On the global in time existence and uniqueness of solutions to the Boltzmann hierarchy

TL;DR

This paper proves the global in time existence and uniqueness of solutions to the Boltzmann hierarchy using innovative combinatorial and $L^{ abla}$-based estimates, advancing the rigorous derivation of the Boltzmann equation.

Contribution

It introduces a novel combination of combinatorial techniques and $L^{ abla}$-based bounds to establish uniqueness and existence of solutions to the Boltzmann hierarchy.

Findings

First to use combinatorial techniques with $L^{ abla}$ estimates for Boltzmann hierarchy

Constructive proof of global in time mild solutions

Employs Hewitt-Savage theorem for initial data analysis

Abstract

In this paper we establish the global in time existence and uniqueness of solutions to the Boltzmann hierarchy, a hierarchy of equations instrumental for the rigorous derivation of the Boltzmann equation from many particles. Inspired by available $L^{\infty}$-based a-priori estimate for solutions to the Boltzmann equation, we develop the polynomially weighted $L^\infty$ a-priori bounds for solutions to the Boltzmann hierarchy and handle the factorial growth of the number of terms in the Dyson's series by reorganizing the sum through a combinatorial technique known as the Klainerman-Machedon board game argument. This paper is the first work that exploits such a combinatorial technique in conjunction with an $L^{\infty}$-based estimate to prove uniqueness of the mild solutions to the Boltzmann hierarchy. Our proof of existence of global in time mild solutions to the Boltzmann hierarchy for admissible initial data is constructive and it employs known global in time solutions to the Boltzmann equation via a Hewitt-Savage type theorem.