On a discrete Boltzmann-Smoluchowski equation with rates bounded in the velocity variables
type: Item
instance of: scholarly article
title: On a discrete Boltzmann-Smoluchowski equation with rates bounded in the velocity variables
zbMATH Open document ID: 1099.82016
author: Nicolas Fournier
author: Stéphane Mischler
published in: Communications in Mathematical Sciences
publication date: 2006-02-08T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.cms/1088777493
review text: The Boltzmann-Smoluchowski equation (BSE) describes a system of particles that undergo both velocity scattering and coalescence phenomena. In the space-homogeneous case, considered by the authors, each particle is characterized by its mass \(m \in (0,\infty)\) and velocity \(v \in {\mathbb R}^3\) and the BSE is written for the time-dependent density \(f(t,m,v) \geq 0\). The equation is characterized by suitable scattering and coagulation rate functions. In the present paper the authors consider the simplified case in which the mass variable is discrete and the rate functions are assumed to be bounded in the velocity variables, deferring the study of the general case to a subsequent work. The BSE is studied in a ``indirect'' way, the paper being rather focused on the associated pure jump Markov process \((M_t,V_t)_{t \geq 0}\). This can be interpreted as the evolution of mass and velocity of a ``typical particle''. The relation to the solution of the BSE is given by \[ f(t,m,v) = \tfrac{1}{m}\,P \left[ M_t = m, V_t \in dv \right]. \] Using such stochastic interpretation of the BSE, the authors achieve two main results: 1. an exact simulation scheme of the process \((M_t,V_t)_{t \geq 0}\) is derived, which yields an immediate and constructive existence proof for the BSE; 2. it is shown that, as \(t\) tends to infinity, \(M_t\) tends almost surely to infinity, which in particular implies that the solution \(f(t,\,\cdot\,,\,\cdot\,)\) of the BSE tends to 0 in \(L^1\).
reviewed by: L. Barletti
Mathematics Subject Classification ID: 82C40
Mathematics Subject Classification ID: 60J75
zbMATH DE Number: 5004776
zbMATH Keywords: ltzmann equations
zbMATH Keywords: Smoluchowski equations
zbMATH Keywords: jump processes
MaRDI profile type: Publication
OpenAlex ID: W2075755146
DOI: 10.4310/CMS.2004.V2.N5.A4
Recommended article: On the discrete velocity models of the Boltzmann equation in several dimensions
Recommended article: Q4854776
Recommended article: Existence, stability, and convergence of solutions of discrete velocity models to the Boltzmann equation
Recommended article: The regularity of discrete models of the Boltzmann equation
Recommended article: A Consistency Result for a Discrete-Velocity Model of the Boltzmann Equation
Recommended article: On the sizes of discrete velocity models of the Boltzmann equation for mixtures
Recommended article: Solutions to the discrete Boltzmann equation with general boundary conditions
Recommended article: Q3482993
Recommended article: Discrete Velocity Models of the Boltzmann Equation: A Survey on the Mathematical ASPECTS of the Theory
Recommended article: Large-time behavior of discrete velocity Boltzmann equations
description: scientific article; zbMATH DE number 5004776