2023/01/23 by R. Esposito, Yan Guo, Esposito, Raffaele +5
Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectroscopy and Quantum Chemical Studies #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2301.09560
openalex publication_date 2023/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider the limit ε→0 of the steady Boltzmann problem v⋅∇x\mathfrakF=ε-1Q[\mathfrakF,\mathfrakF], \mathfrakF|v⋅ nlt;0=Mw∫v'⋅ ngt;0 \mathfrakF(v')|v'⋅ n|dv', where Mw(x0,v):=(1)/(2π(Tw(x0))2) exp(-(|v|2)/(2Tw(x0))) for x0∈∂Ω is the wall Maxwellian in the diffuse-reflection boundary condition. In the natural case of |∇ Tw|=O(1), for any constant P>0, the Hilbert expansion leads to \mathfrakF≈ μ+ε\μ(ρ1+u1⋅ v+T1(|v|2-3T)/(2))-μ(1)/(2)(\mathscrA⋅(∇xT)/(2T2))\ where μ(x,v):=\fracρ(x)(2πT(x))(3)/(2) exp(-(|v|2)/(2T(x))), and (ρ,u1,T) is determined by a Navier-Stokes-Fourier system with "ghost" effect. The goal of this paper is to construct \mathfrakF in the form of \mathfrakF(x,v)=amp;μ+μ(1)/(2)(ε f1+ε2f2)+μw(1)/(2)(ε fB1)+εαμ(1)/(2)R, for interior solutions f1, f2 and boundary layer fB1, where μw is μ computed for T=Tw, and derive equation for the remainder R with some constant α≥1. To prove the validity of the expansion suitable bounds on R are needed, which are provided in the companion paper [Esposito-Guo-Rossana-Wu2023].