Abstract
In this paper, we prove the existence of a unique strong solution to a stochastic tamed 3D Navier–Stokes equation in the whole space as well as in the periodic boundary case. Then, we also study the Feller property of solutions, and prove the existence of invariant measures for the corresponding Feller semigroup in the case of periodic conditions. Moreover, in the case of periodic boundary and degenerated additive noise, using the notion of asymptotic strong Feller property proposed by Hairer and Mattingly (Ann. Math. 164:993–1032, 2006), we prove the uniqueness of invariant measures for the corresponding transition semigroup.
References
Bakhtin Y.: Existence and uniqueness of stationary solutions for 3D Navier–Stokes system with small random forcing via stochastic cascades. J. Stat. Phys. 122(2), 351–360 (2006)
Bensoussan A., Temam R.: Equations stochastiques de type Navier–Stokes. J. Funct. Anal. 13, 195–222 (1973)
Da Prato G., Zabczyk J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)
Da Prato G., Debussche A.: Ergodicity for the 3D stochastic Navier–Stokes equations. J. Math. Pures Appl. 82(8), 877–947 (2003)
Debussche A., Odasso C.: Markov solutions for the 3D stochastic Navier–Stokes equations with state dependent noise. J. Evol. Equ. 6(2), 305–324 (2006)
Mattingly W.E., Mattingly J.C.: Ergodicity for the Navier–Stokes equation with degenerate random forcing: finite-dimensional approximation. Commun. Pure Appl. Math. 54(11), 1386–1402 (2001)
Mattingly W.E., Mattingly J.C., Sinai Y.: Gibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equation. Commun. Math. Phys. 224(1), 83–106 (2001)
Fabes E.B., Jones B.F., Rivière N.M.: The initial value problem for the Navier–Stokes equations with data in L p. Arch. Rational Mech. Anal. 45, 222–240 (1972)
Flandoli F., Gatarek D.: Martingale and stationary solutions for stochastic Navier–stokes equations. Probab. Theory Relat. Fields 102, 367–391 (1995)
Flandoli F., Maslowski B.: Ergodicity of the 2-D Navier–Stokes equation under random perturbations. Commun. Math. Phys. 172(1), 119–141 (1995)
Flandoli F., Romito M.: Markov selections for the 3D stochastic Navier–Stokes equations. Probab. Theory Relat. Fields 140, 407–458 (2008)
Galdi, G.P.: An introduction to the Navier–Stokes initial-boundary value problem. Fundamental directions in mathematical fluid mechanics. Adv. Math. Fluid Mech. pp. 1–70, Birkhäuser, Basel (2000)
Goldys, B., Röckner, M., Zhang, X.: Martingale Solutions and Markov selections for stochastic evolution equations. Bibos-Preprint, 08-041 (2008)
Hairer M., Mattingly J.C.: Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann. Math. 164, 993–1032 (2006)
Kallenberg O.: Foundations of Modern Probability, 2nd edn. Springer, Berlin (2001)
Krylov N.V.: A simple proof of the existence of a solution to the Itô equation with monotone coefficients. Theory Probab. Appl. 35(3), 583–587 (1990)
Ladyzhenskaya, O.A.: The mathematical theory of viscous incompressible flow. Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, vol. 2, pp. xviii+224. Gordon and Breach, New York (1969)
Leray J.: Sur le mouvement d’un liquide visquex emplissant l’espace. Acta Math. 63, 193–248 (1934)
Lions P.L.: Mathematical topics in fluid mechanics: incompressible Models. Oxford Lect. Ser. Math. Appl. 1, 3 (1996)
Malliavin P.: Stochastic Analysis. Springer, Berlin (1995)
Mattingly J.C.: Exponential convergence for the stochastically forced Navier–Stokes equations and other partially dissipative dynamics. Commun. Math. Phys. 230, 421–462 (2002)
Mikulevicius, R., Rozovskii, B.L.: Martingale problems for stochastic PDE’s. In: Stochastic Partial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs, Vol. 64, pp. 185–242, AMS, Providence (1999)
Mikulevicius R., Rozovskii B.L.: Global L 2-solution of Stochastic Navier–Stokes Equations. Ann. Probab. 33(1), 137–176 (2005)
Odasso C.: Exponential mixing for the 3D stochastic Navier–Stokes equations. Commun. Math. Phys. 270(1), 109–139 (2007)
Prévôt C., Röckner M.: A concise course on stochastic partial differential equations. Lecture Notes in Mathematics, vol. 1905, pp. vi+144. Springer, Berlin (2007)
Röckner, M., Zhang, X.: Tamed 3D Navier–Stokes equation: existence, uniqueness and regularity. http://arXiv:math/0703254
Röckner, M., Schmuland, B., Zhang, X.: Yamada–Watanabe theorem for stochastic evolution equations in infinite dimensions. BiBos-Preprint, 07-12-269 (2007)
Romito, M.: Analysis of equilibrium states of Markov solutions to the 3D Navier–Stokes equations driven by additive noise. http://aps.arxiv.org/abs/0709.3267
Rozovskii, B.L.: Stochastic evolution systems: linear theory and applications to nonlinear filtering. Mathematics and its Applications (Soviet Series), vol. 35. Kluwer Academic, Dordrecht (1990)
Stroock D.W., Varadhan S.R.S.: Multidimensional diffusion processes. Springer, Berlin (1979)
Temam, R.: Navier–Stokes equations: theory and numerical analysis. Studies in Mathematics and its Applications, vol. 2, pp. x+500. North-Holland, Amsterdam (1977)
Taira K.: Analytic Semigroups and Semilinear Initial Boundary Value Problems. London Mathematical Society Lecture Note Series, vol 223. Cambridge University Press, London (1995)
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Röckner, M., Zhang, X. Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicity. Probab. Theory Relat. Fields 145, 211–267 (2009). https://doi.org/10.1007/s00440-008-0167-5
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DOI: https://doi.org/10.1007/s00440-008-0167-5
Keywords
- Navier–Stokes equation
- Invariant measure
- Ergodicity
- Asymptotic strong Feller property
Mathematics Subject Classification (2000)
- 60H15
- 37A25