Sytem of Dynamic and Differential Physics Kentu...

Infinite dimensional dynamical systems and the Navier–Stokes equation 131
Inserting (2) and (3) into (1) we find that
∂ w
∂τ
= L w − v · ∇ w . (5)
Here, L is the same operator that we studied in Lecture 2 – namely
L w = ∆
ξ
w +
1
2
∇
ξ
· ( ξ w ) .
Recall that the spectrum of L when acting on functions in L
2
( m ) consists of the
non-positive half integers, plus a half-plane of spectrum { λ ∈ C | Re ( λ )
1
2
−
m
2
} .
Thus, for m > 1 we expect that there will be a one-dimensional invariant manifold
W
c
, tangent at the origin to the eigenspace of the (simple) eigenvalue λ = 0.
Remark 4.1. Verifying the hypotheses (H.1)–(H.4) of the (CHT) invariant manifold
theorem requires combining the ideas of Lectures 2 and 3. Since the linear part of
(2) is is the same as that of (26) verifying (H.1) and (H.2) is exactly the same as
in Lecture 2. Verifying the hypotheses (H.3) and (H.4) on the nonlinearity follows
from estimates very similar to those in Lecture 3 where we estimated the semi-group
for (1) since the form of the nonlinear terms in (1) are the same as those in (5). In