ABSTRACT
In this paper we will review certain recent developments in the optimal stopping and impulse control problems for the stochastic Navier-Stokes equation. One of the main ingredients of this work is a new existence and uniqueness theorem for strong solutions in two dimensions. This result is obtained by utilizing a local monotonicity property of the sum of the Stokes operator and the nonlinearity. This gives a realization of the Markov-Feller process associated with the stochastic Navier-Stokes equation. The dynamic programming equations for the optimal stopping and impulse control problems arise as variational and quasi-variational inequalities respectively in infinite dimensions. These problems are then solved in a weak sense using the semigroup approach.
