76.4. THE GENERAL CASE 2565
Lemma 76.4.6 If Φ ∈ L2(Ω,L∞
([0,T ] ,L2
(Q1/2U,W
)))then∫ T
0
(Φn ◦ J−1)∗Bun ◦ JdW →
∫ T
0
(Φ◦ J−1)∗Bu◦ JdW
Proof:
E(∣∣∣∣∫ T
0
(Φn ◦ J−1)∗Bun ◦ JdW −
∫ T
0
(Φ◦ J−1)∗Bu◦ JdW
∣∣∣∣)
≤ E(∣∣∣∣∫ T
0
(Φn ◦ J−1)∗Bun ◦ JdW −
∫ T
0
(Φ◦ J−1)∗Bun ◦ JdW
∣∣∣∣)+E(∣∣∣∣∫ T
0
(Φ◦ J−1)∗Bun ◦ JdW −
∫ T
0
(Φ◦ J−1)∗Bu◦ JdW
∣∣∣∣)
≤∫
Ω
((∫ T
0∥Φn−Φ∥2
L2⟨Bun,un⟩
)1/2)
dP
+∫
Ω
(∫ T
0∥Φ∥2
L2⟨Bun−Bu,un−u⟩dt
)1/2
dP (76.4.44)
Consider that second term. It is no larger than∫Ω
∥Φ∥L∞([0,T ],L2)
(∫ T
0⟨Bun−Bu,un−u⟩dt
)1/2
dP
≤(∫
Ω
∥Φ∥2L∞([0,T ],L2)
)1/2(∫Ω
∫ T
0⟨Bun−Bu,un−u⟩dtdP
)1/2
Now consider the following. Letting the ei be the special vectors of Lemma 34.4.2, itfollows, ∫
Ω
∫ T
0⟨Bun−Bu,un−u⟩dtdP =
∫Ω
∫ T
0
∞
∑i=1⟨Bun−Bu,ei⟩2 dtdP
=∫
Ω
∫ T
0
∞
∑i=1
lim infp→∞
⟨Bun−Bup,ei
⟩2 dtdP
≤ lim infp→∞
∫Ω
∫ T
0
∞
∑i=1
⟨Bun−Bup,ei
⟩2 dtdP
= lim infp→∞
∫Ω
∫ T
0
⟨Bun−Bup,un−up
⟩dtdP≤ T
2n
The last inequality follows from 76.4.34. Therefore, the second term in 76.4.44 is no largerthan (C (T,Φ)/2n)1/2 which converges to 0 as n→ ∞. Now consider the first term in76.4.44. ∫
Ω
((∫ T
0∥Φn−Φ∥2
L2⟨Bun,un⟩
)1/2)
dP