Methods of Vanishing Viscosity for Nonlinear Conservation Laws

Transcription

1 Methods of Vanishing Viscosity for Nonlinear Conservation Laws Gui-Qiang G. Chen Mathematical Institute, University of Oxford Cleopatra Christoforou (Cyprus) Qian Ding (Northwestern) Marshall Slemrod (Wisconsin-Madison) Constantine Dafermos (Brown) Mikhail Perepelitsa (Houston) Dehua Wang (Pittsburgh) Special Session in Honor of Costas Dafermos Continuum and Kinetic Methods in the Theory Shocks, Fronts, Dislocations and Interfaces Archimedes Center for Modeling, Analysis & Computation (ACMAC) Heraklion, Crete, Greece, June 20 24, 2011 Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

2 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?? Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

3 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), Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

4 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, / 40

5 Scalar Conservation Laws: BV -Estimates t u ε + x f (u ε ) = ε xx u ε Estimates: There exists C independent of ε such that Maximum Principle: BV Estimates: u ε L C Hopf, Oleinik, Lax, Volpert, Kruzhkov, Volpert s Method: x u ε L 1 + t u ε L 1 C t ( x u ε ) + x (f (u ε ) x u ε ) ε xx ( x u ε ), t ( t u ε ) + x (f (u ε ) t u ε ) ε xx ( t u ε ). BV-Compactness Theorem = Strong Convergence of u ε (t, x) Similar arguments to obtain for the L 1 Equicontinuity directly A corollary of the L 1 -stability and the comparison principle via Kruzhkov s Method t u ε + x f(u ε ) = ε x u ε Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

6 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, / 40

7 Scalar Conservation Laws: Compensated Compactness t u ε + x f (u ε ) = ε xx u ε Estimates: There exists C independent of ε such that Maximum Principle: u ε L C (or u ε L p C) Dissipation Estimate: εux ε L 2 C Energy Estimate: ε( x u ε ) 2 = t ( (uε ) 2 u ε 2 ) x( wf (w)dw) + ε xx ( (uε ) 2 2 ) = For any η C 2 with entropy flux q(u) = u η (w)f (w)dw, 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 ) Tartar, Schonbek, DiPerna, Chen-Lu, Tadmor-Rascle-Bagnerini, Scalar Conservation Laws with Memory: Dafermos 1988 Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

8 Hyperbolic Systems: BV -Estimates via Glimm s Scheme (1965) = Wave Interaction Estimates BV-Estimate Techniques: Glimm Functional Global Existence of Solutions in BV when the Total Variation of the Initial Data Is Small Structure of Solutions in BV Dafermos, DiPerna, Asymptotic Behavior: Decay of Periodic Solutions, Glimm-Lax, Dafermos, DiPerna, Liu, Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

9 Hyperbolic Systems: Artificial Viscosity via BV-Estimates I t U ε + x F (U ε ) = ε xx U ε, U(0, x) = U 0 (x) R N Biachini-Bressan (2005): For strictly hyperbolic systems on U in a n.b.h.d. of a compact set K R N, there exist constants δ > 0 and C j, j = 1, 2, 3, such that, if Tot.Var.{U 0 } < δ, lim U 0(x) K, x then, ε > 0, there exists a unique solution U ε (t, ) := S ε t U 0 ( ) s.t. BV Bound: Tot.Var.{S ε t U 0 } C 1 Tot.Var.{U 0 } L 1 -Stability: S ε t U 0 S ε t V 0 L 1 C 2 U 0 V 0 L 1 S ε t U 0 S ε s U 0 L 1 C 3 ( t s + εt εs ) = Strong Convergence and L 1 -Stability of the Limit Solution Even for non-conservative strictly hyperbolic systems The approach requires the artificial viscosity form The total variation of the initial data is sufficiently small Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

10 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, / 40

11 Hyperbolic Systems: Vanishing Viscosity via BV-Estimates Further Problem: BV Estimates and Convergence of vanishing viscosity approximation of the general form: t U ε + x F (U ε ) = ε x (D(U ε ) x U ε ) for general viscosity matrices D(U). Physical Viscosity: Navier-Stokes Viscosity Matrices? Navier-Stokes Equations = Euler Equations?? Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

