200
From Theorem 2.1 we promptly infer the
Theorem 3.1. We assume that the hypotheses (1.2), (1.5)-(1.14), (3.1) and (3.2) are
satisfied and that (2.1) and (2.2) hold with Mo,Ml given by (3.4).
Then the prepared equation (3.4) possesses an inertial manifold M = Me of class
Cl, which is the graph of the function eli = elie E :Fb,t, given by Theorem 2.1.
Remark 3.1. We discuss the relation of Me with an inertial manifold M for (1.15) j
Me has all the properties required of an inertial manifold for (1.15), except that it is
not positively invariant for the semigroup {S(t)}t>Oj it is positively invariant for the
semigroup {Se(t)}t>o.
For (1.15) we n~d a generalization of the inertial manifold, defined on some subset
ofPE.
Definition 3.1. In the non Lipschitz case, M is an inertial manifold for (1.15) (or
{S(t)},>o) if
(i) M is the graph of eli, where q; is a Lipschitz mapping on an open set 0 of PE
into QEj
(i) S(t)M eM, V t ~ OJ
(i) There exist two positive constants ]{,]{', ]{ depending boundedly on luol E , ]{'
independent of uo, such that
dist(S(t)uo, M) :5 K exp( -K'(t)), V t ~ O.
If the open ball B of E centered at 0 of radius p is an absorbing set and a positively
invariant set for both (1.15) and (3.1), then it can be easily verified that for 0 = PB,
the part M of Me above 0 is an inertial manifold for (1.15). If the absorbing set is
closed or is not positively invariant, then M C Me can be obtained, for some suitable
set OJ for the details of the construction of 0 in this case the reader is refereed to C.
Foias, B. Nicolaenko, G. Sell and R. Temam (1988).
3.2. Dimension of the Inertial Manifold
We give some estimate on the dimension of the inertial manifold given by Theorem
2.1 or 3.1 (Section 3.2.3). Before, we show in Sections 3.2.1 and 3.2.2 how the hypotheses
of Theorems 2.1 and 3.1 are verified in cases of interest.
3.2.1. The FUnction Spaces and the Operator A.
We restrict ourselves to the case where the spaces E, F, t: are Hilbert spaces.
In the self-adjoint case we set t: = H, and then A is a self-adjoint unbounded operator in H as classically encountered in the dissipative equations considered of mathematical physics. In this case we set F = D(A1) and E = D(A1+"), for some I ~ 0 and
From Theorem 2.1 we promptly infer the
Theorem 3.1. We assume that the hypotheses (1.2), (1.5)-(1.14), (3.1) and (3.2) are
satisfied and that (2.1) and (2.2) hold with Mo,Ml given by (3.4).
Then the prepared equation (3.4) possesses an inertial manifold M = Me of class
Cl, which is the graph of the function eli = elie E :Fb,t, given by Theorem 2.1.
Remark 3.1. We discuss the relation of Me with an inertial manifold M for (1.15) j
Me has all the properties required of an inertial manifold for (1.15), except that it is
not positively invariant for the semigroup {S(t)}t>Oj it is positively invariant for the
semigroup {Se(t)}t>o.
For (1.15) we n~d a generalization of the inertial manifold, defined on some subset
ofPE.
Definition 3.1. In the non Lipschitz case, M is an inertial manifold for (1.15) (or
{S(t)},>o) if
(i) M is the graph of eli, where q; is a Lipschitz mapping on an open set 0 of PE
into QEj
(i) S(t)M eM, V t ~ OJ
(i) There exist two positive constants ]{,]{', ]{ depending boundedly on luol E , ]{'
independent of uo, such that
dist(S(t)uo, M) :5 K exp( -K'(t)), V t ~ O.
If the open ball B of E centered at 0 of radius p is an absorbing set and a positively
invariant set for both (1.15) and (3.1), then it can be easily verified that for 0 = PB,
the part M of Me above 0 is an inertial manifold for (1.15). If the absorbing set is
closed or is not positively invariant, then M C Me can be obtained, for some suitable
set OJ for the details of the construction of 0 in this case the reader is refereed to C.
Foias, B. Nicolaenko, G. Sell and R. Temam (1988).
3.2. Dimension of the Inertial Manifold
We give some estimate on the dimension of the inertial manifold given by Theorem
2.1 or 3.1 (Section 3.2.3). Before, we show in Sections 3.2.1 and 3.2.2 how the hypotheses
of Theorems 2.1 and 3.1 are verified in cases of interest.
3.2.1. The FUnction Spaces and the Operator A.
We restrict ourselves to the case where the spaces E, F, t: are Hilbert spaces.
In the self-adjoint case we set t: = H, and then A is a self-adjoint unbounded operator in H as classically encountered in the dissipative equations considered of mathematical physics. In this case we set F = D(A1) and E = D(A1+"), for some I ~ 0 and