204
namely
1/p(1-a)
{
(
2 )
lip
}
no = max n., YCfto
r2
•
Then we proceed essentially as in Lemma 3.1. The estimate on N that we obtain depends
explicitly on Y and it depends in a non explicit way on A through n. as explained in
Remark 3.2.
3.2.3. Dimension of the Inertial Manifolds
Estimates on the dimension of the inertial manifolds can be obtained when enough
informations are available on the An, so that one can estimate the values of n for which
(2.1) and (2.2) are satisfied. This may follow from simple arguments like in Lemma 3.1
(and Remark 3.2) or by using specific informations on the eigenvalues; the reader is
referred for details and examples to R. Temam and S. Wang (1993) where the An are
the eigenvalues of the Laplace-Beltrami operator on the sphere or to R. Temam and X.
Wang (1994) which concerns the Kuramoto-Sivashinsky equation.
Typically the dimension n of the inertial manifold is estimated in terms of Mo and
M1 (in (1.3), (1.4) or in (3.1), (3.2)). Usually these numbers are related to physically
relevant quantities, like a Grashoff or a Reynolds number, in particular in the case of
(3.1), (3.2) where these numbers are related to the radius of the absorbing ball of the
semigroup (see (3.7), (3.8)), and the estimates on the size of the absorbing balls.
4. INERTIAL MANIFOLDS AND SLOW MANIFOLDS
Our aim in this section is to present an application of Theorem 2.1 to a case where
the basic linear operator A is not selfadjoint. This example is related to the concept
of slow manifold which appears in meteorology (see e.g. R. Daley (1981), A. Kasahara
(1982), C.E. Leith (1980), B.A. Machenhauer (1977), J.J. Tribbia (1979,1982,1984),
R. Vautard and B. Legras (1986), A. Debussche and R. Temam (1991a,b), R. Temam
(1990)).
4.1. The Motivation
The following situation occurs in meteorology and oceanography. We consider in a
Hilbert space H, and abstract equation like (1.1),
du
dt + Aou = Rou,
( 4.1)
u(O) = Uo,
(4.2)
Here Ao is a linear closed selfadjoint positive unbounded operator in H of the type
encountered in most physically relevant equations ; Ro is a C 1 mapping from D( A a)
namely
1/p(1-a)
{
(
2 )
lip
}
no = max n., YCfto
r2
•
Then we proceed essentially as in Lemma 3.1. The estimate on N that we obtain depends
explicitly on Y and it depends in a non explicit way on A through n. as explained in
Remark 3.2.
3.2.3. Dimension of the Inertial Manifolds
Estimates on the dimension of the inertial manifolds can be obtained when enough
informations are available on the An, so that one can estimate the values of n for which
(2.1) and (2.2) are satisfied. This may follow from simple arguments like in Lemma 3.1
(and Remark 3.2) or by using specific informations on the eigenvalues; the reader is
referred for details and examples to R. Temam and S. Wang (1993) where the An are
the eigenvalues of the Laplace-Beltrami operator on the sphere or to R. Temam and X.
Wang (1994) which concerns the Kuramoto-Sivashinsky equation.
Typically the dimension n of the inertial manifold is estimated in terms of Mo and
M1 (in (1.3), (1.4) or in (3.1), (3.2)). Usually these numbers are related to physically
relevant quantities, like a Grashoff or a Reynolds number, in particular in the case of
(3.1), (3.2) where these numbers are related to the radius of the absorbing ball of the
semigroup (see (3.7), (3.8)), and the estimates on the size of the absorbing balls.
4. INERTIAL MANIFOLDS AND SLOW MANIFOLDS
Our aim in this section is to present an application of Theorem 2.1 to a case where
the basic linear operator A is not selfadjoint. This example is related to the concept
of slow manifold which appears in meteorology (see e.g. R. Daley (1981), A. Kasahara
(1982), C.E. Leith (1980), B.A. Machenhauer (1977), J.J. Tribbia (1979,1982,1984),
R. Vautard and B. Legras (1986), A. Debussche and R. Temam (1991a,b), R. Temam
(1990)).
4.1. The Motivation
The following situation occurs in meteorology and oceanography. We consider in a
Hilbert space H, and abstract equation like (1.1),
du
dt + Aou = Rou,
( 4.1)
u(O) = Uo,
(4.2)
Here Ao is a linear closed selfadjoint positive unbounded operator in H of the type
encountered in most physically relevant equations ; Ro is a C 1 mapping from D( A a)