Abstract
We establish a general criterion which ensures exponential mixing of parabolic stochastic partial differential equations (SPDE) driven by a non additive noise which is white in time and smooth in space. We apply this criterion on two representative examples: 2D Navier–Stokes (NS) equations and Complex Ginzburg–Landau (CGL) equation with a locally Lipschitz noise. Due to the possible degeneracy of the noise, Doob theorem cannot be applied. Hence, a coupling method is used in the spirit of Kuksin and Shirikyan (J. Math. Pures Appl. 1:567–602, 2002) and Mattingly (Commun. Math. Phys. 230:421–462, 2002).
Previous results require assumptions on the covariance of the noise which might seem restrictive and artificial. For instance, for NS and CGL, the covariance operator is supposed to be diagonal in the eigenbasis of the Laplacian and not depending on the high modes of the solutions. The method developed in the present paper gets rid of such assumptions and only requires that the range of the covariance operator contains the low modes.
References
Barton-Smith M. (2004). Invariant measure for the stochastic Ginzburg Landau equation. Nonlinear Diff. Equ. Appl. 11(1): 29–52
Bebouche P. and Jüngel A. (2000). Inviscid limits of the complex Ginzburg–Landau equation. Commun. Math. Phys. 214: 201–226
Bricmont J., Kupiainen A. and Lefevere R. (2002). Exponential mixing for the 2D stochastic Navier–Stokes dynamics. Commun. Math. Phys. 230(1): 87–132
Constantin P. and Foias C. (1988). Navier–Stokes Equations. University of Chicago Press, Chicago
Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. In: London Mathematical Society Lecture Notes, n. 229, Cambridge University Press, Cambridge (1996)
Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. In: London Mathematical Society Lecture Notes, n. 229, Cambridge University Press, Cambridge (1996)
Debussche, A., Odasso, C.: Ergodicity for the weakly damped stochastic non-linear Schrödinger equations. J. Evol. Equ. Birkhser Basel 5(3):317–356
W.E. Mattingly J.C. and Sinai Y.G. (2001). Gibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equation. Commun. Math. Phys. 224: 83–106
Flandoli F. and Gatarek D. (1995). Martingale and stationary solutions for stochastic Navier–Stokes equations. PTRF 102: 367–391
Ginzburg, V., Landau, L.: On the theorie of superconductivity. Zh. Eksp. Fiz. 20, 1064 (1950) English transl. In: Ter Haar (ed.) Men of Physics: L.D. Landau, vol. I, pp. 546–568. Pergammon Press, New York (1965)
Goldys B. and Maslowski B. (2005). Exponential ergodicity for stochastic Burgers and 2D Navier–Stokes equations. J. Funct. Anal. 226: 230–255
Gyöngy I. and Krylov N.V. (1996). Existence of strong solutions for Itô’s stochastic equations via approximations. Probab. Theory Relat. Fields 105: 143–158
Hairer M. (2002). Exponential mixing properties of stochastic PDEs through asymptotic coupling. Proba. Theory Related Fields 124(3): 345–380
Hairer, M., Mattingly, J.: Ergodicity of the 2D Navier–Stokes equations with degenerate forcing. (preprint)
Huber G. and Alstrom P. (1993). Universal Decay of vortex density in two dimensions. Physica A 195: 448–456
Kuksin S. and Shirikyan A. (2000). Stochastic dissipative PDE’s and Gibbs measures. Comm. Math. Phys. 213(2): 291–330
Kuksin S. and Shirikyan A. (2000). Stochastic dissipative PDE’s and Gibbs measures. Commun. Math. Phys. 213: 291–330
Kuksin, S., Shirikyan, A.: Ergodicity for the randomly forced 2D Navier–Stokes equations. Math. Phys. Anal. Geom.4, (2001)
Kuksin S. and Shirikyan A. (2001). A coupling approach to randomly forced randomly forced PDE’s I. Commun. Math. Phys. 221: 351–366
Kuksin S., Piatnitski A. and Shirikyan A. (2002). A coupling approach to randomly forced randomly forced PDE’s II. Commun. Math. Phys. 230(1): 81–85
Kuksin S. and Shirikyan A. (2002). Coupling approach to white-forced nonlinear PDEs. J. Math. Pures Appl. 1: 567–602
Kuksin S. and Shirikyan A. (2004). Randomly forced CGL equation: stationnary measure and the inviscid limit. J. Phys. A 37(12): 3805–3822
Lindvall T. (1992). Lectures on the Coupling Method. Wiley, New York
Mattingly J. (2002). Exponential convergence for the stochastically forced Navier–Stokes equations and other partially dissipative dynamics. Commun. Math. Phys. 230: 421–462
Mattingly, J., Pardoux, E.: Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. (preprint) (2004)
Mikulevicius R. and Rozovskii B.L. (2005). Global L 2-solutions of stochastic Navier–Stokes equations. Ann. Probab. 33(1): 137–176
Newel A. and Whitehead J. (1969). Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38: 279–303
Newel A. and Whitehead J. (1971). Review of the finite bandwidth concept. In: Leipholz, H (eds) Proceedings of the Internat., pp 284–289. Union of Theoretical and Applicable Mathematics. Springer, Berlin
Odasso C. (2006). Ergodicity for the stochastic Complex Ginzburg–Landau equations. Annales de l’Institut Henri Poincare (B). Probab. Stat. 42(4): 417–454
Shirikyan A. (2004). Exponential mixing for 2D Navier–Stokes equation pertubed by an unbounded noise. J. Math. Fluid Mech. 6(2): 169–193
Temam R. (1977). Navier–Stokes Equations. Theory and Numerical Analysis. North-Holland, Amsterdam
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Odasso, C. Exponential mixing for stochastic PDEs: the non-additive case. Probab. Theory Relat. Fields 140, 41–82 (2008). https://doi.org/10.1007/s00440-007-0057-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-007-0057-2
Keywords
- Two-dimensional Navier–Stokes equations
- Complex Ginzburg–Landau equations
- Markov transition semi-group
- Invariant measure
- Ergodicity
- Coupling method
- Girsanov formula
- Expectational Foias–Prodi estimate