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Stochastic tamed 3D Navier-Stokes equations: existence, uniqueness and ergodicity. (English) Zbl 1196.60118
The authors study a stochastic tamed 3D Navier-Stokes equation, where an additional nonlinear term of the type \(g(|u|^2)u\) is added. The equation is considered in the whole space and for periodic boundary conditions with quite general multiplicative and additive noise.
The first main result is the existence of a unique strong solution. Then the Feller property is studied and in the case of periodic boundary conditions the existence of invariant measures for the corresponding Feller semigroup is verified.
The uniqueness of these invariant measures in the case of periodic boundary conditions and degenerated additive noise is proved, using the notion of asymptotic strong Feller property introduced by M. Hairer and J. C. Mattingly [Ann. Math. (2) 164, No. 3, 993–1032 (2006; Zbl 1130.37038)].

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
37A25 Ergodicity, mixing, rates of mixing
37L55 Infinite-dimensional random dynamical systems; stochastic equations
35R60 PDEs with randomness, stochastic partial differential equations
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI arXiv
[1] 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) · Zbl 1089.76012 · doi:10.1007/s10955-005-8014-x
[2] Bensoussan A., Temam R.: Equations stochastiques de type Navier–Stokes. J. Funct. Anal. 13, 195–222 (1973) · Zbl 0265.60094 · doi:10.1016/0022-1236(73)90045-1
[3] Da Prato G., Zabczyk J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992) · Zbl 0761.60052
[4] Da Prato G., Debussche A.: Ergodicity for the 3D stochastic Navier–Stokes equations. J. Math. Pures Appl. 82(8), 877–947 (2003) · Zbl 1109.60047 · doi:10.1016/S0021-7824(03)00025-4
[5] 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) · Zbl 1110.35110 · doi:10.1007/s00028-006-0254-y
[6] 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) · Zbl 1024.76012 · doi:10.1002/cpa.10007
[7] 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) · Zbl 0994.60065 · doi:10.1007/s002201224083
[8] 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) · Zbl 0254.35097 · doi:10.1007/BF00281533
[9] Flandoli F., Gatarek D.: Martingale and stationary solutions for stochastic Navier–stokes equations. Probab. Theory Relat. Fields 102, 367–391 (1995) · Zbl 0831.60072 · doi:10.1007/BF01192467
[10] Flandoli F., Maslowski B.: Ergodicity of the 2-D Navier–Stokes equation under random perturbations. Commun. Math. Phys. 172(1), 119–141 (1995) · Zbl 0845.35080 · doi:10.1007/BF02104513
[11] Flandoli F., Romito M.: Markov selections for the 3D stochastic Navier–Stokes equations. Probab. Theory Relat. Fields 140, 407–458 (2008) · Zbl 1133.76016 · doi:10.1007/s00440-007-0069-y
[12] 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) · Zbl 1108.35133
[13] Goldys, B., Röckner, M., Zhang, X.: Martingale Solutions and Markov selections for stochastic evolution equations. Bibos-Preprint, 08-041 (2008)
[14] Hairer M., Mattingly J.C.: Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann. Math. 164, 993–1032 (2006) · Zbl 1130.37038 · doi:10.4007/annals.2006.164.993
[15] Kallenberg O.: Foundations of Modern Probability, 2nd edn. Springer, Berlin (2001) · Zbl 0892.60001
[16] 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) · Zbl 0735.60061 · doi:10.1137/1135082
[17] 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)
[18] Leray J.: Sur le mouvement d’un liquide visquex emplissant l’espace. Acta Math. 63, 193–248 (1934) · JFM 60.0726.05 · doi:10.1007/BF02547354
[19] Lions P.L.: Mathematical topics in fluid mechanics: incompressible Models. Oxford Lect. Ser. Math. Appl. 1, 3 (1996) · Zbl 0866.76002
[20] Malliavin P.: Stochastic Analysis. Springer, Berlin (1995) · Zbl 0847.60034
[21] Mattingly J.C.: Exponential convergence for the stochastically forced Navier–Stokes equations and other partially dissipative dynamics. Commun. Math. Phys. 230, 421–462 (2002) · Zbl 1054.76020 · doi:10.1007/s00220-002-0688-1
[22] 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)
[23] Mikulevicius R., Rozovskii B.L.: Global L 2-solution of Stochastic Navier–Stokes Equations. Ann. Probab. 33(1), 137–176 (2005) · Zbl 1098.60062 · doi:10.1214/009117904000000630
[24] Odasso C.: Exponential mixing for the 3D stochastic Navier–Stokes equations. Commun. Math. Phys. 270(1), 109–139 (2007) · Zbl 1122.60059 · doi:10.1007/s00220-006-0156-4
[25] Prévôt C., Röckner M.: A concise course on stochastic partial differential equations. Lecture Notes in Mathematics, vol. 1905, pp. vi+144. Springer, Berlin (2007) · Zbl 1123.60001
[26] Röckner, M., Zhang, X.: Tamed 3D Navier–Stokes equation: existence, uniqueness and regularity. http://arXiv:math/0703254 · Zbl 1180.35417
[27] Röckner, M., Schmuland, B., Zhang, X.: Yamada–Watanabe theorem for stochastic evolution equations in infinite dimensions. BiBos-Preprint, 07-12-269 (2007)
[28] 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 · Zbl 1150.82027
[29] 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)
[30] Stroock D.W., Varadhan S.R.S.: Multidimensional diffusion processes. Springer, Berlin (1979) · Zbl 0426.60069
[31] Temam, R.: Navier–Stokes equations: theory and numerical analysis. Studies in Mathematics and its Applications, vol. 2, pp. x+500. North-Holland, Amsterdam (1977) · Zbl 0383.35057
[32] Taira K.: Analytic Semigroups and Semilinear Initial Boundary Value Problems. London Mathematical Society Lecture Note Series, vol 223. Cambridge University Press, London (1995) · Zbl 0861.35001
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