Backward Stochastic Navier - Stokes Equations in Two Dimensions

There are two parts in this dissertation. The backward stochastic Lorenz system is studied in the first part. Suitable a priori estimates for adapted solutions of the backward stochastic Lorenz system are obtained. The existence and uniqueness of solutions is shown by the use of suitable truncations and approximations. The continuity of the adapted solutions with respect to the terminal data is also established. The backward two-dimensional stochastic Navier-Stokes equations (BSNSEs, for short) corresponding to incompressible fluid flow in a bounded domain G are studied in the second part. Suitable a priori estimates for adapted solutions of the BSNSEs are obtained which reveal a surprising pathwise L∞(H) bound on the solutions. The existence of solutions is shown by using a monotonicity argument. Uniqueness is proved by using a novel method that uses finite-dimensional projections, linearization, and truncations. The continuity of the adapted solutions with respect to the terminal data and the external body force is also established.