The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, III: the 3-D Boltzmann equation

TL;DR

This paper proves the global existence and analyzes the large time behavior of smooth solutions to the 3-D Boltzmann equation in an infinitely expanding domain, showing the solution tends to vacuum without forming vacuum regions.

Contribution

It extends previous work on compressible flows to the Boltzmann equation, establishing global existence and detailed asymptotic behavior in an expanding domain.

Findings

Solutions remain smooth for all time

Density bounds confirm tendency towards vacuum

No vacuum formation occurs in finite time

Abstract

This paper is a continuation of the works in \cite{Euler} and \cite{NS}, where the authors have established the global existence of smooth compressible flows in infinitely expanding balls for inviscid gases and viscid gases, respectively. In this paper, we are concerned with the global existence and large time behavior of compressible Boltzmann gases in an infinitely expanding ball. Such a problem is one of the interesting models in studying the theory of global smooth solutions to multidimensional compressible gases with time dependent boundaries and vacuum states at infinite time. Due to the conservation of mass, the fluid in the expanding ball becomes rarefied and eventually tends to a vacuum state meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. In the present paper, we will confirm this physical phenomenon for the Boltzmann equation by obtaining the exact lower and upper bound on the macroscopic density function.

Equations

$${\Omega }_t=\{x=(x_1,x_2,x_3)\in {\mathbb{R}}^3:|x|=\sqrt{x_1^2+x_2^2+x_3^2}<R(t),t\ge 0\},$$

$${\partial }_{t}f+\xi \cdot {\nabla }_{x}f=Q(f,f),$$

$$Q(f,g)=Q(f,g)(t,x,\xi )=\frac{1}{2}{\int }_{{\mathbb{R}}^3\times S^2}|(\xi -{\xi }_{\ast })\cdot \omega |(f^{\prime }{g}_{\ast }^{\prime }+f_{\ast }^{\prime }{g}^{\prime }-f{g}_{\ast }-f_{\ast }g)d{\xi }_{\ast }d\omega$$

$$f_{\ast }=f(t,x,{\xi }_{\ast }),f^{\prime }=f(t,x,{\xi }^{\prime }),f_{\ast }^{\prime }=f(t,x,{\xi }_{\ast }^{\prime }),$$

$${\xi }^{\prime }=\xi -[(\xi -{\xi }_{\ast })\cdot \omega ]\omega ,{\xi }_{\ast }^{\prime }={\xi }_{\ast }+[(\xi -{\xi }_{\ast })\cdot \omega ]\omega .$$

$$\{\begin{array}{rlrl}f(0,x,\xi )& =f_0(x,\xi ),& & \text{x∈Ω0, ξ∈R3},\\ f(t,x,\xi )& =f(t,x,\xi -2(\xi \cdot n_x)n_x+2R^{\prime }(t)n_x),& & \text{x∈∂Ωt, ξ⋅nx<R′(t)},\end{array}$$

$$M(t,x,\xi )=(2\pi {)}^{-3/2}\exp \left(-\frac{|\xi {|}^2}{2}-\frac{h^2}{2}|x-t\xi {|}^2\right)$$

$$\{\begin{array}{rl}\tau & =\frac{1}{h}\arctan (ht),\\ y& =\frac{x}{R(t)},\\ \eta & =R(t)\xi -\frac{h^{2}t}{R(t)}x.\end{array}$$

$$M=M(\tau ,y,\eta )=(2\pi {)}^{-3/2}\exp \left(-\frac{|\eta {|}^2+h^2|y{|}^2}{2}\right).$$

$${\partial }_{\tau }f+\eta \cdot {\nabla }_{y}f-h^{2}y\cdot {\nabla }_{\eta }f={\cos }^2(h\tau )Q(f,f),$$

$$\{\begin{array}{rlrl}f(0,y,\eta )& =f_0(y,\eta ),& & \text{y∈Ω, η∈R3},\\ f(\tau ,y,\eta )& =f(\tau ,y,\eta -2(\eta \cdot n_y)n_y),& & \text{y∈∂Ω, η⋅ny<0}.\end{array}$$

$${\Lambda }_{\tau }:={\partial }_{\tau }+\eta \cdot {\nabla }_y-h^{2}y\cdot {\nabla }_{\eta }.$$

$$f=M+M^{1/2}u.$$

$${\Lambda }_{\tau }u=\tilde{\mu }{\cos }^2(h\tau )Lu+{\tilde{\mu }}^{1/2}{\cos }^2(h\tau )\Gamma (u,u),$$

$$Lu=2{\mu }^{-1/2}Q(\mu ,{\mu }^{1/2}u),$$

$$\Gamma (u,u)={\mu }^{-1/2}Q({\mu }^{1/2}u,{\mu }^{1/2}u).$$

$$\{\begin{array}{rlrl}u(0,y,\eta )& =u_0(y,\eta ),& & \text{y∈Ω, η∈R3},\\ u(\tau ,y,\eta )& =u(\tau ,y,\eta -2(\eta \cdot n_y)n_y),& & \text{y∈∂Ω, η⋅ny<0}.\end{array}$$

$${\int }_{\Omega \times {\mathbb{R}}^3}{u}_0(y,\eta )M^{1/2}dyd\eta =0,$$

$${\int }_{\Omega \times {\mathbb{R}}^3}(|\eta {|}^2+h^2|y{|}^2)u_0(y,\eta )M^{1/2}dyd\eta =0,$$

$${\int }_{\Omega \times {\mathbb{R}}^3}(y\times \eta )u_0(y,\eta )M^{1/2}dyd\eta =0.$$

$${\int }_{\Omega \times {\mathbb{R}}^3}u(\tau ,y,\eta )M^{1/2}dyd\eta =0,$$

$${\int }_{\Omega \times {\mathbb{R}}^3}(|\eta {|}^2+h^2|y{|}^2)u(\tau ,y,\eta )M^{1/2}dyd\eta =0,$$

$${\int }_{\Omega \times {\mathbb{R}}^3}(y\times \eta )u(\tau ,y,\eta )M^{1/2}dyd\eta =0.$$

$${\varphi }_{\beta }(y,\eta )=(1+|\eta {|}^2+h^2|y{|}^2{)}^{\beta /2}.$$

$$\Vert {\varphi }_{\beta }u(\tau ){\Vert }_{\infty }\le C{e}^{-\lambda \tau }\Vert {\varphi }_{\beta }{u}_0{\Vert }_{\infty },$$

$$\begin{array}{rl}& \rho (t,x)={\int }_{{\mathbb{R}}^3}f(t,x,\xi )d\xi ,\\ & \left(\rho v\right)(t,x)={\int }_{{\mathbb{R}}^3}\xi f(t,x,\xi )d\xi ,\\ & \left(\rho \theta \right)(t,x)={\int }_{{\mathbb{R}}^3}\frac{|\xi -v{|}^2}{2}f(t,x,\xi )d\xi .\end{array}$$

$$M=M_{[\bar{\rho },\bar{v},\bar{\theta }]}(\xi )=\frac{\bar{\rho }}{(2\pi r\bar{\theta }{)}^{3/2}}\exp (-\frac{|\xi -\bar{v}{|}^2}{2r\bar{\theta }}),$$

$$f_0=\mu +{\mu }^{1/2}{\tilde{u}}_0$$

$$\Vert {\varphi }_{\beta }{\tilde{u}}_0{\Vert }_{\infty }<{ϵ}_0.$$

$${\int }_{\Omega \times {\mathbb{R}}^3}(f_0-M)dyd\eta =0,$$

Full text

The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, III:

the 3-D Boltzmann equation

Huicheng Yin1,∗,  Wenbin Zhao2111*Huicheng Yin (huicheng$@$nju.edu.cn, 05407@njnu.edu.cn) and Wenbin Zhao (zhaowb1989@gmail.com) are supported by the NSFC (No.11571177) and A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

1. School of Mathematical Sciences, Jiangsu Provincial Key Laboratory for Numerical Simulation

of Large Scale Complex Systems, Nanjing Normal University, Nanjing 210023, China.

2. Department of Mathematics and IMS, Nanjing University, Nanjing 210093, China

Abstract

This paper is a continuation of the works in [34] and [36], where the authors have established the global existence of smooth compressible flows in infinitely expanding balls for inviscid gases and viscid gases, respectively. In this paper, we are concerned with the global existence and large time behavior of compressible Boltzmann gases in an infinitely expanding ball. Such a problem is one of the interesting models in studying the theory of global smooth solutions to multidimensional compressible gases with time dependent boundaries and vacuum states at infinite time. Due to the conservation of mass, the fluid in the expanding ball becomes rarefied and eventually tends to a vacuum state meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. In the present paper, we will confirm this physical phenomenon for the Boltzmann equation by obtaining the exact lower and upper bound on the macroscopic density function.

Keywords: Boltzmann equation, expanding ball, weighted energy estimate, global existence, vacuum state.

