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Remark 1.1 : Hypotheses (1.2)-(1.15) are standing hypotheses in this chapter (and in
the next one). We show in Section 3 below how the hypotheses above can be verified,
especially (1.5)-(1.13) ; in particular we show how to recover the Hilbert case and we
present a nonself-adjoint case related to the slow manifolds.
_
We recall that an inertial manifold (LM.) for equation (1.15) (or for the semigroup
{S(t)h>o) is a finite-dimensional Lipschitz manifold M, which enjoys the following
properties
(i) M is positively invariant for the semigroup (i.e. S(t)M C M, V t ~ 0);
(ii) M attracts all orbits of (1.1) at an exponential rate.
1.2. Construction of the Inertial Manifold
For some fixed n we apply the projectors Pn, Qn to (1.15) and setting Pnu = y,
Qnu = z, and using (1.14), we see that equation (1.15) is equivalent to a coupled system
for y and z,
{
dy
dt +Ay+PnR(y+z)=O,
yeO) = Yo,
{
~; +Az+QnR(y+z)=O,
z(O) = zo,
where Yo = Pnuo and Zo = QnUO'
(1.16)
(1.17)
Our construction of the inertial manifold will be based on the Lyapunov-Perron
method as in Chapter VIII, and we will obtain M as the graph of a function ~ :
PnE -+ QnE. The function cI> will be obtained, as in Chapter VII, as the fixed point of
a mapping T and we now explain the construction of T.
Assuming that cI> is known and that its graph is invariant for {S(t)k~:o, then for
an orbit u(t), t ~ 0, lying on M (i.e. if Uo EM), we have
z(t) = cI>(y(t)),
(1.18)
and we infer from (1.16) and (1.17) that y satisfies
{
dy
dt + Ay = PnR(y + cI>(y)), t > 0,
yeO) = Yo,
(1.19)
and the function t -+ cI>(y, (t)) (= z( t)) satisfies
dcI>(y)
~ + AcI>(y) = QnR(y + cI>(y)), t > O.
(1.20)