Methods of Vanishing Viscosity for Nonlinear ... - ACMAC

Methods of Vanishing Viscosity
Hyperbolic Conservation Laws:
∂ t U(t, x) + ∂ x F (U(t, x)) = 0,
U(t, x) ∈ R N
F : R N → R N Nonlinear: Eigenvalues of ∇ U F (U) are real
∂ t U(t, x) + ∂ x F{U(t, x), U (t) (·, x)} = 0
U (t) (τ, x) := U(t − τ, x)–the past history of U and x r.t. t
Approaches: Honor Physical or Design Artificial N × N matrix function:
D : R N → M N×N , D(U) ≥ 0
∂ t U ε + ∂ x F (U ε ) =ε∂ x (D(U ε )∂ x U ε )
admits a global solution U ε (t, x) for each fixed ε > 0;
U ε (t, x) → U(t, x) topology ?? U(t, x) an entropy solution??
Stokes (1848), Rayleigh (1910), Taylor (1910), Weyl (1949), · · ·
Theory: Entropy Conditions, Nonuniqueness, Existence, Solution Behavior, · · ·
Numerical Methods/Applications: Shock Capturing, Upwind, Kinetic, · · ·
Challenges: Singular limits, · · · =⇒ New Math Ideas/Methods · · ·
*Analogously for Multidimensional Setting
Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20–24, 2011 2 / 40