Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Averaging of incompressible flows on two-dimensional surfaces

Authors: Dmitry Dolgopyat and Leonid Koralov
Journal: J. Amer. Math. Soc. 26 (2013), 427-449
MSC (2010): Primary 70K65, 60J60; Secondary 34E10, 37E35
Published electronically: October 22, 2012
MathSciNet review: 3011418
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a process on a compact two-dimensional surface which consists of the fast motion along the stream lines of an incompressible periodic vector field perturbed by white noise. It gives rise to a process on the graph naturally associated to the structure of the stream lines of the unperturbed flow. It has been shown by Freidlin and Wentzell that if the unperturbed motion is periodic for almost all the initial points, then the corresponding process on the graph weakly converges to a Markov process. We consider the situation when the unperturbed motion is not periodic and the flow has ergodic components of positive measure. We show that the process on the graph still converges to a Markov process, which spends a positive proportion of time in the vertices corresponding to the ergodic components of the flow.

As shown in a companion paper (Deterministic and stochastic perturbations of area preserving flows on a two-dimensional torus, by D. Dolgopyat, M. Freidlin, and L. Koralov), these results allow one to describe the viscosity regularization of dissipative deterministic perturbations of incompressible flows. In the case of surfaces of higher genus, the limiting process may exhibit intermittent behavior in the sense that the time axis can be divided into the intervals of random lengths where the particle stays inside one ergodic component separated by the intervals where the particle moves deterministically between the components.

References [Enhancements On Off] (What's this?)

  • [1] V. I. Arnold, Topological and ergodic properties of closed $ 1$-forms with incommensurable periods, Funktsional. Anal. i Prilozhen. 25 (1991), no. 2, 1-12, 96 (Russian); English transl., Funct. Anal. Appl. 25 (1991), no. 2, 81-90. MR 1142204 (93e:58104),
  • [2] Henri Berestycki, François Hamel, and Nikolai Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys. 253 (2005), no. 2, 451-480. MR 2140256 (2006b:35057),
  • [3] P. Constantin, A. Kiselev, L. Ryzhik, and A. Zlatoš, Diffusion and mixing in fluid flow, Ann. of Math. (2) 168 (2008), no. 2, 643-674. MR 2434887 (2009e:58045),
  • [4] I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ, Ergodic theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. MR 832433 (87f:28019)
  • [5] D. Dolgopyat, M. Freidlin, and L. Koralov, Deterministic and stochastic perturbations of area preserving flows on a two-dimensional torus, Erg. Th. Dyn. Sys. 32, no. 3 (2012), 899-918.
  • [6] Dmitry Dolgopyat and Leonid Koralov, Averaging of Hamiltonian flows with an ergodic component, Ann. Probab. 36 (2008), no. 6, 1999-2049. MR 2478675 (2010d:37105),
  • [7] Brice Franke, Integral inequalities for the fundamental solutions of diffusions on manifolds with divergence-free drift, Math. Z. 246 (2004), no. 1-2, 373-403. MR 2031461 (2005e:58060),
  • [8] Mark Freidlin, Markov processes and differential equations: asymptotic problems, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1996. MR 1399081 (97f:60150)
  • [9] Mark I. Freidlin and Alexander D. Wentzell, Diffusion processes on graphs and the averaging principle, Ann. Probab. 21 (1993), no. 4, 2215-2245. MR 1245308 (94j:60116)
  • [10] Mark I. Freidlin and Alexander D. Wentzell, Random perturbations of Hamiltonian systems, Mem. Amer. Math. Soc. 109 (1994), no. 523, viii+82. MR 1201269 (94j:35064)
  • [11] Ya. G. Sinaĭ and K. M. Khanin, Mixing of some classes of special flows over rotations of the circle, Funktsional. Anal. i Prilozhen. 26 (1992), no. 3, 1-21 (Russian); English transl., Funct. Anal. Appl. 26 (1992), no. 3, 155-169. MR 1189019 (93j:58079),
  • [12] R. Z. Hasminskiĭ, On the principle of averaging the Itô's stochastic differential equations, Kybernetika (Prague) 4 (1968), 260-279 (Russian, with Czech summary). MR 0260052 (41 #4681)
  • [13] Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374 (96c:58055)
  • [14] L. Koralov, Random perturbations of 2-dimensional Hamiltonian flows, Probab. Theory Related Fields 129 (2004), no. 1, 37-62. MR 2052862 (2005e:60179),
  • [15] Carlo Miranda, Partial differential equations of elliptic type, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 2, Springer-Verlag, New York, 1970. Second revised edition. Translated from the Italian by Zane C. Motteler. MR 0284700 (44 #1924)
  • [16] Igor Nikolaev and Evgeny Zhuzhoma, Flows on 2-dimensional manifolds, Lecture Notes in Mathematics, vol. 1705, Springer-Verlag, Berlin, 1999. An overview. MR 1707298 (2001b:37065)
  • [17] S. P. Novikov, The semiclassical electron in a magnetic field and lattice. Some problems of low-dimensional ``periodic'' topology, Geom. Funct. Anal. 5 (1995), no. 2, 434-444. MR 1334874 (96f:58124),
  • [18] Richard B. Sowers, Random perturbations of two-dimensional pseudoperiodic flows, Illinois J. Math. 50 (2006), no. 1-4, 853-959 (electronic). MR 2247849 (2008a:60084)
  • [19] Corinna Ulcigrai, Weak mixing for logarithmic flows over interval exchange transformations, J. Mod. Dyn. 3 (2009), no. 1, 35-49. MR 2481331 (2010b:37005),
  • [20] Andrej Zlatoš, Diffusion in fluid flow: dissipation enhancement by flows in 2D, Comm. Partial Differential Equations 35 (2010), no. 3, 496-534. MR 2748635 (2012c:35178),

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 70K65, 60J60, 34E10, 37E35

Retrieve articles in all journals with MSC (2010): 70K65, 60J60, 34E10, 37E35

Additional Information

Dmitry Dolgopyat
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742

Leonid Koralov
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742

Received by editor(s): February 15, 2011
Received by editor(s) in revised form: July 28, 2012
Published electronically: October 22, 2012
Additional Notes: While working on this article, both authors were supported by the NSF grant DMS-0854982
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society