2022/10/22 by Renjun Duan, Duan, Renjun, Shuangqian Liu +1 · 3 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Mathematical Biology Tumor Growth #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2210.12340
openalex publication_date 2022/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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≤ 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 αm for m>2, we make a rigorous justification of the uniform-in-time asymptotic expansion of solutions up to order α2 under a homoenergetic self-similar scaling that can capture the increase of temperature θ(t)∼ (1+γ\varrho0α2 t)2/γ when time tends to infinity, where \varrho0>0 is a strictly positive constant depending only on the deformation force and the linearized collision operator. Specifically, we establish θ3/2(t)F(t,θ1/2(t)v)= μ+α√μ G1(t,v)+α2 √μG2(t,v)+O(1)αm(1+γ\varrho0α2 t)-2 as t→∞, where μ is a global Maxwellian and G1,G2 are microscopic bounded functions that can be explicitly determined and decay in time as G1∼ (1+γ\varrho0α2 t)-1 and G2∼ (1+γ\varrho0α2 t)-2.