Ghost Effect from Boltzmann Theory: Expansion with Remainder

Abstract

Consider the limit $\varepsilon \rightarrow 0$ of the steady Boltzmann problem

$$\begin{aligned} v\cdot \nabla _{x} \mathfrak {F}=\varepsilon ^{-1}Q[\mathfrak {F},\mathfrak {F}],\quad \mathfrak {F}\big |_{v\cdot n<0}=M_{w}\int _{v'\cdot n>0} \mathfrak {F}(v')|v'\cdot n|\textrm{d}v', \end{aligned}$$

(0.1)

where $M_{w}(x_0,v):=\frac{1}{2\pi (T_{w}(x_0))^2}\exp \big (-\frac{|v|^2}{2T_{w}(x_0)}\big )$ for $x_0\in \partial \Omega$ is the wall Maxwellian in the diffuse-reflection boundary condition. We normalize

$$\begin{aligned} T_{w}=1+O\left( |\nabla T_{w}|_{L^{\infty }}\right) . \end{aligned}$$

In the case of $|\nabla T_{w}|=O(\varepsilon )$, the Hilbert expansion confirms $\mathfrak {F}\approx (2\pi )^{-\frac{3}{2}}\textrm{e}^{-\frac{|v|^2}{2}}+\varepsilon (2\pi )^{-\frac{3}{4}}\textrm{e}^{-\frac{|v|^2}{4}}\big (\rho _1+T_1\frac{|v|^2-3}{2}\big )$ where $(2\pi )^{-\frac{3}{2}}\textrm{e}^{-\frac{|v|^2}{2}}$ is a global Maxwellian and $(\rho _1,T_1)$ satisfies the celebrated Fourier law

$$\begin{aligned} \Delta _xT_1=0. \end{aligned}$$

In the natural case of $|\nabla T_{w}|=O(1)$, for any constant $P>0$, the Hilbert expansion leads to

$$\begin{aligned} \mathfrak {F}\approx \mu +\varepsilon \left\{ \mu \left( \rho _1+u_1\cdot v+T_1\frac{|v|^2-3T}{2}\right) -\mu ^{\frac{1}{2}}\left( \mathscr {A}\cdot \frac{\nabla _{x}T}{2T^2}\right) \right\} , \end{aligned}$$

where $\mu (x,v):=\frac{\rho (x)}{(2\pi T(x))^{\frac{3}{2}}}\exp \big (-\frac{|v|^2}{2T(x)}\big )$, and $(\rho ,u_1,T)$ is determined by a Navier–Stokes–Fourier system with “ghost” effect

$$\begin{aligned} \left\{ \begin{array}{rcl} P&{}=&{}\rho T,\\ \rho (u_1\cdot \nabla _{x}u_1)+\nabla _{x} \mathfrak {p}&{}=&{}\nabla _{x}\cdot \left( \tau ^{(1)}-\tau ^{(2)}\right) ,\\ \nabla _{x}\cdot (\rho u_1)&{}=&{}0,\\ \nabla _{x}\cdot \left( \kappa \frac{\nabla _{x}T}{2T^2}\right) &{}=&{}5P(\nabla _{x}\cdot u_1), \end{array}\right. \end{aligned}$$

(0.2)

with the boundary condition

$$\begin{aligned} T\Big |_{\partial \Omega }= T_{w},\quad u_1\Big |_{\partial \Omega }:=(u_{1,\iota _1},u_{1,\iota _2},u_{1,n})\Big |_{\partial \Omega }=(\beta _0\partial _{\iota _1} T_{w},\beta _0\partial _{\iota _2} T_{w},0). \end{aligned}$$

(0.3)

Here $\kappa [T]>0$ is the heat conductivity, $(\iota _1,\iota _2)$ are two tangential variables and n is the normal variable, $\beta _0=\beta _0[T_{w}]$ is a function of $T_{w}$, $\tau ^{(1)}:=\lambda \left( \nabla _{x}u_1+(\nabla _{x}u_1)^t-\frac{2}{3}(\nabla _{x}\cdot u_1)\textbf{1}\right)$ and $\tau ^{(2)}:=\frac{\lambda ^2}{P}\left( K_1\big (\nabla _{x}^2T-\frac{1}{3}\Delta _x T\textbf{1}\big )+\frac{K_2}{T}\big (\nabla _{x}T\otimes \nabla _{x}T-\frac{1}{3}|\nabla _{x}T|^2\textbf{1}\big )\right)$ for some smooth function $\lambda [T]>0$, the viscosity coefficient, and positive constants $K_1$ and $K_2$. Tangential temperature variation creates non-zero first-order velocity $u_1$ at the boundary (0.3), which plays a surprising “ghost” effect [26, 27] in determining zeroth-order density and temperature field $(\rho ,T)$ in (0.2). Such a ghost effect cannot be predicted by the classical fluid theory, while it has been an intriguing outstanding mathematical problem to justify (0.2) from (0.1) due to fundamental analytical challenges. The goal of this paper is to construct $\mathfrak {F}$ in the form of

$$\begin{aligned} \mathfrak {F}(x,v)=\mu +\mu ^{\frac{1}{2}}\left( \varepsilon f_1+\varepsilon ^2 f_2\right) +\mu _{w}^{\frac{1}{2}}\left( \varepsilon f^{B}_1\right) +\varepsilon ^{\alpha }\mu ^{\frac{1}{2}}R \end{aligned}$$

(0.4)

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 $\alpha \ge 1$. To prove the validity of the expansion suitable bounds on R are needed, which are provided in the companion paper (Esposito 2023).