2688 CHAPTER 79. INCLUDING STOCHASTIC INTEGRALS
Now apply this Ito formula to Theorem 79.1.4 in which we make the assumptionsthere on ∥u0∥ ∈ L2 (Ω) and that f ∈ Lp′ ([0,T ]×Ω;V ′) where the σ algebra is P theprogressively measurable σ algebra, and
Φ ∈ L2(
Ω,L2([0,T ] ,L2
(Q1/2U,W
)))which implies the same is true of Φn. This yields, from the assumed estimates, an expres-sion of the form where δ > 0 is a suitable constant.
12⟨Bun,un⟩(t)−
12⟨Bu0,u0⟩+δ
∫ t
0∥un (s)∥p
V ds
≤ λ
∫ t
0⟨Bun,un⟩(s)ds+
∫ t
0⟨ f ,un⟩V ′,V ds+
∫ t
0c(s,ω)ds
+∫ t
0⟨BΦn,Φn⟩L2
ds+Mn (t) (79.1.13)
where c ∈ L1 ([0,T ]×Ω). Then taking expectations or using that part of the Ito formula,
12
E (⟨Bun,un⟩(t))+δE(∫ T
0∥un (s)∥p
V ds)
≤ λ
∫ t
0E (⟨Bun,un⟩(s))ds+
∫ t
0E(⟨ f ,un⟩V ′,V
)ds+C (Φ,u0)
Then by Gronwall’s inequality and some simple manipulations,
E (⟨Bun,un⟩(t))+E(∫ T
0∥un (s)∥p
V ds)≤C (T, f ,u0,Φ)
Then using obvious estimates and Gronwall’s inequality in 79.1.13, this yields an in-equality of the form
⟨Bun,un⟩(t)−⟨Bu0,u0⟩+∫ t
0∥un (s)∥p
V ds≤C ( f ,λ ,c)+∥B∥∫ t
0∥Φn∥2
L2ds+M∗n (t)
where the random variable C ( f ,λ ,c) is nonnegative and is integrable. Now t →M∗n (t) isincreasing as is the integral on the right. Hence it follows that, modifying the constants,
sups∈[0,t]
⟨Bun,un⟩(s)+∫ t
0∥un (s)∥p
V ds
≤C ( f ,λ ,c,u0)+2∥B∥∫ t
0∥Φn∥2
L2ds+2M∗n (t) (79.1.14)
Next take the expectation of both sides and use the Burkholder Davis Gundy inequalityalong with the description of the quadratic variation of the martingale Mn (t). This yields
E
(sup
s∈[0,t]⟨Bun,un⟩(s)
)+E
(∫ t
0∥un (s)∥p
V ds)
≤ C+2∥B∥E(∫ t
0∥Φn∥2
L2ds)