Annals of Probability

Global L2-solutions of stochastic Navier–Stokes equations

R. Mikulevicius and B. L. Rozovskii

Full-text: Open access


This paper concerns the Cauchy problem in Rd for the stochastic Navier–Stokes equation $$∂_t\mathbf u=Δ\mathbf u−(\mathbf u,∇)\mathbf u−∇p+\mathbf f(\mathbf u)+[(σ,∇)\mathbf u−∇p̃+\mathbf g(\mathbf u)]○\dot W,\mathbf u(0)=\mathbf u_0,  \mathrm{div} \mathbf u=0,$$ driven by white noise . 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.

Article information

Ann. Probab., Volume 33, Number 1 (2005), 137-176.

First available in Project Euclid: 11 February 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 76M35: Stochastic analysis

Stochastic Navier–Stokes Leray solution Kraichnan’s turbulence Wiener chaos strong solutions pathwise uniqueness


Mikulevicius, R.; Rozovskii, B. L. Global L 2 -solutions of stochastic Navier–Stokes equations. Ann. Probab. 33 (2005), no. 1, 137--176. doi:10.1214/009117904000000630.

Export citation


  • Bourbaki, N. (1969). Integration sur les espaces topologiques separes. In {Éléments de Mathématique. Fasc. XXXV.
  • Livre VI: Intégration. Chapter IX. Act. Sci. et Ind.} Hermann, Paris.
  • Brzeźniak, Z., Capiński, M. and Flandoli, F. (1991). Stochastic partial differential equations and turbulence. Math. Models Methods Appl. Sci. 1 41–59.
  • 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.
  • 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.
  • Capiński, M. and Cutland, N. J. (1991). Stochastic Navier–Stokes equations. Acta Appl. Math. 25 59–85.
  • 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.
  • Capiński, M. and Peszat, S. (2001). On the existence of a solution to stochastic Navier–Stokes equations. Nonlinear Anal. 44 141–177.
  • 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.
  • 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.
  • Flandoli, F. (1994). Dissipativity and invariant measures for stochastic Navier–Stokes equations. Nonlinear Differential Equations and Appl. 1 403–423.
  • Flandoli, F. and Gatarek, D. (1995). Martingale and stationary solutions for stochastic Navier–Stokes equations. Probab. Theory Related Fields 102 367–391.
  • Flandoli, F. and Maslowski, B. (1995). Ergodicity of the 2-D Navier–Stokes equation under random perturbations. Comm. Math. Phys. 171 119–141.
  • 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.
  • 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.
  • Gawedzki, K. and Vergassola, M. (2000). Phase transition in the passive scalar advection. Phys. D 138 63–90.
  • 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.
  • Gyongy, I. and Krylov, N. (1996). Existence of strong solutions of Ito's ststochastic equations via approximations. Probab. Theory Related Fields 105 143–158.
  • Hida, T., Kuo, H. H., Potthoff, J. and Streit, L. (1993). White Noise. Kluwer Academic, Dordrecht.
  • Holden, H., Oksendal, B., Uboe, J. and Zhang, T. (1996). Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach. Birkhäuser, Boston.
  • Kraichnan, R. H. (1968). Small-scale structure of a scalar field convected by turbulence. Phys. Fluids 11 945–963.
  • LeJan, Y. and Raimond, O. (2002). Integration of Brownian vector fields. Ann. Probab. 30 826–873.
  • Meyer, P.-A. (1995). Quantum Probability for Probabilists, 2nd ed. Lecture Notes in Math. 1538. Springer, New York.
  • Mikulevicius, R. (2002). On the Cauchy problem for stochastic Stokes equation. SIAM J. Math. Anal. 34 121–141.
  • Mikulevicius, R. and Rozovskii, B. L. (1998). Linear parabolic stochastic PDE's and Wiener chaos. SIAM J. Math. Anal. 29 452–480.
  • 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.
  • 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.
  • 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.
  • 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.
  • Mikulevicius, R. and Rozovskii, B. L. (2004). Stochastic Navier–Stokes equations for turbulent flows. SIAM J. Math. Anal. 35 1250–1310.
  • Rozovskii, B. L. (1990). Stochastic Evolution Systems. Kluwer Academic, Dordrecht.
  • Stein, A. M. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press.
  • Temam, R. (1985). Navier–Stokes Equations. North-Holland, Amsterdam.
  • Viot, M. (1976). Solutions faibles d'équation aux dérivées partielles stochastiques non linéaires. Thése, Univ. Pierre and Marie Curie, Paris.
  • Vishik, M. I. and Fursikov, A. V. (1979). Mathematical Problems of Statistical Hydromechanics. Kluwer Academic, Dordrecht.
  • Yamada, T. and Watanabe, S. (1971). On the uniqueness of solutions of stochastic differential equations I. J. Math. Kyoto Univ. 11 155–167.
  • Yamada, T. and Watanabe, S. (1971). On the uniqueness of solutions of stochastic differential equations II. J. Math. Kyoto Univ. 11 553–563.