1214 CHAPTER 34. GELFAND TRIPLES AND RELATED STUFF

≤C[∥Lu∥2

V ′[0,T ]

+∥u∥2V[0,T ]

]≤C||u||2X . (34.6.54)

This verifies 3 in the case u = v. To obtain the general case,

|⟨Bu,v⟩(t)| ≤ ⟨Bu,u⟩1/2 (t)⟨Bv,v⟩1/2 (t)≤C||u||X ||v||X .

To verify 4, use 34.6.47 to write for t ∈ [0,T ] and I = [−T,2T ] ,∣∣⟨B̃un (t)− B̃um (t) ,un (t)−um (t)⟩∣∣

≤ 2∣∣∣∣∫ 2T

−T⟨(i∗B̃(un−um)

)′(s) ,un (s)−um (s)⟩ds

∣∣∣∣+∫ 2T

−T

∣∣⟨B̃′ (s)(un (s)−um (s)) ,un (s)−um (s)⟩∣∣ds≤ (34.6.55)

C[∥∥∥(i∗B̃un

)′− (i∗B̃um)′∥∥∥

V ′I||un−um||VI

+ ||un−um||2WI

]≡ Enm.

Then from 34.6.48, limn,m→∞ Enm = 0 and so, for t ∈ [0,T ] ,∣∣⟨B̃un (t)− B̃um (t) ,w⟩∣∣≤ E1/2

nm ⟨B(t)w,w⟩1/2 ≤CE1/2nm ||w||W .

It follows that B̃un (·) is uniformly Cauchy in the space of continuous functions C (0,T ;W ′)and so it converges to z ∈C (0,T ;W ′) . But B̃un converges in L2 (0,T ;W ′) to Bu(·) . There-fore B(t)u(t) = z(t) a.e. Letting Bu(·) = z(·) , this shows 4. Formula 5 follows from 3 andthe following argument.

|⟨Bu(t) ,w⟩| ≤ ⟨Bu,u⟩1/2 (t)⟨Bw,w⟩1/2 ≤C ||u||X ||w||W .

Assertion 6 follows easily from the first five parts. It remains to get 7.

Re⟨Ku,u⟩ =∫ T

0Re⟨Lu,u⟩dt + ⟨Bu,u⟩(0)

=∫ T

0

12[⟨Bu,u⟩′ (t)+ ⟨B′ (t)u(t) ,u(t)⟩

]dt + ⟨Bu,u⟩(0)

=12⟨Bu,u⟩(T )+ 1

2⟨Bu,u⟩(0)+ 1

2

∫ T

0⟨B′ (t)u(t) ,u(t)⟩dt

It only remains to verify the last assertion. Let ψn be increasing and piecewise linearsuch that ψn (t) = 1 for t ≥ 2/n and equals 0 on [0,1/n]. Then clearly ψnu→ u in V . Also

(B(ψnu))′ = ψ′nBu+ψn (Bu)′

The latter term converges to (Bu)′ in V ′. Now consider the first term.∫ T

0

∥∥ψ′nBu∥∥p′

V ′ dt ≤∫ 2/n

0n∥∥∥∥∫ t

0(Bu)′ ds

∥∥∥∥p′

dt

≤ n∫ 2/n

0t p′−1

∫ t

0

∥∥(Bu)′∥∥p′

V ′ dsdt ≤∫ 2/n

0

∥∥(Bu)′∥∥p′

V ′ ds1p′(2/n)p′ n

Since p′ > 1, this converges to 0.