A.: Diffusion and mixing in fluid flow. Ann. Math. 168(2), 643–674 (2008)
Debussche, A., Glatt-Holtz, N., Temam, R.: Local martingale and pathwise solutions for an abstract fluids model. Phys. D 240, 1123–1144 (2011)
Da Prato G., Zabczyk J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)
Da Prato G., Zabczyk J.: Ergodicity for Infinite-Dimensional Systems. Cambridge University Press, Cambridge (1996)
Fayad B.: Weak mixing for reparameterized linear flows on the torus. Ergodic Theory Dyn. Syst. 22, 187–201 (2002)
Fayad, B.: Smooth mixing flows with purely singular spectra. Duke Math. J. 132,371–391 (2006)
Flandoli F., Gatarek D.: Martingale and stationary solutions for stochastic Navier–Stokes equations. Probab. Theory Relat. Fields 102, 367–391 (1995)
Freidlin M.: Reaction-diffusion in incompressible fluid: asymptotic problems. J. Differ. Equ. 179, 44–96 (2002)
Freidlin M.I., Wentzell AD: Random perturbations of Hamiltonian systems. Mem. Am. Math. Soc. 109, viii+82 (1994)
Glatt-Holtz N., Sverak V., Vicol V.: On inviscid limits for the stochastic Navier Stokes equations and related models. Arch. Ration. Mech. Anal. 217, 619 (2015)
Hairer, M., Koralov, L., Pajor-Gyulai, Z.: From averaging to homogenization in cellular flows-an exact description of the transition. ArXiv e-prints (2014). arXiv:1407.0982
Kifer Y.: On the principal eigenvalue in a singular perturbation problem with hyperbolic limit points and circles. J. Differ. Equ. 37, 108–139 (1980)
Kifer Y.: Random Perturbations of Dynamical Systems. Birkhäuser Boston, Inc., Boston (1988)
Kifer Y.: Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states. Isr. J. Math. 70, 1–47 (1990)
Kifer, Y.: Random perturbations of dynamical systems: a new approach. Lectures in Appl. Math., Amer. Math. Soc., Providence, RI (1991)
Kolmogorov, A.N.: On dynamical systems with an integral invariant on the torus. Doklady Akad. Nauk SSSR (N.S.). 93 (1953)
Koopman, B.O., Neumann, J.von : Dynamical systems of continuous spectra. Proc. Natl. Acad. Sci. USA 18, 255–263 (1932)
Krylov N.V.: Introduction to the theory of random processes. American Mathematical Society, Providence (2002)
Kuksin S.B.: The Eulerian limit for 2D statistical hydrodynamics. J. Stat. Phys. 115, 469–492 (2004)
Kuksin S.B.: Damped-driven KdV and effective equations for long-time behaviour of its solutions. Geom. Funct. Anal. 20, 1431–1463 (2010)
Kuksin S., Shirikyan A.: Randomly forced CGL equation: stationary measures and the inviscid limit. J. Phys. A 37, 3805–3822 (2004)
Kuksin S., Shirikyan A.: Mathematics of two-dimensional turbulence. Cambridge University Press, Cambridge (2012)
Iyer, G., Novikov, A.: Anomalous diffusion in fast cellular flows at intermediate time scales. Probab. Theory Relat. Fields 164(3–4), 707–740 (2016)
Lin Z., Thiffeault J.-L., Doering C.R.: Optimal stirring strategies for passive scalar mixing. J. Fluid Mech. 675, 465–476 (2011)
Lions J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires; Dunod. Gauthier-Villars, Paris (1969)
Lions J.-L., Magenes E.: Non-homogeneous boundary value problems and applications. Vol. I. Springer-Verlag, New York-Heidelberg (1972)
Mattingly J.C.: Pardoux, Etienne, Invariant measure selection by noise. An example. Discret. Contin. Dyn. Syst. 34, 4223–4257 (2014)
Reed M., Simon B.: Methods of modern mathematical physics. I. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York (1980)
Reed M., Simon B.: Methods of modern mathematical physics. III. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London (1979)
Rhines P.B., Young W.R.: How rapidly is a passive scalar mixed within closed streamlines?. J. Fluid Mech. 133, 133–145 (1983)
Seis C.: Maximal mixing by incompressible fluid flows. Nonlinearity 26, 3279–3289 (2013)
Šklover M.D.: Classical dynamical systems on the torus with continuous spectrum. Izv. Vysš. Učebn. Zaved. Matematika 1967, 113–124 (1967)
Yosida K.: Functional analysis. Vol. 123. Springer-Verlag, Berlin-New York (1980)
Zlatoš A.: Diffusion in fluid flow: dissipation enhancement by flows in 2D. Commun. Partial Differ. Equ. 35, 496–534 (2010)
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Bedrossian, J., Coti Zelati, M. & Glatt-Holtz, N. Invariant Measures for Passive Scalars in the Small Noise Inviscid Limit. Commun. Math. Phys. 348, 101–127 (2016). https://doi.org/10.1007/s00220-016-2758-9
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DOI: https://doi.org/10.1007/s00220-016-2758-9