Methods of Vanishing Viscosity for Nonlinear ... - ACMAC

Artificial Viscosity via Compensated Compactness
∂ t U ε + ∂ x F (U ε ) = ε∂ xx U ε
Assume: The system has a strictly convex entropy function η ∗ (U).
Estimates: There exists C independent of ε such that
Invariant Regions: ‖U ε ‖ L ∞ ≤ C (or ‖U ε ‖ L p ≤ C)
Dissipation Estimate: ‖ √ ε∂ x U ε ‖ L 2 ≤ C
Energy Estimate:
ε(∂ x U ε ) ⊤ ∇ 2 η ∗ (U ε )∂ x U ε = −∂ t η ∗ (U ε ) − ∂ x q ∗ (U ε ) + ε∂ xx η ∗ (U ε )
=⇒ For any η ∈ C 2 with entropy flux q (i.e., ∇q(U) = ∇η(U)∇F (U) ),
∂ t η(U ε ) + ∂ x q(U ε )
is compact in H −1
loc
Compensated Compactness =⇒ Strong Convergence of U ε (t, x)
∂ t U ε + ∂ x F (U ε ) = ε∂ x (D(U ε )∂ x U ε ) for ∇ 2 η ∗ (U)D(U) ≥ c 0 > 0.
Large Initial Data, · · ·
Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20–24, 2011 10 / 40