### A report on the paper "Infinite dimensional dynamical systems and
Navier-Stokes equation" by E. Wayne

Yuri Latushkin

Abstract.
In the series of expository talks, I will try to report in some detail on
the paper by C. Eugene Wayne published in the volume "Hamiltonian Dynamical
Systems and Applications" in 2008. The paper contains many interesting
tricks specific for infinite dimensional PDE dynamical systems. This review
paper is mainly based on the paper
"Invariant Manifolds and the Long-Time Asymptotics of the Navier Stokes
and Vorticity Equations on R^2" by Thierry Gallay & C. Eugene Wayne (Arch.
Rational Mech. Anal. 163 (2002) 209-258).
In addition, I will include some more preparatory material regarding the
Navier-Stokes equations.
The paper by Wayne consists of four parts. The first part is a brief review
of the invariant manifold and invariant foliation theorems and LaSalle
Invariance Principle in infinite dimensional spaces. In the second part we
will discuss a slick rescaling of the nonlinear heat equation in R^d that
allows one to find central manifolds associated with the eigenvalues hidden
under the essential spectrum of the linearization. The third part is a
brief introduction to the Navier-Stokes equation including well-posedness
in R^2. Finally, in the fourth part we will apply the invariant manifold
theorem from the first part, and the rescaling from the second part to
produce an invariant manifold for the two dimensional Navier-Stokes
equation, and to show stability of so called Oseen's vortices, the steady
states solutions of the rescaled Navier-Stokes equations.