×

zbMATH — the first resource for mathematics

Martingale and stationary solutions for stochastic Navier-Stokes equations. (English) Zbl 0831.60072
Consider the stochastic Navier-Stokes equation in \(D\subset \mathbb{R}^d\), \[ \begin{split} {\partial u(t, x)\over \partial t}- \Delta u(t, x)+ (u(t, x)\cdot \nabla) u(t, x)\\ =- \nabla p(t, x)+ f(t, x)+ G(u, \xi)(t, x),\qquad t\in [0, T],\quad x\in D,\end{split} \] with the incompressibility, boundary and initial conditions. Here \(\xi(t, x)\) is a Gaussian random field, white noise in time, and \(G\) is an operator acting on noise and solution \(u\). The authors investigate three distinct cases of \(G\) such as (i) regular diffusion coefficient, (ii) coercive diffusion coefficient, (iii) cylindrical noise; and prove the existence of martingale solutions and of stationary solutions of the corresponding abstract stochastic evolution equation under different assumptions on \(G\). The proofs are due to a new method of compactness. The obtained result extends results of Z. Brzeźniak, M. Capiński and the first author [Math. Models Methods Appl. Sci. 1, No. 1, 41-59 (1991; Zbl 0741.60058), Stochastic Anal. Appl. 10, No. 5, 523-532 (1992; Zbl 0762.35083)] and M. Capiński and N. J. Cutland [Nonlinearity 6, No. 1, 71-78 (1993; Zbl 0765.76018)]. The stationary martingale solutions yield the existence of invariant measures, when the transition semigroup is well-defined. The related idea was presented by A. B. Cruzeiro [Expo. Math. 7, No. 1, 73-82 (1989; Zbl 0665.60066)].
Reviewer: I.Dôku

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Albeverio, S., Cruzeiro, A.B.: Global flow and invariant (Gibbs) measures for Euler and Navier-Stokes two dimensional fluids. Comm. Math. Phys.129, 431-444 (1990) · Zbl 0702.76041 · doi:10.1007/BF02097100
[2] Bensoussan, A., Temam, R.: Equations stochastiques du type Navier-Stokes. J. Funct. Anal.13, 195-222 (1973) · Zbl 0265.60094 · doi:10.1016/0022-1236(73)90045-1
[3] Brzezniak, Z., Capinski, M., Flandoli, F.: Stochastic partial differential equations and turbolence. Math. Models and Methods in Appl. Sc.1(1), 41-59 (1991) · Zbl 0741.60058 · doi:10.1142/S0218202591000046
[4] Brzezniak, Z., Capinski, M., Flandoli, F.: Stochastic Navier-Stokes equations with multiplicative noise. Stoch. Anal. Appl.10(5), 523-532 (1992) · Zbl 0762.35083 · doi:10.1080/07362999208809288
[5] Capinski, M.: A note on uniqueness of stochastic Navier-Stokes equations. Univ. Iagellonicae Acta Math. fasciculus XXX 219-228 (1993) · Zbl 0858.60057
[6] Capinski, M., Cutland, N.J.: Navier-Stokes equations with multiplicative noise. Nonlinearity6, 71-77 (1993) · Zbl 0765.76018 · doi:10.1088/0951-7715/6/1/005
[7] Capinski, M., G?tarek, D.: Stochastic equations in Hilbert space with application to Navier-Stokes equations in any dimension. J. Funct. Anal. (to appear)
[8] Clouet, J.F.: A diffusion approximation theorem in Navier-Stokes equations. Stoch. Anal. Appl. (to appear) · Zbl 0858.35146
[9] Cruzeiro, A.B.: Solutions et mesures invariantes pour des equations stochastiques du type Navier-Stokes. Expositiones Mathematicae7, 73-82 (1989) · Zbl 0665.60066
[10] Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1992 · Zbl 0761.60052
[11] Da Prato, G., G?tarek, D.: Stochastic Burgers equation with correlated noise. Preprint Scuola Normale Superiore, Pisa (1993)
[12] Flandoli, F.: Dissipativity and invariant measures for stochastic Navier-Stokes equations. Preprint no. 24, Scuola Normale Superiore, Pisa, 1993. To appear in Nonlinear Anal., Appl. · Zbl 0820.35108
[13] Fujita Yashima, H.: Equations de Navier-Stokes Stochastiques non Homogenes et Applications. Scuola Normale Superiore, Pisa, 1992 · Zbl 0753.35066
[14] G?tarek, D.: A note on nonlinear stochastic equations in Hilbert space. Statist. and Probab. Lett.17, 387-394 (1993) · Zbl 0786.60089 · doi:10.1016/0167-7152(93)90259-L
[15] G?tarek, D., Goldys, B.: On weak solutions of stochastic equations in Hilbert spaces. Stochastics46, 41-51 (1994)
[16] Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam, 1981 · Zbl 0495.60005
[17] Lions, J.L.: Quelques Methodes de Resolution des Problemes aux Limites non Lineaires. Dunod, Paris, 1969
[18] Metivier: Stochastic Partial Differential Equations in Infinite Dimensional Spaces. Quaderni, Scuola Normale Superiore, Pisa, 1988 · Zbl 0664.60062
[19] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York, 1983 · Zbl 0516.47023
[20] Schmalfuss, B.: Measure attractors of the stochastic Navier-Stokes equations. Univ. Bremen, Report no. 258, (1991) · Zbl 0728.60069
[21] Temam, R.: Navier-Stokes Equations, Theory and Numerical Analysis, North Holland, Amsterdam 1977 · Zbl 0383.35057
[22] Temam, R.: Navier-Stokes Equations and Nonlinear Functional Analysis. SIAM, Philadelphia, 1983 · Zbl 0522.35002
[23] Viot: Solutions Faibles d’Equations aux Derivees Partielles Stochastiques non Lineaires. These de Doctorat, Paris VI, 1976
[24] Visik, M.I., Fursikov, A.V.: Mathematical Problems of Statistical Hydromechanics. Kluver, Dordrecht, 1980
[25] Zabezyk, J.: The fractional calculus and stochastic evolution equations. preprint, 1993
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.