34.4. THE IMPLICIT CASE 1189

Thus, if sn→ t,sn ∈ D, Fatou’s lemma implies

⟨Bu,u⟩(t) = ⟨B(u(t)) ,u(t)⟩=∞

∑k=1|⟨Bu(t) ,ei⟩|2

≤ lim infn→∞

∞

∑k=1|⟨Bu(sn) ,ei⟩|2 ≤C (∥Y∥K′ ,∥u∥K)

and sosupt∈NC⟨Bu,u⟩(t) = sup

t∈NC⟨B(u(t)) ,u(t)⟩ ≤C (∥Y∥K′ ,∥u∥K)

It only remains to verify the claim about weak continuity.Consider now the claim that t→ Bu(t) is weakly continuous on NC. Letting v ∈V,s ∈

NC,

limt→s⟨Bu(t) ,v⟩= ⟨Bu(s) ,v⟩= ⟨Bu(s) ,v⟩ (34.4.24)

The limit follows from the formula 34.4.22 which implies t→ Bu(t) is continuous into V ′.Now for t ∈ NC,

∥Bu(t)∥= sup∥v∥≤1

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

which was shown to be bounded for t,s ∈ NC. Now let w ∈W . Then

|⟨Bu(t) ,w⟩−⟨Bu(s) ,w⟩| ≤ |⟨Bu(t)−Bu(s) ,w− v⟩|+ |⟨Bu(t)−Bu(s) ,v⟩|

Then the first term is less than ε if v is close enough to w and the second converges to 0 so34.4.24 holds for all v ∈W and so this shows the weak continuity on NC.

Now pick t ∈ D, the union of all the mesh points. Then for all k large enough, t ∈Pk.Say t = tm. From Lemma 34.4.4,

−m−1

∑j=0

⟨B(u(t j+1

)−u(t j)

),(u(t j+1

)−u(t j)

)⟩=

⟨Bu(tm) ,u(tm)⟩−⟨Bu0,u0⟩−2m−1

∑j=0

∫ t j+1

t j

⟨Y (r) ,urk (r)⟩dr

Thus, ⟨Bu(tm) ,u(tm)⟩ is constant for all k large enough and the integral term converges to∫ tm

0⟨Y (r) ,u(r)⟩dr

It follows that the term on the left does converge to something as k→ ∞. It just remains toconsider what it does converge to. However, from the equation solved by u,

Bu(t j+1

)−Bu(t j) =

∫ t j+1

t j

Y (r)dr