## Global $$L_2$$-solutions of stochastic Navier-Stokes equations.(English)Zbl 1098.60062

Summary: This paper concerns the Cauchy problem in $$\mathbb{R}^d$$ for the stochastic Navier-Stokes equation $\partial_tu=\Delta u-(u,\nabla)u-\nabla p+f(u)+\bigl[ (\sigma,\nabla)u-\nabla\widetilde p+g(u)\bigr]\circ \dot W,\quad u(0)=u_0,\quad \text{div}\,u=0,$ driven by white noise $$\dot W$$. Under minimal assumptions on regularity of the coefficients and random forces, the existence of a global weak (martingale) solution of the stochastic Navier-Stokes equation is proved. In the two-dimensional case, the existence and pathwise uniqueness of a global strong solution is shown. A Wiener chaos-based criterion for the existence and uniqueness of a strong global solution of the Navier-Stokes equations is established.

### MSC:

 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35Q30 Navier-Stokes equations 35R60 PDEs with randomness, stochastic partial differential equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids 76M35 Stochastic analysis applied to problems in fluid mechanics
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### References:

 [1] Bourbaki, N. (1969). Integration sur les espaces topologiques separes. In Éléments de Mathématique. Fasc. XXXV. Livre VI: Intégration. Chapter IX. Act. Sci. et Ind. 1343 . Hermann, Paris. [2] Brzeźniak, Z., Capiński, M. and Flandoli, F. (1991). Stochastic partial differential equations and turbulence. Math. Models Methods Appl. Sci. 1 41–59. · Zbl 0741.60058 [3] Caffarelli, L., Kohn, R. and Nirenberg, L. (1982). Partial regularity of suitable weak solution of the Navier–Stokes equations. Comm. Pure Appl. Math. 35 771–831. · Zbl 0509.35067 [4] Cameron, R. H. and Martin, W. T. (1947). The orthogonal development of non-linear functionals in a series of Fourier–Hermite functions. Ann. of Math. 48 385–392. JSTOR: · Zbl 0029.14302 [5] Capiński, M. and Cutland, N. J. (1991). Stochastic Navier–Stokes equations. Acta Appl. Math. 25 59–85. · Zbl 0746.60065 [6] Capiński, M. and Gatarek, D. (1994). Stochastic equations in Hilbert spaces with applications to Navier–Stokes equations in any dimensions. J. Funct. Anal. 126 26–35. · Zbl 0817.60075 [7] Capiński, M. and Peszat, S. (2001). On the existence of a solution to stochastic Navier–Stokes equations. Nonlinear Anal. 44 141–177. · Zbl 0976.60063 [8] Da Prato, G. and Debussche, A. (2002). Two-dimensional Navier–Stokes equation driven by a space-time white noise. J. Funct. Anal. 196 180–210. · Zbl 1013.60051 [9] Da Prato, G. and Debussche, A. (2003). Ergodicity for the 3D stochastic Navier–Stokes equations. Inst. de Recherche Mathématique de Rennes, Preprint 03-01 1–65. · Zbl 1067.76023 [10] Flandoli, F. (1994). Dissipativity and invariant measures for stochastic Navier–Stokes equations. Nonlinear Differential Equations and Appl. 1 403–423. · Zbl 0820.35108 [11] Flandoli, F. and Gatarek, D. (1995). Martingale and stationary solutions for stochastic Navier–Stokes equations. Probab. Theory Related Fields 102 367–391. · Zbl 0831.60072 [12] Flandoli, F. and Maslowski, B. (1995). Ergodicity of the 2-D Navier–Stokes equation under random perturbations. Comm. Math. Phys. 171 119–141. · Zbl 0845.35080 [13] Fujiwara, D. and Morimoto, H. (1977). An L$$_r$$ theorem on the Helmholtz decomposition of vector fields. Tokyo Univ. Fac. Sciences J. 24 685–700. · Zbl 0386.35038 [14] Gawedzki, K. and Kupiainen, A. (1996). Universality in turbulence: An exactly solvable model. In Low-Dimensional Models in Statistical Physics and Quantum Field Theory (H. Grosse and L. Pittner, eds.) 71–105. Springer, Berlin. · Zbl 0919.60094 [15] Gawedzki, K. and Vergassola, M. (2000). Phase transition in the passive scalar