Asymptotic Velocity Profiles for Homoenergetic Rayleigh-Boltzmann Flows under Dominant Shear
Nicola Miele, Alessia Nota, Juan J. L. Velázquez
TL;DR
This work analyzes homoenergetic solutions to the Rayleigh–Boltzmann equation under a simple shear in the hyperbolic-dominated regime, where the collision kernel has negative homogeneity $a=|\\gamma|\in(0,1)$. By introducing an approximate adjoint model with $B(n\cdot\omega,|w|)=|w|^{-a}$ and a delta-background for the Maxwellian, the authors derive a tractable framework that leads to a formal long-time asymptotic profile for the velocity distribution. The analysis hinges on a matched scaling $f(w,t)=t^{-(3/a-2)}F(\xi,\tau)$, a reduced 2D problem for $G$ via integration over $\xi_3$, and a boundary-value (delay) formulation that yields explicit asymptotics and a mass-conservation constraint. They also provide a probabilistic interpretation where the tagged particle dynamics correspond to a Markov process with alternating free flights and collisions, yielding flight, jump, and transition probabilities. In the regime $-1<\gamma<0$, the results reveal a non-self-similar, multi-scale mass distribution driven by shear, while for $\\gamma<-1$ a frozen-collision scenario is expected, highlighting rich dynamical behavior of open Boltzmann-type systems under strong deformation.
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 $0 \leq γ< 1$, 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 $γ< 0$; in particular, we focus on the range $γ\in (-1, 0)$. 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 $γ< -1$. 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.