A Four Step Scheme Approach to the Forward-Backward Stochastic Navier-Stokes Equations

In this paper, the authors presented a novel fluid dynamics system, the forward-backward stochastic Navier-Stokes equations in two dimensions for incompressible fluid flows. The well-posedness of the system is obtained through a two-step process. First, certain projections of the system to the finite dimensions are employed, and the existence and uniqueness of solutions in finite dimensions are proved via the four step scheme. Then the Galerkin approximation is used to show the existence and uniqueness of a solution to the system in an infinite dimensional functional setup.