2704 CHAPTER 79. INCLUDING STOCHASTIC INTEGRALS
≤∫ t
0
(C+C |v|2H
)ds+C ( f ,w0,ω)+M∗ (t)
Using compactness of the embedding of V into W, we can simplify this to an inequality ofthe following form where this involves modifying the constants C.
12|v(t)|2H +2α
∫ t
0∥v∥2
V ds+12∥w(t)∥2
V
≤∫ t
0
(C+C |v|2H
)ds+C ( f ,w0,ω)+M∗ (t)
Then, since the functions on the right are increasing in t, we can modify the constants againand obtain an inequality of the form
sups≤t|v(s)|2H + sup
s≤t∥w(t)∥2
V +α
∫ t
0∥v∥2
V ds
≤∫ t
0
(C+C |v|2H
)ds+C ( f ,w0,ω)+CM∗ (t)
Take expectations using the Burkholder Davis Gundy inequality and estimates for σ andobtain
E(
sups≤t|v(s)|2H
)+E
(sups≤t∥w(t)∥2
V
)+E
(α
∫ t
0∥v∥2
V ds)
≤∫ t
0E(
C+C(
supτ≤s|v(τ)|2H
))ds
+C ( f ,w0)+C∫
Ω
(∫ t
0
((C+C |v|H)
2 supτ≤s|v(τ)|2H
))1/2
dP
≤∫ t
0E(
C+C(
supτ≤s|v(τ)|2H
))ds+C ( f ,w0)
+C∫
Ω
supτ≤t|v(τ)|H
(∫ t
0
(C+C |v|2H
))1/2
dP
Using Cauchy Schwarz inequality one can simplify the above to
E(
sups≤t|v(s)|2H
)+E
(sups≤t∥w(t)∥2
V
)+E
(α
∫ t
0∥v∥2
V ds)
≤ C ( f ,w0,T )+C∫ t
0E(
supτ≤s|v(τ)|2H
)ds
Then Gronwall’s inequality implies
E(
sups≤t|v(s)|2H
)+E
(sups≤t∥w(t)∥2
V
)+E
(α
∫ t
0∥v∥2
V ds)≤C ( f ,w0,T ) (79.3.39)