Weak Convergence Methods for Nonlinear Partial Differential ...

We will try to find solutions by first looking at the “singularly perturbed”
system {
∂ t u ε + ∂ x F(u ε ) − ε∂ xx u ε = 0 in R × (0, ∞)
u ε = g on R × {0}.
Physically the term ε∂ xx u ε takes account of (small) viscosity effects. Mathematically
it regularizes the solutions.
The scalar equation in one space dimension:
We now restrict out attention to the special case of a scalar conservation law
in 1 + 1 dimensions. Note that then for any convex Φ there exists a Ψ such that
Φ ′ (y) F ′ (y) = Ψ ′ (y) ∀ y ∈ R,
so that (Φ, Ψ) is an entropy/entropy-flux pair.
We follow the approach by L. Tartar. The main ingredient into the existence
proof in the following theorem:
Theorem 3.4 Let Ω ⊂ R 2 be open and bounded, F : R → R in C 1 . Suppose
(u ε ) ⊂ L ∞ (Ω) satisfies
u ε ∗ ⇀ u in L ∞
and that for every entropy/entropy-flux pair (Φ, Ψ) there is a compact subset of
W −1,2
loc
containing
∂ t Φ(u ε ) + ∂ x Ψ(u ε )
for every ε > 0. Then
F(u ε ) ∗ ⇀ F(u) in L ∞ ,
F ′ (u ε ) → F ′ (u)
in L p ∀ p < ∞
and moreover, if there is no interval on which F is affine, then even
F(u ε ) ∗ ⇀ F(u) in L p ∀ p < ∞.
Proof. Fix an entropy/entropy-flux pair (Φ, Ψ) and consider the bounded sequence
(u ε , v ε , w ε , z ε ) := (u ε , F(u ε ), Φ(u ε ), Ψ(u ε )) ∈ L ∞ (Ω; R 4 ).
Passing to a subsequence we have
(u ε , v ε , w ε , z ε ) ∗ ⇀ (u, v, w, z) in L ∞ (Ω; R 4 )
for suitable u, v, w, z ∈ L ∞ (Ω).
In order to prove the first assertion, namely, F(u ε ) ∗ ⇀ F(u), it thus remains
to be seen that v = F(u) a.e.
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