Weak Convergence Methods for Nonlinear Partial Differential ...

Now note that the Laplace operator −∆ with Dirichlet boundary conditions
induces an isomorphism between W 1,2
0 (˜Ω) and W −1,2 (˜Ω). Without proof we will
use the fact that this is indeed true for the spaces W 1,p
0 (˜Ω) and W −1,p (˜Ω) for any
p ∈ (1, ∞).
It follows that (v (ν)
1 ) is compact in W 1,2
0 (˜Ω). As for v (ν)
2 , we note that M(˜Ω)
embeds compactly into W −1,p (˜Ω) for p <
n by Theorem 2.32 and so (v(ν)
n−1 2 )
is precompact in W 1,p (˜Ω). So v (ν) := v (ν)
1 + v (ν)
2 is precompact in W 1,p (˜Ω) for
p < n . n−1
But for any q < ∞ we have
−∆v (ν) = g (ν)
1 + g (ν)
2 = g (ν) in ˜Ω, v (ν) = v (ν)
1 + v (ν)
2 = 0 on ∂˜Ω,
whence v (ν) is bounded in W 1,q (˜Ω). For q > 2 this boundedness together with
precompactness in W 1,p (˜Ω) for some p ≥ 1 proves that (v (ν) ) is precompact in
W 1,2 (˜Ω) and so (g (ν) ) is precompact in W −1,2 (˜Ω).
□
Now consider the approximating equations
∂ t u ε + ∂ x F(u ε ) − ε∂ xx u ε = 0 in R × (0, ∞), u ε (x, 0) = g(x). (3.8)
Lemma 3.6 There exist classical solutions u ε of (3.8) such that both u ε and
ε∂ x u ε are bounded in L ∞ .
We will not prove this lemma. But note that it suffices to provide a solution
ū for ε = 1 and initial conditions ū(x, 0) = g(εx) and then set u ε (x) := ū( x ε , t ε ).
Lemma 3.7 If u ε is a solution of (3.8) as given by Lemma 3.6, then √ ε∂ x u ε is
bounded in L 2 (Ω) for each Ω ⊂ R × [0, ∞) bounded.
Proof. Let Ω ⊂ R×[0, ∞) be bounded and, w.l.o.g., of the form Ω = (a, b)×[0, T).
By testing (3.8) with u ε itself we obtain
∫
∫
ε u ε xx u ε dxdt = u ε t u ε + F(u ε ) x u ε dxdt
and so, after partial integrations,
∫ T
ε
0
= 1 2
Ω
[u ε x u ε ] x=b
x=a dt − ε ∫ T
∫ T
0
∫ b
∂ t (u ε ) 2 dxdt +
a
0
∫ b
a
∫ T
0
Ω
u ε x u ε x dxdt
[F(u ε ) u ε ] x=b
x=a dt − ∫ T
0
∫ b
a
F(u ε ) u ε x dxdt.
Now since u ε and εu ε x are uniformly bounded, the first term on the left hand
side and the second term on the right hand side of this equation are bounded
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