1206 CHAPTER 34. GELFAND TRIPLES AND RELATED STUFF

Also recall that t → B(t)u(t) was shown to be weakly continuous into W ′ on NC. Is itcontinuous on NC? Suppose t ∈ NC and let sn → t where sn ∈ D. Then u(sn) = ur

mn (t)because sn is one of the mesh points. Since sn → t one can assume that mn → ∞. Henceu(sn) = ur

mn (t)→ u(t) by the pointwise convergence implied by 34.5.29. Then obviously

B(sn)u(sn) = B(sn)ulmn (t)→ B(t)u(t)

Now suppose you just have tn→ t where each of tn, t are in NC. Does it always follow thatB(tn)u(tn)→ B(t)u(t)? Suppose not. Then there exists such a sequence tn→ t of pointsin NC and ε > 0 such that

∥B(tn)u(tn)−B(t)u(t)∥ ≥ ε

However, from the density of D and what was just shown, there exists sn ∈ D such that|sn− tn|< 1

2n and

∥B(sn)u(sn)−B(tn)u(tn)∥<12n

Then

ε ≤ ∥B(tn)u(tn)−B(sn)u(sn)∥+∥B(sn)u(sn)−B(t)u(t)∥

<12n +∥B(sn)u(sn)−B(t)u(t)∥

Since sn→ t, what was just shown implies both terms on the right converge to 0. This is acontradiction. Thus t→ B(t)u(t) must be continuous on NC into W ′.

Consider the case that t → B(u(t)) has a weak derivative, denoted as (Bu)′ (t) whichis in Lp′ (0,T ;V ′) . Then as shown above, there is a continuous function, denoted as Bu(t)which equals B(t)(u(t)) for a.e. t and

Bu(t) = Bu(0)+∫ t

0(Bu)′ (s)ds

Then the above theorem applies. Then one obtains the following corollary.

Corollary 34.5.5 Let V ⊆W,W ′ ⊆V ′ be separable Banach spaces, and B(t)∈L (W,W ′)is nonnegative and self adjoint, B ∈C1 ([0,T ] ;W ′). Also suppose t→ B(u(t)) has a weakderivative (Bu)′ ∈ Lp′ (0,T ;V ′) for u ∈ Lp ([0,T ] ;V )∩L2 ([0,T ] ;W ). Then there is a con-tinuous function denoted as Bu(t) which equals B(t)(u(t)) a.e. t. Say for t /∈ N. SupposeBu(0) = Bu0, u0 ∈W. Then

Bu(t) = Bu0 +∫ t

0(Bu)′ (s)ds in V ′ (34.5.38)

Then t→ Bu(t) is in C(NC,W ′

)and also for such t,

12⟨Bu(t) ,u(t)⟩+ 1

2

∫ t

0

⟨B′ (s)u(s) ,u(s)

⟩ds =

12⟨Bu0,u0⟩+

∫ t

0

⟨(Bu)′ (s) ,u(s)

⟩ds