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Consider the limit $\varepsilon\rightarrow0$ 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 $\displaystyleM_w(x_0,v):=\frac{1}{2\pi\big(T_w(x_0)\big)^2}\exp\bigg(-\frac{|v|^2}{2T_w(x_0)}\bigg)$ for $x_0\in\partial\Omega$ is thewall Maxwellian in the diffuse-reflection boundary condition. In the naturalcase of $|\nabla T_w|=O(1)$, for any constant $P>0$, the Hilbert expansionleads to \begin{align}\label{expansion}  \mathfrak{F}\approx \mu+\varepsilon\bigg\{\mu\bigg(\rho_1+u_1\cdotv+T_1\frac{|v|^2-3T}{2}\bigg)-\mu^{\frac{1}{2}}\left(\mathscr{A}\cdot\frac{\nabla_xT}{2T^2}\right)\bigg\}\end{align} where $\displaystyle\mu(x,v):=\frac{\rho(x)}{\big(2\piT(x)\big)^{\frac{3}{2}}} \exp\bigg(-\frac{|v|^2}{2T(x)}\bigg)$, 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)=&\mu+\mu^{\frac{1}{2}}\Big(\varepsilonf_1+\varepsilon^2f_2\Big)+\mu_w^{\frac{1}{2}}\Big(\varepsilonf^B_1\Big)+\varepsilon^{\alpha}\mu^{\frac{1}{2}}R, \end{align} for interiorsolutions $f_1$, $f_2$ and boundary layer $f^B_1$, where $\mu_w$ is $\mu$computed for $T=T_w$, and derive equation for the remainder $R$ with someconstant $\alpha\geq1$. To prove the validity of the expansion suitable boundson $R$ are needed, which are provided in the companion paper[Esposito-Guo-Rossana-Wu2023].

Ghost Effect from Boltzmann Theory: Expansion with Remainder