On the singularities of the global small solutions of the full Boltzmann equation
type: Item
instance of: scholarly article
title: On the singularities of the global small solutions of the full Boltzmann equation
zbMATH Open document ID: 0984.76076
author: Laurent Boudin
author: Laurent Desvillettes
published in: Monatshefte für Mathematik
publication date: 2001-06-12T00:00:00Z
review text: The authors consider global solutions of initial value problem for Boltzmann equation describing the expansion of a rare gas cloud in vacuum (i.e., the mean free path has to be sufficiently large, or the initial value has to be sufficiently small), solutions which are known to exist since the mid 1980s. Here these solutions are shown to have a representation \[ f(t,x,v)=f_0(x-tv,v) \Gamma_1(t,x,v) + \Gamma_2(t,x,v) \] with \(\Gamma_1,\Gamma_2 \in H^{\alpha}_{\text{loc}}( \Re_+\times \Re^3_x \times \Re_v^3)\), \(\alpha \in (0,1/25)\). This says in essence that initial singularities propagate with the free flow and decay (because \(\Gamma_1\) decays exponentially in time). The proof of this result is based on Duhamel principle and on regularity estimates for various components of collision operator. A result on the propagation of regularity is also given. The original reference for global existence of solutions of Boltzmann equation for a rare gas cloud is misquoted as [\textit{S. Kaniel} and \textit{M. Shinbrot}, Commun. Math. Phys. 58, 65-84 (1978; Zbl 0371.76061)] (a paper in which local existence and uniqueness were proved). The correct reference is [\textit{R. Illner} and \textit{M. Shinbrot}, Commun. Math. Phys. 95, 217-226 (1984; Zbl 0599.76088)].
Mathematics Subject Classification ID: 76P05
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 82B40
zbMATH DE Number: 1563301
zbMATH Keywords: inhomogeneous Boltzmann equation
zbMATH Keywords: initial value problem
zbMATH Keywords: expansion of gas cloud in vacuum
zbMATH Keywords: singularity
zbMATH Keywords: Duhamel principle
zbMATH Keywords: regularity estimates
zbMATH Keywords: collision operator
zbMATH Keywords: propagation of regularity
zbMATH Keywords: global existence of solutions
reviewed by: Reinhard Illner
MaRDI profile type: Publication
full work available at URL: s006050070015
OpenAlex ID: W2075515041
DOI: 10.1007/S006050070015
description: scientific article; zbMATH DE number 1563301