Stochastic 2D Navier-Stokes Equation and Applications to 2D Turbulence
We will consider the 2-dimensional Navier-Stokes equation for an incompressible fluid with periodic boundary condition, and with a random perturbation that is in the form of white noise in time and a deterministic perturbation due to the large deviation principle. Our ultimate goal is to find appropriate conditions on the initial data and the forcing terms so that global existence and uniqueness of a mild solution is guaranteed. We will use the Picard's iteration method to prove existence of local mild solution and then prove the existence of a maximal solution which then leads to global existence. The result is applied to the backward Kolmogorov-Obukhov energy cascade and the forward Kraichnan enstrophy cascade in 2D turbulence.