Mathematical Subject Classification 2000: 35L70, 35L65, 35L67, 76N15

1 Introduction

The compressibility of gases plays a basic role in gas dynamics. When one squeezes a soft container filling with gases, the gases will become denser and the corresponding temperature will get higher in the adiabatic process. In this paper, as in [34] and [36], we consider an opposite situation for the compressible gases filling a 3-D expansive ball. It is assumed that the expansive ball is described by

[TABLE]

at the time $t$, where $R(t)=(1+h^{2}t^{2})^{1/2}$ for some positive constant $h$. From the expression of $\Omega_{t}$, we know that the expansive ball at time $t$ is formed by pulling out the initial unit ball $\Omega_{0}=\{x:|x|<1\}$ with smooth speed and acceleration (see Figure 1 below). The pulling speed on the boundary is $R^{\prime}(t)=h^{2}t(1+h^{2}t^{2})^{-1/2}$, which increases smoothly from [math] to $h$. We denote the time-space domain by $S=\{(t,x):\,t>0,|x|<R(t)\}$. Suppose that the movement of the gases in $\Omega_{t}$ is described by the 3-D Boltzmann equation:

[TABLE]

where $f=f(t,x,\xi)$ stands for the distribution function of gas particles at time $t$, position $x\in\Omega_{t}$ and velocity $\xi\in\mathbb{R}^{3}$, the collision operator $Q(f,g)$ with hard-sphere interaction is given by

[TABLE]

with $\omega\in S^{2}$ being the unit sphere in $\mathbb{R}^{3}$, and

[TABLE]

[TABLE]

In view of the physical property for the gas flow in $\Omega_{t}$, it is plausible to pose the following initial-boundary conditions for equation (1.1),

[TABLE]

where $n_{x}$ is the unit outer normal direction of $\{x\in\mathbb{R}^{3}:|x|=R(t)\}$ in $\mathbb{R}^{3}$, $R^{\prime}(t)n_{x}$ is the velocity of the expansive boundary. Note that the boundary value condition in (1.3) just only corresponds to the specular-reflection boundary condition. It is easy to check that the “traveling Maxwellian”

[TABLE]

is a special solution to equation (1.1) despite the initial data.

To solve problem (1.1) together with (1.3), we first make a change of variables $(t,x,\xi)$ such that the expansive domain $\Omega_{t}$ becomes a fixed domain $\Omega=\{y=(y_{1},y_{2},y_{3})\in\mathbb{R}^{3}:|y|<1\}$. For this purpose, we set

[TABLE]

In this case, $t\in(0,\infty)$ turns into a finite interval $\tau\in(0,\pi/2h)$, and the special solution $M=M(t,x,\xi)$ becomes

[TABLE]

Denote by $\mu=\mu(\eta):=(2\pi)^{-3/2}e^{-|\eta|^{2}/2}$ and $\tilde{\mu}=\tilde{\mu}(y):=e^{-h^{2}|y|^{2}/2}$, we then have $M=\mu\tilde{\mu}$ for (1.6). Under the new coordinates $(\tau,y,\eta)$, Boltzmann equation (1.1) becomes

[TABLE]

and the initial-boundary data (1.3) become

[TABLE]

Note that (1.7) is a kind of Boltzmann equation containing a potential term. Denote the transport operator by

[TABLE]

As in [32], we define the standard perturbation around $M$ of $f$ by

[TABLE]

In this case, one obtains the Boltzmann equation of $u$ as follows:

[TABLE]

where $L$ and $\Gamma$ are the Boltzmann operators in a bounded domain that can be expressed as

[TABLE]

[TABLE]

Correspondingly, the initial-boundary data of $u$ are

[TABLE]

Suppose that the initial perturbation satisfies the following conservations:

[TABLE]

[TABLE]

and

[TABLE]

Since the specular-reflection boundary conserves both mass, energy and angular momentum, as in [16], without loss of generality, we may assume that the mass, energy and angular momentum conservation laws hold for all the time. That is, for all $\tau\in[0,\pi/2h)$,

[TABLE]

[TABLE]

[TABLE]

In order to state our results conveniently, we introduce the following weight function for $\beta>3/2$,

[TABLE]

The main theorem in this paper is

Theorem 1.1. For small $\epsilon_{0}>0$, suppose that the initial data $f_{0}=M+M^{1/2}u_{0}\geq 0$ satisfying (1.14)-(1.16) with $\|\phi_{\beta}u_{0}\|_{\infty}\leq\epsilon_{0}$, then there exists a constant $\lambda>0$ and a unique mild solution $f=M+M^{1/2}u\geq 0$ to problem (1.10) together with (1.13) and (1.17)-(1.19) such that for $\tau\in(0,\pi/2h)$,

[TABLE]

*where $C>0$ is a constant independent of $\tau$. Moreover, if the initial data $f_{0}(y,\eta)$ is continuous except on $\gamma_{01}$, then $f(\tau,y,\eta)$ is continuous in $[0,\pi/2h)\times\{\bar{\Omega}\times\mathbb{R}^{3}\setminus\gamma_{01}\}$, where $\gamma_{01}=\{(y,\eta)\in\gamma_{0}:|\eta|\geq h\}$ with $\gamma_{0}=\{(y,\eta)\in\partial\Omega\times\mathbb{R}^{3}:\eta\cdot n_{y}=0\}$ and $\partial\Omega=\{x\in\mathbb{R}^{3}:|x|=1\}$. *

Remark 1.1. We can also consider the case that the expanding speed of the ball is exactly the constant number $h>0$ in Theorem 1.1. At this time, the radius of the expanding ball at time $t$ is $R_{1}(t)=1+ht$. Correspondingly, $M_{1}(t,x,\xi)=e^{-|\xi-h(x-t\xi)|^{2}}$ is the background solution of problem (1.10). As long as we modify the pulling speed near the time $t=0$ to let the speed increase smoothly from [math] to $h$, then we can obtain the analogous result to Theorem 1.1.

Next, we study the global physical phenomenon of problem (1.7) together with (1.8). Return to the original coordinates $(t,x,\xi)$ and equation (1.1). Let $\rho=\rho(t,x)$, $v=v(t,x)$ and $\theta=\theta(t,x)$ be the mass density, velocity and temperature of the gases, respectively, i.e,

[TABLE]

For the traveling Maxwellian (1.4), under transformation (1.5), we have

[TABLE]

where $r=2/3$, $\bar{\rho}(t,x)=\frac{2^{-3/2}}{R(t)^{3/2}}e^{-\frac{h^{2}|x|^{2}}{2R(t)^{2}}}$, $\bar{v}(t,x)=\frac{R^{\prime}(t)}{R(t)}x$, and $\bar{\theta}(t,x)=\frac{3/4}{R(t)^{2}}$.

Theorem 1.2. For $\epsilon_{0}>0$ small, suppose that the pulling speed $h\in(0,\epsilon_{0}^{1/2})$, and the initial data around the equilibrium $\mu$ satisfies

[TABLE]

with

[TABLE]

In addition, $f_{0}$ also satisfies the conservation laws

[TABLE]

[TABLE]

[TABLE]

Then there exists a unique mild solution $f=M+M^{1/2}u\geq 0$ to problem (1.7) together with (1.8). Moreover, scaling back to the original coordinates $(t,x,\xi)$ by (1.5), we have the following decay estimates

[TABLE]

where $c_{0}$ and $C_{0}$ are two positive constants independent of $t$.

*In addition, if the initial data $f_{0}(x,\xi)$ is continuous except on $\gamma_{01}$, then $f(t,x,\xi)$ is continuous in $\{S\times\mathbb{R}^{3}\}\setminus\gamma_{\mathrm{sing}}$, where $\gamma_{\mathrm{sing}}=\{(t,x,\xi)\in\partial S\times\mathbb{R}^{3}:\,t>0,\,|x|=R(t),\,\xi\cdot n_{x}=R^{\prime}(t),\,|\xi|\geq h\}$, $n_{x}$ stands for the unit outer normal at $x\in\partial\Omega_{t}$, and $\gamma_{01}$ has been defined in Theorem 1.1. *

Remark 1.2. In [34] and [36], the movements of gases in an expanding ball $\Omega_{t}$ are globally described by the Euler equations and the Navier-Stokes equations, respectively, and the authors have shown that the vacuum will not appear in $\Omega_{t}$.