12 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, / 40

13 Compactness: Young Measure and Commutation Identity Div-Curl Lemma [Tartar (1979), Murat (1978)] Young Measure Rep. [Tartar (1979), Ball (1989), Alberti-Müller (2001)] = ν(λ), η1 (λ)q 2 (λ) q 1 (λ)η 2 (λ) = ν(λ), η 1 (λ) ν(λ), q 2 (λ) ν(λ), q 1 (λ) ν(λ), η 2 (λ) for any entropy-entropy flux pairs (η j, q j ), j = 1, 2, where ν = ν t,x (λ) is the associated Young measure (probability measure) for the sequence U ε (t, x). Issue: Is ν a Dirac measure? = Compactness of U ε (t, x) in L 1 Remarks: The viscosity matrix 2 η (U)D(U) > 0 = Dissipation Estimate 2 2 Strict Hyperbolicity: = Rich Entropy-Entropy Flux Pairs DiPerna, Dafermos, Serre, Morawetz, Perthame-Tzavaras, Chen-Li, Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

14 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, / 40

15 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, / 40

16 Limit System: Isentropic Euler Equations?? { t ρ + x m = 0, (m = ρu) t m + x ( m2 ρ + p(ρ)) = 0 Strict hyperbolicity: Fails when ρ 0 (vacuum) Entropy Pair (η, q): q(u) = η(u) F (U) Convex Entropy: 2 η(u) > 0 Weak Entropy: η(ρ, ρu) ρ=0 = 0 Weak entropy pairs are represented as η ψ (ρ, ρu) = χ(s)ψ(s) ds, q ψ (ρ, ρu) = (θs + (1 θ)u)χ(s)ψ(s) ds R R by C 2 test functions ψ(s), where χ(s) is the weak entropy kernel: χ(s) := [ρ 2θ (u s) 2] λ +, θ = γ 1, λ = 3 γ 2 2(γ 1). Physical Convex Entropy: Mechanical energy-energy flux pair (η, q ): η (ρ, m) = 1 m 2 2 ρ + ρe(ρ), q (ρ, m) = 1 2 m 3 ρ 2 + m(e(ρ) + p ρ ) Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

17 Vanishing Artificial/Numerical Viscosity Methods I { t ρ + x m = ε 2 Artificial Viscosity: xρ, t m + x ( m2 + p(ρ)) = ρ ε 2 xm, Numerical Viscosity: Lax-Friedrichs Scheme, Godunov Scheme, Advantages: There exists C > 0, independent of ε > 0, such that Invariant Regions = L Estimate: 0 ρ ε (t, x) C, 0 m ε (t, x) Cρ ε (t, x) a.e. Dissipation Estimate: ε x (ρ ε, m ε ) L 2 ([0,T ] R) C via the mechanical energy-entropy flux pairs (η, q ) that is strictly convex for 1 < γ 2 (and convex for γ > 2: weighted dissipation estimate). = For any C 2 weak entropy-entropy flux pair (η, q), t η(ρ ε, m ε ) + x q(ρ ε, m ε ) is compact in H 1 loc Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

18 Vanishing Artificial/Numerical Viscosity Methods II Physical Convex Entropy: Mechanical Energy-Energy Flux Pair (η, q ): η (ρ, m) = 1 m 2 2 ρ + e(ρ), q (ρ, m) = 1 m 3 2 ρ 2 + me (ρ) = 2 U η (U) > 0 for U = (ρ, m) when 1 < γ 2. W.O.L.G. assume that lim x U ε = Ū = U ±. Set Φ (U) = η (U) η (Ū) η (Ū)(U Ū) 0, Ψ (U) = q (U) q (Ū) η (Ū)(F (U) F (Ū)) Multiply Φ (U ε ), both sides of the system: t Φ (U ε ) + x Ψ (U ε ) = ε x ( Φ (U ε ) x U ε ) ε( x U ε ) 2 η (U ε ) x U ε. = ε T 0 ( xu ε ) 2 η (U ε ) x U ε dxdt Φ (U 0 (x)) dx < Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

19 Isentropic Euler Equations: Reduction of a Measure-Valued Solution ν t,x : supp ν Is Bounded DiPerna: γ = N+2, N Ding-Chen-Luo, Chen: γ (1, 5] 3 Lions-Perthame-Tadmor: γ 3 Lions-Perthame-Souganidis: γ ( 5, 3) = γ (1, 3) 3 Chen-LeFloch: General Pressure Laws Conclusion: If supp ν is bounded, then ν t,x = ν (ρ(t,x),m(t,x)), that is, (ρ ε (t, x), m ε (t, x)) (ρ(t, x), m(t, x)) a.e. (t, x). Key Point: Employ Effectively the Weak Entropy-Entropy Flux Pairs Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

