2.4. Proof of Theorem 2.1
3. CompleUlents and Applications
3.1. The Locally Lipschitz Case
3.2. Dimension of the Inertial Manifold
4. Inertial Manifolds and Slow Manifolds
4.1.The Motivation
4.2. The Abstract Equation
4.3. An Equation of Navier-Stokes Type.
1. THE FUNCTIONAL SETTING
The general form of the abstract equation that we consider is the following
du
dt + Au = R(u).
(Ll)
In Section 1.1 we present the general hypotheses related to equation (1.1), then, in
Section 1.2, we describe our approach to the construction of inertial manifolds.
1.1. Notations and Hypotheses
We are given three Banach spaces E, F, E, such that
E c FeE,
(1.2)
each space being dense in the following one, the injections being continuous. The norms
on E,E and F are denoted by I . Ie, I . IE and I . IF.
We assume that R is a C 1 nonlinear function from E into F which satisfies the
following boundedness and Lipschitz properties
IR(u)IF s:: Mo,
Vu E E,
(1.3)
IR(u)-R(v)IFS::Mllu-vIE, VU,vEE;
(1.4)
here Mo and Ml are positive constants.
Concerning the linear operator A in (1.1), we assume that A is a linear densely
defined operator in E. Furthermore we assume that the linear equation
y+Au =0,
{
du
u~O) = Uo,
(1.5)