Stochastic Navier–Stokes Equations

Let us begin with the abstract evolution form of the controlled stochastic Navier-Stokes equation [9] in the divergence free subspace H of square integrable vector fields which are parallel to the boundary

du(t) + (νAu(t) +B(u(t)))dt = U(t)dt + dW (t). (22.1)

Here ν is the coefficient of kinematic viscosity, A is the Stokes operator and B(·) is the nonlinear inertia term with well-known properties. U(t) is a distributed control with possible local support and W (t) is an H-valued Wiener process with covariance operator Q. Here both the cases of degenerate noise (where Q is of trace class) and nondegenerate noise (where, for example, Q = I) are of importance. Moreover, flow problems in two and threedimensional bounded, periodic as well as unbounded physical regions are of interest.