A random process governed by a two-dimensional, stochastic Navier-Stokes system with a white noise under periodic boundary conditions is constructed. The solution of the corresponding Kolmogorov forward equation is obtained in the class of exponentially decreasing measures. Furthermore, the generalized solution of the corresponding Kolmogorov backward equation is also constructed. This solution possesses a probabilistic representation as path integral and satisfies a Hölder condition. The generalized solution is defined by the Green identity for the solution of the Kolmogorov forward and backward equations. On any small interval of time this generalized solution of the Kolmogorov backward equation is the classical one.
Cite this article
Marko I. Vishik, Alexander I. Komech, Strong solutions of a two-dimensional stochastic Navier-Stokes system and corresponding Kolmogorov equations (in Russian). Z. Anal. Anwend. 1 (1982), no. 3, pp. 23–52DOI 10.4171/ZAA/18