Weak Convergence Methods for Nonlinear Partial Differential ...

contained in a compact subset of W −1,2 (Ω). In fact, it even is a null-sequence in
that space: Viewing the smooth function εΦ(u ε ) xx ∈ W −1,2 (Ω) as a distribution
extended to a linear functional on W 1,2
0 which acts on test functions ϕ (and hence
all ϕ ∈ W 1,2
0 ) through ϕ ↦→ ∫ εΦ(u ε ) xx ϕ, we can estimate
{∫
}
‖εΦ(u ε ) xx ‖ W −1,2 = sup ε Φ(u ε ) xx ϕ : ϕ ∈ W 1,2
0 , ‖ϕ‖ W
1,2 = 1
0
{ ∫
}
= sup ε Φ(u ε ) x ϕ x : ‖ϕ‖ W
1,2 = 1
0
{ √ε
≤ sup ‖Φ ′ (u ε )‖ L ∞‖ √ εu ε x ‖ L 2‖ϕ x‖ L 2 : ‖ϕ‖ W
1,2
0
≤ C √ ε,
}
= 1
where we have used Φ(u ε ) x = Φ ′ (u ε ) u ε x in the second step and Lemma 3.7 in the
fourth step. Lemma 3.5 now implies the claim.
u ε being a solution of (3.8) we have
∫ ∞ ∫ ∞
0
−∞
u ε ϕ t + F(u ε ) ϕ x + εu ε ϕ xx dxdt +
∫ ∞
−∞
g(x) ϕ(x, 0) dx = 0
for any ϕ ∈ Cc ∞ (R × [0, ∞)). As a consequence of the above claim we can
now deduce from Theorem 3.4 that F(u ε ) ⇀ ∗ F(u). With Ω bounded such that
supp ϕ ⊂ Ω we finally get that indeed
∫ ∞ ∫ ∞
0
−∞
u ϕ t + F(u) ϕ x dxdt +
∫ ∞
−∞
g(x) ϕ(x, 0) dx = 0,
which shows that u is an integral solution.
If there is no interval on which F is affine, then also u ε → u strongly in any
L p , p < ∞. But then
∫ ∞
∫ ∞
0
−∞
∫ ∞
Φ(u) ϕ t + Ψ(u) ϕ x dxdt = lim
ε→0
0
∫ ∞
−∞
Φ(u ε ) ϕ t + Ψ(u ε ) ϕ x dxdt ≥ 0.
Remark 3.9 Other methods show that in fact there always exists an entropy solution.
The proof by compensated compactness, however, appears most amenable
to systems of conservation laws. But little is known rigorously for such systems.
We conclude the section by making the remark that the entropy condition
does indeed single out a unique integral solution:
Theorem 3.10 There exists at most one entropy solution of (3.2).
A proof of this statement can be found, e.g., in [Ev 98]. (XXX)
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