Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains

Brzeźniak, Zdzisław; Li, Yuhong

doi:10.1090/S0002-9947-06-03923-7

Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains
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Trans. Amer. Math. Soc. 358 (2006), 5587-5629 Request permission

Abstract:

We introduce a notion of an asymptotically compact (AC) random dynamical system (RDS). We prove that for an AC RDS the $\Omega$-limit set $\Omega _B(\omega )$ of any bounded set $B$ is nonempty, compact, strictly invariant and attracts the set $B$. We establish that the $2$D Navier Stokes Equations (NSEs) in a domain satisfying the Poincaré inequality perturbed by an additive irregular noise generate an AC RDS in the energy space $\mathrm {H}$. As a consequence we deduce existence of an invariant measure for such NSEs. Our study generalizes on the one hand the earlier results by Flandoli-Crauel (1994) and Schmalfuss (1992) obtained in the case of bounded domains and regular noise, and on the other hand the results by Rosa (1998) for the deterministic NSEs.

References

Frédéric Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Differential Equations 83 (1990), no. 1, 85–108. MR 1031379, DOI 10.1016/0022-0396(90)90070-6
Ludwig Arnold, Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1723992, DOI 10.1007/978-3-662-12878-7
Peter Baxendale, Gaussian measures on function spaces, Amer. J. Math. 98 (1976), no. 4, 891–952. MR 467809, DOI 10.2307/2374035
Zdzisław Brzeźniak, On analytic dependence of solutions of Navier-Stokes equations with respect to exterior force and initial velocity, Univ. Iagel. Acta Math. 28 (1991), 111–124. MR 1136785
Zdzisław Brzeźniak, On Sobolev and Besov spaces regularity of Brownian paths, Stochastics Stochastics Rep. 56 (1996), no. 1-2, 1–15. MR 1396751, DOI 10.1080/17442509608834032
Zdzisław Brzeźniak, On stochastic convolution in Banach spaces and applications, Stochastics Stochastics Rep. 61 (1997), no. 3-4, 245–295. MR 1488138, DOI 10.1080/17442509708834122
Zdzisław Brzeźniak, Some remarks on Itô and Stratonovich integration in 2-smooth Banach spaces, Probabilistic methods in fluids, World Sci. Publ., River Edge, NJ, 2003, pp. 48–69. MR 2083364, DOI 10.1142/9789812703989_{0}004
Zdzisław Brzeźniak, Marek Capiński, and Franco Flandoli, Pathwise global attractors for stationary random dynamical systems, Probab. Theory Related Fields 95 (1993), no. 1, 87–102. MR 1207308, DOI 10.1007/BF01197339
Z. Brzeźniak and Y. Li, Asymptotic behaviour of solutions to the 2D stochastic Navier-Stokes equations in unbounded domains – new developments, Proceedings of the First Sino-German Conference in Stochastic Analysis, 28 August-3 September, 2002, Beijing, in press.
Zdzisław Brzeźniak and Szymon Peszat, Maximal inequalities and exponential estimates for stochastic convolutions in Banach spaces, Stochastic processes, physics and geometry: new interplays, I (Leipzig, 1999) CMS Conf. Proc., vol. 28, Amer. Math. Soc., Providence, RI, 2000, pp. 55–64. MR 1803378
Zdzisław Brzeźniak and Szymon Peszat, Stochastic two dimensional Euler equations, Ann. Probab. 29 (2001), no. 4, 1796–1832. MR 1880243, DOI 10.1214/aop/1015345773
Zdzisław Brzeźniak and Jan van Neerven, Space-time regularity for linear stochastic evolution equations driven by spatially homogeneous noise, J. Math. Kyoto Univ. 43 (2003), no. 2, 261–303. MR 2051026, DOI 10.1215/kjm/1250283728
Henri Cartan, Differential calculus, Hermann, Paris; Houghton Mifflin Co., Boston, Mass., 1971. Exercises by C. Buttin, F. Rideau and J. L. Verley; Translated from the French. MR 0344032
Lamberto Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova 31 (1961), 308–340 (Italian). MR 138894
C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310
Hans Crauel, Random probability measures on Polish spaces, Stochastics Monographs, vol. 11, Taylor & Francis, London, 2002. MR 1993844
Hans Crauel, Arnaud Debussche, and Franco Flandoli, Random attractors, J. Dynam. Differential Equations 9 (1997), no. 2, 307–341. MR 1451294, DOI 10.1007/BF02219225