20 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: Vanishing Viscosity 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, / 40

21 Navier-Stokes Equations Euler Equations New Difficulties: No Invariant Regions: Only Energy Norms No Direct Derivative Estimates: ε x (ρ ε, m ε ) L2 ([0,T ] R) C although ε x u ε L 2 ([0,T ] R) C No A-Priori Bounded Support of the Measure-Valued Solution ν t,x Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

22 Navier-Stokes Equations Euler Equations New Difficulties: No Invariant Regions: Only Energy Norms No Direct Derivative Estimates: ε x (ρ ε, m ε ) L2 ([0,T ] R) C although ε x u ε L 2 ([0,T ] R) C No A-Priori Bounded Support of the Measure-Valued Solution ν t,x Strategies: Finite-Energy Bounds+Higher Integrability Bounds (replacing L bound) New Derivative Estimate for ε x ρ ε H 1 -Compactness of Weak Entropy Dissipation Measures Only for Weak Entropy Pairs with Compactly Supported C 2 Test Functions Prove that Any Connected Component of Support of the Measure-Valued Solution ν t,x Must Be Bounded Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

23 Navier-Stokes Equations Euler Equations New Difficulties: No Invariant Regions: Only Energy Norms No Direct Derivative Estimates: ε x (ρ ε, m ε ) L2 ([0,T ] R) C although ε x u ε L 2 ([0,T ] R) C No A-Priori Bounded Support of the Measure-Valued Solution ν t,x Strategies: Finite-Energy Bounds+Higher Integrability Bounds (replacing L bound) New Derivative Estimate for ε x ρ ε H 1 -Compactness of Weak Entropy Dissipation Measures Only for Weak Entropy Pairs with Compactly Supported C 2 Test Functions Prove that Any Connected Component of Support of the Measure-Valued Solution ν t,x Must Be Bounded Related Earlier Work: Serre-Shearer 1994: 2 2 system of elasticity with severe growth conditions LeFloch-Westdickenberg 2009: Energy Norms, γ (1, 5 3 ], full dissipation Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

24 Navier-Stokes Equations = Euler Equations Theorem (Chen-Perepelitsa: CPAM 2010) Let the initial functions (ρ 0, u 0 ) satisfy the finite-energy conditions. Let (ρ ε, m ε ), m ε = ρ ε u ε, be the solution of the Cauchy problem for the Navier-Stokes equations for each fixed ε > 0. = When ε 0, there exists a subsequence of (ρ ε, m ε ) that converges strongly almost everywhere to a finite-energy entropy solution (ρ, m) to the Cauchy problem for the isentropic Euler equations for any γ > 1. Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

25 Navier-Stokes Equations: Key Estimates I Let the initial functions (ρ 0, u 0 ) satisfy Φ (U ε 0 (x))dx E 0 <, ( ε 2 ρε 0,x (x) 2 ρ ε 0 (x)3 + 2ε ρε 0,x (x)uε 0 (x) ) ρ ε 0 (x) dx E 1 <, where Φ (U) = η (U) η (Ū) η (Ū)(U Ū) 0, η (U) = 1 m 2 2 ρ + ρe(ρ), and E 0, E 1 are independent of ε. Then there exists C > 0 independent of ε, t > 0 such that, for any t > 0, Energy Estimate: Φ (U ε (t, x)) dx + t 0 ε u ε x 2 dxdτ E 0 ; Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

26 Navier-Stokes Equations: Key Estimates I Let the initial functions (ρ 0, u 0 ) satisfy Φ (U ε 0 (x))dx E 0 <, ( ε 2 ρε 0,x (x) 2 ρ ε 0 (x)3 + 2ε ρε 0,x (x)uε 0 (x) ) ρ ε 0 (x) dx E 1 <, where Φ (U) = η (U) η (Ū) η (Ū)(U Ū) 0, η (U) = 1 m 2 2 ρ + ρe(ρ), and E 0, E 1 are independent of ε. Then there exists C > 0 independent of ε, t > 0 such that, for any t > 0, Energy Estimate: Φ (U ε (t, x)) dx + t 0 ε u ε x 2 dxdτ E 0 ; New Derivative Estimate for Density: ε 2 ρ ε x(t,x) 2 ρ ε (t,x) dx +ε t 3 0 (ρ ε ) γ 3 ρ ε x 2 dxdτ C(E 0 +E 1 ). Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

