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Asymptotic compactness and absorbing sets for stochastic Burgers’ equations driven by space-time white noise and for some two-dimensional stochastic Navier-Stokes equations on certain unbounded domains. (English) Zbl 1186.60059
Da Prato, Giuseppe (ed.) et al., Stochastic partial differential equations and applications – VII. Papers of the 7th meeting, Levico, Terme, Italy, January 5–10, 2004. Boca Raton, FL: Chapman & Hall/CRC (ISBN 0-8247-0027-9/pbk). Lecture Notes in Pure and Applied Mathematics 245, 35-52 (2006).
Author’s introduction: In the last decade there has been a growing interest in the ergodic properties of infinite dimensional systems governed by stochastic partial differential equations (SPDEs). In particular, existence of attractors for two-dimensional (2-D) Navier-Stokes equations (NSEs)in bounded domains both driven by real and additive noise has been established [see e.g., Z. Brzezniak, M. Capinski and F. Flandoli, Probability Theory and Relat. Fields 95, No. 1, 87–102 (1993; Zbl 0791.58056), H. Crauel and F. Flandoli, ibid. 100, No. 3, 365–393 (1994; Zbl 0819.58023); and 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, Reitmann, T. Riedrich and N. Koksch (eds.) (1992)]. Recently, in a joint work with Y. Li we have generalized the results from Crauel and Flandoli [loc. cit.] and Schmalfuss [loc. cit.] to the case of unbounded domains. We observed there that the method of asymptotic compactness used by us should work also for equations in bounded domains with much rougher noise than the original methods could handle. The main motivation of this chapter is to show that this is indeed the case for the one dimensional (‘D) stochastic Burgers equations with additive space-time white noise. Even for readers mainly interested in the Navier-Stokes equations it could be useful to study the Burgers equations case. Contrary to some recent works on stochastic Burgers equations (see G. Da Prato and J. Zabczyk), our approach is very similar to the approach we use for the NSEs in [Z. Brzezniak and Y. Li, Trans. Am. Math. Soc. 358, No. 12, 5587–5629 (2006; Zbl 1113.60062)] the second motivation is to show that it also works for a certain special form of two-dimensional (2D) stochastic NSEs with multiplicative noise. In fact, using a generalization of a recent result [M. Capinski and N. J. Cutland, Probab. Theor. Relat. Fields 115, No. 1, 121–151 (1999; Zbl 0938.35202)] to bounded and unbounded domains, we show the existence of a compact invariant set for such problems. Full proofs of the results presented in this section will be published elsewhere.
Reviewer: Ying Hu (Rennes)
MSC:
60H15Stochastic partial differential equations
37H10Generation, random and stochastic difference and differential equations
37L30Attractors and their dimensions, Lyapunov exponents
76D05Navier-Stokes equations (fluid dynamics)
76M35Stochastic analysis (fluid mechanics)