Hyperbolic Scaling Limits in a Regime of Shock Waves A Synthesis of Probabilistic and PDE Techniques Vivát TADAHISA!!!

Hyperbolic Scaling Limits in a Regime of Shock Waves A Synthesis of Probabilistic and PDE Techniques Vivát TADAHISA!!! József Fritz, TU Budapest Kochi: December 5, 2011

Historical Notes (Hyperbolic systems) C.Morrey (1955): Idea of scaling limits for mechanical models. R.L.Dobrushin + coworkers (1980 85): One-dimensional hard rods and harmonic oscillators. Continuum of conservation laws. H.Rost (1981): Asymmetric exclusion rarefaction waves. F.Rezakhanlou (1991): Coupling techniques for general attractive systems in a regime of shock waves. Single conservation laws only. H.-T. Yau (1991) + Olla - Varadhan - Yau (1993): Preservation of local equilibrium in a smooth regime via the method of relative entropy. Hamiltonian dynamics with conservative, diffusive noise. JF (2001 ): Stochastic theory of compensated compactness. Further results with B. Tóth, Kati Nagy and C. Bahadoran. Shocks, non - attractive models, couples of conservation laws.

Models and Methods The Anharmonic Chain with Conservative Noise. Physical and Artificial Viscosity. Ginzburg - Landau perturbation.

Models and Methods The Anharmonic Chain with Conservative Noise. Physical and Artificial Viscosity. Ginzburg - Landau perturbation. Replacement of Microscopic Currents by their Equilibrium Estimators via Logarithmic Sobolev Inequalities. Imitation of the Vanishing Viscosity Limit: L = L 0 + σs. Stochastic Theory of Compensated Compactness

Hyperbolic Systems of Conservation Laws t 0, x R, u = u(t, x), u, Φ(u) R d : t u(t, x) + x Φ(u(t, x)) = 0 ; Φ := Φ has distinct real eigenvalues.

Hyperbolic Systems of Conservation Laws t 0, x R, u = u(t, x), u, Φ(u) R d : t u(t, x) + x Φ(u(t, x)) = 0 ; Φ := Φ has distinct real eigenvalues. Lax Entropy Pairs (h, J) : t h(u(t, x)) + x J(u(t, x)) = 0 along classical solutions if J = h Φ.

The Vanishing Viscosity Limit Parabolic Approximation: t u σ (t, x) + x Φ(u σ (t, x)) = σ 2 x u σ (t, x) ; u σ u as 0 < σ 0??

The Vanishing Viscosity Limit Parabolic Approximation: t u σ (t, x) + x Φ(u σ (t, x)) = σ x 2 u σ (t, x) ; u σ u as 0 < σ 0?? A Priori Bound for Entropy Production: X (h, u) = σ x (h (u) x u) σ( x u h (u) x u) whence σ 1/2 x u is bounded in L 2 if h is convex, but σ 0!!

The Vanishing Viscosity Limit Parabolic Approximation: t u σ (t, x) + x Φ(u σ (t, x)) = σ 2 x u σ (t, x) ; u σ u as 0 < σ 0?? A Priori Bound for Entropy Production: X (h, u) = σ x (h (u) x u) σ( x u h (u) x u) whence σ 1/2 x u is bounded in L 2 if h is convex, but σ 0!! The bound does not vanish!! WE DO NOT HAVE ANY STRONG COMPACTNESS ARGUMENT!!

Compensated Compactness Young Measure: dθ := dt dx θ t,x (dy) represents u if θ t,x is the Dirac mass at u(t, x). Hence u σ is relative compact in a space of measures.

Compensated Compactness Young Measure: dθ := dt dx θ t,x (dy) represents u if θ t,x is the Dirac mass at u(t, x). Hence u σ is relative compact in a space of measures. F. Murat: Sending σ 0, the above decomposition implies θ t,x (h 1 J 2 ) θ t,x (h 2 J 1 ) = θ t,x (h 1 )θ t,x (J 2 ) θ t,x (h 2 )θ t,x (J 1 ) for all couples of entropy pairs.