Hans Crauel and Franco Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields 100 (1994), no. 3, 365–393. MR 1305587, DOI 10.1007/BF01193705
Giuseppe Da Prato and Arnaud Debussche, Two-dimensional Navier-Stokes equations driven by a space-time white noise, J. Funct. Anal. 196 (2002), no. 1, 180–210. MR 1941997, DOI 10.1006/jfan.2002.3919
Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. MR 1207136, DOI 10.1017/CBO9780511666223
G. Da Prato and J. Zabczyk, Ergodicity for infinite-dimensional systems, London Mathematical Society Lecture Note Series, vol. 229, Cambridge University Press, Cambridge, 1996. MR 1417491, DOI 10.1017/CBO9780511662829
J.-P. Eckmann and M. Hairer, Invariant measures for stochastic partial differential equations in unbounded domains, Nonlinearity 14 (2001), no. 1, 133–151. MR 1808628, DOI 10.1088/0951-7715/14/1/308
Franco Flandoli, Dissipativity and invariant measures for stochastic Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl. 1 (1994), no. 4, 403–423. MR 1300150, DOI 10.1007/BF01194988
Franco Flandoli and Vincenzo M. Tortorelli, Time discretization of Ornstein-Uhlenbeck equations and stochastic Navier-Stokes equations with a generalized noise, Stochastics Stochastics Rep. 55 (1995), no. 1-2, 141–165. MR 1382289, DOI 10.1080/17442509508834022
Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
Daisuke Fujiwara and Hiroko Morimoto, An $L_{r}$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 3, 685–700. MR 492980
Jean-Michel Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations, J. Differential Equations 110 (1994), no. 2, 356–359. MR 1278375, DOI 10.1006/jdeq.1994.1071
Leonard Gross, Measurable functions on Hilbert space, Trans. Amer. Math. Soc. 105 (1962), 372–390. MR 147606, DOI 10.1090/S0002-9947-1962-0147606-6
Martin Hairer, Ergodicity of stochastic differential equations driven by fractional Brownian motion, Ann. Probab. 33 (2005), no. 2, 703–758. MR 2123208, DOI 10.1214/009117904000000892
John G. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions, Indiana Univ. Math. J. 29 (1980), no. 5, 639–681. MR 589434, DOI 10.1512/iumj.1980.29.29048
G. Stephen Jones, Stability and asymptotic fixed-point theory, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1262–1264. MR 180728, DOI 10.1073/pnas.53.6.1262
H. Keller and B. Schmalfuss, Attractors For Stochastic Sine Gordon Equations Via Transformation into Random Equations, preprint, the University of Bremen, 1999.
Olga Ladyzhenskaya, Attractors for semigroups and evolution equations, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1991. MR 1133627, DOI 10.1017/CBO9780511569418
J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR 0350177
Jacques-Louis Lions and Giovanni Prodi, Un théorème d’existence et unicité dans les équations de Navier-Stokes en dimension 2, C. R. Acad. Sci. Paris 248 (1959), 3519–3521 (French). MR 108964
A.L. Neidhardt, Stochastic Integrals in 2-uniformly smooth Banach Spaces, University of Wisconsin, 1978.
Ricardo Rosa, The global attractor for the $2$D Navier-Stokes flow on some unbounded domains, Nonlinear Anal. 32 (1998), no. 1, 71–85. MR 1491614, DOI 10.1016/S0362-546X(97)00453-7
Jacques Rougemont, Space-time invariant measures, entropy, and dimension for stochastic Ginzburg-Landau equations, Comm. Math. Phys. 225 (2002), no. 2, 423–448. MR 1889231, DOI 10.1007/s002200100586
B. Schmalfuss, Backward cocycles and attractors of Stochastic Differential Equations", pp. 185–192 in International Seminar on Applied Mathematics–Nonlinear Dynamics: Attractor Approximation and Global Behaviour, eds. Reitmann, T. Riedrich and N. Koksch, 1992.
Roger Temam, Navier-Stokes equations, Revised edition, Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam-New York, 1979. Theory and numerical analysis; With an appendix by F. Thomasset. MR 603444
Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, 2nd ed., Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997. MR 1441312, DOI 10.1007/978-1-4612-0645-3
John A. Toth and Steve Zelditch, $L^p$ norms of eigenfunctions in the completely integrable case, Ann. Henri Poincaré 4 (2003), no. 2, 343–368. MR 1985776, DOI 10.1007/s00023-003-0132-x