27 Navier-Stokes Equations: Key Estimates II Case I: u + = u = 0. Set v = 1 ρ. Then the conservation of mass implies v t + uv x = vu x. Differentiating it in x and then multiplying by 2v x to obtain ( vx 2) t + u( v x 2) x = 2v x(vu x ) x 2u x v x 2. Multiplying by ρ and using the conservation of mass yield ( ρ vx 2) t + ( ρu v x 2) x = 2v x u xx = 2 ε v xp x + 2 ε v ( x (ρu)t + (ρu 2 ) ) x = (γ 1)2 ρ γ 3 ρ x ε (ρuv x) t + R, where R = 2 ε( ρu(uvx ) x ρu(vu x ) x + v x (ρu 2 ) ) x. Note that R dx = 2 ε u x 2 dx. Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

28 Navier-Stokes Equations: Key Estimates III Then That is, ε 2 ρx (t, x) 2 (γ t 1)2 dx + ε ρ γ 3 ρ ρ(t, x) 3 x 2 dxdτ 2 0 [ ] ρx u t t = 2ε dx + 2ε u x 2 dxdτ ρ τ=0 0 ε2 ρx (t, x) 2 ρ0,x (x) 2 2 ρ(t, x) 3 dx + ε2 dx + C. ρ 0 (x) 3 ε 2 ρx (t, x) 2 ρ(t, x) 3 C t dx + ε 0 (ε 2 ρ0,x (x) 2 ρ 0 (x) 3 dx + 1 ). ρ γ 3 ρ x 2 dxdτ Case II: u + u : More technically involved. Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

29 Navier-Stokes Equations: Higher Integrability Estimates Let the initial data (ρ 0, u 0 ) satisfy Φ (U ε 0 (x))dx E 0 <, ( ε 2 ρε 0,x (x) 2 ρ ε 0 (x)3 where E 0, E 1, M 0 are independent of ε. ρ ε 0(x) u ε 0(x) dx M 0 < + 2ε ρε 0,x (x)uε 0 (x) ) ρ ε 0 (x) dx E 1 <, Then, for any compact set K R and t > 0, There exists C 1 = C 1 (K, E 0, γ, t) > 0, indept. of ε > 0, such that t ( ρ ε (t, x) ) γ+1 dxdτ C1 ; 0 K There exists C 2 = C 2 (E 0, E 1, M 0, K, γ, t), indept. of ε > 0, such that t ( ρ ε u ε 3 + (ρ ε ) γ+θ) dxdτ C 2. 0 K *Perthame-Lions-Tadmor 1994, LeFloch-Westdickenberg 2009 Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

30 Entropy and Measure-Valued Solution ν t,x : γ-law Weak entropy pairs are represented as η ψ (ρ, ρu) = χ(s)ψ(s) ds, q ψ (ρ, ρu) = (θs + (1 θ)u)χ(s)ψ(s) ds R by C 2 test functions ψ(s), for χ(s) := [ρ 2θ (u s) 2] λ +, θ = γ 1 2, λ = 3 γ 2(γ 1). Let ν t,x be the Young measure determined by the solutions of the Navier-Stokes equations. Then ν t,x is a measure-valued solution of the Euler equations: For the test functions ψ {±1, ±s, s 2 }, ν t,x, η ψ t + ν t,x, q ψ x 0, ν t,x, η ψ (0, ) = η ψ (ρ 0, ρ 0 u 0 ), in the sense of distributions in R 2 +. In addition, denoting f (s) := ν t,x, f (s; ρ, u), ν t,x is confined by: θ(s 2 s 1 ) ( χ(s 1 )χ(s 2 ) χ(s 1 ) χ(s 2 ) ) = (1 θ) ( uχ(s 2 ) χ(s 1 ) uχ(s 1 ) χ(s 2 ) ) for a.e. s 1, s 2 R New Difficulty: supp ν t,x Is Unbounded. Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40 R