So far there exists extensive literature on the study of the Boltzmann equation. For the Cauchy problem of the Boltzmann equation, under some special assumptions on the initial data, T.Carleman [5] proved the local existence of solutions for the spatially homogeneous case, while H.Grad [12] and S.Ukai [30] established the local and global existence of solutions for the spatially inhomogeneous case. After that, by our knowledge, the Boltzmann equation is mainly discussed in three different frameworks. In $L^{\infty}$ framework, by the spectral analysis of the linearized Boltzmann equation, various initial value problems and initial-boundary value problems were considered (see [31]-[33] and so on). In $L^{1}$ framework, R.J.DiPerna and P.L.Lions [8] constructed the renormalized solutions, which were based on the velocity-averaging lemma and entropy dissipation (see also [1] and [7]). In $L^{2}$ framework, based on macro-micro decomposition, the authors in [26] and [35] developed energy methods for the Boltzmann equation, and subsequently different wave patterns were considered by utilizing the energy methods (e.g. [28], [27], [20] and so on). In addition, the $L^{2}$ energy methods were generalized by Y.Guo and applied to various systems in [14]-[15] and [17] respectively. On the other hand, for the Boltzmann equation without angular cut-off, there also exist many results (see [3], [2], [13] and references therein).

For the initial-boundary problems of Boltzmann equation, Y.Guo developed the $L^{2}-L^{\infty}$ procedure in [16] and solved the problems for all the four kinds of boundary conditions (namely, the inflow, reverse-reflection, specular-reflection and diffuse-reflection boundary conditions). For the case of specular-reflection boundary condition, the domain was required to be strictly convex and analytic in [16] (when the domain is only strictly convex, the analyticity assumption of boundary has been removed recently in [24]). With respect to the regularities of solutions, the authors in [19] proved that the solution is $C^{1}$ away from the grazing set for the Boltzmann equation in a convex domain. While if the domain is non-convex, singularities of solution may propagate from the grazing set to the interior of domain, see [23] and [18] for more details. The corresponding results have been generalized to the Boltzmann equation with soft potential and angular cut-off in [25].

To derive the $L^{2}$ decay in the $L^{2}-L^{\infty}$ procedure, the author in [16] applied the compactness method to a finite time interval. Since the Boltzmann operator forms a semigroup, the long time $L^{2}$ decay of solutions was derived by iteration in [16]. Subsequently, by writing the linearized Boltzmann equation in a weak formulation, a constructive method of $L^{2}$ estimate was constructed in [9] for the diffuse-reflection boundary condition. By choosing the test functions suitably, the $L^{2}$ estimates of the macro-components could be controlled by the micro-components, while the remaining boundary terms were controlled well by the dissipation property of diffuse-reflection boundary. The methods in [16] and [9] were generalized to the boundary condition which was a linear combination of the diffuse-reflection condition and specular-reflection condition in [4]. However, the case of pure specular-reflection boundary condition was not considered in these papers.

In this paper, we consider the Boltzmann equation in an expanding ball with pure specular-reflection boundary condition, which leads to a new form (1.7) of Boltzmann equation whose coefficient depends on the time variable $\tau$. In this case, the resulting linearized operator no longer forms a semigroup and we can not use the implicit method directly to get the $L^{2}$ decay of solutions as in [16]. Motivated by [9], we will apply the constructive method to the pure specular-reflection boundary problem (1.7) although there is no dissipation property on the boundary. Through choosing the Burnette functions as orthogonal bases in micro-components, we can reformulate the boundary integral in a more delicate way and look for suitable test functions to handle the resulting boundary terms. As a byproduct, we give a constructive method to prove the $L^{2}$ decay for the Boltzmann equation in a bounded domain with specular-reflection boundary conditions. On the other hand, reverse-reflection boundary condition is ill-posed here because of the potential term in (1.7) (see Remark 2.1 below)

Here we point out that we have used the so-called “traveling Maxwellian” in (1.4) to treat the Boltzmann equation (1.1) with (1.3) in the expanding ball $\Omega_{t}$. The global Cauchy problem of the traveling Maxwellian was studied by R.Illner and M.Shinbrot in [21]. For the extremely rarefied gases, the authors in [21] applied the iteration scheme in [22] to obtain a global mild solution of Boltzmann equation (one can also see Chapter 5 of [6]). In the present paper, since the gases are not extremely rarefied in the ball at the beginning and lie in a bounded domain at any finite time, we are required to give some different treatments from those in [21].

The paper is organized as follows: In Section 2, we discuss the properties of backward trajectories of operator $\Lambda_{\tau}$ and reformulate the Velocity Lemma (see Lemma 2.4 below). In Section 3, we list some basic properties of Boltzmann operators which will be applied later on. In Section 4, we establish the $L^{2}$-estimates of solutions to linear Boltzmann equations. In Section 5, an explicit formula of solution to the transport equation is given, and subsequently the $L^{\infty}$-estimate of solutions to a class of linear weighted Boltzmann equations is established. In Section 6, at first, by the Duhamel’s principle, one can write out the implicit expression of the solution to full Boltzmann equation. Based on this, by iteration, we derive the existence and uniqueness of the solution to problem (1.7) with (1.8). And then the proofs of Theorem 1.1 and Theorem 1.2 are completed.

Notations: In the following sections, to simplify the notations, we denote by $\|u\|=\|u\|_{L^{2}(\Omega\times\mathbb{R}^{3})}$, $\|u\|_{\infty}=\|u\|_{L^{\infty}(\Omega\times\mathbb{R}^{3})}$ and $\|u\|_{\nu}=\|\sqrt{\nu}u\|_{L^{2}(\Omega\times\mathbb{R}^{3})}$, where $\Omega=\{y=(y_{1},y_{2},y_{3})\in\mathbb{R}^{3}:|y|<1\}$ and $\nu=\nu(\eta)=\sqrt{2\pi}[(|\eta|+|\eta|^{-1})\int_{0}^{|\eta|}e^{-s^{2}}ds+e^{-|\eta|^{2}}]$. We say $A\lesssim B$ if $A\leq CB$ holds for some positive constant $C$ independent of the quantities $A$ and $B$.

2 Backward trajectories of operator $\Lambda_{\tau}$

Recall that $\Lambda_{\tau}=\partial_{\tau}+\eta\cdot\nabla_{y}-h^{2}y\cdot\nabla_{\eta}$ in (1.9). We now consider the trajectory $(Y,H)(\tau)=(Y,H)(\tau;\tau_{0},y_{0},\eta_{0})$ of $\Lambda_{\tau}$ in the whole phase space $\mathbb{R}^{3}_{y}\times\mathbb{R}^{3}_{\eta}$ starting from the point $(y_{0},\eta_{0})$ and time $\tau_{0}$:

[TABLE]

Direct computation yields

[TABLE]

This means that $Y(\tau)$ is an ellipse in $\mathbb{R}^{3}_{y}$ with the origin $O$ as its center. Since the related time interval is $(0,2\pi/h)$, the trajectory only takes a quarter of the ellipse. Moreover, we have the conservations of energy and angular momentum for $(H,Y)$:

[TABLE]

and

[TABLE]

From this, we can define the numbers $e_{0}$ and $m_{0}$ as follows

[TABLE]

[TABLE]

Denote by $l_{\max}$ and $l_{\min}$ the major and minor semi-axis of ellipse $Y(\tau)$, respectively. It is easy to know that

[TABLE]

[TABLE]

Next we study the properties of operator $\Lambda_{\tau}$ in the domain $\bar{\Omega}\times\mathbb{R}^{3}_{\eta}$, which will be divided into two cases of $|y_{0}|<1$ and $|y_{0}|=1$.

2.1 Backward trajectory $Y(\tau)$ in the interior of $\Omega$

For $(y_{0},\eta_{0})\in\Omega\times\mathbb{R}^{3}$, the backward trajectory $Y(\tau)$ may hit the boundary $\partial\Omega$ and change its velocity, then travels along another ellipse. This can be precisely stated as follows

Lemma 2.1. For $(y_{0},\eta_{0})\in\Omega\times\mathbb{R}^{3}$, we have

(a) If $l_{\min}<1<l_{\max}$, i.e., $e_{0}-m_{0}>h^{2}$, then the backward trajectory $Y(\tau)$ will hit the boundary $\partial\Omega$ at some point, then reflect specularly and travel along another ellipse. In particular, this is the case when $|\eta_{0}|\geq 2h$;

(b) If $l_{\min}<l_{\max}=1$, i.e., $e_{0}-m_{0}=h^{2}$, then the backward trajectory $Y(\tau)$ will graze the boundary $\partial\Omega$ at some point and travel along the same ellipse;

*(c) If $l_{\min}\leq l_{\max}<1$, i.e., $e_{0}-m_{0}<h^{2}$, then the backward trajectory $Y(\tau)$ remains in the interior of $\Omega$ and travels along the same ellipse. *

Proof. Note that $l_{\min}<1$ holds for $y_{0}\in\Omega$. Thus it only suffices to consider the relation between $l_{\max}$ and $1$. In fact, Lemma 2.1 holds by direct verifications and observation.           $\square$

Definition 2.2. For $(y_{0},\eta_{0})\in\Omega\times\mathbb{R}^{3}$ satisfying $e_{0}-m_{0}>h^{2}$ (corresponding to Case (a) of Lemma 2.1), we define

[TABLE]

Here we point out that $(\tau_{0}-\tau_{b},y_{b},\eta_{b})$ is just the point when the backward trajectory $H(\tau;\tau_{0},y_{0},\eta_{0})$ starting from $(\tau_{0},y_{0},\eta_{0})$ first hits the boundary $\partial\Omega$. In order to define the backward trajectory piece by piece, we introduce the following notation.

