Effects of Soft Interaction and Non-isothermal Boundary Upon Long-Time Dynamics of Rarefied Gas

Abstract

In the paper, assuming that the motion of rarefied gases in a bounded domain is governed by the angular cutoff Boltzmann equation with diffuse reflection boundary, we study the effects of both soft intermolecular interaction and non-isothermal wall temperature upon the long-time dynamics of solutions to the corresponding initial boundary value problem. Specifically, we are devoted to proving the existence and dynamical stability of stationary solutions whenever the boundary temperature has suitably small variations around a positive constant. For the proof of existence, we introduce a new mild formulation of solutions to the steady boundary-value problem along the speeded backward bicharacteristic, and develop the uniform estimates on approximate solutions in both $L^2$ and $L^\infty$. Such mild formulation proves to be useful for treating the steady problem with soft potentials even over unbounded domains. In showing the dynamical stability, a new point is that we can obtain the sub-exponential time-decay rate in $L^\infty$ without losing any velocity weight, which is actually quite different from the classical results, such as those in Caflisch (Commun Math Phys 74:97–109, 1980) and Strain and Guo (Arch Ration Mech Anal 187:287–339, 2008), for the torus domain and essentially due to the diffuse reflection boundary and the boundedness of the domain.

1 Introduction

1.1 Boltzmann Equation

Let a rarefied gas be contained in a bounded domain $\Omega$ in ${\mathbb {R}}^3$, and let $F=F(t,x,v)$ denote the density distribution function of gas particles with position $x\in \Omega$ and velocity $v\in {\mathbb {R}}^3$ at time $t>0$. We assume that F is governed by the Boltzmann equation

$$\begin{aligned} \partial _tF+v\cdot \nabla _x F=Q(F,F). \end{aligned}$$

(1.1)

The Boltzmann collision term on the right-hand takes the non-symmetric bilinear form of

$$\begin{aligned} Q(F_1,F_2)=\int _{{\mathbb {R}}^3}\int _{{\mathbb {S}}^2} B(|v-u|,\omega )[F_1(u')F_2(v')-F_1(u)F_2(v)]\,\mathrm{d}\omega \mathrm{d}u, \end{aligned}$$

(1.2)

where the velocity pair $(v',u')$ is defined by the velocity pair (v, u) as well as the parameter $\omega \in {\mathbb {S}}^2$ in terms of the relation

$$\begin{aligned} v'=v-[(v-u)\cdot \omega ]\omega ,\quad u'=u+[(v-u)\cdot \omega ]\omega , \end{aligned}$$

according to conservation laws of momentum and energy

$$\begin{aligned} v'+u'=v+u,\quad |v'|^2+|u'|^2=|v|^2+|u|^2, \end{aligned}$$

due to the elastic collision of two particles. To the end, the Boltzmann collision kernel $B(|v-u|,\omega )$, depending only on the relative velocity $|v-u|$ and $\cos \phi =\omega \cdot (v-u)/|v-u|$, is assumed to satisfy

$$\begin{aligned} B(|v-u|,\omega )=|v-u|^{\kappa }b(\phi ), \end{aligned}$$

(1.3)

with

$$\begin{aligned} -3<\kappa <0,\quad 0\leqq b(\phi )\leqq C|\cos \phi | \end{aligned}$$

(1.4)

for a generic constant $C>0$, namely, we consider in this paper the full range of soft potentials under the Grad’s angular cutoff assumption.

1.2 Diffuse Reflection Boundary Condition

We assume that $\Omega =\{\xi (x)<0\}$ is connected and bounded with $\xi (x)$ being a smooth function in ${\mathbb {R}}^3$. At each boundary point with $\xi (x)=0$, we assume that $\nabla \xi (x)\ne 0$. The outward unit normal vector is therefore given by $n(x)=\nabla \xi (x)/|\nabla \xi (x)|$. We define that $\Omega$ is strictly convex if there is $c_{\xi }>0$ such that $\sum _{ij}\partial _{ij}\xi (x)\eta ^{i}\eta ^{j}\geqq c_{\xi }|\eta |^{2}$ for all $x \in {\bar{\Omega }}$ and all $\eta \in {\mathbb {R}}^{3}$.

