The above approach is pretty interesting, but there is a quicker way to do it discussed in
this section. I am also including the case where the operator B is actually a function
of t. I have never had a reason to use this level of generality, but it is here if
it is of any interest. Also, this is presented in the context of complex Banach
spaces. In addition, it is shown that by including i^{∗} in various formulas, you don’t
need to have V dense in W. Of course, this is typically not of any interest, but
for the sake of generality, it is included. The approach is due to Lions. It is
assumed for convenience that p ≥ 2. This was apparently not needed in the last
section. It may be that this approach can also be generalized to not require
this.
Let B
(t)
∈ℒ
′
(W, W )
satisfy
---------
〈B (t)u,v〉 = 〈B (t)v,u〉,u,v ∈ W (32.6.39)
(32.6.39)
〈B (t)u,u〉 ≥ 0 (32.6.40)
(32.6.40)
∫
t ′
B (t) = B (0)+ 0 B (s)ds (32.6.41)
(32.6.41)
where B^{′}∈ L^{∞}
(0,T;ℒ (W,W ′))
. Here W is a Banach space such that V ⊆ W. Also
V_{I}≡ L^{p}
(I;V )
and W_{I}≡ L^{2}
(I;W )
.
Now let I = [a,b] and c < a < b < d.Here and in what follows ϕ_{n}
(t)
= nϕ
(nt)
where
ϕ ≥ 0,ϕ ∈ C_{0}^{∞}
(− 1,1)
, and ∫ϕdt = 1. The following proposition is known and the
essential features of its proof may be found in [?]. We give a proof for the convenience of
the reader.
Proposition 32.6.1Suppose D
(t)
∈ℒ
′
(W,W )
and D
(t)
= 0 if t
∈∕
(c,d). Suppose alsothat
∫
t ′ ′ ∞ ′
D (t) = c D (s)ds,D ∈ L (c,d;ℒ (W, W )).
For u ∈W_{I}and a − n^{−1}> c,b + n^{−1}< d, define
′ ′
Tnu = (D (u∗ ϕn))− ((Du )∗ϕn) (32.6.42)
(32.6.42)
where we let u = 0 off I. Then
||Tnu ||W′I → 0 (32.6.43)
(32.6.43)
Proof: First, we show that
||T ||
n
is uniformly bounded. Letting w = 0 off
I,
|∫ ∫ |
|〈T u,w〉| = || 〈D′(t) u (s)ϕ (t − s)ds,w (t)〉dt||
n | ℝ ℝ n |