Definition 2.3. Let $(y_{0},\eta_{0})\in\Omega\times\mathbb{R}^{3}$ satisfy $e_{0}-m_{0}>h^{2}$, for $k\geq 0$, we define

[TABLE]

In this case, the backward trajectory $(y(\tau),\eta(\tau))$ starting from $(\tau_{0},y_{0},\eta_{0})$ can be expressed as

[TABLE]

where $1_{(\tau_{k+1},\tau_{k}]}(\tau)$ stands for the characteristic function of interval $(\tau_{k+1},\tau_{k}]$.

For Case (b) of Lemma 2.1, the representations in (2.27) for the backward trajectory are still plausible. The only difference from Case (a) is that the trajectory grazes the boundary $\partial\Omega$. This means that there is actually no change of velocity $u$ at the grazing point due to the specular-reflection boundary condition.

For Case (c) of Lemma 2.1, the backward exit time $\tau_{b}(y_{0},\eta_{0})$ cannot be defined since the trajectory remains in the interior of domain $\Omega$. But we still use the representation (2.27) with only $k=0$.

To derive the $L^{\infty}$ decay of solutions to linearized Boltzmann equations, for small $\kappa>0$, large $N>0$, we need to define the following set

[TABLE]

Lemma 2.4 (Velocity Lemma) For $(\tau_{0},y_{0},\eta_{0})\in(0,\pi/2h)\times A_{\kappa,h,N}$, denote the backward trajectory $(y(\tau),\eta(\tau))$ by (2.27), then we have

(a) The time interval $\Delta\tau$ between two adjacent reflections point is

[TABLE]

and admits the upper and lower bounds as

[TABLE]

(b) For $k>\frac{\pi N^{2}}{h\kappa}$, we have $\tau_{k}<0$, which means that the summation of $k$ in (2.27) is finite.

(c) For $k\leq\frac{\pi N^{2}}{h\kappa}$, set $\delta=\frac{h^{2}\kappa}{N^{2}}$. When $\tau\in(\tau_{k}+\delta,\tau_{k-1}-\delta)$, we have that for all $\eta^{\prime}\in\mathbb{R}^{3}$ satisfying $2h\leq|\eta^{\prime}|\leq 2N$,

[TABLE]

(d) The measure of the set $(0,\tau_{0})\backslash\cup_{k}(\tau_{k}-\delta,\tau_{k}+\delta)$ is less than $2\pi h$.

*(e) $\tau_{k},\,y_{k}$ and $\eta_{k}$ are analytic functions of $(\tau_{0},y_{0},\eta_{0})$. *

Proof. (a) For $(\tau_{0},y_{0},\eta_{0})\in(0,\pi/2h)\times A_{\kappa,h,N}$, from (2.23) we have

[TABLE]

Then we get

[TABLE]

where

[TABLE]

Let $|Y(\tau)|=1$ in (2.32) yield

[TABLE]

This derives

[TABLE]

Hence, the backward exit time $\tau_{b}$ is

[TABLE]

and the time interval between two adjacent reflection is

[TABLE]

For the upper bound of $\Delta\tau$, by $e_{0}^{2}\geq\eta_{0}^{2}\geq 4h^{2}$, we have

[TABLE]

In addition, by $e_{0}-m_{0}\geq h^{2}+\kappa^{2}$, we have

[TABLE]

This yields

[TABLE]

(b) Since $\tau_{0}\in(0,\frac{\pi}{2h})$ and $\Delta\tau\geq\frac{\kappa}{2N^{2}}$ from (a), the number of reflections along the backward trajectory is less than $\frac{\pi N^{2}}{h\kappa}$. Hence we have

[TABLE]

(c) Since $\delta=\frac{h^{2}\kappa}{N^{2}}<\Delta\tau$, we have

[TABLE]

From (2.32), $|y(\tau)|^{2}$ is a convex function of $\tau\in(\tau_{k}+\delta,\tau_{k-1}-\delta)$. Thus,

[TABLE]

It follows from $e_{0}-m_{0}\geq h^{2}+\kappa^{2}$ that

[TABLE]

This, together with (2.34), yields

[TABLE]

Note that for any $\eta^{\prime}\in\mathbb{R}^{3}$ satisfying $2h\leq|\eta^{\prime}|\leq 2N$, we have

[TABLE]

Then for $\tau\in(\tau_{k}+\delta,\tau_{k-1}-\delta)$, $2h\leq|\eta^{\prime}|\leq 2N$, we conclude that

[TABLE]

(d) The measure of the set $(0,\tau_{0})/\cup_{k}(\tau_{k+1}+\delta,\tau_{k}-\delta)=\cup_{k}[\tau_{k}-\delta,\tau_{k}+\delta]$ is less than

[TABLE]

(e) To prove the analyticity of $\tau_{k},\,y_{k}$ and $\eta_{k}$ with respect to the variable $(\tau_{0},y_{0},\eta_{0})$, for $(y_{0},\eta_{0})\in A_{\kappa,h,N}$, we only need to study the functions $\tau_{b}$, $y_{b}$ and $\eta_{b}$ defined in (2.26). Since $|y_{b}|=1$, $y_{b}\cdot\eta_{b}<0$ and

[TABLE]

we have

[TABLE]

When solving $|y(-\tau_{b})|^{2}=1$, by the fact that

[TABLE]

then we can see that $\tau_{b}$ is locally solvable. Similarly, direct computation yields that the derivatives of $|y(-\tau_{b})|^{2}$ with respect to variables $y$ and $\eta$ are

[TABLE]

Note that $|y(\tau)|^{2}$ is analytic. Then by implicit function theorem, we know that $\tau_{b}$, $y_{b}$ and $\eta_{b}$ are analytic with respect to the variable $(\tau_{0},y_{0},\eta_{0})$.                         $\square$

2.2 Backward trajectory near boundary $\partial\Omega$

For $(y_{0},\eta_{0})\in\partial\Omega\times\mathbb{R}^{3}$, the property of backward trajectory is more subtle. We denote the phrase boundary as $\gamma=\partial\Omega\times\mathbb{R}^{3}$, and split $\gamma$ into the outgoing boundary $\gamma_{+}$, the incoming boundary $\gamma_{-}$, and the grazing set $\gamma_{0}$ as follows:

[TABLE]

Compared with [16] where $\gamma_{0}$ is a singular set, in the present paper, only some part of $\gamma_{0}$ is singular. Since the potential force $-h^{2}y$ in $\Lambda_{\tau}$ is pointing to the center of the unit ball, particles on part of $\gamma_{0}$ will depart from the boundary and go to the interior of the ball. We should further split $\gamma_{0}$ into non-singular set $\gamma_{00}$, and singular set $\gamma_{01}$ as follows:

[TABLE]

The action of non-singular set $\gamma_{00}$ is similar to that of the interior domain of $\Omega\times\mathbb{R}^{3}$ while the singular set $\gamma_{01}$ acts as the singular grazing set. More precisely, we have the following conclusion

Lemma 2.5. The backward trajectory $(y,\eta)(\tau;\tau_{0},y_{0},\eta_{0})$ is continuous for all

[TABLE]

Proof. To prove Lemma 2.5, it only suffices to study the situation around the grazing set $\gamma_{00}$ where there is at most one collision with the boundary. In this case, we require to consider three classes of points, $(y_{j},\eta_{j})$, $j=1,2,3$, representing the three cases discussed in Lemma 2.1 (see Figure 2 below) respectively.

For $j=1,2,3$, set

[TABLE]

[TABLE]

Then

[TABLE]

In the first two cases of Lemma 2.1, when $|(\tau_{1},y_{1},\eta_{1})-(\tau_{2},y_{2},\eta_{2})|<\varepsilon$ sufficiently small, it follows from (2.33) that for $j=1,2$,

[TABLE]

Together with the fact

[TABLE]

this yields that $\nabla_{y,\eta}\tau_{b}$ is locally finite around $(y_{j},\eta_{j})$ and $|\tau_{b1}-\tau_{b2}|\leq C\varepsilon$ holds. After specular reflection, it is easy to know $|y_{1}(\tau)-y_{2}(\tau)|\leq C\varepsilon$.

