# Long time asymptotics for homoenergetic solutions of the Boltzmann equation. Hyperbolic-dominated case

## TL;DR

This paper investigates the long-time behavior of homoenergetic solutions to the Boltzmann equation in the hyperbolic-dominated regime, providing explicit formulas for key physical quantities and discussing conjectures for complex asymptotics.

## Contribution

It extends previous analyses by focusing on the hyperbolic-dominated case, offering explicit relationships between physical quantities and formulating conjectures for complex asymptotic behaviors.

## Key findings

- Explicit formulas for density, temperature, and entropy in hyperbolic-dominated solutions.
- Discussion of physical situations and formulation of conjectures for long-time asymptotics.
- Differentiation of behavior from equilibrium and collision-dominated cases.

## Abstract

In this paper we continue the formal analysis of the long-time asymptotics of the homoenergetic solutions for the Boltzmann equation that we began in [18]. They have the form $f\left( x,v,t\right) =g\left(v-L\left( t\right) x,t\right) $ where $L\left( t\right) =A\left(I+tA\right) ^{-1}$ where $A$ is a constant matrix. Homoenergetic solutions satisfy an integro-differential equation which contains, in addition to the classical Boltzmann collision operator, a linear hyperbolic term. Depending on the properties of the collision kernel the collision and the hyperbolic terms might be of the same order of magnitude as $t\to\infty$, or the collision term could be the dominant one for large times, or the hyperbolic term could be the largest. The first case has been rigorously studied in [17]. Formal asymptotic expansions in the second case have been obtained in [18]. All the solutions obtained in this case can be approximated by Maxwellian distributions with changing temperature. In this paper we focus in the case where the hyperbolic terms are much larger than the collision term for large times (hyperbolic-dominated behavior). In the hyperbolic-dominated case it does not seem to be possible to describe in a simple way all the long time asymptotics of the solutions, but we discuss several physical situations and formulate precise conjectures. We give explicit formulas for the relationship between density, temperature and entropy for these solutions. These formulas differ greatly from the ones at equilibrium.