Self-similar profiles for homoenergetic solutions of the Boltzmann equation: particle velocity distribution and entropy

TL;DR

This paper investigates homoenergetic solutions of the Boltzmann equation, establishing their existence, long-term behavior, and self-similar profiles, revealing non-Maxwellian distributions and entropy relations akin to equilibrium in certain cases.

Contribution

It proves existence and characterizes the asymptotics of homoenergetic solutions, including the construction of non-Maxwellian self-similar solutions with explicit entropy formulas.

Findings

Existence of homoenergetic solutions for broad initial data.

Characterization of long-time asymptotics for different flow types.

Explicit formulas for the H-function in Maxwell molecules.

Abstract

In this paper we study a class of solutions of the Boltzmann equation 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. These solutions are known as homoenergetic solutions. We prove existence of homoenergetic solutions for a large class of initial data. For different choices for the matrix $A$ and for different homogeneities of the collision kernel, we characterize the long time asymptotics of the velocity distribution for the corresponding homoenergetic solutions. For a large class of choices of $A$ we then prove rigorously the existence of self-similar solutions of the Boltzmann equation. The latter are non Maxwellian distributions and describe far-from-equilibrium flows. For Maxwell molecules we obtain exact formulas for the $H$-function for some of these flows. These formulas show that in some cases, despite being very far from equilibrium, the relationship between density, temperature and entropy is exactly the same as in the equilibrium case. We make conjectures about the asymptotics of homoenergetic solutions that do not have self-similar profiles.