Weak Convergence Methods for Nonlinear Partial Differential ...

Summarizing our discussion, we have found that if Φ is convex and Ψ ′ = F ′ Φ ′
then
in regions where u is smooth and
∂ t Φ(u) + ∂ x Ψ(u) = 0 (3.3)
[[Ψ(u)]] ≥ [[F(u)]] [[Φ(u)]]
[[u]]
along a jump discontinuity curve C. By the Rankine-Hugoniot condition [[F(u)]]
[[u]]
=
σ, this inequality can be written as
[[Ψ(u)]] ≥ σ[[Φ(u)]]. (3.4)
Now a calculation similar to the derivation of the Rankine-Hugoniot condition
shows that conditions (3.3) and (3.4) are implied by the inequality
∂ t Φ(u) + ∂ x Ψ(u) ≤ 0.
(In the distributional sense, i.e., after testing with arbitrary v ∈ Cc
∞
v ≥ 0.)
such that
Remark 3.2 In physical applications, −Φ can be an entropy as, e.g., for a mass
density u = ρ with
−Φ(ρ) = −ρ log ρ.
(Indeed, (−Φ(ρ)) ′′ = (− log ρ − 1) ′ = − 1 ρ ≤ 0.)
This motivates the following definition:
Definition 3.3 1. (Φ, Ψ) is called an entropy/entropy-flux pair if Φ is convex
and DΨ = DF DΦ.
2. An integral solution u of (3.2) is called an entropy solution if
∂ t Φ(u) + ∂ x Ψ(u) ≤ 0
holds true for every entropy/entropy-flux pair (Φ, Ψ).
This negativity condition is to be understood in the distributional sense, i.e.
∫ ∞ ∫ ∞
0 −∞
Φ(u) ∂ t v + Ψ(u)∂ x v dxdt ≥ 0 ∀ v ∈ C ∞ c
(R × (0, ∞)), v ≥ 0.
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