We denote the phase boundary of the phase space $\Omega \times {\mathbb {R}}^{3}$ as $\gamma =\partial \Omega \times {\mathbb {R}}^{3}$, and split $\gamma$ into three disjoint parts, outgoing boundary $\gamma _{+},$ the incoming boundary $\gamma _{-},$ and the singular boundary $\gamma _{0}$ for grazing velocities : 

$$\begin{aligned} \gamma _{+}&=\{(x,v)\in \partial \Omega \times {\mathbb {R}}^{3}: n(x)\cdot v>0\}, \\ \gamma _{-}&=\{(x,v)\in \partial \Omega \times {\mathbb {R}}^{3}: n(x)\cdot v<0\}, \\ \gamma _{0}&=\{(x,v)\in \partial \Omega \times {\mathbb {R}}^{3}: n(x)\cdot v=0\}. \end{aligned}$$

We supplement the Boltzmann equation (1.1) with the diffuse reflection boundary condition

$$\begin{aligned} F(t,x,v)|_{\gamma _-}=\mu _{\theta }(v) \int _{v'\cdot n(x)>0} F(t,x,v') \{v'\cdot n(x)\}\,\mathrm{d}v', \end{aligned}$$

(1.5)

where $\mu _{\theta }(v)$ is a local Maxwellian with a non-isothermal wall temperature $\theta =\theta (x)>0$:

$$\begin{aligned} \mu _{\theta }(v)=\frac{1}{2\pi \theta ^2(x)} {e^{-\frac{|v|^2}{2\theta (x)}}}. \end{aligned}$$

Throughout this paper, we assume that $\theta (x)$ has a small variation around a fixed postive temperature $\theta _0>0$. Without loss of generality, we assume $\theta _0=1$, and for brevity we denote the global Maxwellian

$$\begin{aligned} \mu =\mu (v)\equiv \mu _{\theta _0}(v)=\frac{1}{2\pi }{e^{-\frac{|v|^2}{2}}}. \end{aligned}$$

(1.6)

1.3 Main Results

Note that

$$\begin{aligned} \int _{v\cdot n(x)>0}\mu _{\theta }(v) \{v\cdot n(x)\}\mathrm{d}v=1 \end{aligned}$$

(1.7)

for any $x\in \partial \Omega$, and hence $\mu _{\theta }(v)$ satisfies the boundary condition (1.5). However, it is straightforward to see that the stationary local Maxwellian $\mu _{\theta }(v)$ does not satisfy the Boltzmann equation (1.1) because of spatial variation unless $\theta (x)$ is constant on $\partial \Omega$. One may expect that the long-time behavior of solutions to (1.1) and (1.5) could be determined by the time-independent steady equation with the same boundary condition. Thus the study of this paper includes two parts. In the first part we investigate the steady problem in order to obtain the existence of stationary solutions, and in the second part we are devoted to showing the dynamical stability of the obtained stationary solutions under small perturbations and further under a class of large perturbations in velocity weighted $L^\infty$ spaces.

In what follows we present the main results of this paper. The first one is to clarify the well-posedness of the boundary-value problem on the Boltzmann equation with diffuse reflection boundary condition

$$\begin{aligned} \left\{ \begin{aligned}&v\cdot \nabla _x F=Q(F,F),\quad \ (x,v)\in \Omega \times {\mathbb {R}}^3,\\&F(x,v)|_{\gamma _-}=\mu _{\theta }(v) \int _{v'\cdot n(x)>0} F(x,v') \{v'\cdot n(x)\}\,\mathrm{d}v'. \end{aligned}\right. \end{aligned}$$

(1.8)

We define a velocity weight function

$$\begin{aligned} w=w(v):= (1+|v|^2)^{\frac{\beta }{2}} e^{\varpi |v|^\zeta }, \end{aligned}$$

(1.9)

where $\beta >0$ and $0<\zeta \leqq 2$ are given constants, and $(\varpi ,\zeta )$ belongs to

$$\begin{aligned} \left\{ \zeta =2,\,0<\varpi<\frac{1}{8}\right\} \cup \{0<\zeta <2,\, \varpi >0\}. \end{aligned}$$

(1.10)

Here and in the sequel, for brevity we have omitted the explicit dependence of w on all parameters $\beta$, $\varpi$ and $\zeta$.

