Proof of a McKean conjecture on the rate of convergence of Boltzmann-equation solutions

• 成果类型: Article
• 署名作者: Dolera, Emanuele; Regazzini, Eugenio
• 署名单位: Universita di Modena e Reggio Emilia; University of Pavia; Consiglio Nazionale delle Ricerche (CNR); Istituto di Matematica Applicata e Tecnologie Informatiche Enrico Magenes (IMATI-CNR)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-013-0530-z
• 发表日期: 2014
• 页码: 315-389
• 关键词: CENTRAL-LIMIT-THEOREM; entropy production; maxwellian molecules; spectral gap; equilibrium; energy; uniqueness; explosion; speed

摘要

The present work provides a definitive answer to the problem of quantifying relaxation to equilibrium of the solution to the spatially homogeneous Boltzmann equation for Maxwellian molecules. Under really mild conditions on the initial datum and a weak, physically consistent, angular cutoff hypothesis, our main result (Theorem 1) contains the first precise statement that the total variation distance between the solution and the limiting Maxwellian distribution admits an upper bound of the form , being the least negative eigenvalue of the linearized collision operator and a constant depending only on the initial datum. The validity of this quantification was conjectured, about fifty years ago, by Henry P. McKean. As to the proof of our results, we have taken as point of reference an analogy between the problem of convergence to equilibrium and the central limit theorem of probability theory, highlighted by McKean.