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Uniform shear flow via the Boltzmann equation with hard potentials

Year: 2025
Type: article
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 this 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$ \begin{aligned} &\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} \end{aligned} $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} $.
DOI: 10.3934/dcds.2025049
Language: English
Open Access status: diamond
OpenAlex ID: W4409418935
FWCI: 3.753
Cites: 48
Cited by: 2
Related to: 10