76.4. THE GENERAL CASE 2567

Hence we obtain

Bu(t)−Bu0 (ω)+∫ t

0A(ω)(u)ds =

∫ t

0f (s,ω)ds+B

∫ t

0ΦdW

This is a solution for a given ω /∈ N. Also, a stopping time argument like the above and thecoercivity estimates for A along with the implicit Ito formula show that u ∈ V . This yieldsthe existence part of the following existence and uniqueness theorem.

Theorem 76.4.7 Suppose V ≡ Lp ([0,T ]×Ω,V ) where p ≥ 2,with the σ algebra of pro-gressively measurable sets and Vω = Lp ([0,T ] ,V ).

Φ ∈ L2([0,T ]×Ω,L2

(Q1/2U,W

))∩L2

(Ω,L∞

([0,T ] ,L2

(Q1/2U,W

))),

f ∈ V ′ ≡ Lp′ ([0,T ]×Ω,V ′)

and both are progressively measurable. Suppose that

λB+A(ω) : Vω → V ′ω , λB+A : V → V ′

are monotone hemicontinuous and bounded where

A(ω)u(t)≡ A(t,u(t) ,ω)

and (t,u,ω)→ A(t,u,ω) is progressively measurable. Also suppose for p ≥ 2, the coer-civity, and the boundedness conditions

λ ⟨Bu,u⟩+ ⟨A(t,u,ω) ,u⟩V ≥ δ ∥u∥pV − c(t,ω) (76.4.46)

where c ∈ L1 ([0,T ]×Ω) for all λ large enough. Also,

∥A(t,u,ω)∥V ′ ≤ k∥u∥p−1V + c1/p′ (t,ω) (76.4.47)

also suppose the monotonicity condition for all λ large enough.

⟨(λB+A(ω))(u)− (λB+A(ω))(v) ,u− v⟩ ≥ δ ∥u− v∥α

U (76.4.48)

Then if u0 ∈ L2 (Ω,W ) with u0 F0 measurable, there exists a unique solution u(·,ω) ∈ Vω

with u ∈ V (Lp ([0,T ]×Ω,V ) and progressively measurable) such that for ω off a set ofmeasure zero,

Bu(t,ω)−Bu0 (ω)+∫ t

0A(s,u(s,ω) ,ω)ds =

∫ t

0f ds+B

∫ t

0ΦdW.

It is also assumed that V is a reflexive separable real Banach space.

Proof: The uniqueness assertion follows easily from the monotonicity condition.Now we remove the assumption that Φ ∈ L2

(Ω,L∞

([0,T ] ,L2

(Q1/2U,W

))). Every-

thing is the same except for the need for a different argument to show that∫ T

0

(Φn ◦ J−1)∗Bun ◦ JdW →

∫ T

0

(Φ◦ J−1)∗Bu◦ JdW