Inhomogeneous Continuity Equation with Application to Hamiltonian - PowerPoint PPT Presentation

Inhomogeneous Continuity Equation with Application to Hamiltonian ODE (joint work with L. Chayes & W. Gangbo) Helen K. Lei California Institute of Technology October 12, 2011

Mathematical Background ✆ Evolution of Measure ✆ Continuity Equation “Physical” Motivations ✆ Hamiltonian ODE with Interaction ✆ Mass Reaching Infinity in Finite Time ✆ Regularization: Fade With Arc Length Inhomogeneous Continuity Equation ✆ Inhomogeneous Continuity Equation ✆ Deficient Hamiltonian ODE Limiting Equation and Dynamical Considerations ✆ Dynamical Hypothesis ✆ Closeness of Trajectories & Representation Formula ✆ Validity of Regularization: Convergence of Mass

Evolution of Measure Eulerian: v t µ 1 µ 0 Given v t , have flow equation: ✩ ✾ X t ✏ v t ♣ X t q ✫ X 0 ✏ id ✪ X t

Continuity Equation I △ in mass = flux in/out of infinitesimal volume: ❇ ρ ❇ t � ∇ ☎ ♣ ρ v q ✏ 0 V ρ = (probability) density v = velocity field Integrated version for macroscopic volume: dM V ❇ ρ ➩ ➩ ➩ ✏ ❇ t dx ✏ ✁ V ∇ ☎ ♣ ρ v q dx ✏ ✁ ❇ V ρ v ☎ ˆ n dS dt V

Continuity Equation II Mass of particle constant along trajectories (incompressible): dt r ρ ♣ X t , t qs ✏ ❇ ρ d ❇ t � ∇ ρ ☎ v ✏ 0 . Therefore, ∇ ρ ☎ v ✏ ∇ ☎ ♣ ρ v q ù ñ ∇ ☎ v ✏ 0 and have weak formulation for measures : ❇ t µ t � ∇ ☎ ♣ µ t v t q ✏ 0 means ➺ T ➺ c ♣ R d ✂ ♣ 0 , T qq ❅ ϕ P C ✽ ❇ t ϕ � ① v t , ∇ ϕ ② d µ t dt ✏ 0 0

Weak Formulation or for any test function ϕ P L 1 ♣ d ν q (Here T # µ ✏ ν if for Define ➩ ➩ any measurable A ϕ ♣ y q d ν ♣ y q ✏ ϕ ♣ T ♣ x qq d µ ♣ x q ) µ t ✏ X t # µ 0 ν ♣ A q ✏ µ ♣ T ✁ 1 ♣ A qq Then (formally) ❇ t µ t � ∇ ☎ ♣ v t µ t q ✏ 0: c ♣ R d ✂ ♣ 0 , T qq ; ϕ P C ✽ Ψ ♣ x , t q ✏ ϕ ♣ X t ♣ x q , t q ➺ T ➺ R d ❇ t ϕ ♣ x q � ① v t ♣ x q , ∇ ϕ ♣ x q② d µ t ♣ x q dt 0 ➺ T ➺ ✏ R d ❇ t ϕ ♣ X t ♣ x q , t q � ① v t ♣ X t ♣ x q , ∇ ϕ ♣ X t ♣ x qq② d µ 0 ♣ x q dt 0 ➺ T ➺ d Ψ ✏ dt ♣ x , t q d µ 0 ♣ x q dt R d 0 ➺ ✏ R d ϕ ♣ X T ♣ x q , T q ✁ ϕ ♣ x , 0 q d µ 0 ♣ x q ✏ 0

Hamiltonian Dynamics I Let R 2 d ◗ x ✏ ♣ p , q q ✏ ♣ momentum, position q H ♣ p , q q ✏ 1 2 ⑤ p ⑤ 2 � Ψ ♣ q q ✏ kinetic � potential Then ✄ ☛ ✄ ☛ ✄ ☛ 0 ✁ Id p ✾ H p x ✏ ✾ ✏ ✏ J ∇ H q ✾ Id 0 H q Start with measure, infinite dimensional Hamiltonian system? H ♣ µ q ✏ 1 Φ ♣ q q d µ � 1 ➺ ➺ ➺ ⑤ p ⑤ 2 d µ � ♣ W ✝ µ q♣ q q d µ 2 2 ✾ X t ✏ J r ∇ H ♣ µ qs♣ p , q q ✏ ♣✁ ∇ ♣ W ✝ µ � Φ q♣ q q , p q ✍ interaction means velocity field has non–trivial dependence on µ t ✍

Finite Range Interactions

Hamiltonian Dynamics II ✆ Infinitesimal conservation of mass certainly holds ✆ ∇ H ❑ J ∇ H ù ñ ∇ ☎ ♣ J ∇ H q ✏ 0 Should describe by continuity equation: ❇ t µ t � ∇ ☎ ♣ J ∇ H ♣ µ t q µ t q ✏ 0 . ✆ Energy not pointwise conserved: d H ♣ µ t q ✒ ① ∇ H , J ∇ H ② � ❇ H ✚ ♣ p , q q ✏ 1 ♣ p , q q ✏ 2 ❇ t ♣ W ✝ µ t q . dt ❇ t ✍ Formally, using continuity equation and supposing ⑤ ∇ W ⑤ ↕ B ⑤❇ t ♣ W ✝ µ t q⑤ ✏ ⑤ d ➺ ➺ W ♣ x ✁ y q d µ t ♣ y q⑤ ↕ B ⑤ J ∇ H ♣ µ t q⑤ d µ t dt is locally bounded ✍ Total energy (integrated over µ t ) should still be conserved.

