Methods of Vanishing Viscosity for Nonlinear ... - ACMAC

Limit System: Isentropic Euler Equations ??
{
∂t ρ + ∂ x m = 0, (m = ρu)
∂ t m + ∂ x ( m2
ρ + p(ρ)) = 0
Strict hyperbolicity: Fails when ρ → 0 (vacuum)
Entropy Pair (η, q): ∇q(U) = ∇η(U)∇F (U)
Convex Entropy: ∇ 2 η(U) > 0 Weak Entropy: η(ρ, ρu)| ρ=0 = 0
Weak entropy pairs are represented as
∫
∫
η ψ (ρ, ρu) = χ(s)ψ(s) ds, q ψ (ρ, ρu) = (θs + (1 − θ)u)χ(s)ψ(s) ds
R
R
by C 2 test functions ψ(s), where χ(s) is the weak entropy kernel:
χ(s) := [ρ 2θ − (u − s) 2] λ
+ , θ = γ − 1 , λ = 3 − γ
2 2(γ − 1) .
Physical Convex Entropy: Mechanical energy-energy flux pair (η ∗ , q ∗ ):
η ∗ (ρ, m) = 1 m 2
2
ρ + ρe(ρ), q ∗(ρ, m) = 1 2
m 3
ρ 2 + m(e(ρ) + p ρ )
Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20–24, 2011 14 / 40