In the last two case, when $|(\tau_{2},y_{2},\eta_{2})-(\tau_{3},y_{3},\eta_{3})|<\varepsilon$ sufficiently small, we have $y_{j}(\tau)=Y(\tau;\tau_{j},y_{j},\eta_{j})$, $j=2,3$. This derives $|y_{2}(\tau)-y_{3}(\tau)|\leq C\varepsilon$.                $\square$

Remark 2.1. When considering the same problem for the reverse-reflection boundary, we cannot get any continuity result as in Lemma 2.5. As Figure 3 shows, if we trace back along the trajectories of $(y_{1},\eta_{1})$ and $(y_{2},\eta_{2})$, then the backward trajectory of $(y_{2},\eta_{2})$ stays on the same ellipse. But the backward trajectory of $(y_{1},\eta_{1})$ hits the boundary $\partial\Omega$ and then reflects reversely, which leads to the fact that these two trajectories of $(y_{1},\eta_{1})$ and $(y_{2},\eta_{2})$ can no longer stay close to each other.

3 Basic properties of the Boltzmann operators

Before establishing the $L^{2}$ estimate of solutions to the linear Boltzmann equation, we list some properties of the Boltzmann operators. The proofs are elementary and can be found in [11] and Chapter 7 of [6].

The linearized collision operator $L$, given by (1.11), is a self-adjoint nonnegative operator in $L^{2}$. The null space $\mathcal{N}$ of $L$ is spanned by

[TABLE]

which are known as the collision invariants. After normalization, we write

[TABLE]

In addition, we denote the projection to the null space $\mathcal{N}$ by $P$ as follows

[TABLE]

where $<g,h>=\int_{\mathbb{R}^{3}}ghd\eta$. As in [35], we shall use the following Burnette functions of the space $\mathcal{N}^{\perp}$:

[TABLE]

where $\delta_{kl}=1$ if $k=l$ and $\delta_{kl}=0$ if $k\neq l$. Direct verification yields

Lemma 3.1. For $i,j,k,l=1,2,3$,

(a) $PA_{j}=0$ and $PB_{kl}=0$.

(b) $<A_{j},A_{i}>=\delta_{ji}$ and $<A_{j},B_{kl}>=0$.

*(c) $<B_{ij},B_{kl}>=\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-\frac{2}{3}\delta_{ij}\delta_{kl}$. *

On the other hand, the operator $L$ can be split as $-\nu(\eta)I+K$, where

[TABLE]

and $K$ is an integral operator with the symmetric kernel function $k(\eta,\eta_{*})$ given by

[TABLE]

Obviously, $\nu(\eta)$ satisfies

[TABLE]

where $\nu_{0}$ and $\nu_{1}$ are certain positive constants. And the kernel function $k(\eta,\eta_{*})$ admits the following properties:

Lemma 3.2. $k(\eta,\eta_{*})$* is integrable and square integrable with respect to the variable $\eta_{*}\in\mathbb{R}^{3}$, moreover, the integral is bounded by a positive constant independent of $\eta$. *

Proof. Following the expression of $k(\eta,\eta_{*})$ in (3.38), we have

[TABLE]

where $C>0$ is a constant independent of $\eta$. Then

[TABLE]

Similar result holds for the square norm of $k(\eta,\eta_{*})$ with respect to variable $\eta_{*}$.          $\square$

Let the weight function $\phi(y,\eta)=(1+|\eta|^{2}+h^{2}|y|^{2})^{\beta/2}$ with $\beta>3/2$. Then we have

Lemma 3.3. The operator $K_{\phi}$ is a bounded operator in $L^{\infty}(\mathbb{R}^{3})$ space, where $K_{\phi}w=\phi K\left(\phi^{-1}w\right)$.

Proof. The kernel of the operator $K_{\phi}$ is

[TABLE]

Note that for fixed $y\in\Omega$,

[TABLE]

Then we have

[TABLE]

Hence $K_{\phi}$ is a bounded operator in $L^{\infty}$.                      $\square$

It is well-known that the operator $L$ satisfies

[TABLE]

where $\|h\|_{\nu}=\|\nu(\eta)^{1/2}h\|$. As for the bilinear operator $\Gamma(g,h)$, we have

Lemma 3.4. There exists a constant $C>0$ independent of $\eta$ such that

[TABLE]

Proof. From the definitions in (1.12) and (1.2), we can split $\Gamma(g,h)$ as

[TABLE]

Note $\mu\mu_{*}=\mu^{\prime}\mu^{\prime}_{*}$, we then have

[TABLE]

where in the last inequality we have used the property of $\nu(\eta)$ in (3.39). In addition,

[TABLE]

Similar estimates hold for $\Gamma_{2}$ and $\Gamma_{4}$. Thus, Lemma 3.4 is proved.            $\square$

Lemma 3.4 together with the fact of $\phi(\eta)\leq\phi(\eta^{\prime})\phi(\eta^{\prime}_{*})$, yields

Lemma 3.5. There exists a constant $C>0$ independent of $\eta$ such that

[TABLE]

*where $\Gamma_{\phi}(g,h)=\phi\Gamma\left(\phi^{-1}g,\phi^{-1}h\right)$. *

4 $L^{2}$-estimate of solutios to the linear Boltzmann equation

Consider the following linear Boltzmann equation

[TABLE]

where $g=g(\tau,y,\eta)$ is a smooth function. The initial-boundary condition of $u$ is given by

[TABLE]

In addition, we assume that $g$ also satisfies the following condition to assure the conservation of mass, energy and angular momentum:

[TABLE]

where the operator $P$ is defined in (3.36). Set

[TABLE]

We will prove the following $L^{2}$ decay estimates of solution $u$ to (4.42).

Proposition 4.1. Let $u\in L^{2}$ be the weak solution of (4.42) with initial-boundary value condition (4.43), then there exists a constant $\lambda>0$ such that for $\tau\in(0,\pi/2h)$,

[TABLE]

The rest of this section is devoted to the proof of Proposition 4.1. In terms of the macro-micro decomposition, set

[TABLE]

where $\tilde{\mu}=e^{-h^{2}|y|^{2}/2}$ and $q=\frac{h^{2}|y|^{2}}{\sqrt{6}}a$. Let

[TABLE]

It follows from the conservation laws of mass and energy in (1.17) and (1.18) that

[TABLE]

and

[TABLE]

In addition,

[TABLE]

Next, we derive the estimates of $a,b=(b^{1},b^{2},b^{3}),c$ in terms of $d$ and $g$. Rewrite the linear Boltzmann equation (4.42) in the following weak formulation:

[TABLE]

where $d\gamma=(\eta\cdot n_{y})dS_{y}d\eta$, and $\psi\in C^{\infty}\cap L^{2}((0,\pi/2h)\times\Omega\times\mathbb{R}^{3})$ is a test function.

We now focus on the treatment of (4.50), which is divided into the following six steps. In Step 1-Step 3, some useful estimates are derived for different choices of test function $\psi$. In Step 4-Step 6, based on Step 1-Step 3, the functions $a,b,c$ in (4.46) are dealt with.

Step 1. Choosing the test function $\psi=\varphi(y)\tilde{\mu}^{1/2}\chi_{0}$ in (4.50)

Direct computation yields

[TABLE]

Let $[s,\tau]=[\tau,\tau+\varepsilon]$ in (4.50). Then it follows from (4.50), (4.51) and (4.46) that the left-hand side (will be briefly written as LHS) of equation (4.50) now becomes

[TABLE]

By (4.51),

[TABLE]

And $I_{2}=0$, $I_{4}=0$ by the boundary condition in (4.43) and the assumption of $g$ in (4.44), respectively. In addition, the fact of $<\chi_{0},Lu>=0$ yields $I_{3}=0$. In this case, taking the difference quotient in (4.50) as $\varepsilon\to 0$ and using (4.52) yield

[TABLE]

Let $\varphi\equiv 1$ in (4.53), then

[TABLE]

On the other hand, for $\varphi\in H^{1}(\Omega)$, we have that from (4.53),

[TABLE]

This leads to

[TABLE]

For fixed $\tau\in(0,\pi/2h)$, by (4.54)-(4.55) and the standard elliptic theory, there exists a unique weak solution $\Phi_{a}$ to the problem

[TABLE]

Moreover,

[TABLE]

Step 2. Choosing the test function $\psi_{j}=\varphi(y)\tilde{\mu}^{1/2}\chi_{j}$ ($j=1,2,3$) in (4.50)

In this case, we have

[TABLE]

When $[s,\tau]=[\tau,\tau+\varepsilon]$ is chosen in (4.50), the left-hand side of (4.50) is

[TABLE]

Meanwhile, by (4.57),

[TABLE]

and $I_{4}=0$. In addition, from the fact that $<Lu,\chi_{j}>=0$, we have $I_{3}=0$. Consequently, taking the difference quotient as $\varepsilon\to 0$ in (4.50) and using (4.58)-(4.59) yield

[TABLE]

For fixed $\tau>0$, let $\Phi_{b}=(\Phi_{b}^{1},\Phi_{b}^{2},\Phi_{b}^{3})$ satisfy

[TABLE]

The existence of $\Phi_{b}$ is given in Appendix A. Choosing $\varphi=\Phi_{b}^{j}$ in (4.60), after summation of $j=1,2,3$, the last term on the right hand of (4.60) is

[TABLE]

So

[TABLE]

This, together with $\|\Phi_{b}^{j}\|\lesssim\|\nabla_{y}\Phi_{b}^{j}\|$, yields

[TABLE]

Step 3. Choosing the test function $\psi=\varphi(y)\tilde{\mu}^{1/2}\chi_{4}$ in (4.50)

In this case, one has

[TABLE]

Let $[s,\tau]=[\tau,\tau+\varepsilon]$ in (4.50). Then it follows from (4.50) and (4.46) that

[TABLE]

Meanwhile, by (4.63),

[TABLE]

and $I_{2}=I_{4}=0$. In addition, the fact that $<Lu,\chi_{4}>=0$ derives $I_{3}=0$. Consequently, as in Step 1 and Step 2, we have

[TABLE]

Choosing $\varphi\equiv 1$ in (4.64) yields

[TABLE]

Thus, for fixed $\tau>0$, we can choose $\varphi=\Phi_{c}$ such that

[TABLE]

This leads to

[TABLE]

Thus, we have

[TABLE]

Before continuing to Step 4-6, we rewrite (4.50) in the following form

[TABLE]

We will consider the weak formulation (4.67) instead of (4.50) in the following.

