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

2021/02/10 by Théophile Dolmaire, Dolmaire, Théophile
Engineering · Mathematics · #35Q20 (Primary) 82C40 #82D05 (Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Gas Dynamics and Kinetic Theory #Mathematical Physics (math-ph) #Radiative Heat Transfer Studies #Thermoelastic and Magnetoelastic Phenomena

paper · doi:10.48550/arxiv.2102.05513

openalex publication_date 2021/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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