Hamiltonian ODE on Wasserstein Space W. Gangbo, H. K. Kim, and T. Pacini. Differ- L. Ambrosio and W. Gangbo. Hamiltonian ODE’s in the Wasserstein Space of Probability ential forms on Wasserstein space and infinite Measures. Comm. in Pure and Applied Math., dimensional Hamiltonian systems. To appear 61 , 18–53 (2007). in Memoirs of AMS. Definition (Hamiltonian ODE). H : P 2 ♣ R 2 d q Ñ ♣✁✽ , ✽s (proper, lowersemicontinuous) . A.C. curve t µ t ✉ r 0 , T s is a Hamiltonian ODE w.r.t. H if ❉ v t P L 2 ♣ d µ t q , ⑥ v t ⑥ L 2 ♣ d µ t q P L 1 ♣ 0 , T q ✩ such that ❇ t µ t � ∇ ☎ ♣ J v t µ t q ✏ 0 , t P ♣ 0 , T q ✫ v t P T µ t P 2 ♣ R 2 d q ❳ ❇ H ♣ µ t q for a.e., t ✪ Theorem. (Ambrosio, Gangbo) Suppose H : P 2 ♣ R 2 d q Ñ R satisfies ♣ ⑤ ∇ H ♣ x q⑤ ↕ C ♣ 1 � ⑤ x ⑤q ✆ If µ n ✏ ρ n L 2 d , µ ✏ ρ L 2 d and µ n á µ then ∇ H ♣ µ n k q µ n k á ∇ H ♣ µ q µ Then given µ 0 ✏ ρ 0 L 2 d : ✆ The Hamiltonian ODE admits a solution for t P r 0 , T s ✆ t ÞÑ µ t is L ♣ T , µ 0 q –Lipschitz (with respect to the Wasserstein distance) ✆ If H is λ –convex, then H ♣ µ t q ✏ H ♣ µ q .

Wasserstein Distance M ✝ M (flow map) (density) Φ# ρ 0 ✏ ρ Ð Φ : Π ρ 0 Φ ρ g ✝ ➩ s ✏ ✁ ∇ ☎ ♣ ρ ∇ p q ; g ρ ♣ s 1 , s 2 q ✏ ➩ ρ ∇ ρ 1 ☎ ρ 2 Φ ♣ v 1 , v 2 q ✏ ♣ v 1 ☎ v 2 q ρ 0 (flat) (non-flat) (Induced distance: x 0 x 1 d ♣ x 0 , x 1 q 2 ✏ inf t ➩ 1 0 g x ♣ t q ♣ dx dt , dx dt q dt : t ÞÑ x ♣ t q P M ✶ , x ♣ 0 q ✏ x 0 , x ♣ 1 q ✏ x 1 ✉ ) Upshot: F. Otto. The geometry of dissipative evolution eqns: d ♣ ρ 0 , ρ q 2 ✏ inf Φ: ρ ✏ Φ# ρ 0 the porous medium equation. ➩ ρ 0 ⑤ id ✁ Φ ⑤ 2 Comm. PDE, 26 (2001), 101-174.

A.C. Curves and the Continuity Equation Definition. Let P 2 ♣ R d , W 2 q denote the space of probability measures with bounded second moment equipped with the Wasserstein distance ★➺ ✰ W 2 R d ✂ R d ⑤ x ✁ y ⑤ 2 d γ ♣ x , y q : γ P Γ ♣ µ, ν q 2 ♣ µ, ν q ✏ min and Γ ♣ µ, ν q ✏ t γ : γ ♣ A ✂ R d q ✏ µ ♣ A q and γ ♣ R d ✂ B q ✏ ν ♣ B q , for all measurable A and B ✉ Theorem. There is a correspondence: t A.C. curves in P 2 ♣ R d , W 2 q✉ ð ñ t velocity fields v t P L 2 ♣ d µ t q✉ via 1 ❇ t µ t � ∇ ☎ ♣ v t µ t q ✏ 0 and lim ⑤ h ⑤ W 2 ♣ µ t � h , µ t q♣↕q ✏ ⑥ v t ⑥ L 2 ♣ µ t q h Ñ 0 Thus ✧➺ 1 ✯ W 2 ⑥ v t ⑥ 2 2 ♣ µ 0 , µ 1 q ✏ min L 2 ♣ d µ t q : ❇ t µ t � ∇ ☎ ♣ v t µ t q ✏ 0 0 and L 2 ♣ d µ q T µ P 2 ♣ R d , W 2 q ✏ t ∇ ϕ : ϕ P C ✽ c ♣ R d q✉

