About Lanford's theorem in the half-space with specular reflection

TL;DR

This paper rigorously derives the Boltzmann equation in a half-space with specular reflection, extending Lanford's theorem by defining collision operators and controlling recollisions in this domain.

Contribution

It provides a rigorous definition of the collision operator and functional space, and adapts recollision control techniques to the half-space setting for Lanford's theorem.

Findings

Convergence of the particle system to the Boltzmann equation in the half-space.

Rigorous formulation of the collision operator and functional space.

Extension of recollision control methods to the half-space domain.

Abstract

The present article proposes a rigorous derivation of the Boltzmann equation in the half-space. We show an analog of the Lanford's theorem in this domain, with specular reflection boundary condition, stating the convergence in the low density limit of the first marginal of the density function of a system of N hard spheres towards the solution of the Boltzmann equation associated to the initial data corresponding to the initial state of the one-particle-density function. The original contributions of this work consist in two main points: the rigorous definition of the collision operator and of the functional space in which the BBGKY hierarchy is solved in a strong sense; and the adaptation to the case of the half-space of the control of the recollisions performed by Gallagher, Saint-Raymond and Texier, which is a crucial step to obtain the Lanford's theorem.