Methods of Vanishing Viscosity for Nonlinear ... - ACMAC

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Further Challenges Nonstrictly Hyperbolic Systems Initial Data of Large Oscillation without Bounded Variation Viscosity Matrix with ∇ 2 η ∗ (U)D(U) ≥ 0 but Not Positive Definite ?? No L ∞ -Uniform Bound: Only Energy Bounds · · · · · · Important Problems: The Compressible Navier-Stokes Equations =⇒ the Compressible Euler Equations Global Spherically Symmetric Solutions to the Compressible Euler Equations Global Solutions to the Compressible Euler Equations with Random Forcing Terms Global Solutions to Nonlocal Conservation Laws with Memory · · · · · · Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20–24, 2011 12 / 40
Navier-Stokes Equations: Inviscid Limit with the initial conditions: ρ(0, x) = ρ 0 (x), u(0, x) = u 0 (x), { ρt + (ρu) x = 0, (ρu) t + (ρu 2 + p) x = εu xx , lim (ρ 0(x), u 0 (x)) = (ρ ± , u ± ), x→±∞ (ρ ± , u ± ) are constant end-states with ρ ± > 0 The viscosity coefficient ε ∈ (0, ε 0 ] for some fixed ε 0 ρ – Density, u – Velocity of Fluid when ρ > 0 p = p(ρ) = ρ 2 e ′ (ρ) – Pressure with internal energy e(ρ) For a polytropic perfect gas, p(ρ) = κρ γ , e(ρ) = κ γ−1 ργ−1 , γ > 1 Existence of C 2 solutions (ρ ε , u ε )(t, x) for Large Data: Ya Kanel 1968, Hoff 1998 Problem: Inviscid Limit of (ρ ε , u ε )(t, x) when ε → 0?? Gilberg (1951), Hoff-Liu (1989), Gùes-Métivier-Williams-Zumbrun (2006), · · · · · · Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20–24, 2011 13 / 40
Page 1 and 2: Methods of Vanishing Viscosity for
Page 3 and 4: Methods of Vanishing Viscosity Hype
Page 5 and 6: Scalar Conservation Laws: BV -Estim
Page 7 and 8: Scalar Conservation Laws: Compensat
Page 9 and 10: Hyperbolic Systems: Artificial Visc
Page 11 and 12: Hyperbolic Systems: Vanishing Visco
Page 13: Compactness: Young Measure and Comm
Page 17 and 18: Vanishing Artificial/Numerical Visc
Page 19 and 20: Isentropic Euler Equations: Reducti
Page 21 and 22: Navier-Stokes Equations ⇒ Euler E
Page 23 and 24: Navier-Stokes Equations ⇒ Euler E
Page 25 and 26: Navier-Stokes Equations: Key Estima
Page 27 and 28: Navier-Stokes Equations: Key Estima
Page 29 and 30: Navier-Stokes Equations: Higher Int
Page 31 and 32: Measure-Valued Solution: Reduction
Page 33 and 34: Measure-Valued Solution with Unboun
Page 35 and 36: Measure-Valued Solution with Unboun
Page 37 and 38: Measure-Valued Solution: Any Connec
Page 39 and 40: Measure-Valued Solution: Any Connec
Page 41 and 42: Multi-D Isentropic Euler Eqns with
Page 43 and 44: Focusing: Imploding Spherically Sym
Page 45 and 46: Spherically Symmetric Solutions for
Page 47 and 48: Conservation Laws of Viscoelastic M
Page 49 and 50: Vanishing Viscosity for Further Fun
Page 51: Happy 70th Birthday Costas! Gui-Qia

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