A special evolution equation used in the analysis of the stochastic Navier-Stokes equation (Q2722274)

Stochastic Navier-Stokes equations have important physical and technical applications. They describe the behaviour of a viscous velocity field of an incompressible liquid influenced by random internal and external perturbations. Several approximation methods were developed for these equations, for example, the Galerkin method. The latter is useful to prove the existence of a solution, but it is complicated for numerical developments because it involves nonlinear terms. In this paper a new approximation method is proposed by making use of linear evolution equations which are easier to study. It is also proved that the approximations converge in mean square to a solution of the stochastic Navier-Stokes equation. This approximation method makes use of stochastic evolution equations of the special type. Existence of a solution of this type of equations is proved. This is the basic result for the development of the approximation method. A weak convergence property is also given for solutions of equations of this type. This result is used to prove the existence of optimal controls as soon as in the formulation of a stochastic minimum principle for the control problem of the stochastic Navier-Stokes equation.