31 Measure-Valued Solution: Reduction for γ = 3 (cf. LPT) When γ = 3, then θ = 1 and the commutation relation becomes χ(s 1 )χ(s 2 ) = χ(s 1 ) χ(s 2 ), which implies by taking s 1 = s 2. That is, χ(s) 2 = χ(s) 2, ν, ( χ(s) χ(s) ) 2 = 0 for any s R This implies that ν must be a Dirac mass on the set {ρ > 0} or be supported completely in the vacuum V = {ρ = 0}, that is, the measure-valued solution ν t,x is a Dirac mass in the phase coordinates (ρ, m): ν t,x (ρ, m) = δ (ρ(t,x),m(t,x)) (ρ, m). Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

32 Measure-Valued Solution with Unbounded Support: γ > 3 Let A := {(u ρ θ, ρ θ + u) : (ρ, u) supp ν}. Let J = (s, s + ) be any connected component of A. Note that supp χ(s) = {(ρ, u) : u ρ θ s u + ρ θ }. Claim: Any connected component J of the support is bounded for γ > 3 Strategy: On the contrary, let inf{s : s J} =. Fix M 0 such that M J and restrict s 2 (M 0, M 0 + 1); Choose sufficiently small s 1 2 M 0 to reach the contradiction. New Observation: Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

33 Measure-Valued Solution with Unbounded Support: γ > 3 Let A := {(u ρ θ, ρ θ + u) : (ρ, u) supp ν}. Let J = (s, s + ) be any connected component of A. Note that supp χ(s) = {(ρ, u) : u ρ θ s u + ρ θ }. Claim: Any connected component J of the support is bounded for γ > 3 Strategy: On the contrary, let inf{s : s J} =. Fix M 0 such that M J and restrict s 2 (M 0, M 0 + 1); Choose sufficiently small s 1 2 M 0 to reach the contradiction. New Observation: M0+1 χ(s 1)χ(s 2) M 0 χ(s 1) ds 2 C(λ) s 1 λ, λ < 0. Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

34 Measure-Valued Solution with Unbounded Support: γ > 3 Let A := {(u ρ θ, ρ θ + u) : (ρ, u) supp ν}. Let J = (s, s + ) be any connected component of A. Note that supp χ(s) = {(ρ, u) : u ρ θ s u + ρ θ }. Claim: Any connected component J of the support is bounded for γ > 3 Strategy: On the contrary, let inf{s : s J} =. Fix M 0 such that M J and restrict s 2 (M 0, M 0 + 1); Choose sufficiently small s 1 2 M 0 to reach the contradiction. New Observation: M0+1 χ(s 1)χ(s 2) M 0 χ(s 1) Lions-Perthame-Tadmor s argument: = M 0 +1 χ(s 1 )χ(s 2 ) M 0 χ(s 1 ) ds 2 C(λ) s 1 λ, λ < 0. χ(s 1)χ(s 2) χ(s 1) χ(s 2 ) a.e. s 1, s 2 J, s 1 < s 2. ds 2 M 0 +1 M 0 χ(s 2 )ds 2 = C(M 0, λ) > 0 Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

35 Measure-Valued Solution with Unbounded Support: γ > 3 Let A := {(u ρ θ, ρ θ + u) : (ρ, u) supp ν}. Let J = (s, s + ) be any connected component of A. Note that supp χ(s) = {(ρ, u) : u ρ θ s u + ρ θ }. Claim: Any connected component J of the support is bounded for γ > 3 Strategy: On the contrary, let inf{s : s J} =. Fix M 0 such that M J and restrict s 2 (M 0, M 0 + 1); Choose sufficiently small s 1 2 M 0 to reach the contradiction. New Observation: M0+1 χ(s 1)χ(s 2) M 0 χ(s 1) Lions-Perthame-Tadmor s argument: = M 0 +1 χ(s 1 )χ(s 2 ) M 0 χ(s 1 ) = 0 < C(M 0, λ) = M 0+1 χ(s 1)χ(s 2) M 0 χ(s 1) ds 2 C(λ) s 1 λ, λ < 0. χ(s 1)χ(s 2) χ(s 1) χ(s 2 ) a.e. s 1, s 2 J, s 1 < s 2. ds 2 M 0 +1 M 0 χ(s 2 )ds 2 = C(M 0, λ) > 0 ds 2 C(λ) s 1 λ 0 *The case when J is unbounded from above can be treated similarly. = Lions-Perthame-Tadmor s case for γ > 3. when s 1. Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

