The Annals of Probability

Invariant measure for the stochastic Navier–Stokes equations in unbounded 2D domains

Zdzisław Brzeźniak, Elżbieta Motyl, and Martin Ondrejat

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Abstract

Building upon a recent work by two of the authors and J. Seidler on $bw$-Feller property for stochastic nonlinear beam and wave equations, we prove the existence of an invariant measure to stochastic 2-D Navier–Stokes (with multiplicative noise) equations in unbounded domains. This answers an open question left after the first author and Y. Li proved a corresponding result in the case of an additive noise.

Article information

Source
Ann. Probab., Volume 45, Number 5 (2017), 3145-3201.

Dates
Received: February 2015
Revised: June 2016
First available in Project Euclid: 23 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1506132035

Digital Object Identifier
doi:10.1214/16-AOP1133

Mathematical Reviews number (MathSciNet)
MR3706740

Zentralblatt MATH identifier
06812202

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 37L40: Invariant measures
Secondary: 76M35: Stochastic analysis 60J25: Continuous-time Markov processes on general state spaces

Keywords
Invariant measure $bw$-Feller semigroup stochastic Navier–Stokes equations

Citation

Brzeźniak, Zdzisław; Motyl, Elżbieta; Ondrejat, Martin. Invariant measure for the stochastic Navier–Stokes equations in unbounded 2D domains. Ann. Probab. 45 (2017), no. 5, 3145--3201. doi:10.1214/16-AOP1133. https://projecteuclid.org/euclid.aop/1506132035


