玻尔兹曼理论中的幽灵效应：带余项展开
Ghost Effect from Boltzmann Theory: Expansion with Remainder
AI总结：
本文研究稳态玻尔兹曼方程在小克努森数极限下的希尔伯特展开，构造包含内部解、边界层和余项的展开形式，并推导带幽灵效应的纳维-斯托克斯-傅里叶系统及余项方程，为后续建立余项估计奠定基础。
摘要：
考虑稳态玻尔兹曼问题在 $\varepsilon\rightarrow0$ 极限下的情形：\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} 其中 $\displaystyle M_w(x_0,v):=\frac{1}{2π\big(T_w(x_0)\big)^2}\exp\bigg(-\frac{|v|^2}{2T_w(x_0)}\bigg)$（$x_0\in\partialΩ$）是漫反射边界条件中的壁面麦克斯韦分布。在自然情形 $|\nabla T_w|=O(1)$ 下，对任意常数 $P>0$，希尔伯特展开给出 \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} 其中 $\displaystyleμ(x,v):=\frac{ρ(x)}{\big(2πT(x)\big)^{\frac{3}{2}}}\exp\bigg(-\frac{|v|^2}{2T(x)}\bigg)$，且 $(ρ,u_1,T)$ 由带“幽灵”效应的纳维-斯托克斯-傅里叶系统确定。本文的目标是将 $\mathfrak{F}$ 构造为 \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} 其中 $f_1$、$f_2$ 为内部解，$f^B_1$ 为边界层，$μ_w$ 是将 $μ$ 中 $T$ 取为 $T_w$ 所得，并针对某个常数 $α\geq1$ 推导余项 $R$ 所满足的方程。为证明该展开的有效性，需要对 $R$ 建立适当的界，这些界在伴随论文 [Esposito-Guo-Rossana-Wu2023] 中给出。
英文摘要：
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 $\displaystyle M_w(x_0,v):=\frac{1}{2π\big(T_w(x_0)\big)^2} \exp\bigg(-\frac{|v|^2}{2T_w(x_0)}\bigg)$ for $x_0\in\partialΩ$ 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 $\displaystyleμ(x,v):=\frac{ρ(x)}{\big(2πT(x)\big)^{\frac{3}{2}}} \exp\bigg(-\frac{|v|^2}{2T(x)}\bigg)$, and $(ρ,u_1,T)$ 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 $f^B_1$, where $μ_w$ is $μ$ computed for $T=T_w$, and derive equation for the remainder $R$ with some constant $α\geq1$. To prove the validity of the expansion suitable bounds on $R$ are needed, which are provided in the companion paper [Esposito-Guo-Rossana-Wu2023].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
