Asymptotic Velocity Profiles for Homoenergetic Rayleigh-Boltzmann Flows under Dominant Shear

TL;DR

This paper analyzes the long-time asymptotic velocity profiles of homoenergetic solutions to the Rayleigh-Boltzmann equation under dominant shear flow, focusing on collision kernels with negative homogeneity, and provides explicit asymptotic forms and probabilistic interpretations.

Contribution

It derives the explicit asymptotic velocity profile for homoenergetic solutions in the hyperbolic-dominated regime with b3 c (-1, 0), extending understanding of long-time behavior in shear flows.

Findings

Explicit asymptotic velocity profile derived for b3 c (-1, 0)

Probabilistic interpretation of particle velocity as a Markov process

Discussion of asymptotic behavior for b3 < -1

Abstract

In this paper, we study a particular class of solutions to the Rayleigh--Boltzmann equation, known in the nonlinear setting as \emph{homoenergetic solutions}. These solutions take the form $ g(x, v, t) = f(v - L(t)x, t),$ where the matrix $L(t)$ represents a shear flow deformation. We began our analysis in \cite{MNV}, where we rigorously proved the existence of a stationary non-equilibrium solution and established different behaviours of the solutions depending on the size of the shear parameter, for cut-off collision kernels with homogeneity parameter $0 \leq \gamma < 1$, thus including Maxwell molecules and hard potentials. In the present work, we focus on the regime in which the deformation term dominates the collision term for large times (hyperbolic-dominated regime). This scenario occurs for collision kernels with $\gamma < 0$; in particular, we focus on the range $\gamma \in (-1, 0)$. In this regime, it is challenging to obtain a clear and direct description of the long-time asymptotic behaviour of the solutions. Here we present a formal analysis of the velocity distribution's long-time asymptotics and derive for the first time the explicit form of the corresponding asymptotic profile. We also discuss the different asymptotic behaviour expected in the case of homogeneity $\gamma < -1$. In addition, we provide a probabilistic interpretation involving a stochastic process combining collisions with shear flow. The tagged particle velocity $\{v(t)\}_{t\geq 0}$ is a Markov process that arises from the combination of free flights in a shear flow along with random jumps caused by collisions.