Step 4. Estimates of $c$

For fixed $\tau>0$, by (4.48) let $\varphi_{c}$ be a solution of the following problem

[TABLE]

Choosing

[TABLE]

in (4.67). In addition, direct computation yields

[TABLE]

Thus, the left-hand side of (4.67) is

[TABLE]

where, for any $\varepsilon>0$,

[TABLE]

and

[TABLE]

On the other hand, by the fact that $\Phi_{c}=\partial_{\tau}\varphi_{c}$ and estimate of $\Phi_{c}$ in (4.66), we have that for any $\varepsilon>0$,

[TABLE]

By the boundary conditions of $\varphi_{c}$ and $u$, one has that

[TABLE]

Therefore,

[TABLE]

In addition, we obtain that for any $\varepsilon>0$,

[TABLE]

By choosing $\varepsilon>0$ and $h>0$ small, it follows from (4.69)-(4) that

[TABLE]

Step 5. Estimate of $b$

For fixed $\tau>0$, let $\varphi_{b}=(\varphi_{b}^{1},\varphi_{b}^{2},\varphi_{b}^{3})$ be the solution of the following problem:

[TABLE]

where the existence of solution $\varphi_{b}$ is proved in Appendix A. Set

[TABLE]

in (4.67), we then have that by direct computation,

[TABLE]

Thus, the left-hand side of (4.67) becomes

[TABLE]

where, for any $\varepsilon>0$,

[TABLE]

Next we estimate $J_{i}$ $(1\leq i\leq 4)$ in (4.67). From the fact that $\Phi_{b}=\partial_{\tau}\varphi_{b}$ and the estimate of $\Phi_{b}$ in (4.62), we have that for any $\varepsilon>0$,

[TABLE]

In addition,

[TABLE]

For fixed $y\in\partial\Omega$, set

[TABLE]

Through a coordinate rotation, we may assume that $n_{y}=(1,0,0)$. Then

[TABLE]

From the boundary condition of $u$ and $n_{y}=(1,0,0)$, we have $u(\eta^{1},\eta^{2},\eta^{3})=u(-\eta^{1},\eta^{2},\eta^{3})$. So $J_{21}=J_{24}=0$. On the other hand, since the second boundary condition of $\varphi_{b}$ in (4.76) gives $\partial_{1}\varphi_{b}^{i}=0$ for $i\neq 1$, $J_{22}=0$ holds; since the first boundary condition of $\varphi_{b}$ in (4.76) gives $\varphi_{b}^{1}=0$ on $\partial\Omega$, $\partial_{j}\varphi_{b}^{1}=0$ for $j\neq 1$ and further $J_{23}=0$ hold. Thus by (4.80) and the expression of $J_{2}$ we obtain

[TABLE]

Meanwhile, it follows from direct computation that

[TABLE]

and

[TABLE]

Substituting (4.77)-(4.83) into (4.67) yields that for small $\varepsilon>0$,

[TABLE]

Step 6. Estimates of $a$

For fixed $\tau>0$, by (4.47) let $\varphi_{a}$ be the solution of the following problem:

[TABLE]

Choosing

[TABLE]

Then direct computation yields

[TABLE]

Thus, the left-hand side of (4.67) becomes

[TABLE]

where

[TABLE]

Next we estimate $J_{i}$ $(1\leq i\leq 4)$ in (4.67). From the fact that $\Phi_{a}=\partial_{\tau}\varphi_{a}$ and the estimate of $\Phi_{a}$ in (4.56), we have

[TABLE]

Similar to the treatment in Step 4, we have

[TABLE]

and thus

[TABLE]

In addition, we have

[TABLE]

and

[TABLE]

It follows from (4.86)-(4.90) that

[TABLE]

Using the fact that $\|d\|\leq\|d\|_{\nu}$ and the asymptotic behavior of $\nu$ in (3.39), we have

[TABLE]

Combining (4.91), (4.84) and (4.75) with (4.92) yields

[TABLE]

where

[TABLE]

Next we derive the $L^{2}$-decay of solution $u$ to the linear Boltzmann equation (4.42). For some constant $\lambda>0$ to be determined later on, set

[TABLE]

Then $U$ satisfies

[TABLE]

Multiplying both sides of (4.95) by $U$ and integrating with respect to the variable $(\tau,y,\eta)$, and using the dissipation property of $L$ in (3.41), we arrive at

[TABLE]

For $n\geq 0$, set

[TABLE]

Let $s=\tau_{n}$ and $\tau=\tau_{n+1}$ in (4.96). Then we have

[TABLE]

Furthermore, (4.97)$+\{$(4.93) $\times\varepsilon m_{n}$} for some small $\varepsilon>0$ gives

[TABLE]

Using the fact that $m_{n}\leq M_{n}\leq 4\epsilon^{4\lambda}m_{n}\leq 4\epsilon^{4}m_{n}$, we can choose $\varepsilon\lesssim 1/2$ and $\lambda\lesssim 1/16\epsilon^{4}$ in the above. In addition, by the estimate of $G$ in (4.94), we then get

[TABLE]

Summing $n$ over $0,1,2...$ in the above, we conclude that

[TABLE]

Thus, the proof of Proposition 4.1 is completed.                     $\square$

5 $L^{\infty}$ decay of solutions to linear weighted Boltzmann equations

Let $\phi$ be the weight function:

[TABLE]

where $\beta>3/2$ is large enough.

Consider the equation of $w=\phi u$:

[TABLE]

where

[TABLE]

The initial-boundary data of $w$ are

[TABLE]

Recall that $\alpha(\tau)=\int_{0}^{\tau}\cos^{2}(hs)ds$ and $\nu_{0}$ is defined in (3.39), we will prove the following conclusion.

Proposition 5.1. Let $w=w(\tau,y,\eta)$ be the mild solution of the linear Boltzmann equation (5.98) with (5.99). Then for some constant $\lambda\in(0,\nu_{0}/2)$, we have

[TABLE]

To prove Proposition 5.1, we will express the solution $w(\tau)$ of (5.98) by an integral along the backward trajectory. After splitting the integration into several parts, we can bound the main part by the $L^{2}$ norm of $u(\tau)$ established in Section 4, and bound the remaining parts by $L^{\infty}$ norm of $w$. Such a kind of estimate is called the $L^{2}-L^{\infty}$ estimate in [16].

For any fixed $(\tau,y,\eta)\in(0,\pi/2h)\times\Omega\times\mathbb{R}^{3}$, set $(\tau_{0}^{\prime},y_{0}^{\prime},\eta_{0}^{\prime})=(\tau,y,\eta)$. For $\tau_{1}\in(0,\tau)$, by the formula (2.27), the backward trajectory $(y_{1}(\tau_{1}),\eta_{1}(\tau_{1}))$ is

[TABLE]

To simplify the notation, along the trajectory $(y_{1}(\tau_{1}),\eta_{1}(\tau_{1}))$, we set

[TABLE]

By integrating along the trajectory, we have

[TABLE]

Recall that in (2.28), for some $N>0$ large and $\kappa>0$ small, we have defined the set

[TABLE]

Next we start to estimate $w(\tau)$ in (5.102).

