Abstract
We consider the 3D Navier–Stokes system in the Fourier space with regular forcing given by a stationary in time stochastic process satisfying a smallness condition. We explicitly construct a stationary solution of the system and prove a uniqueness theorem for this solution in the class of functions with Fourier transform majorized by a certain function h. Moreover we prove the following “one force—one solution” principle: the unique stationary solution at time t is presented as a functional of the realization of the forcing in the past up to t. Our explicit construction of the solution is based upon the stochastic cascade representation.
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REFERENCES
R. N. Bhattacharya, L. Chen, S. Dobson, R. B. Guenther, C. Orum, M. Ossiander, E. Thomann and E. C. Waymire, Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations. Trans. Am. Math. Soc. 355(12):5003–5040 (2003).
Y. Bakhtin, E. Dinaburg and Y. Sinai, On solutions of the Navier-Stokes system of infinite energy and enstrophy. In memory of A. A. Bolibrukh. Uspekhi Mat. Nauk 59(6):55–72 (2004).
J. Bricmont, A. Kupiainen and R. Lefevere, Ergodicity of the 2D Navier-Stokes equations with random forcing. Commun. Math. Phys. 224(1):65–81 (2001).
P.-L. Chow and R. Z. Khasminskii, Stationary solutions of nonlinear stochastic evolution equations. Stochastic Anal. Appl. 15(5):671–699 (1997).
G. Da Prato and A. Debussche, Ergodicity for the 3D stochastic Navier-Stokes equations. J. Math. Pures Appl. 82(8):877–947 (2003).
G. Da Prato and J. Zabczyk, Ergodicity for infinite dimensional systems. London Mathematical Society Lecture Note Series. 229 (Cambridge Univ. Press, Cambridge, 1996) xi, p. 339.
E. Weinan, J. C. Mattingly and Ya. Sinai, Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation. Comm. Math. Phys. 224(1):83–106 (2001). Dedicated to Joel L. Lebowitz.
B. Ferrario, Ergodic results for stochastic Navier-Stokes equation. Stochastics Stochastics Rep. 60(3–4):271–288 (1997).
F. Flandoli and B. Maslowski, Ergodicity of the 2-D Navier-Stokes equation under random perturbations. Commun. Math. Phys. 172(1):119–141 (1995).
F. Flandoli and M. Romito, Statistically stationary solutions to the 3-D Navier-Stokes equation do not show singularities. Electron. J. Probab. 6(5):15 (2001). electronic only.
M. Hairer and J. C. Mattingly, Ergodic properties of highly degenerate 2D stochastic Navier-Stokes equations. Comptes Rendus Mathematique 339(12):879–882 (2004).
S. Kuksin, A. Piatniski and A. Shirikyan, A coupling approach to randomly forced nonlinear PDE's. II. Commun. Math. Phys. 230(1):81–85, (2002).
S. Kuksin and A. Shirikyan, Stochastic dissipative PDE's and Gibbs measures. Commun. Math. Phys. 213(2):291–330, (2000).
S. Kuksin and A. Shirikyan, Coupling approach to white-forced nonlinear PDEs. J. Math. Pures Appl. 81:567–602 (2002).
Y. Le Jan and A. S. Sznitman, Stochastic cascades and 3-dimensional Navier-Stokes equations. Prob. Theory and Rel. Fields 109(3):343–366 (1997).
J. C. Mattingly, On Recent Progress for the Stochastic Navier–Stokes Equations. Journées “Equations aux Dérivées Partielles” (Forges-les-Eaux, 2003), XV, Summer 2003.
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Bakhtin, Y. Existence and Uniqueness of Stationary Solutions for 3D Navier–Stokes System with Small Random Forcing via Stochastic Cascades. J Stat Phys 122, 351–360 (2006). https://doi.org/10.1007/s10955-005-8014-x
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DOI: https://doi.org/10.1007/s10955-005-8014-x