Theorem 1.1

Let $-3<\kappa <0$, $\beta >3+|\kappa |$, and $(\varpi ,\zeta )$ belong to (1.10). For given $M>0$, there exist $\delta _0>0$ and $C>0$ such that if

$$\begin{aligned} \delta :=|\theta -\theta _0|_{L^\infty (\partial \Omega )}\leqq \delta _0, \end{aligned}$$

(1.11)

then there exists a unique nonnegative solution $F_*(x,v)=M\mu (v)+\mu ^{\frac{1}{2}}(v)f_*(x,v)\geqq 0$ to the steady problem (1.8), satisfying the mass conservation

$$\begin{aligned} \int _\Omega \int _{{\mathbb {R}}^3} f_*(x,v)\mu ^{\frac{1}{2}}(v)\,\mathrm{d}v \mathrm{d}x=0, \end{aligned}$$

and the estimate

$$\begin{aligned} \Vert wf_*\Vert _{L^\infty }+|wf_*|_{L^\infty {(\gamma )}}\leqq C\delta . \end{aligned}$$

(1.12)

Moreover, if $\Omega$ is strictly convex and $\theta (x)$ is continuous on $\partial \Omega$, then $F_*$ is continuous on $(x,v)\in {\bar{\Omega }}\times {\mathbb {R}}^3\backslash \gamma _0$.

For simplicity, through the paper we would take $M=1$ in Theorem 1.1 without loss of generality, namely, the stationary solution $F_*(x,v)$ has the same total mass as $\mu (v)$ in $\Omega$. Note that when there is no spatial variation on the boundary temperature, that is, $\delta =0$, the stationary solution is reduced to the global Maxwellian.

The second result is concerned with the dynamical stability of $F_*(x,v)$ under small perturbations in $L^\infty$. We assume that (1.1) is also supplemented with initial data

$$\begin{aligned} F(t,x,v)|_{t=0}=F_0(x,v). \end{aligned}$$

(1.13)

The goal is to show the large-time convergence of solutions of the initial-boundary value problem (1.1), (1.5) and (1.13) to the stationary solution $F_*(x,v)$, whenever they are sufficiently close to each other in some sense at initial time.

Theorem 1.2

Let $-3<\kappa <0$, $\beta >3+|\kappa |$ and $(\varpi ,\zeta )$ belong to (1.10). Assume (1.11) with $\delta _0>0$ chosen to be further small enough. There exist constants ${\varepsilon }_0>0$, $C_0>0$ and $\lambda _0>0$ such that if $F_0(x,v)={F_*}(x,v)+\mu ^{\frac{1}{2}}(v)f_{0}(x,v)\geqq 0$ satisfies the mass conservation

$$\begin{aligned} \int _{\Omega }\int _{{\mathbb {R}}^{3}}f_0(x,v)\mu ^{\frac{1}{2}}(v)\,\mathrm{d}v \mathrm{d}x=0, \end{aligned}$$

(1.14)

and

$$\begin{aligned} \Vert wf_0\Vert _{L^{\infty }}\leqq {\varepsilon }_0, \end{aligned}$$

(1.15)

then the initial-boundary value problem (1.1), (1.5) and (1.13) on the Boltzmann equation admits a unique solution $F(t,x,v)={F_*(x,v)}+\mu ^{\frac{1}{2}}(v)f(t,x,v)\geqq 0$ satisfying

$$\begin{aligned} \int _{\Omega }\int _{{\mathbb {R}}^{3}}f(t,x,v)\mu ^{\frac{1}{2}}(v)\,\mathrm{d}v \mathrm{d}x=0, \end{aligned}$$

(1.16)

and

$$\begin{aligned} \Vert wf(t)\Vert _{L^{\infty }}+|wf(t)|_{L^{\infty }(\gamma )}\leqq C_0e^{-\lambda _0 t^\alpha }\Vert wf_0\Vert _{L^{\infty }}, \end{aligned}$$

