On the monotonicity of the Fisher information for the Boltzmann equation

• 成果类型: Article; Early Access
• 署名作者: Imbert, Cyril; Silvestre, Luis; Villani, Cedric
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite PSL; Ecole Normale Superieure (ENS); University of Chicago; Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01376-3
• 发表日期: 2025
• 关键词: spatially homogeneous boltzmann; entropy dissipation; equilibrium; range; trend

摘要

We prove that the Fisher information is monotone decreasing in time along solutions of the space-homogeneous Boltzmann equation for a large class of collision kernels covering all classical interactions derived from systems of particles. For general collision kernels, a sufficient condition for the monotonicity of the Fisher information along the flow is related to the best constant for an integro-differential inequality for functions on the sphere, which belongs in the family of the Log-Sobolev inequalities. As a consequence, we establish the existence of global smooth solutions to the space-homogeneous Boltzmann equation in the main situation of interest where this was not known, namely the regime of very soft potentials. This is opening the path to the completion of both the classical program of qualitative study of space-homogeneous Boltzmann equation, initiated by Carleman, and the program of using the Fisher information in the study of the Boltzmann equation, initiated by McKean. From the proofs and discussion emerges a strengthened picture of the links between kinetic theory, information theory and log-Sobolev inequalities.