Methods of Vanishing Viscosity for Nonlinear ... - ACMAC

Nonlocal Scalar Conservation Laws: BV -Estimates
∂ t u + ∂ x
(
f (u) +
∫ t
0 k δ(t − τ)f (u(τ))dτ) = 0
Memory kernel: κ δ (t) ⇀ (α − 1) δ(t) when δ → 0
Artificial Viscosity Method (C-Christoforou 2007):
∂ t u ε + ∂ x f (u ε ) + ∫ t
0 k δ(t − τ)f (u ε (τ)) x dτ
= ε∂ xx (u ε + ∫ t
k 0 δ(t − τ)u ε (τ)dτ)
Idea: Employ the resolvent kernel r δ (t) of k δ (t): r δ + k δ ∗ r δ = −k δ
(MacCamy 1977, Dafermos 1988, Nohel-Rogers-Tzavaras 1988)
=⇒ ∂ t u ε + ∂ x f (u ε ) + r δ (0)u ε = r δ (t)u 0 − ∫ t
r ′ 0 δ (t − τ)uε (τ) + ε∂ xx u ε
Estimates: There exists C independent of ε, δ such that
BV ∩ L ∞ Estimates:
‖u ε,δ ‖ L ∞ + ‖∂ x u ε,δ ‖ L 1 + ‖∂ t u ε,δ ‖ L 1 ≤ C
BV-Compactness Theorem =⇒ Strong Convergence of u ε,δ (t, x)
Existence and Stability of Entropy Solutions u δ (t, x) to hyperbolic
conservation laws with memory as the limit of ε → 0
When κ δ (t) ⇀ (α − 1) δ(t) as δ → 0, u δ (t, x) → u(t, x) in L 1 , a
solution to the scalar conservation laws ∂ t u + α∂ x f (u) = 0
Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20–24, 2011 4 / 40