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References

  • [1] Albeverio, S., Brzeźniak, Z. and Wu, J.-L. (2010). Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients. J. Math. Anal. Appl. 371 309–322.
  • [2] Albeverio, S. and Röckner, M. (1989). Dirichlet forms, quantum fields and stochastic quantization. In Stochastic Analysis, Path Integration and Dynamics (Warwick, 1987). Pitman Res. Notes Math. Ser. 200 1–21. Longman Sci. Tech., Harlow.
  • [3] Avez, A. (1968). Propriétés ergodiques des endomorphisms dilatants des variétés compactes. C. R. Math. Acad. Sci. Paris Sér. A-B 266 A610–A612.
  • [4] Ball, J. M. (1997). Continuity properties and global attractors of generalized semiflows and the Navier–Stokes equations. J. Nonlinear Sci. 7 475–502.
  • [5] Brzeźniak, Z. and Ga̧tarek, D. (1999). Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces. Stochastic Process. Appl. 84 187–225.
  • [6] Brzeźniak, Z., Goldys, B. and Jegaraj, T. (2017). Large deviations and transitions between equilibria for stochastic Landau–Lifshitz–Gilbert equation. Arch. Ration. Mech. Anal. Available at https://doi-org.libproxy.york.ac.uk/10.1007/s00205-017-1117-0.
  • [7] Brzeźniak, Z., Hausenblas, E. and Razafimandimby, P. Martingale solutions for stochastic equations of reaction diffusion type driven by Lévy noise or Poisson random measure. Preprint. Available at arXiv:1010.5933v2.
  • [8] Brzeźniak, Z. and Li, Y. (2006). Asymptotic compactness and absorbing sets for 2D stochastic Navier–Stokes equations on some unbounded domains. Trans. Amer. Math. Soc. 358 5587–5629 (electronic).
  • [9] Brzeźniak, Z. and Motyl, E. (2013). Existence of a martingale solution of the stochastic Navier–Stokes equations in unbounded 2D and 3D domains. J. Differential Equations 254 1627–1685.
  • [10] Brzeźniak, Z. and Ondreját, M. (2011). Weak solutions to stochastic wave equations with values in Riemannian manifolds. Comm. Partial Differential Equations 36 1624–1653.
  • [11] Brzeźniak, Z. and Ondreját, M. (2013). Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces. Ann. Probab. 41 1938–1977.
  • [12] Brzeźniak, Z., Ondreját, M. and Seidler, J. (2016). Invariant measures for stochastic nonlinear beam and wave equations. J. Differential Equations 260 4157–4179.
  • [13] Chow, P.-L. and Khasminskii, R. Z. (1997). Stationary solutions of nonlinear stochastic evolution equations. Stoch. Anal. Appl. 15 671–699.
  • [14] Constantin, P. and Ramos, F. (2007). Inviscid limit for damped and driven incompressible Navier–Stokes equations in $\mathbb{R}^{2}$. Comm. Math. Phys. 275 529–551.
  • [15] Daners, D. (2005). Perturbation of semi-linear evolution equations under weak assumptions at initial time. J. Differential Equations 210 352–382.
  • [16] Da Prato, G. and Debussche, A. (2002). Two-dimensional Navier–Stokes equations driven by a space–time white noise. J. Funct. Anal. 196 180–210.
  • [17] Da Prato, G. and Debussche, A. (2003). Strong solutions to the stochastic quantization equations. Ann. Probab. 31 1900–1916.
  • [18] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge.
  • [19] Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite-Dimensional Systems. London Mathematical Society Lecture Note Series 229. Cambridge Univ. Press, Cambridge.
  • [20] Flandoli, F. (1994). Dissipativity and invariant measures for stochastic Navier–Stokes equations. NoDEA Nonlinear Differential Equations Appl. 1 403–423.
  • [21] Flandoli, F. and Ga̧tarek, D. (1995). Martingale and stationary solutions for stochastic Navier–Stokes equations. Probab. Theory Related Fields 102 367–391.
  • [22] Holly, K. and Wiciak, M. (1995). Compactness method applied to an abstract nonlinear parabolic equation. In Selected Problems of Mathematics. 50th Anniv. Cracow Univ. Technol. Anniv. Issue 6 95–160. Cracow Univ. Technol., Kraków.
  • [23] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library 24. North-Holland, Amsterdam.
  • [24] Jakubowski, A. (1997). The almost sure Skorokhod representation for subsequences in nonmetric spaces. Teor. Veroyatn. Primen. 42 209–216.
  • [25] Kim, J. U. (2005). Invariant measures for the stochastic von Karman plate equation. SIAM J. Math. Anal. 36 1689–1703 (electronic).
  • [26] Kryloff, N. and Bogoliouboff, N. (1937). La théorie générale de la mesure dans son application à l’étude des systèmes dynamiques de la mécanique non linéaire. Ann. of Math. (2) 38 65–113.
  • [27] Kuratowski, C. (1952). Topologie, Vol. I. 3ème ed. Monografie Matematyczne XX. Polskie Towarzystwo Matematyczne, Warsawa.
  • [28] Lasota, A. and Pianigiani, G. (1977). Invariant measures on topological spaces. Boll. Unione Mat. Ital. (5) 14 592–603.
  • [29] Maslowski, B. and Seidler, J. (1999). On sequentially weakly Feller solutions to SPDE’s. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 10 69–78.
  • [30] Motyl, E. (2014). Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains—abstract framework and applications. Stochastic Process. Appl. 124 2052–2097.
  • [31] Ondreját, M. (2004). Existence of global mild and strong solutions to stochastic hyperbolic evolution equations driven by a spatially homogeneous Wiener process. J. Evol. Equ. 4 169–191.
  • [32] Ondreját, M. (2004). Uniqueness for stochastic evolution equations in Banach spaces. Dissertationes Math. (Rozprawy Mat.) 426 63.
  • [33] Oxtoby, J. C. and Ulam, S. M. (1939). On the existence of a measure invariant under a transformation. Ann. of Math. (2) 40 560–566.
  • [34] Pardoux, E. (1979). Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3 127–167.
  • [35] Parisi, G. and Wu, Y. S. (1981). Perturbation theory without gauge fixing. Sci. China Ser. A 24 483–496.
  • [36] Parthasarathy, K. R. (1967). Probability Measures on Metric Spaces. Probability and Mathematical Statistics 3. Academic Press, New York.
  • [37] Peszat, S. and Zabczyk, J. (2007). Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach. Encyclopedia of Mathematics and Its Applications 113. Cambridge Univ. Press, Cambridge.
  • [38] Röckner, M. and Sobol, Z. (2006). Kolmogorov equations in infinite dimensions: Well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations. Ann. Probab. 34 663–727.
  • [39] Rosa, R. (1998). The global attractor for the $2$D Navier–Stokes flow on some unbounded domains. Nonlinear Anal. 32 71–85.
  • [40] Schmidt, T. (2015). Strict interior approximation of sets of finite perimeter and functions of bounded variation. Proc. Amer. Math. Soc. 143 2069–2084.
  • [41] Strauss, W. A. (1966). On continuity of functions with values in various Banach spaces. Pacific J. Math. 19 543–551.
  • [42] Temam, R. (1995). Navier–Stokes Equations and Nonlinear Functional Analysis, 2nd ed. CBMS-NSF Regional Conference Series in Applied Mathematics 66. SIAM, Philadelphia, PA.
  • [43] Temam, R. (2001). Navier–Stokes Equations: Theory and Numerical Analysis. AMS Chelsea, Providence, RI. Reprint of the 1984 edition.
  • [44] Vakhania, N. N., Tarieladze, V. I. and Chobanyan, S. A. (1987). Probability Distributions on Banach Spaces. Mathematics and Its Applications (Soviet Series) 14. Reidel, Dordrecht. Translated from the Russian and with a preface by Wojbor A. Woyczynski.
  • [45] Vishik, M. J. and Fursikov, A. V. (1988). Mathematical Problems of Statistical Hydromechanics. Kluwer Academic, Dordrecht.