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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
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##### 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
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