Abstract
Stochastic Navier-Stokes equations are investigated. Preliminary results of existence of solutions are summarized, and open questions on the well posedness are discussed. Uniqueness and ergodicity of the asymptotic regime (invariant measure) is also presented.
Similar content being viewed by others
References
Albeverio, S. andCruzeiro, A.B.,Global flow and invariant (Gibbs) measures for Euler and Navier-Stokes two dimensional fluids. Comm. Math. Phys.,129 (1990), 431–444.
Bensoussan, A. andTemam, R.,Equations stochastiques du type Navier-Stokes. J. Funct. Anal.,13 (1973), 195–222.
Chorin, A.J., Vorticity and Turbulence, Springer-Verlag, New York, 1994.
Chorin, A.J. andMarsden, J.E., A Mathematical Introduction to Fluid Mechanics, second ed., Springer-Verlag, New York, 1990.
Da Prato, G. andZabczyk, J., Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992.
Doob, J.L.,Asymptotic properties of Markov transition probability, Trans. Amer. Math. Soc.,64 (1948), 393–421.
Ferrario, B.,Ergodicity for the 2-dimensional stochastic Navier-Stokes equation, preprint (1996).
Flandoli, F.,Dissipativity and invariant measures for stochastic Navier-Stokes equations, Nonlinear Anal. and Appl.,1 (1994), 403–423.
Flandoli, F. andGątarek, D.,Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Relat. Fields,102 (1995), 367–391.
Flandoli, F., andMaslowski, B.,Ergodicity of the 2-D Navier-Stokes equation under random perturbations. Comm. Math. Phys.,171 (1995), 119–141.
Flandoli, F. andSchmalfuss, B.,Attractors for 3-D Navier-Stokes equations with non-regular or stochastic force, preprint (1996).
Fujita Yashima H., Equations de Navier-Stokes stochastiques non homogeneous et applications, Scuola Normale Superiore, Pisa (1992).
Holmes, P.,Can dynamical systems approach turbulence?, In: Whiter turbulence? Turbulence at the crossroad, Lumley ed., Springer-Verlag, Berlin, 1991.
Ikeda, N. andWatanabe, S., Stochastic Differential Equations and Diffusion Processes, North-Holland, Kodanska, 1981.
Khas'minskii, R.Z.,Ergodic properties of recurrent diffusion processes and stabilization of the solutions to the Cauchy problem for parabolic equations, Theory of Prob. and its Appl.,5 (1960), 179–196.
Kraichnan, R.H.,Turbulent cascade and intermittency growth, Proc. R. Soc. London,434 (1991), 65–78.
Landau, L.D. andLifschitz, E.M., Fluid Mechanics, Pergamon Press, Oxford, 1959.
Lions, J.L. andMagenes, E., Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin, 1972.
Monin, A.S. andYaglom, A.M., Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. II, M.I.T. Press, Cambridge, 1975.
Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983.
Revuz, D. andYor, M., Continuous Martingales and Brownian Motion, second ed., Springer-Verlag, Berlin, 1994.
Seidler, J.,Ergodic behaviour of stochastic parabolic equations, Math. Nachrichten (1995).
Stettner, L.,Remarks on ergodic conditions for Markov processes on Polish spaces, Bull. Polish Acad. Sci. Math.,42 (1994), 103–114.
Stroock, D.W. andVaradhan, S.R.S., Multidimensional Diffusion Processes, Springer-Verlag, New-York, 1979.
Temam, R., The Navier-Stokes Equations, North-Holland, 1977.
Viot, Solution Faibles d'Equations aux Derivees Partielles Stochastiques non Lineaires, These de Doctorat, Paris VI, 1976.
Vishik, M.I. andFursikov, A.V., Mathematical Problems of Statistical Hydromechanics, Kluwer, Dordrecht, 1980.
Author information
Authors and Affiliations
Additional information
Conferenza tenuta il 7 maggio 1996
Rights and permissions
About this article
Cite this article
Flandoli, F. Stochastic differential equations in fluid dynamics. Seminario Mat. e. Fis. di Milano 66, 121–148 (1996). https://doi.org/10.1007/BF02925357
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02925357