Uniform shear flow via the Boltzmann equation with hard potentials

TL;DR

This paper analyzes the uniform shear flow of rarefied gases using the Boltzmann equation with hard potentials, proving global existence, large-time behavior, and asymptotic expansions of solutions under small deformation forces.

Contribution

It provides a rigorous justification of the uniform-in-time asymptotic expansion of solutions up to order  for the Boltzmann equation with hard potentials under shear flow, including temperature growth.

Findings

Proves global existence of solutions for small deformation force .

Establishes large-time asymptotic behavior of solutions with explicit decay rates.

Derives uniform-in-time asymptotic expansion up to order  for solutions under self-similar scaling.

Abstract

The motion of rarefied gases for uniform shear flow at the kinetic level is governed by the spatially homogeneous Boltzmann equation with a deformation force. In the paper we study the corresponding Cauchy problem with initial data of finite mass and energy for the collision kernel in case of hard potentials $0<\gamma\leq 1$ under the cutoff assumption. We prove the global existence and large time behavior of solutions provided that the force strength $\alpha>0$ is small enough. In particular, when the initial perturbation is of order $\alpha^m$ for $m>2$, we make a rigorous justification of the uniform-in-time asymptotic expansion of solutions up to order $\alpha^2$ under a homoenergetic self-similar scaling that can capture the increase of temperature $\theta(t)\sim (1+\gamma \varrho_0\alpha^2 t)^{2/\gamma}$ when time tends to infinity, where $\varrho_0>0$ is a strictly positive constant depending only on the deformation force and the linearized collision operator. Specifically, we establish $$ \theta^{3/2}(t)F(t,\theta^{1/2}(t)v)= \mu+\alpha \sqrt{\mu} G_1(t,v)+\alpha^2 \sqrt{\mu}G_2(t,v)+O(1)\alpha^m(1+\gamma \varrho_0\alpha^2 t)^{-2} $$ as $t\to\infty$, where $\mu$ is a global Maxwellian and $G_1,G_2$ are microscopic bounded functions that can be explicitly determined and decay in time as $G_1\sim (1+\gamma \varrho_0\alpha^2 t)^{-1}$ and $G_2\sim (1+\gamma \varrho_0\alpha^2 t)^{-2}$.