The Navier-Stokes equations are a system of nonlinear, partial differential equations which describe the motion of a viscous, incompressible fluid. I will describe an approach based on ideas from dynamical systems theory to understand the long time behavior of solutions of these equations. For systems of ordinary differential equations invariant manifold theorems have been a powerful tool for identifying the modes of the system which are most important for governing the long-time behavior of solutions as well as providing a way of computing the asymptotics of these solutions.
On the other hand, if one studies the Navier-Stokes equations on R2 or R3 the phase space of the linearization of the equation does not seem to possess the "center", "stable", and "unstable" subspaces needed to apply the invariant manifold theory. However, rewriting the equations in terms of similarity variables we are able identify finite dimensional, invariant manifolds in the phase space which control the long-time behavior of solutions in a neighborhood of the origin and in the neighborhood of certain special vortex solutions, the Oseen or Burger's vortices. In two dimensions this leads to a very complete picture of the evolution of solutions -- any solution whose initial vorticity distribution is somewhat localized will converge, as time tends to infinity, to an explicit vortex solution at a computable (and optimal) rate.
This is joint work with Prof. Thierry Gallay of the University of Grenoble.