Step 1. Estimate of the main part $\|w(\tau)1_{A_{\kappa,h,N}}\|_{\infty}$

For any $(\tau,y,\eta)\in(0,\pi/2h)\times A_{\kappa,h,N}$, by $2\lambda\leq\nu_{0}\leq\nu(\eta_{1})$ in (3.39), we can bound $I_{1}$ in (5.102) by

[TABLE]

For $I_{3}$, by the fact that

[TABLE]

we have

[TABLE]

Next we focus on the estimate of $I_{2}$. Note that

[TABLE]

Set $(\tau_{0}^{\prime\prime},y_{0}^{\prime\prime},\eta_{0}^{\prime\prime})=(\tau_{1},y_{1},\eta^{\prime})$, along the following backward trajectory $(y_{2}(\tau_{2}),\eta_{2}(\tau_{2}))$

[TABLE]

$w(\tau,y_{1},\eta^{\prime})$ can be expressed as

[TABLE]

here we have used the simplified notations:

[TABLE]

One can see Figure 4 for these two trajectories in double integration in (5.106).

Substituting (5.106) into the expression of $I_{2}$ in (5.102), we have $I_{2}=I_{21}+I_{22}+I_{23}$, just as in (5.102). It follows from Lemma 3.3 that $K_{\phi}$ is a bounded operator, and then the following estimates of $I_{21}$ and $I_{23}$ hold (similar to $I_{1}$ in (5.103) and $I_{3}$ in (5.104)),

[TABLE]

and

[TABLE]

The remaining part $I_{22}$ is

[TABLE]

where we have split the time-velocity integration into three cases, i.e., $I_{221}$ contains the integral domain: $|\eta|\geq N$ for some $N>0$ large enough; $I_{222}$ contains the integral domain: $|\eta|\leq N$, $|\eta^{\prime}|\geq 2N$ or $|\eta^{\prime}|\leq 2N$, $|\eta^{\prime\prime}|\geq 3N$; $I_{223}$ contains the integral domain: $|\eta|\leq N$, $|\eta^{\prime}|\leq 2N$ and $|\eta^{\prime\prime}|\leq 3N$. We next treat $I_{221}$, $I_{222}$ and $I_{223}$, respectively.

(a) The estimate of $I_{221}$

In this case, from the conservation of energy in (2.24), one has

[TABLE]

By Lemma 3.2 and (3.40), we have that for $N$ large enough,

[TABLE]

The double time integration can be estimated as follows

[TABLE]

which derives

[TABLE]

(b) The estimate of $I_{222}$

In this case, similar to (5.110), we have $|\eta_{1}-\eta^{\prime}|\geq N-1$ or $|\eta^{\prime\prime}-\eta_{2}|\geq N-1$. Then either of the following holds

[TABLE]

or

[TABLE]

From Lemma 3.2 and (3.40), we arrive at

[TABLE]

which yields

[TABLE]

(c) The estimate of $I_{223}$

To handle the singular factor $|\eta_{1}-\eta^{\prime}|^{-1}$ in the kernel $k_{\phi}(\eta_{1},\eta^{\prime})$, we choose a smooth function with compact support $k_{N}$ such that

[TABLE]

and then split

[TABLE]

Using the fact that

[TABLE]

and

[TABLE]

and noting that the smooth function $k_{N}$ is bounded by some constant $C_{N}>0$, we obtain that

[TABLE]

where

[TABLE]

Choosing

[TABLE]

in Lemma 2.4 (c). We separate the time intervals $(0,\tau)$ and $(0,\tau_{1})$ into the following parts:

[TABLE]

[TABLE]

By (d) of Lemma 2.4, we obtain

[TABLE]

where

[TABLE]

Applying Lemma 2.4 (a) for set $A_{\kappa,h,N}$, we know that $(y,\eta)\in A_{\kappa,h,N}$ and $(y_{1},\eta_{1})\in A_{h^{2}\kappa/N,h,N}$. This derives that the time intervals $\Delta\tau^{\prime}$ and $\Delta\tau^{\prime\prime}$ between two adjacent bounces on each trajectory are

[TABLE]

Moreover, the summations of $k$ and $j$ in (5.101) and (5.105) are finite:

[TABLE]

To apply the $L^{2}$ decay of $u(\tau)$ to derive the bound of $J_{kj}$, we will make a transformation of coordinate: $y_{2}\mapsto\eta^{\prime}$, where $y_{2}=y_{2}(\tau_{2};\tau_{1},y_{1}(\tau_{1};\tau,y,\eta),\eta^{\prime})$. As in Lemma 22 of [16], we establish the following result on $|\det\left(\frac{\partial y_{2}}{\partial\eta^{\prime}}\right)|$.

Lemma 5.2. For fixed $k$ and $j$, $|\det\left(\frac{\partial y_{2}}{\partial\eta^{\prime}}\right)|$ is an analytic function. For any $\varepsilon>0$ small enough, there are a number $\sigma=\sigma(\kappa,N,\varepsilon,k,j)>0$ and an open covering $\cup_{i=1}^{m}B(\tau_{i},y_{i},\eta_{i};r_{i})$ of $(0,\pi/2h)\times A_{\kappa,h,N}$, and corresponding open sets $O_{\tau_{i},y_{i},\eta_{i}}$ related to $[\tau_{k+1}^{\prime}+\delta^{\prime},\tau_{k}^{\prime}-\delta^{\prime}]\times[\tau_{j+1}^{\prime\prime}+\delta^{\prime\prime},\tau_{j}^{\prime\prime}-\delta^{\prime\prime}]\times\mathbb{R}^{3}_{\eta^{\prime}}$ with $|O_{\tau_{i},y_{i},\eta_{i}}|<h\varepsilon$, such that

[TABLE]

*holds for $(\tau,y,\eta)\in(0,\pi/2h)\times A_{\kappa,h,N}$ and $(\tau_{1},\tau_{2},\eta^{\prime})\in O_{\tau_{i},y_{i},\eta_{i}}^{c}\cap[\tau_{k+1}^{\prime}+\delta^{\prime},\tau_{k}^{\prime}-\delta^{\prime}]\times[\tau_{j+1}^{\prime\prime}+\delta^{\prime\prime},\tau_{j}^{\prime\prime}-\delta^{\prime\prime}]\times\{|\eta^{\prime}|\leq 2N\}$. *

Proof. We only require to prove Lemma 5.2 for $h=1$. Indeed, for general $h>0$, by choosing new coordinates,

[TABLE]

then the backward trajectory equations now turn to

[TABLE]

which just corresponds to the case of $h=1$. Hence, we have the estimates

[TABLE]

and the open sets $|O_{\tau_{i},y_{i},\eta_{i}}|=h|O_{\theta_{i},y_{i},\zeta_{i}}|\leq h\varepsilon$.

Next we assume $h=1$. For fixed $k$, when $\tau_{1}\in[\tau_{k+1}^{\prime}+\delta^{\prime},\tau_{k}^{\prime}-\delta^{\prime}]$, we have $y_{1}=y_{1}(\tau_{1};\tau,y,\eta)$ satisfying $(y_{1},\eta^{\prime})\in A_{h^{2}\kappa/N,h,N}$. So for fixed $j$, when $\tau_{2}\in[\tau_{j+1}^{\prime\prime}+\delta^{\prime\prime},\tau_{j}^{\prime\prime}-\delta^{\prime\prime}]$, $y_{2}$ is an analytic function of $(\tau_{1},\tau_{2},\eta^{\prime})$. This can be seen from the explicit formula of $y_{2}$ and Lemma 2.4 (e).

We now start to show that $|\det\left(\frac{\partial y_{2}}{\partial\eta^{\prime}}\right)|$ is not identically zero. To do so, choose $\tau_{2}$ such that $|\tau_{2}-\tau_{1}|\leq\frac{h^{2}\kappa}{2N^{3}}\leq\Delta\tau^{\prime\prime}$, then

[TABLE]

So

[TABLE]

That is, $y_{2}$ is an analytic function of $\tau_{1},\tau_{2},\eta^{\prime}$ and is not identically zero. From Lemma 22 of [16] or p.240 of [10], we conclude the proof of Lemma 5.2.

