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

TL;DR

This paper develops a multi-scale convergence theory for Hamilton-Jacobi PDEs in probability measure spaces, linking hydrodynamic limits of particle dynamics with infinite-dimensional PDE analysis using weak K.A.M. theory.

Contribution

It introduces a novel approach to analyze Hamilton-Jacobi equations in measure spaces, combining variational methods and weak K.A.M. theory in an infinite-dimensional setting.

Findings

Established convergence of solutions in probability measure spaces.

Developed new viscosity solution techniques for metric space PDEs.

Provided a rigorous framework for hydrodynamic limits of action minimizing dynamics.

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.