Mass Reaching Infinity in Finite Time Condition ( ♣ ). What about other We are solving Hamiltonians? E.g., ❇ t µ t � ∇ ☎ ♣ J ∇ H µ t q ✏ 0; v t : ✏ J ∇ H ♣ µ t q Φ ♣ q q Recall characteristics ✾ q X t ✏ v t ♣ X t q ; X 0 ✏ id ñ ⑤ X t ⑤ ➚ e Ct ♣ 1 � ⑤ X 0 ⑤q : ⑤ v t ♣ x q⑤ ↕ C ♣ 1 � ⑤ x ⑤q ù preserves compact support, second moment... Explicit Computation. ⑤ v t ♣ X t q⑤ ✏ C ♣ 1 � ⑤ X t ⑤q R , R → 1 ☛ R ✁ 1 ✄ ⑤ X t ⑤ 1 ✏ 1 ✁ t ♣ R ✁ 1 q⑤ X 0 ⑤ R ✁ 1 ⑤ X 0 ⑤ 1 x � ✽ at time τ ♣ x q ✏ ♣ R ✁ 1 q⑤ x ⑤ R ✁ 1 ➔ ✽

Continuity Equation in “Finite Volume” Particles that have ever been in finite region during r 0 , t s : blue = good pink = negligible red = bad yellow = gone. Expect. Under reasonable dynamical conditions, still have ❇ t µ t � ∇ ☎ ♣ J ∇ H ♣ µ t q µ t q ✏ 0 distributionally.

Example: Quadratic Velocity in 1D Consider the velocity field and associated trajectories x 0 v t ♣ x q ✏ x 2 , x t ✏ 1 ✁ tx 0 and densities ρ 0 ✏ 1 r 0 , 1 s , ρ t ✏ x t # ρ 0 . By change of variables, have ρ t ♣ y q ✏ ρ 0 ♣ x ✁ 1 ♣ y qq♣ x ✁ 1 q ✶ ♣ y q t t 1 ✏ ♣ 1 � yt q 2 . We have then ✁ 2 y 2 y ♣ ρ t v t q ✶ ✏ ❇ t ρ t ✏ and ♣ 1 � yt q 3 ♣ 1 � yt q 3 and so ❇ t ρ t � ♣ ρ t v t q ✶ ✏ 0 .

Regularization: Fade With Arc Length ➩ t ✾ M t ✏ M 0 e ✁ 0 C s ♣ X s q⑤ v s ♣ X s q⑤ ds X t ✏ v t ♣ X t q For simplicity, C s ✑ ε ; later, send ε Ñ 0.

Inhomogeneous Continuity Equation ♣ ♠ q ❇ t µ ε t � ∇ ☎ ♣ v t µ ε t q ✏ ✁ ε ⑤ v t ⑤ µ ε t Given µ 0 , v t , define µ ✝ t t q ✝ ✏ X ε ♣ µ ε t # µ 0 µ t ➺ t R ε t ♣ X ε t q ✏ exp ♣✁ ε ⑤ v t ♣ X ε s q⑤ ds q µ 0 0 then t q ✝ µ ε t ✏ R ε t ♣ µ ε satisfies ♣ ♠ q . Proposition. ( ♠ ) preserves α –exponential moments for α ↕ ε , since distance traveled ↕ arclength ☞ ✍ directly gives global (in space) regularization ✍

Existence of ε –Dynamics Lemma. Let µ 0 P M ✽ ,ε . Suppose we have prescribed (time–dependent) velocity fields v ε t satisfying ⑤ v ε t ♣ x q⑤ ↕ C ♣ 1 � ⑤ x ⑤q R for some constants C , R → 0. Then for 0 ➔ T ➔ ✽ ✆ ❉ distributional solution ♣ µ ε t q t Pr 0 , T s to ❇ t µ ε t � ∇ ☎ ♣ v ε t µ ε t q ✏ ✁ ε ⑤ v ε t ⑤ µ ε t c ♣ R 2 d ✂ r 0 , T sqq , ➩ T ➩ T ❅ ϕ P C ✽ ➩ ➩ R 2 d ⑤ v ε R 2 d ♣❇ t ϕ � ① v t , ∇ x ϕ ②q d µ t dt ✏ ✁ ε t ⑤ ϕ d µ t dt 0 0 realized as a linear functional such that ➺ ➺ ❅ ϕ P C c ♣ R 2 d q . R 2 d ϕ d µ ε ♣ R ε t ϕ q ✆ X ε t ✏ t d µ 0 , S ε t ✆ ♣ µ ε t q t Pr 0 , T s is narrowly continuous. ✆ Preservation of moments. ☞