Weak Convergence Methods for Nonlinear Partial Differential ...

independently of ε. The first term on the right hand side is bounded, too, because
it is equal to
1
2
∫ b
a
[
(u ε ) 2] t=T
t=0 dx.
But also the last term is bounded as we can choose G(y) := ∫ y
0
F(s) ds and write
∫ T ∫ b
0
a
F(u ε ) u ε x dxdt =
∫ T ∫ b
In summary, it follows that
‖ √ ∣ ∫ ∣∣∣ T
εu ε x‖ 2 L 2 (Ω) = ε
0
0
a
G(u ε ) x dxdt =
∫ b
a
∫ T
|u ε x| 2 dxdt
∣ ≤ C.
0
[G(u ε )] x=b
x=a dt.
□
Theorem 3.8 Suppose F : R → R is smooth, g ∈ C 1 ∩ L 1 ∩ L ∞ . Then
u t + F(u) x = 0 in R × (0, ∞),
u(x, 0) = g(x)
has an integral solution. If there is no interval on which F is affine, then there
even exists an entropy solution.
Proof. Let (u ε ) be a sequence of solutions of (3.8) as given by Lemma 3.6. W.l.o.g.
we have
u ε ∗ ⇀ u
for some u ∈ L ∞ . Let Φ be convex and choose Ψ such that Ψ ′ = F ′ Φ ′ . Let
Ω ⊂ R × [0, ∞) be bounded.
Claim: Φ(u ε ) t + Ψ(u ε ) x lies in a compact subset of W −1,2
loc
(Ω).
In order to prove this claim, we first note that w.l.o.g. we may assume that Φ
and hence Ψ is smooth: By mollification one can approximate Φ by convex smooth
functions uniformly such that also Φ ′ is approximated boundedly in measure.
Now calculate
Φ(u ε ) t + Ψ(u ε ) x = Φ ′ (u ε ) u ε t + Ψ′ (u ε ) u ε x
= Φ ′ (u ε ) (εu ε xx − F(uε ) x ) + Φ ′ (u ε ) F ′ (u ε ) u ε x
= ε Φ ′ (u ε ) u ε xx
= ε Φ(u ε ) xx − ε Φ ′′ (u ε ) (u ε x )2 .
Clearly, Φ(u ε ) t + Ψ(u ε ) x is bounded in W −1,∞ because Φ(u ε ) and Ψ(u ε ) are
bounded in L ∞ . Also, Φ ′′ (u ε ) is bounded in L ∞ and ε(u ε x )2 is bounded in L 1
by Lemma 3.7 and so εΦ ′′ (u ε )(u ε x) 2 is bounded in M(Ω). Finally, εΦ(u ε ) xx is
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