Abstract
For a certain class of stochastic evolution equations in a Banach space, a sufficient condition is given to ensure the evolution equation of Itô type has a stationary strong solution corresponding to some invariant initial distribution. It is then shown that the general result can be applied to the two-dimensional Navier-Stokes equations perturbed by a Gaussian white-noise and provides a simple proof that such a system of stochastic equations has a stationary solution.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Albeverio, S. and Cruzerio, A.-B., Global Flows with Invariant (Gibbs) Measures for Euler and Navier-Stokes Two Dimensional Fluids, 129, Comm. Math. Phys., 1990, pp. 432–444.
Breiman, L., Probability Theory,Addison-Wesley, Reading, Mass.,1968.
Chow, P.-L. and Khasminskii, R.Z., Stationary Solutions of Nonlinear Stochastic Evolution Equation;, Preprint, 1994.
Daprato, G. and Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, England, 1992.
Hida, T., Kuo, H.-H., Potthoff, J., and Streit, L., White Noise: An Infinite Dimensional Calculus, Kluwer Academic Pub., Dordrecht, The Netherlands, 1993.
Marcus, R., Parabolic Itô equations, Trans. Amer. Math. Soc. 198, 1974, pp. 177–190.
Marcus, R., Parabolic Itô Equations with Monotone nonlinearities, 29, J. Funct. Analy., 1978, pp. 257–287.
Temam, R., Navier-Stokes Equations and Nonlinear Functional Analysis, CBMSNSF Regional Conf. Series in Appl. Math., SIAM Pub., Philadelphia, 1983.
Vishik, M.J. and Fursikov, A.V., Mathematical Problems in Statistical Hydromechanics, Kluwer Academic Pub. Dordrecht, The Netherlands, 1988.
Walsh, J.B., An Introduction to Stochastic Partial Differential Equations, Ecole d’Eté de Probabilitiés de Saint Flour XIV, 1180, Lectures Notes in Mathematics, Springer-Verlag, Berlin, 1984, pp. 265–435.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Chow, P.L. (1996). Stationary Solutions of Two-Dimensional Navier-Stokes Equations with Random Perturbation. In: Funaki, T., Woyczynski, W.A. (eds) Nonlinear Stochastic PDEs. The IMA Volumes in Mathematics and its Applications, vol 77. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8468-7_13
Download citation
DOI: https://doi.org/10.1007/978-1-4613-8468-7_13
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-8470-0
Online ISBN: 978-1-4613-8468-7
eBook Packages: Springer Book Archive