The Anharmonic Chain Configurations: ω = {(p k, r k ) : k Z}, p k, r k R are the momentum and the deformation at site k Z. Dynamics: p k = V (r k ) V (r k 1 ) and r k = p k+1 p k, V (x) x 2 /2 at infinity, sub - exponential growth of p k, r k. Generator: the Liouville operator L 0, t ϕ(ω(t)) = L 0 ϕ(ω). Hyperbolic scaling: π ε (t, x) := p k (t/ε), ρ ε (t, x) := r k (t/ε) if kε x < ε/2, as 0 < ε 0. Lattice approximation to t π = x V (ρ), t ρ = t π?? Classical conservation laws: p k, r k and H k := p 2 k /2 + V (r k) ; t H k = p k+1 V (r k ) p k V (r k 1 ). Is there any other?? Stationary product measures: λ β,π,γ, p k N(π, 1/β), Lebesgue density of r k e γx βv (x). HDL: Compressible Euler equations? Strong ergodic hypothesis: Description of all stationary states and conservation laws!!

Physical Viscosity The anharmonic chain can be regularized by adding stochastic perturbations to the equations of momenta. Random exchange of momenta: L = L 0 + σs ep, S ep ϕ(ω) = k Z(ϕ(ω k,k+1 ) ϕ(ω)), ω ω k,k+1 means p k p k+1. The classical conservation laws are OK, and the strong ergodic hypothesis (F - Funaki - Lebowitz 1994) implies the triplet of compressible Euler equations in a smooth regime with periodic boundary conditions. The asymptotic preservation of local equilibrium follows by the relative entropy argument.

Ginzburg - Landau perturbation. Stochastic dynamics: dp k = (V (r k ) V (r k 1 )) dt + σ (p k+1 + p k 1 2p k ) dt + 2σ (dw k dw k 1 ), dr k = (p k+1 p k ) dt, k Z, σ > 0 is fixed, {w k : k Z} are independent Wiener processes. Energy is not conserved, λ π,γ := λ 1,π,γ are stationary. We have convergence to classical solutions of the nonlinear sound equation of elastodynamics: t π = x S (ρ) and t ρ = x π as V (r k ) dλ π,γ = γ = S (ρ) if r k dλ π,γ = ρ = F (γ), S(ρ) := sup{γρ F (γ)}, F (γ) := log exp(γx V (x)) dx. γ

Artificial Viscosity In a regime of shock waves the randomness must be very strong: dp k = (V (r k ) V (r k 1 )) dt + σ(ε) (p k+1 + p k 1 2p k ) dt + 2σ(ε) (dw k dw k 1 ), k Z, dr k = (p k+1 p k ) dt + σ(ε) (V (r k+1 ) + V (r k 1 ) 2V (r k )) dt + 2σ(ε) (d w k+1 d w k ), k Z, where {w k } and { w k } are independent families of independent Wiener processes. The macroscopic viscosity: εσ(ε) 0, but εσ 2 (ε) + as ε 0.

Conditions on V. The substitution of the microscopic currents V (r k ) by their equilibrium expectation S (v) is done by means of a logarithmic Sobolev inequality, thus V must be strictly convex.

Conditions on V. The substitution of the microscopic currents V (r k ) by their equilibrium expectation S (v) is done by means of a logarithmic Sobolev inequality, thus V must be strictly convex. The genuine nonlinearity of its flux is a condition for existence of weak solutions to the sound equation, that is S (v) = 0 can not have more that one root. In terms of V this follows from the same property of V, but there are other examples, too. In particular if V is symmetric then V should be strictly convex or concave on the half - line R +. A technical condition of asymptotic normality is also needed: V (x) converges at an exponential rate as x ±.

Main Result. The distributions P ε of the empirical process (π ε, ρ ε ) form a tight family with respect to the weak topology of the C space of trajectories, and its limit distributions are all concentrated on a set of weak solutions to the sound equation. The notion of weak convergence above changes from step to step of the argument. We start with the Young measure of the block - averaged empirical process (ˆπ ε, ˆρ ε ), finally we get tightness in the strong local L p (R 2 +) topology if p < 2. Compensated compactness is the most relevant keyword of the proofs, results by J. Shearer (1994) and Serre - Shearer (1994) are applied at the end.

On the ideas of the proof The Main Steps. We follow the argumentation of the vanishing viscosity approach. There is a rich family of Lax entropy pairs (h, J), entropy production X ε := t h(ˆπ ε, ˆρ ε ) + x J(ˆπ ε, ˆρ ε ) is considered as a generalized function. First difficulty: to identify the macroscopic flux J in the microscopic expression of L 0 h, and to show that the remainders do vanish in the limit. Replace block averages of the microscopic currents of momenta with their equilibrium expectations via LSI. It is based on our a priori bounds on relative entropy and its Dirichlet form.

The a Priori Bounds The Entropy Bound and LSI. The initial condition implies that t S[µ t,ε,n λ 0,0 ] + σ(ε) D[µ s,ε,n λ 0,0 ] ds C (t + n) 0 for all t, n, ε with the same constant C ; f n := dµ t,ε,n /dλ 0,0, n 1 n 1 D := ( 1 k fn ) 2 dλ + ( 1 k fn ) 2 dλ, k= n k= n l ξ k := (1/l)(ξ k+l ξ k ), k := / p k and k := / r k. Since V is convex, the following LSI holds true. Given r l,k = v, let µ ρ l,k and λρ l,k denote the conditional distributions of the variables r k, r k 1,..., r k l+1, and set fl,k v := dµρ l,k /dλρ l,k, then log f ρ l,k dµρ l,k l2 C lsi k 1 j=k l+1 ( 1 k (f ρ l,k )1/2) 2 dλ ρ l,k.

Replacement of the microscopic flux. Combining LSI and the entropy inequality ϕ dµ S[µ λ] + log e ϕ dλ we get k <n t 0 ( V l,k S ( r l,k ) ) ( ) 2 nt dµs,ε ds C 1 l + nl2, σ(ε) where ξ l,k := (ξ k + ξ k 1 + + ξ k l+1 )/l, e.g. V k = V (r k ). Similar bounds control the differences r l,k+l r l,k and ˆr l,k r l,k. Entropy production X ε is written in terms of the mollified block averages ˆξ l,k, these are defined by means of a triangular weight function. Mesoscopic blocks of size l = l(ε) are used: lim ε 0 l(ε) = 0 and lim σ(ε) ε 0 εl 3 (ε) σ(ε) = +.

Lax Entropy Pairs One critical term of X ε can be computed as X 0,k := L 0 h(ˆp l,k, ˆr l,k ) + J(ˆp l,k+1, ˆr l,k+1 ) J(ˆp l,k, ˆr l,k ) h u(ˆp l,k, ˆr l,k )( ˆV l,k ˆV l,k 1 ) + h v (ˆp l,k, ˆr l,k )(ˆp l,k+1 ˆp l,k ) + J u(ˆp l,k, ˆr l,k )(ˆp l,k+1 ˆp l,k ) + J v (ˆp l,k, ˆr l,k )(ˆr l,k+1 ˆr l,k ). Since h π(π, ρ)s (ρ) + J ρ(π, ρ) = h ρ(π, ρ) + J π(π, ρ) = 0, X 0,k h u(ˆπ l,k, ˆρ l,k ) ( ˆV l,k ˆV l,k 1 S (ˆr l,k )(ˆr l,k+1 ˆr l,k )). Observe now that ˆξ l,k+1 ˆξ l,k = (1/l)( ξ l,k+l ξ l,k ), thus the substitution V l,k S ( r l,k ) results in X 0,k 1/l. In fact (εl(ε)σ(ε)) 1 is the order of the replacement error; that is why we need?? εσ 2 (ε) + and the sharp explicit bounds provided by the logarithmic Sobolev inequality.

Stochastic Compensated Compactness For Lax entropy pairs (h, J) set X ε (ψ, h) := 0 ( h(ûε )ψ t(t, x) + J(û ε )ψ x(t, x) ) dx dt, where u ε = (π, ρ), and the test function ψ is compactly supported in the interior of R 2 +. Another test function φ localizes X. An entropy pair (h, J) is well controlled if X ε decomposes as X ε (ψ, h) = Y ε (ψ, h) + Z ε (ψ, h), and we have two random functionals A ε (φ, h) and B ε (φ, h) such that Y ε (ψφ, h) A ε (φ, h) ψ + and Z ε (ψ, h) B ε (φ, h) ψ : lim EA ε (φ, h) = 0 and lim sup EB ε (φ, h) < + as ε 0.

The Young family Θ is defined as dθ ε := dt dx θ ε,t,x (du), where θ ε,t,x is the Dirac mass at the actual value of û ε.

The Young family Θ is defined as dθ ε := dt dx θ ε,t,x (du), where θ ε,t,x is the Dirac mass at the actual value of û ε. If (h 1, J 1 ) and (h 2, J 2 ) are well controlled entropy pairs with bounded second derivatives then the Div - Curl lemma holds true: θ t,x (h 1 J 2 ) θ t,x (h 2 J 1 ) = θ t,x (h 1 )θ t,x (J 2 ) θ t,x (h 2 )θ t,x (J 1 ) a.s. with respect to any limit distribution of P ε as ε 0. Now we are in a position to refer to the papers by J. Shearer and Serre - Shearer on an L p theory of compensated compactness. The Dirac property of the Young measure follows by means of their moderately increasing entropy families.

Interacting Exclusions, Tóth - Valkó (2002) Charged particles: ω = (ω k = 0, ±1 : k Z), η k := ωk 2. L 0 ϕ(ω) = 1 (η k + η k+1 + ω k ω k+1 )(ϕ(ω k,k+1 ) ϕ(ω)). 2 k Z

Interacting Exclusions, Tóth - Valkó (2002) Charged particles: ω = (ω k = 0, ±1 : k Z), η k := ωk 2. L 0 ϕ(ω) = 1 (η k + η k+1 + ω k ω k+1 )(ϕ(ω k,k+1 ) ϕ(ω)). 2 k Z Conservation laws: L 0 ω k = j ω k 1 jω k and L 0η = j η k 1 jη k, where j ω k : = (1/2) (η k + η k+1 2ω k ω k+1 + ω k η k+1 η k ω k+1 ) + (1/2)(η k η k+1 ), j η k : = (1/2) (ω k + ω k+1 ω k η k+1 η k ω k+1 + η k η k+1 ). All Bernoulli measures λ u,ρ are stationary: ω k dλ u,ρ = u and ηk dλ u,ρ = ρ. j ω k dλ u,ρ = ρ u 2 and j η k dλ u,ρ = u uρ.

Creation and Annihilation The action ω ω k+ of creation means that (ω k, ω k+1 ) (1, 1) if ω k = ω k+1 = 0, while annihilation ω ω k is defined by (ω k, ω k+1 ) (0, 0) if ω k = 1 and ω k+1 = 1. The generator of the composed process reads as L = L 0 + β G, where β > 0 and G ϕ(ω) := k Z(1 η k )(1 η k+1 )(ϕ(ω k+ ) ϕ(ω)) + 1 4 (η k + ω k )(η k+1 ω k+1 )(ϕ(ω k ) ϕ(ω)). k Z Creation - annihilation violates the conservation of particle number. The product measure λ u,ρ will be stationary if λ u,ρ [ω k = 0, ω k+1 = 0] = λ u,ρ [ω k = 1, ω k+1 = 1], whence ρ = F (u) := (1/3)(4 4 3u 2 ) is the criterion of equilibrium because the second root: F (u) := (1/3)(4 + 4 3u 2 ) 5/3 > 1.

Substitution Equilibrium Expectations. λ u := λ u,f (u), u < 1 is the family of our stationary product measures: ωk dλ u = u, η k dλ u = F (u) and j ω k dλ u = F (u) u 2.

Substitution Equilibrium Expectations. λ u := λ u,f (u), u < 1 is the family of our stationary product measures: ωk dλ u = u, η k dλ u = F (u) and j ω k dλ u = F (u) u 2. G ω k = j ω k 1 jω k is also a difference of currents: j ω k (ω) := (1/4)(η k + ω k )(η k+1 ω k+1 ) (1 η k )(1 η k+1 ), and j ω k dλ u,ρ = C(u, ρ) := (3/4)(ρ F (u))( F (u) ρ). Therefore t u(t, x) + x (F (u) u 2 ) = 0 (CreAnni) is the expected result of the hyperbolic scaling limit.

Main Result Since we do not want to postulate the smoothness of the macroscopic solution, the basic process is regularized by an overall stirring S e, the full generator reads as L := L + σ(ε) S e, and the theory of compensated compactness is applied. Assume that the initial distributions satisfy lim ε ϕ(εk)ω k (0) = ψ(x)u 0 (x) dx ε 0 k Z in probability for all compactly supported ϕ C(R). The artificial viscosity σ(ε) and the size l = l(ε) of the mesoscopic block averages are the same as before. Our empirical process is defined as û ε (t, x) := ˆω l,k (t/ε) if εk x < ε/2.

With C. Bahadoran and K. Nagy (EJP 2011) we prove that τ r lim E u(t, x) û ε (t, x) dx dt = 0 ε 0 0 r for all r, τ > 0, where u is the uniquely specified weak entropy solution to the CreAnni equation with initial value u 0. The coefficient β > 0 needs not be a constant, it is sufficient to assume that εσ 2 (ε)β 4 (ε) + and σ(ε)β(ε) + as ε 0. The proof follows the standard technology of the stochastic theory of compensated compactness, the entropy production for entropy pairs (h, J) of equation CreAnni has to be evaluated. However, the present logarithmic Sobolev inequality is not sufficient for the identification of x J in the stochastic equation of h.

Main Steps of the Proof Entropy Production. The local bound on relative entropy and an LSI involving S e allow us to do the replacements j l,k J( ω l,k, η l,k ) : J = ρ u 2, J = u uρ and J = C(u, ρ) if j = j ω, j = j η and j = j ω, respectively. The explicit form of the bounds is the same as for V above. Our entropy pairs (h, J) satisfy J (u) = (F (u) 2u)h (u). Since G is reversible, one critical component of X ε reds as X 0,k := L 0h(ˆω l,k ) + J(ˆω l,k+1 ) J(ˆω l,k ) (1/l)h (ˆω l,k ) ( η l,k η l,k+l + F ( ω l,k )( ω l,k+l ω l,k ) ), whence the required X0,k 1/l would follow by η l,k F ( ω l,k ). Since we do not have the appropriate logarithmic Sobolev inequality, another tool must be found.

Relaxation in action η k appears with a negative sign in the formula of G η k = j ω k 1 jω k and G η k dλ u,ρ = 2C(u, ρ), thus we hope to find a relaxation scheme. The approximate identities below reflect the underlying structure: dũ ε + x ( ρ ε ũ 2 ε) dt + β x C(ũ ε, ρ ε ) dt 0, d ρ ε + x (ũ ε ũ ε ρ ε ) + (2β/ε) C(ũ ε, ρ ε ) dt 0, where ũ ε ω l,k and ρ ε η l,k by mollification. Since (ρ F (u))c(u, ρ) Ψ(u, ρ) := (1/2) (ρ F (u)) 2, even the trivial Liapunov function Ψ can be applied to conclude that η l,k F ( ω l,k ). This trick works well if εσ 2 (ε)β 2 (ε) + as ε 0, a slightly better result can be proven by replacing Ψ with a clever Lax entropy.

The End. The Div - Curl lemma is now a consequence of our a priori bounds including η l,k F ( ω l,k ). The uniqueness of the hydrodynamic limit follows by the Lax entropy inequality: lim sup X ε (ψ, h) 0 for ψ 0 and convex h. The bound on Z ε of the decomposition X ε = Y ε + Z ε does never vanish. Open problems: Lax inequality for the anharmonic chain with artificial viscosity. Uniqueness of HDL to the Leroux system, say. Relaxation of εσ 2 (ε) + by a careful non - gradient analysis. Derivation of the compressible Euler equations with physical viscosity by adding energy and momentum preserving noise to the equations of the anharmonic chain. Navier - Stokes correction for creation and annihilation.