209
where n = (0,£)( is the f-cube.
By classical results, (4.28)-(4.31) reduce to an evolution equation for u of the form
(1.1). Let H;!r(n) denote the restriction to n of the n-periodicfunctions v from JRl into
JR which are locally in Hm(JRl). Let H;!r(n) denote the subspace of H;!r(n) consisting
of the functions v which satisfy In v(x)dx = o. Then the spaces H;!r(n) and H;!r(n)
are both Hilbert subspaces of Hm(n).
For the application of Theorem 2.1, we define F = & = H to be the subspace of
L2(n)l consisting of the restrictions to n of the locally L2 vector functions with free
divergence and vanishing average on n. We set D(Ao) = H~!r(n) n H and D(A~/2) =
·u
Hper(n) n H. Then
Aou = c( -~t u, for all u E D(Ao),
and Ro( u) is defined by
(Ro(u), v) = -y 10 ~u· v dx + 10 ((u· 'V)u)v dx - j, If u, v E D(A),
where j is given in H.
We choose E = D(A~) and
1
Q: = 4 for f = 2,3,
It is easy to check that, for r = 2£, R is a C 1 mapping from D(A~) into H.
Then equation
du
dt + Aou = Ro(u),
u(O) = Uo,
(4.32)
(4.33)
satisfies all the hypotheses of Theorem 2.1 except (1.3) and (1.4) which are replaced by
(3.1) and (3.2). Hence Theorem 2.1 applies to the prepared form of this equation, as in
Theorem 3.1. It is noteworthy that several other choices are possible here for r and for
E and F ; E = D(A~+i'), F = & = D(AJ), with another value of Q: and suitable values
of 'Y. Theorems 2.1 and 3.1 give then inertial manifolds in D(A~+i'); this is left as an
exercise to the reader.
Now, let us discuss the nonself-adjoint case corresponding to the slow manifold
motivation as in Section 4.1.
We consider a stationary solution ii,p of (4.28), (4.29), i.e.
c( -~rii - Y~ii + (ii . 'V)ii + 'Vp = j,
( 4.34)
'V. ii = 0,
(4.35)