### Dynamical Systems and the Navier-Stokes Equations

Gene Wayne

Boston University

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 **R**^{2}
or **R**^{3} 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.

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