Methods of Vanishing Viscosity for Nonlinear ... - ACMAC

Hyperbolic Systems: Artificial Viscosity via BV-Estimates II
Using the heat kernel to estimate the solution for t ∈ [0, τ ε ]:
‖∂ x U ε (t, ·)‖ L 1 ≤ κ δ, κ indept. of ε and δ
Decompose ∂ x U ε along a suitable basis of unit vectors {r 1 , · · · , r N }:
∂ x U ε = ∑ v ε
i r i
(Sum of gradients of viscous travelling waves)
Obtain a system of N equations for these scalar components:
∂ t v ε
i
+ ∂ x (˜λ i v ε
i )−ε∂ xx v ε
i = φ ε i , i = 1, · · · , N.
Then, as the scalar case, we obtain that, for all t ≥ τ ε ,
∫ ∞ ∫
‖vi ε (t, ·)‖ L 1 ≤ ‖vi ε (τ ε , ·)‖ L 1 + |φ ε i (t, x)|dxdt.
Construct the basis {r 1 , · · · , r N } in a clever way so that, for t ≥ τ ε ,
∫ ∞ ∫
|φ ε i (t, x)|dxdt ≤ C, C > 0 independent of ε > 0.
τ ε
=⇒ Tot. Var.{U ε (t, ·)} = ‖U ε x (t, ·)‖ L 1 ≤ ∑ i
τ ε
‖v ε
i (t, ·)‖ L 1 ≤ C
Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20–24, 2011 8 / 40