(1.17)

for all $t\geqq 0$, where $\alpha \in (0,1)$ is given by

$$\begin{aligned} \alpha :=\frac{\zeta }{\zeta +|\kappa |}. \end{aligned}$$

(1.18)

Moreover, if $\Omega$ is strictly convex, $F_0(x,v)$ is continuous except on $\gamma _0$ and satisfying

$$\begin{aligned} F_0(x,v)|_{\gamma _-}=\mu _{\theta }(v) \int _{v'\cdot n(x)>0} F_0(x,v') \{v'\cdot n(x)\} \,\mathrm{d}v', \end{aligned}$$

(1.19)

and $\theta (x)$ is continuous on $\partial \Omega$, then F(t, x, v) is continuous in $[0,\infty )\times \{{\bar{\Omega }}\times {\mathbb {R}}^{3}\setminus \gamma _0\}$.

We remark that the value of $\alpha$ in (1.18), which is optimal in terms of the exponential velocity weighted function space, can be formally determined as in [11]; we will come back to this point later. By (1.17), we have obtained the global existence and large-time behavior of solutions simultaneously in the velocity-weighted $L^\infty$ space which is the same as that initial data belong to.

One may notice from (1.15) in Theorem 1.2 above that the initial perturbation $f_0(x,v)$ is required to generally have a small amplitude in the velocity-weighted $L^\infty$ space. The goal of the third result is to relax such restriction by allowing $F_0(x,v)$ to have large oscillations around the stationary solution $F_*(x,v)$ with the price that the initial perturbation $f_0(x,v)$ is small enough in some $L^p$ norm for $1<p<\infty$.

Theorem 1.3

Assume that all conditions in Theorem 1.2 are satisfied, and additionally, let

$$\begin{aligned} \max \left\{ \frac{3}{2},\frac{3}{3+\kappa }\right\}<p<\infty ,\quad \beta >\max \{3+|\kappa |, 4\}. \end{aligned}$$

(1.20)

Assume (1.11) with $\delta _0>0$ chosen to be further small enough and initial data $F_0(x,v)=F_*(x,v)+\mu ^{\frac{1}{2}}(v)f_0(x,v)\geqq 0$ satisfies the mass conservation (1.14). There exist constants ${\varepsilon }_1>0$, $C_1>{1}$ and $C_2>1$ such that if $f_0(x,v)$ satisfies

$$\begin{aligned} M_0:=\Vert wf_0\Vert _{L^\infty }\leqq C_1|\log \delta |, \end{aligned}$$

(1.21)

and

$$\begin{aligned} \Vert f_0\Vert _{L^p}\leqq {\varepsilon }_1, \end{aligned}$$

(1.22)

then the initial-boundary value problem (1.1), (1.5) and (1.13) on the Boltzmann equation admits a unique solution $F(t,x,v)={F_*(x,v)}+\mu ^{\frac{1}{2}}(v)f(t,x,v)\geqq 0$ satisfying (1.16) and

$$\begin{aligned} \Vert wf(t)\Vert _{L^{\infty }}+|wf(t)|_{L^{\infty }{(\gamma )}}\leqq C_2 e^{C_2M_0}e^{-\lambda _0 t^{\alpha }}\Vert wf_0\Vert _{L^{\infty }}, \end{aligned}$$

(1.23)

for all $t\geqq 0$, where $\alpha$ is the same as in (1.18) and $\lambda _0$ is the same as in (1.17). Moreover, if $\Omega$ is strictly convex, $F_0(x,v)$ is continuous except on $\gamma _0$ satisfying (1.19), and $\theta (x)$ is continuous over $\partial \Omega$, then F(t, x, v) is continuous in $[0,\infty )\times \{{\bar{\Omega }}\times {\mathbb {R}}^{3}\setminus \gamma _0\}$.

Remark 1.4

We give a few remarks in order on the above theorem.