36 Measure-Valued Solution: Any Connected Component J of the Support Is Bounded for γ (1, 3), I: Strategy On the contrary, suppose that J is unbounded from below. Let M 0 = sup{s : s J} (, ]. Let s 1, s 2, s 3 (, M 0 ) with s 1 < s 2 < s 3. The commutation relation = (s 2 s 1 ) χ(s 1)χ(s 2 ) χ(s 1 ) + (s 3 s 2 ) χ(s 3)χ(s 2 ) χ(s 3 ) = (s 3 s 1 )χ(s 2 ) χ(s 1)χ(s 3 ) χ(s 1 ) χ(s 3 ). Differentiating this equation in s 2 and dividing by (s 3 s 1 ), we obtain χ (s 2 ) χ(s 1)χ(s 3 ) χ(s 1 ) χ(s 3 ) = s 2 s 1 χ(s 1 )χ (s 2 ) + s 3 s 2 χ(s 3 )χ (s 2 ) s 3 s 1 χ(s 1 ) s 3 s 1 χ(s 3 ) + 1 χ(s 1 )χ(s 2 ) 1 χ(s 3 )χ(s 2 ). s 3 s 1 χ(s 1 ) s 3 s 1 χ(s 3 ) Strategy: Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

37 Measure-Valued Solution: Any Connected Component J of the Support Is Bounded for γ (1, 3), I: Strategy On the contrary, suppose that J is unbounded from below. Let M 0 = sup{s : s J} (, ]. Let s 1, s 2, s 3 (, M 0 ) with s 1 < s 2 < s 3. The commutation relation = (s 2 s 1 ) χ(s 1)χ(s 2 ) χ(s 1 ) + (s 3 s 2 ) χ(s 3)χ(s 2 ) χ(s 3 ) = (s 3 s 1 )χ(s 2 ) χ(s 1)χ(s 3 ) χ(s 1 ) χ(s 3 ). Differentiating this equation in s 2 and dividing by (s 3 s 1 ), we obtain χ (s 2 ) χ(s 1)χ(s 3 ) χ(s 1 ) χ(s 3 ) = s 2 s 1 χ(s 1 )χ (s 2 ) + s 3 s 2 χ(s 3 )χ (s 2 ) s 3 s 1 χ(s 1 ) s 3 s 1 χ(s 3 ) + 1 χ(s 1 )χ(s 2 ) 1 χ(s 3 )χ(s 2 ). s 3 s 1 χ(s 1 ) s 3 s 1 χ(s 3 ) Strategy: Take s 1 and show that the left-hand side has a smaller order than the right-hand side = Contradiction. Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

38 Measure-Valued Solution: Any Connected Component J of the Support Is Bounded for γ (1, 3), II: Steps As before: For any s 1, s 3 J, χ(s 1)χ(s 3) χ(s 1) χ(s 3) 1; χ(s) 0 is not identically zero and χ(s) 0 as s inf J, sup J, = there exists s 2 such that χ (s 2 ) > 0, χ(s 2 ) > 0. Let s 3 > s 2 be points such that χ(s 3 ) > 0 and let s 1. From the 1st identity, χ(s1)χ(s2) χ(s 1) = χ(s 2 ) χ(s1)χ(s3) χ(s 1) χ(s 3) + o(1), as s 1. [χ (s)] + 2λ s s 1 χ(s). From the 2nd equation, by throwing away the negative terms, we obtain χ (s 2 ) χ(s 1)χ(s 3 ) χ(s 1 ) χ(s 3 ) 2λ + 1 χ(s 1 )χ(s 2 ) + o(1). s 3 s 1 χ(s 1 ) = Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

