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On a Hamilton-Jacobi PDE theory for hydrodynamic limit of action minimizing collective dynamics

Jin Feng

TL;DR

The paper develops a rigorous, multi-scale Hamilton-Jacobi framework for hydrodynamic limits of action-minimizing collective dynamics in the space of probability measures ${\mathcal P}_2(\mathbb{R}^d)$. By combining Alexandrov metric-space calculus with Wasserstein-space tools and a weak-KAM averaging mechanism, it derives an effective Hamiltonian $\bar{{\mathsf H}}$ through a cell problem and proves convergence of finite-particle Hamiltonians ${\mathsf H}_N$ to the limiting operator ${\mathbb H}$ in a suitable sense, yielding convergence of resolvents and action functionals. The work introduces novel techniques—projection of Hamilton-Jacobi equations via submetry, tangent-cone calculus, and Barles–Perthame-type comparison principles in metric spaces—that enable two-scale homogenization and a rigorous connection between microscopic action-minimizing dynamics and macroscopic measure-valued PDEs. Although focused on weakly interacting particles, the framework establishes a solid pathway toward more complex, strongly interacting regimes and offers a principled variational route to hydrodynamic limits in nonlinear PDEs for probability measures. Collectively, these results advance a rigorous bridge between microscopic Lagrangian dynamics and macroscopic measure-valued Hamilton-Jacobi equations with potential broad implications for variational approaches to kinetic and fluid-type limits.

Abstract

We establish multi-scale convergence theory for a class of Hamilton-Jacobi PDEs in space of probability measures. They arise from context of hydrodynamic limit of N-particle deterministic action minimizing (global) Lagrangian dynamics. From a Lagrangian point of view, this can also be viewed as a limit result on two scale convergence of action minimizing probability-measure-valued paths. However, we focus on the Hamiltonian formulation here mostly. We derive and study convergence of the associated abstract but scalar Hamilton-Jacobi equations, defined in space of probability measures. There is an infinite dimensional singular averaging structure within these equations. We develop an indirect variational approach to apply finite dimensional weak K.A.M. theory to such infinite dimensional setting here. With a weakly interacting particle assumption, the averaging step only involves that of individual particles, which is implicitly but rigorously treated using the weak K.A.M. theory. Consequently, we can close the above mentioned averaging step by identifying limiting Hamiltonian, and arrive at a rigorous convergence result on solutions of the nonlinear PDEs in space of probability measures. In technical development parts of the paper, we devise new viscosity solution techniques regarding projection of equations with a submetry structure in state space, multi-scale convergence for certain abstract Hamilton-Jacobi equations in metric spaces, as well as comparison principles for equations in space of probability measures. The space of probability measure we consider is a special case of Alexandrov metric space with curvature bounded from below. Since some results are better explained in such metric space setting, we also develop some techniques in the general settings which are of independent interests.

On a Hamilton-Jacobi PDE theory for hydrodynamic limit of action minimizing collective dynamics

TL;DR

The paper develops a rigorous, multi-scale Hamilton-Jacobi framework for hydrodynamic limits of action-minimizing collective dynamics in the space of probability measures . By combining Alexandrov metric-space calculus with Wasserstein-space tools and a weak-KAM averaging mechanism, it derives an effective Hamiltonian through a cell problem and proves convergence of finite-particle Hamiltonians to the limiting operator in a suitable sense, yielding convergence of resolvents and action functionals. The work introduces novel techniques—projection of Hamilton-Jacobi equations via submetry, tangent-cone calculus, and Barles–Perthame-type comparison principles in metric spaces—that enable two-scale homogenization and a rigorous connection between microscopic action-minimizing dynamics and macroscopic measure-valued PDEs. Although focused on weakly interacting particles, the framework establishes a solid pathway toward more complex, strongly interacting regimes and offers a principled variational route to hydrodynamic limits in nonlinear PDEs for probability measures. Collectively, these results advance a rigorous bridge between microscopic Lagrangian dynamics and macroscopic measure-valued Hamilton-Jacobi equations with potential broad implications for variational approaches to kinetic and fluid-type limits.

