FINITENESS OF ENTROPY FOR THE HOMOGENEOUS BOLTZMANN EQUATION WITH MEASURE INITIAL CONDITION

• 成果类型: Article
• 署名作者: Fournier, Nicolas
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Sorbonne Universite; Universite Paris Cite
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/14-AAP1012
• 发表日期: 2015
• 页码: 860-897
• 关键词: long-range interactions; kac equation; maxwellian molecules; angular singularity; part i; cutoff; REGULARITY; uniqueness; EXISTENCE; STABILITY

摘要

We consider the 3D spatially homogeneous Boltzmann equation for (true) hard and moderately soft potentials. We assume that the initial condition is a probability measure with finite energy and is not a Dirac mass. For hard potentials, we prove that any reasonable weak solution immediately belongs to some Besov space. For moderately soft potentials, we assume additionally that the initial condition has a moment of sufficiently high order (8 is enough) and prove the existence of a solution that immediately belongs to some Besov space. The considered solutions thus instantaneously become functions with a finite entropy. We also prove that in any case, any weak solution is immediately supported by R-3.