FINITE-DIMENSIONAL INERTIAL FORMS FOR THE 2D NAVIER-STOKES EQUATIONS

In this paper we explain how the long-time dynamics of 2D Navier-Stokes (N-S) equations with periodic boundary conditions on a suitable bounded region OMEGA in R2 can be described completely by a finite dimensional system of ordinary differential equations. Our approach is to imbed the 2D N-S equations into a reactions diffusion system which possesses exactly the same asymptotic dynamics. We then prove the existence of an inertial manifold for the transformed equations and we interpret the dynamics of N-S equations via the inertial form of the transformed equations.