advection. Phys. D 138 63–90. · Zbl 0967.76041 [16] Grigelionis, B. and Mikulevicius, R. (1983). Stochastic evolution equations and densities of conditional distributions. Lecture Notes in Control and Inform. Sci. 49 49–86. Springer, New York. · Zbl 0513.60058 [17] Gyongy, I. and Krylov, N. (1996). Existence of strong solutions of Ito’s ststochastic equations via approximations. Probab. Theory Related Fields 105 143–158. · Zbl 0847.60038 [18] Hida, T., Kuo, H. H., Potthoff, J. and Streit, L. (1993). White Noise . Kluwer Academic, Dordrecht. · Zbl 0771.60048 [19] Holden, H., Oksendal, B., Uboe, J. and Zhang, T. (1996). Stochastic Partial Differential Equations . A Modeling , White Noise Functional Approach . Birkhäuser, Boston. · Zbl 0860.60045 [20] Kraichnan, R. H. (1968). Small-scale structure of a scalar field convected by turbulence. Phys. Fluids 11 945–963. · Zbl 0164.28904 [21] LeJan, Y. and Raimond, O. (2002). Integration of Brownian vector fields. Ann. Probab. 30 826–873. · Zbl 1037.60061 [22] Meyer, P.-A. (1995). Quantum Probability for Probabilists , 2nd ed. Lecture Notes in Math. 1538 . Springer, New York. · Zbl 0773.60098 [23] Mikulevicius, R. (2002). On the Cauchy problem for stochastic Stokes equation. SIAM J. Math. Anal. 34 121–141. · Zbl 1015.60054 [24] Mikulevicius, R. and Rozovskii, B. L. (1998). Linear parabolic stochastic PDE’s and Wiener chaos. SIAM J. Math. Anal. 29 452–480. · Zbl 0911.60045 [25] Mikulevicius, R. and Rozovskii, B. L. (1999). Martingale problems for stochastic PDE’s. In Stochastic Partial Differential Equations : Six Perspectives (R. Carmona and B. L. Rozovskii, eds.). Mathematical Surveys and Monographs 64 243–325. Amer. Math. Soc., Providence, RI. · Zbl 0938.60047 [26] Mikulevicius, R. and Rozovskii, B. L. (2001). Stochastic Navier–Stokes equations. Propagation of chaos and statistical moments. In Optimal Control and Partial Differential Equations. In Honor of Professor Alain Bensoussan’s 60th Birthday (J. L. Menaldi, E. Rofmann and A. Sulem, eds.) 258–267. IOS Press, Amsterdam. · Zbl 1054.60512 [27] Mikulevicius, R. and Rozovskii, B. L. (2001). On equations of stochastic fluid mechanics. In Stochastics in Finite and Infinite Dimensions : In Honor of Gopinath Kallianpur (T. Hida et al., eds.) 285–302. Birkhäuser, Boston. · Zbl 1003.76017 [28] Mikulevicius, R. and Rozovskii, B. L. (2002). On martingale problem solutions of stochastic Navier–Stokes equations. In Stochastic Partial Differential Equations and Applications (G. Da Prato and L. Tubaro, eds.) 405–416. Dekker, New York. · Zbl 1009.76018 [29] Mikulevicius, R. and Rozovskii, B. L. (2004). Stochastic Navier–Stokes equations for turbulent flows. SIAM J. Math. Anal. 35 1250–1310. · Zbl 1062.60061 [30] Rozovskii, B. L. (1990). Stochastic Evolution Systems . Kluwer Academic, Dordrecht. · Zbl 0724.60070 [31] Stein, A. M. (1970). Singular Integrals and Differentiability Properties of Functions . Princeton Univ. Press. · Zbl 0207.13501 [32] Temam, R. (1985). Navier–Stokes Equations . North-Holland, Amsterdam. · Zbl 0572.35083 [33] Viot, M. (1976). Solutions faibles d’équation aux dérivées partielles stochastiques non linéaires. Thése, Univ. Pierre and Marie Curie, Paris. [34] Vishik, M. I. and Fursikov, A. V. (1979). Mathematical Problems of Statistical Hydromechanics . Kluwer Academic, Dordrecht. · Zbl 0503.76045 [35] Yamada, T. and Watanabe, S. (1971). On the uniqueness of solutions of stochastic differential equations I. J. Math. Kyoto Univ. 11 155–167. · Zbl 0236.60037 [36] Yamada, T. and Watanabe, S. (1971). On the uniqueness of solutions of stochastic differential equations II. J. Math. Kyoto Univ. 11 553–563. · Zbl 0229.60039
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