Paper Info
Title
Long-Time Asymptotics for Homoenergetic Solutions of the Boltzmann Equation: Collision-Dominated Case.
Abstract
In this paper, we present a formal analysis of the long-time asymptotics of a particular class of solutions of the Boltzmann equation, known as homoenergetic solutions, which have the form $$f\left( x,v,t\right) =g\left( v-L\left( t\right) x,t\right) $$ where $$L\left( t\right) =A\left( I+tA\right) ^{-1}$$ with the matrix A describing a shear flow or a dilatation or a combination of both. We began this study in James et al. (Arch Ration Mech Anal 231(2):787–843, 2019). Homoenergetic solutions satisfy an integro-differential equation which contains, in addition to the classical Boltzmann collision operator, a linear hyperbolic term. In James et al. (2019), it has been proved rigorously the existence of self-similar solutions which describe the change of the average energy of the particles of the system in the case in which there is a balance between the hyperbolic and the collision term. In this paper, we focus in homoenergetic solutions for which the collision term is much larger than the hyperbolic term (collision-dominated behavior). In this case, the long-time asymptotics for the distribution of velocities is given by a time-dependent Maxwellian distribution with changing temperature.
Year
DOI
Venue
2019
10.1007/s00332-019-09535-6
Journal of Nonlinear Science
Keywords
DocType
Volume
Kinetic theory, Boltzmann equation, Homoenergetic solutions, Non-equilibrium, Hilbert expansion, 35Q20, 82C40, 74A25
Journal
29
Issue
ISSN
Citations 
5
0938-8974
0
PageRank 
References 
Authors
0.34
0
3
Name
Order
Citations
PageRank
Richard D. James110.96
Alessia Nota200.34
J. J. L. Velázquez3138.41