Methods of Vanishing Viscosity for Nonlinear ... - ACMAC

Vanishing Artificial/Numerical Viscosity Methods II
Physical Convex Entropy: Mechanical Energy-Energy Flux Pair (η ∗ , q ∗ ):
η ∗ (ρ, m) = 1 m 2
2 ρ + e(ρ), q ∗(ρ, m) = 1 m 3
2 ρ 2 + me′ (ρ)
=⇒ ∇ 2 U η ∗(U) > 0 for U = (ρ, m) when 1 < γ ≤ 2.
W.O.L.G. assume that lim |x|→∞ U ε = Ū = U ± . Set
Φ ∗ (U) = η ∗ (U) − η ∗ (Ū) − ∇η∗(Ū)(U − Ū) ≥ 0,
Ψ ∗ (U) = q ∗ (U) − q ∗ (Ū) − ∇η∗(Ū)(F (U) − F (Ū))
Multiply ∇Φ ∗ (U ε ), both sides of the system:
∂ t Φ ∗ (U ε ) + ∂ x Ψ ∗ (U ε ) = ε∂ x (∇Φ ∗ (U ε )∂ x U ε ) − ε(∂ x U ε ) ⊤ ∇ 2 η ∗ (U ε )∂ x U ε .
=⇒ ε ∫ T
0
∫ ∞
−∞ (∂ xU ε ) ⊤ ∇ 2 η ∗ (U ε )∂ x U ε dxdt
≤ ∫ ∞
−∞ Φ ∗(U 0 (x)) dx < ∞
Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20–24, 2011 16 / 40