76.7. OTHER EXAMPLES, INCLUSIONS 2593
where Φ ∈ L∞((0,T )×Ω,L2
(Q1/2U,H
))for H = L2 (0,1). As in the previous example,
simple considerations involving maximal monotone operators imply that there exists Ψ ∈L∞((0,T )×Ω,L2
(Q1/2U,W
))such that Φ = BΨ. We also assume that f ∈ V ′ and u0,v0
are in L2 (Ω,V ) and L2 (Ω,W ) respectively, both being F0 measurable. The above equationdoes not fit the general theory developed earlier because it is second order in time and isa stochastic version of a hyperbolic equation rather than a parabolic one. We consider itusing a parabolic regularization which can be studied with the above general theory alongwith a simple fixed point argument.
Consider the approximate problem which is to find a solution to
Bv(t)−Bv0 + ε
∫ t
0Avds+
∫ t
0A(u)ds =
∫ t
0f ds+
∫ t
0ΦdW (76.7.71)
where u is given above as an integral of v. First we argue that there exists a unique solutionto the above integral equation and then we pass to a limit as ε → 0. Let u ∈ V be given.
From Corollary 76.4.9 there exists a unique solution v to 76.7.71. Now suppose u1,u2are two given in V and denote by vi the corresponding v which solves the above. Thenfrom the Ito formula or standard considerations,
12
E ⟨B(v1 (t)− v2 (t)) ,v1 (t)− v2 (t)⟩+ εE∫ t
0∥v1− v2∥2
V ds
≤ ε
2E∫ t
0∥v1− v2∥2
V ds+Cε E∫ t
0∥u1−u2∥2
V ds
Now define a mapping θ from V to V as follows. Begin with v then obtain
u(t)≡ u0 +∫ t
0v(s)ds (76.7.72)
Use this u in 76.7.71. Then θv is the solution to 76.7.71 which corresponds to u. Then theabove inequality shows that∫ t
0
∫Ω
∥θv1 (s)−θv2 (s)∥2 dPds≤ Cε
ε
∫ t
0
∫Ω
∥u1−u2∥2V dPds
≤ Cε
εCT
∫ t
0
∫ s
0
∫Ω
∥v1 (r)− v2 (r)∥2V dPdrds
It follows that a high enough power of θ is a contraction map on L2(0,T,L2 (Ω,V )
)and
so there exists a unique fixed point. This yields a unique solution to the above approximateproblem 76.7.71 in which u,v are related by 76.7.72.
Next we let ε → 0. Index the above solution with ε . By the Ito formula again,
12
E ⟨Bvε (t) ,vε (t)⟩−12
E ⟨Bv0,v0⟩+ ε
∫ t
0E ∥vε∥2
V ds
+12
E ∥uε (t)∥2V −
12
E ∥u0∥2V =
∫ t
0E ⟨ f ,vε⟩ds