39 Measure-Valued Solution: Any Connected Component J of the Support Is Bounded for γ (1, 3), II: Steps As before: For any s 1, s 3 J, χ(s 1)χ(s 3) χ(s 1) χ(s 3) 1; χ(s) 0 is not identically zero and χ(s) 0 as s inf J, sup J, = there exists s 2 such that χ (s 2 ) > 0, χ(s 2 ) > 0. Let s 3 > s 2 be points such that χ(s 3 ) > 0 and let s 1. From the 1st identity, χ(s1)χ(s2) χ(s 1) = χ(s 2 ) χ(s1)χ(s3) χ(s 1) χ(s 3) + o(1), as s 1. [χ (s)] + 2λ s s 1 χ(s). From the 2nd equation, by throwing away the negative terms, we obtain χ (s 2 ) χ(s 1)χ(s 3 ) χ(s 1 ) χ(s 3 ) 2λ + 1 χ(s 1 )χ(s 2 ) + o(1). s 3 s 1 χ(s 1 ) ( ) = χ (s 2 ) 2λ+1 s 3 s 1 χ(s 2 ) χ(s1)χ(s 3) o(1). χ(s 1) χ(s 3) Contradiction as s 1. *Another different proof is given by LeFloch-Westdickenberg 2009 for 1 < γ 5 3. Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

40 Navier-Stokes Equations = Euler Equations Theorem (Chen-Perepelitsa: CPAM 2010) Let the initial functions (ρ 0, u 0 ) satisfy the finite-energy conditions. Let (ρ ε, m ε ), m ε = ρ ε u ε, be the solution of the Cauchy problem for the Navier-Stokes equations for each fixed ε > 0. = When ε 0, there exists a subsequence of (ρ ε, m ε ) that converges strongly almost everywhere to a finite-energy entropy solution (ρ, m) to the Cauchy problem for the isentropic Euler equations for any γ > 1. Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

41 Multi-D Isentropic Euler Eqns with Spherical Symmetry { ρ t + x (ρv) = 0, (ρv) t + x (ρv v) + x p = 0 x = (x 1,..., x d ) R d, x Gradient w.r.t. x R d ρ Density, v = (v 1,..., v d ) R d Velocity, p Pressure Pressure-density constitutive relation (by scaling): Spherically Symmetric Solutions: p = p(ρ) = ρ γ /γ, γ > 1. ρ(t, x) = ρ(t, r), v(t, x) = u(t, r) x r, r = x. Then the functions (ρ, m) = (ρ, ρu) are governed by { ρt + m r + d 1 m = 0, r m t + ( m2 ρ + p(ρ)) r + d 1 r m 2 ρ = 0 Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

42 Defocusing: Expanding Spherically Symmetric Solutions Chen: Proc. Roy. Soc. Edinburgh, 127A (1997), : 0 ρ(t, x) γ 1 2 u(t, x) C <. Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

43 Focusing: Imploding Spherically Symmetric Solutions Guderley 1942, Courant-Fridrichs 1945: Singularity of Self-Similar Solutions Rauch 1986: No BV or L Bounds Longstanding Problem: Does the Concentration Phenomenon Occur? Does the Density Develop a Measure at the Origin? Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

44 Viscosity Methods for the Euler Equations with Spherical Symmetry { ρt + m r + d 1 r m t + ( m2 ρ m = ε ( ) ρ rr + d 1 ρ r r, m 2 = ε( m ρ rr + d 1 m ), r r + p) r + d 1 r with the initial conditions: ρ(0, r) = ρ 0 (r), m(0, r) = m 0 (r), with the bdry condition: (ρ r, m) r=a(ε) = (0, 0), a(ε) 0 as ε 0. Chen-Perepelitsa 2011(Unform Estimates): For any compact set K R + and T > 0, there exists C > 0 independent of ε > 0 such that a(ε) (1 2 ρu2 + ρe(ρ) ) r d 1 dr +ε T 0 a(ε) ( ρ γ 2 ρ r 2 + ρ u r 2 + ρu2 2r 2 ) r d 1 drdt C; t ( ρ ε u ε 3 + (ρ ε ) (3γ 1)/2 + (ρ ε ) γ+1) r d 1 drdt C. 0 K Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

45 Spherically Symmetric Solutions for the Euler Equations via Vanishing Viscosity Limit { ρt + m r + d 1 r m t + ( m2 ρ m = ε ( ) ρ rr + d 1 ρ r r, m 2 = ε( m ρ rr + d 1 m ), r r + p) r + d 1 r with the initial conditions: ρ(0, r) = ρ 0 (r), m(0, r) = m 0 (r), with the boundary condition: (ρ r, m) r=a(ε) = (0, 0) Chen-Perepelitsa 2011: Let the initial functions (ρ 0, m 0 ) satisfy the finite-energy conditions. Then For each fixed ε > 0, there exists a global viscous solution (ρ ε, m ε ); When ε 0, there exists a subsequence of (ρ ε, m ε ) that converges strongly almost everywhere to a finite-energy spherically symmetric entropy solution (ρ, m) for any γ > 1. *Answer to the Longstanding Problem: No Concentration Occurs at the Origin!! *Vanishing Physical Viscosity Limit? Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

