Methods of Vanishing Viscosity for Nonlinear ... - ACMAC

Methods of Vanishing Viscosity 

for Nonlinear Conservation Laws 

Gui-Qiang G. Chen 

Mathematical Institute, University of Oxford 

Gui-Qiang.Chen@maths.ox.ac.uk 

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, 2011 1 / 40


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?? 




∂ t U(t, x) + ∂ x F (U(t, x)) = 0, 

U(t, x) ∈ R N 


∂ t U(t, x) + ∂ x F{U(t, x), U (t) (·, x)} = 0 



D : R N → M N×N , D(U) ≥ 0 




Stokes (1848), Rayleigh (1910), Taylor (1910), Weyl (1949), · · · 




∂ t U(t, x) + ∂ x F (U(t, x)) = 0, 

U(t, x) ∈ R N 


∂ t U(t, x) + ∂ x F{U(t, x), U (t) (·, x)} = 0 



D : R N → M N×N , D(U) ≥ 0 




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 


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 


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 


Scalar Conservation Laws: Compensated Compactness 

∂ t u ε + ∂ x f (u ε ) = ε∂ xx u ε 


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 


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, · · · 

· · · · · · 


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 


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 


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?? 


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). 


Invariant Regions: ‖U ε ‖ L ∞ ≤ C (or ‖U ε ‖ L p ≤ C) 

Dissipation Estimate: ‖ √ ε∂ x U ε ‖ L 2 ≤ C 


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 ) 


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, · · · 


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, · · · · · · 


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 

· · · · · · 


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), · · · · · · 


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 ) 


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 ) 


loc 


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 < ∞ 


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 


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), · · · · · · 


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 








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 


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. 


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, 


∫ ∞ 

−∞ Φ ∗(U ε (t, x)) dx + ∫ t ∫ 

0 ε|u 

ε 

x | 2 dxdτ ≤ E 0 ; 


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, 


∫ ∞ 

−∞ 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 ). 


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 | 2 + 2 4 

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. 


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 ∫ 

2 ρ(t, x) 3 dx + |ρ0,x (x)| 2 

ε2 dx + C. 

ρ 0 (x) 3 

∫ 

ε 2 |ρx (t, x)| 2 ∫ t ∫ 

ρ(t, x) 3 dx + ε ρ γ−3 |ρ x | 2 dxdτ 

0 

∫ 

≤ C 

(ε 2 |ρ0,x (x)| 2 ) 

ρ 0 (x) 3 dx + 1 . 

Case II: u + ≠ u − : More technically involved. 


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 


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, 2011 25 / 40 

R

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). 


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 0 + 1 ∈ 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. 











∫ 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 











∫ 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 → −∞. 


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: 



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. 



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 ) 

=⇒ 



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 . 


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. 


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 


Defocusing: Expanding Spherically Symmetric Solutions 

Chen: Proc. Roy. Soc. Edinburgh, 127A (1997), 243–259: 

0 ≤ ρ(t, x) γ−1 

2 ≤ u(t, x) ≤ C < ∞. 


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? 


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 


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? 


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 


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∂ xx w + 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 


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 


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 

· · · · · · 


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.... 


Happy 70th Birthday 

Costas!

Methods of Vanishing Viscosity for Nonlinear ... - ACMAC

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