Ghost Effect from Boltzmann Theory: Expansion with Remainder

TL;DR

This paper develops an expansion for the steady Boltzmann equation near the zero Knudsen number limit, incorporating boundary effects and a 'ghost' correction, with a focus on constructing solutions and analyzing the remainder.

Contribution

It introduces a detailed expansion of the Boltzmann solution including boundary layers and ghost effects, and derives equations for the remainder term with bounds for validation.

Findings

Constructed an expansion including boundary layer and ghost effects.

Derived equations for the remainder term with specific bounds.

Validated the expansion's accuracy through a companion analysis.

Abstract

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 $\displaystyle M_w(x_0,v):=\frac{1}{2\pi\big(T_w(x_0)\big)^2} \exp\bigg(-\frac{|v|^2}{2T_w(x_0)}\bigg)$ for $x_0\in\partial\Omega$ is 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 \mu+\varepsilon\bigg\{\mu\bigg(\rho_1+u_1\cdot v+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 $\displaystyle\mu(x,v):=\frac{\rho(x)}{\big(2\pi T(x)\big)^{\frac{3}{2}}} \exp\bigg(-\frac{|v|^2}{2T(x)}\bigg)$, 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(\varepsilon f_1+\varepsilon^2f_2\Big)+\mu_w^{\frac{1}{2}}\Big(\varepsilon f^B_1\Big)+\varepsilon^{\alpha}\mu^{\frac{1}{2}}R, \end{align} for interior solutions $f_1$, $f_2$ and boundary layer $f^B_1$, where $\mu_w$ is $\mu$ computed for $T=T_w$, and derive equation for the remainder $R$ with some constant $\alpha\geq1$. To prove the validity of the expansion suitable bounds on $R$ are needed, which are provided in the companion paper [Esposito-Guo-Rossana-Wu2023].