Abstract

We establish multi-scale convergence theory for a class of Hamilton-Jacobi PDEs in space of probability measures. They arise from context of hydrodynamic limit of N-particle deterministic action minimizing (global) Lagrangian dynamics. From a Lagrangian point of view, this can also be viewed as a limit result on two scale convergence of action minimizing probability-measure-valued paths. However, we focus on the Hamiltonian formulation here mostly. We derive and study convergence of the associated abstract but scalar Hamilton-Jacobi equations, defined in space of probability measures. There is an infinite dimensional singular averaging structure within these equations. We develop an indirect variational approach to apply finite dimensional weak K.A.M. theory to such infinite dimensional setting here. With a weakly interacting particle assumption, the averaging step only involves that of individual particles, which is implicitly but rigorously treated using the weak K.A.M. theory. Consequently, we can close the above mentioned averaging step by identifying limiting Hamiltonian, and arrive at a rigorous convergence result on solutions of the nonlinear PDEs in space of probability measures. In technical development parts of the paper, we devise new viscosity solution techniques regarding projection of equations with a submetry structure in state space, multi-scale convergence for certain abstract Hamilton-Jacobi equations in metric spaces, as well as comparison principles for equations in space of probability measures. The space of probability measure we consider is a special case of Alexandrov metric space with curvature bounded from below. Since some results are better explained in such metric space setting, we also develop some techniques in the general settings which are of independent interests.
Paper Structure (109 sections, 156 theorems, 999 equations, 1 figure)

This paper contains 109 sections, 156 theorems, 999 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Models of collective dynamics and their limits, formal derivations
  3. A particle model
  4. One scale mean-field limit
  5. Hydrodynamic limit
  6. Relations with other approaches of hydrodynamic limit
  7. Notations, assumptions and main results
  8. Actions and equations induced by finite particle collective Lagrangian dynamics
  9. Actions and equations by effective collective dynamics of infinite particles
  10. Main results
  11. Comments on physical interests of the main results
  12. A toy example of one dimensional ideal gas
  13. Relation to micro-canonical ensemble in statistical mechanics
  14. Comments on technical developments
  15. First order calculus in Wasserstein space as an Alexandrov metric space
  16. ...and 94 more sections

Key Result

Theorem 1.6

[Limit theorem to Hamilton-Jacobi PDEs] Let $\alpha >0$ be arbitrary but fixed. Let $h \in C({\mathsf X})$ with $\sup_{\mathsf X} h<+\infty$, and have at most sub-linear growth to $-\infty$. Moreover, we assume that the $h$ has a modulus of continuity with respect to the ${\mathsf d}$-metric, on eve as well as a super-solution, in the point-wise strong viscosity sense, to equation Moreover, such

Figures (1)

  • Figure 1: A graphical representation of marginal probability measures as projection, through a submetry map ${\mathsf p}$, of square integrable random variables, where the probability space $(\Omega, {\mathcal{F}}, {\mathbb P}):= ([0,1], {\mathcal{B}}_{[0,1]}, \text{Leb})$ and random variables $\Xi:=\xi(X_\epsilon,Y_\epsilon, Y_1,\ldots, Y_K)$, $P:=\sum_k \alpha_k (X_\epsilon -Y_k)$ and $V:= p(X_\epsilon, \Xi)$. The measure $\widehat{\widehat{{\boldsymbol N}}}$ is the joint distribution of $(X_\epsilon, Y_\epsilon, Y_1,\ldots, Y_K, \Xi,P, V)$, which can be viewed as a "section" in the graph.

Theorems & Definitions (324)

  • Definition 1.2
  • Theorem 1.6
  • Definition 2.1: Semi-concavity / convexity
  • Definition 2.2: Velocity of curve
  • Definition 2.3: Differential
  • Lemma 2.4: Proposition 6.16 of AKP19
  • Example 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 314 more