UPPER BOUNDS FOR THE FUNCTION SOLUTION OF THE HOMOGENEOUS 2D BOLTZMANN EQUATION WITH HARD POTENTIAL

• 成果类型: Article
• 署名作者: Bally, Vlad
• 署名单位: Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Universite Gustave-Eiffel; Centre National de la Recherche Scientifique (CNRS); Inria
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/18-AAP1451
• 发表日期: 2019
• 页码: 1929-1961
• 关键词: kac equation; REGULARITY; EXISTENCE; densities; entropy; driven; cutoff; sdes

摘要

We deal with f(t)(dv), the solution of the homogeneous 2D Boltzmann equation without cutoff. The initial condition f(0)(dv) may be any probability distribution (except a Dirac mass). However, for sufficiently hard potentials, the semigroup has a regularization property (see Probab. Theory Related Fields 151 (2011) 659-704): f(t)(dv) = f(t)(v) dv for every t > 0. The aim of this paper is to give upper bounds for f(t)(v), the most significant one being of type f(t)(v) <= Ct(-eta)e(-vertical bar v vertical bar lambda) for some eta, lambda > 0.