Ghost Effect from Boltzmann Theory: Expansion with Remainder

Abstract

Consider the limit $\epsilon \to 0$ of the steady Boltzmann problem \begin{align} v\cdot\nabla_x\mathfrak{F}=\varepsilon^{-1}Q[\mathfrak{F},\mathfrak{F}],\quad \mathfrak{F}\big|_{v\cdot n<0}=M_w\displaystyle\int_{v'\cdot n>0} \mathfrak{F}(v')|v'\cdot n|\mathrm{d}{v'}, \end{align} where $M_w(x_0,v):=\frac{1}{2\pi (T_w(x_0){)}^2}\exp (-\frac{|v{|}^2}{2T_w(x_0)})$ for $x_0\in \partial \Omega$ is the wall Maxwellian in the diffuse-reflection boundary condition. In the natural case of $|\nabla T_w|=O(1)$, for any constant $P>0$, the Hilbert expansion leads to \begin{align}\label{expansion} \mathfrak{F}\approx μ+\varepsilon\bigg\{μ\bigg(ρ_1+u_1\cdot v+T_1\frac{|v|^2-3T}{2}\bigg)-μ^{\frac{1}{2}}\left(\mathscr{A}\cdot\frac{\nabla_xT}{2T^2}\right)\bigg\} \end{align} where $\mu (x,v):=\frac{\rho (x)}{(2\pi T(x){)}^{\frac{3}{2}}}\exp (-\frac{|v{|}^2}{2T(x)})$, and $(\rho ,u_1,T)$ is determined by a Navier-Stokes-Fourier system with "ghost" effect. The goal of this paper is to construct $\mathfrak{F}$ in the form of \begin{align}\label{aa 08} \mathfrak{F}(x,v)=&μ+μ^{\frac{1}{2}}\Big(\varepsilon f_1+\varepsilon^2f_2\Big)+μ_w^{\frac{1}{2}}\Big(\varepsilon f^B_1\Big)+\varepsilon^αμ^{\frac{1}{2}}R, \end{align} for interior solutions $f_1$, $f_2$ and boundary layer $f_1^B$, where ${\mu }_w$ is $\mu$ computed for $T=T_w$, and derive equation for the remainder $R$ with some constant $\alpha \ge 1$. To prove the validity of the expansion suitable bounds on $R$ are needed, which are provided in the companion paper [Esposito-Guo-Rossana-Wu2023].

27 pages; references updated. arXiv admin note: text overlap with arXiv:2301.09427