The Cauchy problem for the linear Maxwell-Boltzmann equation
type: Item
instance of: scholarly article
title: The Cauchy problem for the linear Maxwell-Boltzmann equation
zbMATH Open document ID: 0655.35068
DOI: 10.1016/0022-0396(87)90115-X
published in: Journal of Differential Equations
publication date: 1987-01-01T00:00:00Z
review text: It is supposed that two kinds of particles, labelled M and N are present in physical space. Type M particles, not influenced by external forces, are in equilibrium and condition the physical behaviour of the type N particles without being conditioned themselves. It is also supposed that an external force \(\underset \tilde{} F\), dependent on the position vector \(\underset \tilde{} r\) and on the velocity \(\underset \tilde{} v\) but not on time t, acts on each N particle and that the collision of N particles amongst themselves can be neglected compared with those of the M particles. The one-particle distribution function f(\(\underset \tilde{} r,\underset \tilde{} v,t)\) for N particles satisfies the linear Maxwell-Boltzmann equation. The author discusses the critical value problem for this equation in the Banach space L' using the theory of linear semigroups, under the following assumptions: (1) the external force \(\underset \tilde{} F\) is defined on the whole phase space and the mapping F: \(R^ 6\to \tilde R^ 3\) is Lipschitz continuous, (2) the collision frequency \(\nu\) also depends on the state variables and the function \(\nu\) : \(R^ 6\to R\) is measurable and non-negative, (3) \(\nu\) is either bounded or Lipschitz continuous, (4) the collision kernel k is measurable and a non-negative real valued function in \(R^ 9\). For each (\(\underset \tilde{} r,\underset \tilde{} v)\in R^ 6\) it satisfies the condition \[ \int_{R^ 3}k(\underset \tilde{} r,\underset \tilde{} v',v)dv'=\nu (\underset \tilde{} r,\underset \tilde{} v). \]
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 4069457
zbMATH Keywords: Cauchy problem
zbMATH Keywords: two kinds of particles
zbMATH Keywords: collision
zbMATH Keywords: one-particle distribution function
zbMATH Keywords: Maxwell-Boltzmann equation
zbMATH Keywords: critical value problem
zbMATH Keywords: Banach space
zbMATH Keywords: linear semigroups
zbMATH Keywords: Lipschitz continuous
zbMATH Keywords: collision kernel
author: L. Arlotti
reviewed by: T. R. Faulkner
MaRDI profile type: Publication
cites work: Equivalent Norms for Sobolev Spaces
cites work: Abstract time-dependent transport equations
cites work: Q3861989
cites work: Q4188958
cites work: Q3906660
cites work: On the Boltzmann Equation with External Forces
cites work: Abstract Evolution Equations
cites work: Q3337042
cites work: Spectral methods in linear transport theory
cites work: On the spectrum of an unsymmetric operator arising in the transport theory of neutrons
cites work: Q4100575
cites work: Existence, uniqueness and properties of the solutions of the Boltzmann kinetic equation for a weakly ionized gas. I.
cites work: Q5593890
cites work: Q5535601
cites work: On the classical solutions of the boltzmann equation
cites work: Linearized Boltzmann equation in \(L^ 1\) space
cites work: Q5775491
cites work: On the perturbation theory for strongly continuous semigroups
cites work: Sulla derivabilità, rispetto a valori iniziali ed a parametri, delle soluzioni dei sistemi di equazioni differenziali ordinarie del primo ordine
cites work: Sul problema di Cauchy per una equazione lineare alle derivate parziali del primo ordine
description: scientific article; zbMATH DE number 4069457