Large-time behavior of discrete velocity Boltzmann equations
type: Item
instance of: scholarly article
title: Large-time behavior of discrete velocity Boltzmann equations
zbMATH Open document ID: 0637.76070
DOI: 10.1007/BF01463401
published in: Communications in Mathematical Physics
publication date: 1986-01-01T00:00:00Z
review text: The asymptotic behaviour of equations representing one-dimensional motions in a gas with a discrete set of velocities is studied. This fictitious gas is described by the following system of equations (discrete velocity Boltzmann equations): \[ u_{i,t}+c_ iu_{i,x}=\sum^{n}_{j,k=1}a_ i^{jk}u_ ju_ k\equiv F_ i(u_ 1,..,u_ n),\quad i=1,...,n, \] where \(u_ i(x,t)\) is the density of particles with speeds \(c_ i\), \(a_ i^{jk}\) are the interaction coefficients (they are constant satisfying conditions for mass and momentum conservation and an entropy condition; \(a_ i^{jk}=a_ i^{kj})\). The initial condition \(u_ i(x,0)=u_{0i}(x)\), \(i=1,..,n\), with \(u_{0i}\geq 0\) in \(L^ 1\cap L^{\infty}\) is prescribed. The solution \(u_ i(x,t)\) is in class \(C(0,T;L^ p({\mathbb{R}}))\) for \(1\leq p\leq \infty\), with \(T>0\) arbitrary. \(u_ i(x,t)\) as \(t\to \infty\) have the form \(u_ i^{\infty}(x-c_ it)\) it corresponds to a state in which the interaction is negligible. 3 theorems are proved \((L^ 1\)-asymptotics; boundedness and \(L^ p\)- asymptotics; uniform asymptotics).
Mathematics Subject Classification ID: 76P05
Mathematics Subject Classification ID: 82B40
zbMATH DE Number: 4039503
zbMATH Keywords: discrete velocity Boltzmann equations
zbMATH Keywords: interaction coefficients
zbMATH Keywords: entropy condition
zbMATH Keywords: initial condition
zbMATH Keywords: boundedness
zbMATH Keywords: uniform asymptotics
author: J. Thomas Beale
MaRDI profile type: Publication
cites work: Large-time behavior of the Broadwell model of a discrete velocity gas
cites work: Q3912268
cites work: Q3710267
cites work: ON DISCRETE MODELS OF THE KINETIC BOLTZMANN EQUATION
cites work: Q3742987
cites work: Q3661190
cites work: Q3904280
cites work: Global solution of the initial value problem for a discrete velocity model of the Boltzmann equation
cites work: On the Broadwell's model for a simple discrete velocity gas
description: scientific article; zbMATH DE number 4039503