Global Solution of a Functional Hamilton-Jacobi Equation associated with a Hard Sphere Gas

TL;DR

This paper develops a global-in-time solution for a functional Hamilton-Jacobi equation related to hard sphere gases, extending previous short-time results and analyzing long-term behavior through coupled Boltzmann equations.

Contribution

It introduces a method to construct a global solution of the Hamilton-Jacobi equation for hard sphere gases, advancing the understanding of long-term dynamics.

Findings

Global solution converges to a stationary state

Extends short-time results to long-time behavior

Analyzes coupled Boltzmann equations for stability

Abstract

In recent years it has been shown for hard sphere gas that, by retaining the correlation information, dynamical fluctuation and large deviation of empirical measure around Boltzmann equation could be proved, in addition to the classical kinetic limit result by Lanford. After taking low-density limit, the correlation information can be encoded into a functional Hamilton-Jacobi equation. The results above are restricted to short time. This paper establishes global-in-time construction of a solution of the Hamilton-Jacobi equation, by analyzing a system of coupled Boltzmann equations. The global solution converges to a non-trivial stationary solution of the Hamilton-Jacobi equation in the long-time limit under proper assumptions.