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 \cite{JNV1}. Homoenergetic solutions satisfy an integro-differential equation which contains, in addition to the classical Boltzmann collision operator, a linear hyperbolic term. In \cite{JNV1} 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.

29 pages, 1 figure. arXiv admin note: text overlap with arXiv:1710.03653