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

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.