76.5. REPLACING Φ WITH σ (u) 2575
and let u be the solution which results from placing w in σ . Then from the estimates,
⟨Bu,u⟩(t)−⟨Bu0,u0⟩+δ
∫ t
0∥u∥p
V ds = 2∫ t
0⟨ f ,u⟩ds+C (b3,b4,b5)
+λ
∫ t
0⟨Bu,u⟩ds+
∫ t
0⟨Bσ (w) ,σ (w)⟩L2
ds+2M∗ (t)
≤ 2∫ t
0⟨ f ,u⟩ds+C (b3,b4,b5)+λ
∫ t
0⟨Bu,u⟩ds+
∫ t
0
(C+C∥w∥2
W
)ds+2M∗ (t)
where M∗ (t) = sups∈[0,t] |M (s)| and the quadratic variation of M is no larger than∫ t
0∥σ (w)∥2 ⟨B(u) ,u⟩ds
Then using Gronwall’s inequality, one obtains an inequality of the form
sups∈[0,T ]
⟨Bu,u⟩(s)≤C+C(
M∗ (t)+∫ t
0∥w∥2
W ds)
where C = C (u0, f ,δ ,λ ,b3,b4,b5,T ) and is integrable. Then take expectation. By Burk-holder Davis Gundy inequality and adjusting constants as needed,
E
(sup
s∈[0,T ]⟨Bu,u⟩(s)
)
≤ C+C∫
Ω
∫ T
0∥w∥2
W dsdP+C∫
Ω
(∫ T
0∥σ (w)∥2 ⟨B(u) ,u⟩ds
)1/2
dP
≤ C+C∫
Ω
∫ T
0∥w∥2
W dsdP+C∫
Ω
sups∈[0,T ]
⟨Bu,u⟩1/2 (s)(∫ T
0∥σ (w)∥2 ds
)1/2
dP
≤C+C∫
Ω
∫ T
0∥w∥2
W dsdP+12
E
(sup
s∈[0,T ]⟨Bu,u⟩(s)
)+C
∫Ω
∫ T
0
(C+C∥w∥2
W
)Thus
E (⟨Bu,u⟩(t))≤ E
(sup
s∈[0,T ]⟨Bu,u⟩(s)
)≤C+C
∫Ω
∫ T
0∥w∥2
W dsdP
and so
∥u∥2L∞([0,T ],L2(Ω,W )) ≤C+C
∫Ω
∫ T
0∥w∥2
W dsdP
which implies u ∈ L∞([0,T ] ,L2 (Ω,W )
)and is progressively measurable.
Using the monotonicity assumption, there is a suitable λ such that
12⟨B(u1−u2) ,u1−u2⟩(t)+ r
∫ t
0∥u1−u2∥2
W ds
−λ
∫ t
0⟨B(u1−u2) ,u1−u2⟩ds