学术文献库

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 $0<γ\leq 1$ under the cutoff assumption. We prove the global existence and large time behavior of solutions provided that the force strength $α>0$ is small enough. In particular, when the initial perturbation is of order $α^m$ for $m>2$, we make a rigorous justification of the uniform-in-time asymptotic expansion of solutions up to order $α^2$ under a homoenergetic self-similar scaling that can capture the increase of temperature $θ(t)\sim (1+γ\varrho_0α^2 t)^{2/γ}$ 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 $$ θ^{3/2}(t)F(t,θ^{1/2}(t)v)= μ+α\sqrtμ G_1(t,v)+α^2 \sqrtμG_2(t,v)+O(1)α^m(1+γ\varrho_0α^2 t)^{-2} $$ as $t\to\infty$, where $μ$ is a global Maxwellian and $G_1,G_2$ are microscopic bounded functions that can be explicitly determined and decay in time as $G_1\sim (1+γ\varrho_0α^2 t)^{-1}$ and $G_2\sim (1+γ\varrho_0α^2 t)^{-2}$.