2410 CHAPTER 70. MEASURABILITY WITHOUT UNIQUENESS

What happens with the nonlinear operator N̂? Let w ∈ L∞ ([0,T ] ;V ) . A computationshows then that∣∣∣∣∫ T

0⟨Nun(t)−Nu(t),w(t)⟩dt

∣∣∣∣=

∣∣∣∣∫ T

0

∫U(uni(t)un j(t)−ui(t)u j(t))w j,idxdt

∣∣∣∣≤ ∥w∥L∞([0,T ],V )

∫ T

0

∫U(|un(t)|+ |u(t)|)(|u(t)−un(t)|)dxdt

≤ ∥w∥L∞([0,T ],V )

(∫ T

0

∫U(|un|+ |u|)2 dxdt

)1/2(∫ T

0

∫U(|u−un|)2 dxdt

)1/2

.

This converges to 0 thanks to the estimates and the strong convergence 70.4.10. Similarconvergence holds for the other nonlinear terms B(un (t) ,q) ,B(q,un (t)).

We have shown that for any n≥ m, and w ∈Vm,⟨u′n (t) ,w

⟩V ′,V + ⟨A(un(t)),w⟩V ′,V +

⟨N̂(un(t)),w

⟩= ⟨f(t),w⟩ . (70.4.12)

Let ζ ∈C∞ ([0,T ]) be such that ζ (T ) = 0. Then⟨u′n (t) ,wζ (t)

⟩V ′,V + ⟨A(un(t)),wζ (t)⟩V ′,V +

⟨N̂(un(t)),wζ (t)

⟩= ⟨f(t),wζ (t)⟩ .

Integrating this equation from 0 to T we obtain

−(u0n (ω) ,wζ (0))H −∫ T

0ζ′ (s)(un (s,ω) ,w)H ds

= −∫ T

0⟨A(un (s,ω)),wζ (s)⟩ds−

∫ T

0

⟨N̂(un (s,ω)),wζ (s)

⟩ds

+∫ T

0⟨f(s,ω) ,wζ (s)⟩ds.

Now letting n→ ∞, from the above list of convergent sequences,

−(u0 (ω) ,wζ (0))H −∫ T

0ζ′ (s)(u(s,ω) ,w)H ds

= −∫ T

0⟨A(u(s,ω)),wζ (s)⟩ds−

∫ T

0

⟨N̂(u(s,ω)),wζ (s)

⟩ds

+∫ T

0⟨f(s) ,wζ (s)⟩ds.

It follows that in the sense of V ′ valued distributions,

u′(ω)+A(u(ω))+ N̂(u(ω)) = f(ω) (70.4.13)

along with the initial conditionu(0) = u0. (70.4.14)

This has proved most of the following lemma: