Random Perturbations Compensated Compactness Relaxation of Interacting Exclusions Hyperbolic Scaling Limits in a Regime of Shock Waves A Synthesis of Probabilistic and PDE Techniques Viv´ at TADAHISA!!! J´ ozsef Fritz, TU Budapest Kochi: December 5, 2011
Random Perturbations Compensated Compactness Relaxation of Interacting Exclusions 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. T´ oth, Kati Nagy and C. Bahadoran. Shocks, non - attractive models, couples of conservation laws.
Random Perturbations Compensated Compactness Relaxation of Interacting Exclusions Models and Methods ◮ The Anharmonic Chain with Conservative Noise. Physical and Artificial Viscosity. Ginzburg - Landau perturbation.
Random Perturbations Compensated Compactness Relaxation of Interacting Exclusions 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.
Random Perturbations Compensated Compactness Relaxation of Interacting Exclusions 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 + σ S .
Random Perturbations Compensated Compactness Relaxation of Interacting Exclusions 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 + σ S . ◮ Stochastic Theory of Compensated Compactness
Random Perturbations Compensated Compactness Relaxation of Interacting Exclusions 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 + σ S . ◮ Stochastic Theory of Compensated Compactness ◮ Interacting Exclusions with Creation and Annihilation: Relaxation Scheme Replaces the missing LSI.
Random Perturbations Compensated Compactness Relaxation of Interacting Exclusions 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.
Random Perturbations Compensated Compactness Relaxation of Interacting Exclusions 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 ′ .
Random Perturbations Compensated Compactness Relaxation of Interacting Exclusions 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 ′ . ◮ Entropy Production: X ( h , u ) := ∂ t h ( u ) + ∂ x J ( u ) ≈ 0 ?? beyond shocks in the sense of distributions.
Random Perturbations Compensated Compactness Relaxation of Interacting Exclusions 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 ??
Random Perturbations Compensated Compactness Relaxation of Interacting Exclusions 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 !!
Random Perturbations Compensated Compactness Relaxation of Interacting Exclusions 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!!
Random Perturbations Compensated Compactness Relaxation of Interacting Exclusions Compensated Compactness ◮ Young Measure: d Θ := 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.
Random Perturbations Compensated Compactness Relaxation of Interacting Exclusions Compensated Compactness ◮ Young Measure: d Θ := 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.
Random Perturbations Compensated Compactness Relaxation of Interacting Exclusions Compensated Compactness ◮ Young Measure: d Θ := 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. ◮ L. Tartar - R. DiPerna: The limiting θ is Dirac, therefore it represents a weak solution.
Random Perturbations Compensated Compactness Relaxation of Interacting Exclusions 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 | k ε − x | < ε/ 2 , as 0 < ε → 0 .
Random Perturbations Compensated Compactness Relaxation of Interacting Exclusions 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 | k ε − x | < ε/ 2 , as 0 < ε → 0 . ◮ Lattice approximation to ∂ t π = ∂ x V ′ ( ρ ) , ∂ t ρ = ∂ t π ??
Random Perturbations Compensated Compactness Relaxation of Interacting Exclusions 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 | k ε − 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 γ x − β V ( x ) . HDL: Compressible Euler equations? Strong ergodic hypothesis: Description of all stationary states and conservation laws!!
Recommend
More recommend
