We study the long time behavior of the solution X(t, s, x) of a 2D-Navier–Stokes equation subjected to a periodic time dependent forcing term. We prove in particular that as \({t \to \infty}\) , \({\mathbb{E}[\varphi(X(t, s, x))]}\) approaches a periodic orbit independently of s and x for any continuous and bounded real function \({\varphi}\).
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Albeverio S. and Cruzeiro A.B. (1990). Global flows with invariant (Gibbs) measures for Euler and Navier–Stokes two dimensional fluids. Commun. Math. Phys. 129: 431–444
Bensoussan A. and Temam R. (1973). Équations stochastiques du type Navier-Stokes. J. Funct. Anal. 13: 195–222
Bricmont J., Kupiainen A. and Lefevere R. (2002). Exponential mixing for the 2D Navier–Stokes dynamics. Comm. Math. Phys. 230(1): 87–132
Brzezniak Z., Capinski M. and Flandoli F. (1992). Stochastic Navier-Stokes equations with multiplicative noise. Stoch. Anal. Appl. 10: 523–532
Capinski M. and Gatarek D. (1994). Stochastic equations in Hilbert spaces with applications to Navier-Stokes equations in any dimension. J. Funct. Anal. 126: 26–35
Da Prato G. and Debussche A. (2003). Ergodicity for the 3D stochastic Navier-Stokes equations. J. Math. Pures et Appl. 82: 877–947
Da Prato G. and Röckner M. (2006). Dissipative stochastic equations in Hilbert space with time dependent coefficients. Rend. Lincei Mat. Appl. 17: 397–403
Da Prato, G., and Zabczyk, J. (1996). Ergodicity for infinite dimensional systems, London Mathematical Society Lecture Notes, Vol. 229, Cambridge University Press: Cambridge.
Debussche A. and Odasso C. (2006). Markov solutions for the 3D stochastic Navier–Stokes equations with state dependent noise. J. Evol. Equ. 6(2): 305–324
Dudley, R. M. (2002). Real analysis and probability, Cambridge Studies in Avanced Maathematics.
Weinan 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. (1994). Dissipativity and invariant measures for stochastic Navier-Stokes equations. Nonlin. Diff. Eq. Appl. 1: 403–423
Flandoli, F. (2005). An introduction to 3D stochastic Fluid Dynamics, CIME lecture notes 2005.
Flandoli F. and Gatarek D. (1995). Martingale and stationary solutions for stochastic Navier-Stokes equations. Prob. Theory Relat. Fields 102: 367–391
Flandoli F. and Maslowski B. (1995). Ergodicity of the 2D Navier-Stokes equations under random perturbations. Comm. Math. Phys. 172(1): 119–141
Flandoli, F., and Romito, M. (to appear). Markov selections for the 3D stochastic Navier-Stokes equation.
Hairer, M., and Mattingly, J. C. (2006). Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann. Math. 164(3).
Hairer, M., and Mattingly, J. C. (2006). Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations.
Kuksin, S. B. (2006). Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions. Europear Mathematical Society Publishing House.
Kuksin S. and Shirikyan A. (2001). Ergodicity for the radomly forced 2D Navier-Stokes equations. Math. Phys. Anal. Geom. 4(2): 147–195
Kuksin S. and Shirikyan A. (2001). A coupling approach to randomly forced nonlinear PDEs. I. Comm. Math. Phys. 221: 351–366
Mattingly J.C. (2002). Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics. Commun. Math. Phys. 230(3): 421–462
Mattingly J.C. and Pardoux E. (2006). Malliavin calculus for the stochastic 2D Navier-Stokes equation. Comm. Pure Appl. Math. 59(12): 1742–1790
Mattingly, J. C. personal communication.
Odasso, C. (2005). (to appear). Exponential mixing for the 3D Navier-Stokes equation. 2005. To appear in Commun. Math. Phys.
Shirikyan, A. (2006). Qualitative properties of stationary measures for three-dimensional Navier-Stokes equations. Preprint.
Villani, C. (2003). Topics in optimal transportation, vol. 58, on Graduate Studies in Mathematics. American Mathematikcal Society, Providence RI.
Vishik, M. I. (1992). Asymptotic Behaviour of Solutions of Evolutionary Equations. In Lezioni Lincee, Radicati edited by L.A., Cambridge University Press: Cambridge.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Da Prato, G., Debussche, A. 2D Stochastic Navier–Stokes Equations with a Time-Periodic Forcing Term. J Dyn Diff Equat 20, 301–335 (2008). https://doi.org/10.1007/s10884-007-9074-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-007-9074-1