46 Conservation Laws of Viscoelastic Materials with Memory { t u(t, x) x v(t, x) = 0, t v(t, x) x σ(t, x) = 0 Elastic Medium: σ(t, x) = f (u(t, x)) Nohel-Rogers-Tzavaras 1988: σ(t, x) = f (u(t, x)) + t k(t τ)f (u(τ, x))dτ 0 C-Dafermos 1997: σ(t, x) = f ( u(t, x) + t k(t τ)g(u(τ, x))dτ) 0 When k(t) = 1exp( t ), the system is equivalent to δ δ t u x v = 0, t v x f (w) = 0, ( ) t w x v + 1 δ w (g(u) + u) = 0 w(t, x) = u(t, x) + t k(t τ)g(u(τ, x))dτ 0 Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

47 Conservation Laws of Viscoelastic Materials with Memory Method of Vanishing Viscosity: C-Dafermos 1997 Set { t u x v = ε xx (u + t k(t τ)g(u(τ, ))dτ), 0 t v x f (u + t k(t τ)g(u(τ, ))dτ) = ε 0 xxv w(t, x) = u(t, x) + t 0 k(t τ)g(u(τ, x))dτ J[u] = k(0)g(u) + t 0 k (t τ)g(u(τ, x))dτ = { t w x v = ε xxw + J[u], t v x f (w) = ε xx v Estimates: M T > 0 such that (w ε, v ε ) L L 2 (R 2 + ) + ε(w ε x, v ε x ) L 2 (R 2 + ) M T for 0 t T Compensated Compactness = *Strong Convergence of (u ε (t, x), v ε (t, x)) to (u(t, x), v(t, x)) *(u(t, x), v(t, x)) is an entropy solution Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

48 Vanishing Viscosity for Further Fundamental Problems Stochastic Compressible Euler Equations with Random Forcing: { ρt + (ρu) x = 0, (ρu) t + (ρu 2 + p) x = σ(ρ, u, t, x) t W (t), *W (t) is a Gaussian white noise martigale random measure Chen & Qian Ding 2011: Method of Artificial Viscosity: = Global Existence of Stochastic Entropy Solutions Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

49 Vanishing Viscosity for Further Fundamental Problems Stochastic Compressible Euler Equations with Random Forcing: { ρt + (ρu) x = 0, (ρu) t + (ρu 2 + p) x = σ(ρ, u, t, x) t W (t), *W (t) is a Gaussian white noise martigale random measure Chen & Qian Ding 2011: Method of Artificial Viscosity: = Global Existence of Stochastic Entropy Solutions Transonic Flow: C-Slemrod-Wang: ARMA 2008 { v x u y = ε (σ 1 (ρ) θ), (ρu) x + (ρv) y = ε (σ 2 (ρ) ρ) q 2 = u 2 + v 2, (u, v) = (q cos θ, q sin θ) ρ = ρ(q) = (1 γ 1 q 2 ) 1 γ 1, γ > 1 2 = Invariant Regions& Compensated Compactness Framework Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

50 Concluding Remarks Methods of Vanishing Viscosity are fundamental approaches. *Theory of Hyperbolic Conservation Laws: Entropy Conditions, Nonuniqueness, Existence, Solution Behavior *Nonuniqueness: In general, the limit solutions of vanishing viscosity limit may depend on the viscosity matrices D in the systems under consideration: MHD, multiphase flow models, *Numerical Methods: Shock capturing methods, unwind, kinetic, Design correct numerical viscosity,, Challenges: Singular limits, = New Math Ideas/Methods In this talk, we have presented several examples to show you how various methods of vanishing viscosity can be developed to obtain global entropy solutions. These examples show that every solution of a vanishing viscosity limit problem will lead to new ideas, techniques, approaches, frameworks, etc. which are useful for solving other important problems... Many fundamental problems are widely open and require new further methods of vanishing viscosity and related new ideas, techniques and approaches... Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40

51 Happy 70th Birthday Costas! Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20 24, / 40