Now we derive the estimates of $J_{kj}$. We split the integral in $J_{kj}$ as two parts: including $O_{\tau_{i},y_{i},\eta_{i}}$ and including $O_{\tau_{i},y_{i},\eta_{i}}^{c}$ respectively. Since $|O_{\tau_{i},y_{i},\eta_{i}}|<h\varepsilon$, we have

[TABLE]

For the second part, we have

[TABLE]

Combining (5)-(5) with the $L^{2}$ decay of $u$ in (4.45) yields

[TABLE]

Collecting all the estimates (5.103)-(5.104), (5.107)-(5.108), (5.111)-(5.112) and (5.117), we conclude that

[TABLE]

Step 2. Estimate of $\|w(\tau)\|_{\infty}$

By expression (5.102) of $w(\tau)$, for any $(\tau,y,\eta)\in(0,\pi/2h)\times\Omega\times\mathbb{R}^{3}$, $I_{1}$ and $I_{3}$ satisfy the same estimates (5.103) and (5.104) as in Step 1. We only need to estimate $I_{2}$. It follows from direct computation that

[TABLE]

We rewrite $K_{\phi}w(\tau_{1})$ as

[TABLE]

Similar to the treatments in Step 1, we have

[TABLE]

To estimate $\int_{0}^{\tau}e^{\lambda\alpha(\tau_{1})}K_{1}d\tau_{1}$, we need to consider the following three cases of the integration variable $\eta^{\prime}$: $|\eta^{\prime}|<4h$, $|\eta^{\prime}|>2N$ and $4h\leq|\eta^{\prime}|\leq 2N$. In this case,

[TABLE]

By the measure $|\{\eta^{\prime}\in\mathbb{R}^{3}:|\eta^{\prime}|<4h\}|\lesssim h^{3}$, we have

[TABLE]

In addition, it follows from Lemma 3.2 and (3.40) that for large $N$,

[TABLE]

Note that by the definition of $A_{\kappa,h,N}$ in (2.28), one has

[TABLE]

This means that

[TABLE]

which yields $|y_{1}|^{2}\geq 1-\kappa^{2}/h^{2}$ and $|\eta^{\prime}\cdot y_{1}|\leq\kappa$. For fixed $y_{1}\in\Omega$, we arrive at

[TABLE]

Choosing $\kappa<1/N^{3}$, then we have

[TABLE]

Collecting (5.121)-(5.123) and (5), and Choosing $N$ large enough and $\varepsilon>0$ small enough, we eventually get

[TABLE]

Then we complete the proof of Proposition 5.1.                     $\square$

6 Proofs of Theorem 1.1 and 1.2

Set $w=\phi u$. Then we have

[TABLE]

We will prove Theorem 1.1 by the standard Picard iteration.

Proof of Theorem 1.1. The proof will be divided into the following four steps.

Step 1. Existence of solution $w$ to (6.124)

Let $w^{0}\equiv 0$. For $m\geq 0$, we define the following iteration:

[TABLE]

with the initial-boundary data

[TABLE]

As in Section 5, a mild solution $w(\tau,y,\eta)$ of (6.125) with (6.126) can be explicitly constructed. Moreover, by Proposition 5.1 and Lemma 3.4 for $\Gamma_{\phi}$, we have

[TABLE]

Assume that $\|w^{m}(\tau)\|_{\infty}\leq 2Ce^{-\lambda\alpha(\tau)}\|w_{0}\|_{\infty}$, which is true for $m=0$, then it follows from (6.127) that

[TABLE]

If the initial data satisfies that $\|w_{0}\|_{\infty}\leq 1/4C^{2}$, we then conclude that

[TABLE]

Thus, by induction method, we have that for all $m\geq 0$,

[TABLE]

On the other hand, setting $g^{m+1}=w^{m+1}-w^{m}$ yields

[TABLE]

Note that

[TABLE]

Then by the analogous estimate in Proposition 5.1 for (6.130), one has

[TABLE]

By assuming $\|w_{0}\|_{\infty}\leq 1/4C^{2}$ as in the above, we arrive at

[TABLE]

Thus there exists a function $w(\tau)$ such that $w^{m}(\tau)\to w(\tau)$ in $L^{\infty}$ and $w$ is a mild solution to (6.124) with (6.126).

Step 2. Uniqueness of solution $w$ to (6.124)

Assume that there is another solution $\bar{w}$ to (6.124) with the same initial-boundary data as $w$, and also assume that $\sup_{\tau}\{e^{\lambda\alpha(\tau)}\|\bar{w}(\tau)\|_{\infty}\}$ is small. Then

[TABLE]

with the vanishing initial data for $w-\bar{w}$. As in Step 1, we can derive that $\|\{w-\bar{w}\}(\tau)\|_{\infty}\equiv 0$. Therefore, the uniqueness of solution $w$ to (6.124) is shown.

Step 3. Positivity of solution $f$ to (1.1)

Let $f^{0}=f_{0}$, by (1.1), we solve $f^{m+1}$ for $m\geq 0$ as follows

[TABLE]

where

[TABLE]

and

[TABLE]

Set $f^{m}=M+M^{1/2}u^{m}$ and $f^{m+1}=M+M^{1/2}u^{m+1}$. Then it follows from (6.131) that

[TABLE]

One can check that $w^{m}=\phi u^{m}$ converges in $L^{\infty}$ to the solution $w$ of (6.124) as in Step 1.

Assume $f^{m}\geq 0$. Let

[TABLE]

Then by integrating along the backward trajectories $(Y(\tau),H(\tau))$ of (6.131), we have that for $(\tau,y,\eta)\in S_{0}\times\mathbb{R}^{3}$,

[TABLE]

This derives $f^{m+1}\geq 0$ and we further deduce that the solution $f\geq 0$ of (1.1) by the uniqueness in Step 2.

Step 4. Continuity of solution $f$ to (1.1)

The continuity of the solution $f$ is obvious since we have obtained the continuity of the backward trajectory in Lemma 2.5.                               $\square$

Based on Theorem 1.1, we start to prove Theorem 1.2.

Proof of Theorem 1.2. The initial data $f_{0}$ can be reformulated as

[TABLE]

where

[TABLE]

and

[TABLE]

When $\|\phi\tilde{u_{0}}\|_{\infty}<\epsilon$ and $h<\epsilon^{1/2}$, we have

[TABLE]

In this case, all the assumptions in Theorem 1.1 are fulfilled. Therefore, by Theorem 1.1, there exists a unique mild solution $f=M+M^{1/2}u\geq 0$ to problem (1.7) with (1.8). Going back to the original coordinates $(t,x,\xi)$, the perturbation solution $u$ satisfies

[TABLE]

Next we derive the decay property of the mass density $\rho$ of gases. Note that

[TABLE]

Hence, when $\epsilon>0$ is small, by $\rho(t,x)=\rho_{1}(t,x)+\rho_{2}(t,x)$ we have

[TABLE]

where $C_{0}>1$ is a constant. With respect to the continuity of the solution $f$ to problem (1.7) with (1.8), we only need to change $\gamma_{01}$ in Theorem 1.1 to the corresponding set in coordinates $(t,x,\xi)$. Consequently, we complete the proof of Theorem 1.2.                   $\square$

Appendix A The study on elliptic system (4.61) and (4.76)

In this section, we will prove the existence of the solution to the elliptic system (4.61) and (4.76). For a vector function $b(y)$ defined in $\Omega=\{|y|<1\}$, consider the following elliptic system of $\varphi=(\varphi_{1},\varphi_{2},\varphi_{3})$:

[TABLE]

The second boundary condition in (A.132) can be rewritten as

[TABLE]

Note that the rank of the coefficient matrix in (A.139) is $2$.

Define the following Banach space $\mathcal{V}$

[TABLE]

with the norm

[TABLE]

Obviously,

[TABLE]

and the dual space of $\mathcal{V}^{*}$ satisfies

[TABLE]

For $\varphi,\psi\in\mathcal{V}$ and $\varphi\in H^{2}$, we have

[TABLE]

By the first boundary condition of $\psi\in\mathcal{V}$ in (A.132), the boundary term in (A.140) turns to

[TABLE]

This means that the boundary term in (A.132) vanishes if $\varphi$ satisfies the second boundary condition in (A.132). Therefore, we can define the operator $\mathcal{L}:\,\mathcal{V}\to\mathcal{V}^{*}$ as follows

[TABLE]

It is easy to check that $\mathcal{L}$ is a bounded self-adjoint operator. We now show that $\mathcal{L}$ has a closed range. In fact, since

[TABLE]

we have

[TABLE]

Together with the fact that the mapping $\mathrm{id}:\,\mathcal{V}\to L^{2}$ is compact, we know that $\mathcal{L}$ has a closed range by Proposition 6.7 in Appendix A of [29].

Next we prove that $\mathcal{L}$ is a one-to-one and onto mapping. For $\varphi_{0}\in\text{Ker}\mathcal{L}$, due to

[TABLE]

$\varphi_{0}$ is a constant vector in $\Omega$. This, together with the boundary condition $\varphi_{0}\cdot n=0$, yields $\varphi_{0}=0$. So $\mathcal{L}$ is a one-to-one mapping. In addition, since $\mathcal{L}$ is self-adjoint and has a closed range, we obtain the range $\mathcal{R}(\mathcal{L})=(\text{Ker}\mathcal{L})^{\perp}=\mathcal{V}^{*}$. This means that $\mathcal{L}$ is also an onto mapping. Consequently, the bounded bijective linear operator $\mathcal{L}$ has a bounded inverse $\mathcal{L}^{-1}$. So problem (A.132) is uniquely solved by $\varphi=-\mathcal{L}^{-1}b\in\mathcal{V}$ in a weak sense. On the other hand, the existence of a classical solution $\varphi$ to problem (A.132) can be obtained by the standard methods in Chapter 5 of [29]. We omit the proof here.

Acknowledgement. The authors wish to express their gratitude to Professor Yang Tong, the City University of Hong Kong, for his interests in this problem and many very fruitful discussions in the past. In particular, Professor Yang Tong gave many suggestions and comments in this topic.