Kac's program in kinetic theory

• 成果类型: Article
• 署名作者: Mischler, Stephane; Mouhot, Clement
• 署名单位: Universite PSL; Universite Paris-Dauphine; University of Cambridge
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-012-0422-3
• 发表日期: 2013
• 页码: 1-147
• 关键词: homogeneous boltzmann-equation; spectral gap; probability metrics; part i; entropy; energy; propagation; CONVERGENCE; equilibrium; derivation

摘要

This paper is devoted to the study of propagation of chaos and mean-field limits for systems of indistinguishable particles, undergoing collision processes. The prime examples we will consider are the many-particle jump processes of Kac and McKean (Kac in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, Vol. III, pp. 171-197, 1956; McKean in J. Comb. Theory 2:358-382, 1967) giving rise to the Boltzmann equation. We solve the conjecture raised by Kac (Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, Vol. III, pp. 171-197, 1956), motivating his program, on the rigorous connection between the long-time behavior of a collisional many-particle system and the one of its mean-field limit, for bounded as well as unbounded collision rates. Motivated by the inspirative paper by Grunbaum (Arch. Ration. Mech. Anal. 42:323-345, 1971), we develop an abstract method that reduces the question of propagation of chaos to that of proving a purely functional estimate on generator operators (consistency estimates), along with differentiability estimates on the flow of the nonlinear limit equation (stability estimates). This allows us to exploit dissipativity at the level of the mean-field limit equation rather than the level of the particle system (as proposed by Kac). Using this method we show: (1) Quantitative estimates, that are uniform in time, on the chaoticity of a family of states. (2) Propagation of entropic chaoticity, as defined by Carlen et al. (Kinet. Relat. Models 3:85-122, 2010). (3) Estimates on the time of relaxation to equilibrium, that are independent of the number of particles in the system. Our results cover the two main Boltzmann physical collision processes with unbounded collision rates: hard spheres and true Maxwell molecules interactions. The proof of the stability estimates for these models requires significant analytic efforts and new estimates.