• (a) Note that $C_1$ is independent of $\delta$. Then, from (1.21), $M_0=\Vert wf_0\Vert _{L^\infty }$ can be arbitrarily large, provided that both $\delta$ and $\Vert f_0\Vert _{L^p}$ are sufficiently small. Particularly, if one takes $\delta =0$ corresponding to the isothermal boundary temperature, there is no restriction on the upper bound of $M_0$. However, it is unclear how to remove the condition (1.21) whenever $\delta >0$.

• (b) From (1.20), p has to be large enough as $\kappa$ gets close to $-3$. The condition (1.22) for the smallness of $f_0$ in $L^p$ is different from that in [18, 19] where $L^1$ norm and $L^2$ norm were used respectively. Note that (1.22) can be also guaranteed by the smallness of $L^1$ or $L^2$ norm of $f_0$ and the velocity-weighted $L^\infty$ bound with the help of the interpolation.

• (c) As already mentioned for Theorem 1.2, by (1.23) we have obtained the global existence and large-time behavior of solutions simultaneously in the velocity-weighted $L^\infty$ space which is the same as that initial data belong to. Estimate (1.23) also implies that the solution may grow with an exponential rate of $M_0$ within a short time.

1.4 Comments and Literature

The focus of this paper is on the effects of both the soft intermolecular interaction and the non-isothermal wall temperature on the large-time behaviour of solutions to the initial-boundary value problem on the Boltzmann equation. In what follows we review some known results related to our results and also give comments on how such effects occur.

(a) Effect of soft potentials. First of all, we discuss the effect of soft potentials on the global well-posedness of the Boltzmann equation in perturbation framework. Compared to the hard potentials, the main difficulty is the lack of the spectral gap of the linearized Boltzmann operator L, for instance, the multiplication operator $\nu (v)\sim \langle v\rangle ^\kappa$ has no strictly positive lower bound over large velocities |v| for $\kappa <0$.

In the spatially periodic domain ${\mathbb {T}}^3$, Caflish [10, 11] first constructed the global-in-time solution for $-1<\kappa <0$ and also studied the large-time behavior of solutions, where the proof is based on the time-decay property of the linearised equation together with the bootstrap argument on the nonlinear equation. One important observation by Caflish is that the function $\exp \{-\langle v\rangle ^\kappa t -c|v|^2\}$, obtained as the solution to the spatially homogeneous equation $\partial _t f+\langle v\rangle ^\kappa f=0$ with initial data $f(0,v)=\exp \{-c|v|^2\}$, decays in time with a rate $\exp \{-\lambda t^\beta \}$ with $\beta =2/(2+|\kappa |)$ by taking the infimum of $\langle v\rangle ^\kappa t +c|v|^2$ in $v\in {\mathbb {R}}^3$. We remark that such sub-exponential time-decay is ensured essentially by adding more exponential velocity weight at initial time; see [11, equations (3.1) and (3.2) of Theorem 3.1 on page 76].

Independently, Ukai–Asano [44] developed the semigroup theory in the case of soft potentials $-1<\kappa <0$, and also obtained the global solution as well as the large-time behavior of solutions for the problem in the whole space ${\mathbb {R}}^3$. As pointed out by [44, Theorem 9.1 and Remark 9.1 on page 96], no solutions have been found in the large in time if initial data and solutions belong to the function space with the same velocity weights. We remark that it is the same situation if one adopts the approach of [44] to treat the case of ${\mathbb {T}}^3$, for instance, one can obtain the arbitrarily large algebraic time-decay rate by postulating more polynomial velocity weights on initial data.

By the pure energy method in high-order Sobolev spaces, Guo [28] constructed the global solutions over ${\mathbb {T}}^3$ for the full range of soft potentials $-3<\kappa <0$, but the large-time behavior of solutions was left. This problem was later completely solved by Strain–Guo in [42, 43] in terms of the same spirit as in [11, 44] by putting additional polynomial or exponential velocity weights on initial data. Such approach was also applied by Strain [40] to study the asymptotic stability of the relativistic Boltzmann equation for the soft potentials in ${\mathbb {T}}^3$.

In the case of ${\mathbb {R}}^3$, we also mention Duan–Yang–Zhao [20] and Strain [41] to treat the optimal large-time behavior of solutions for $-3<\kappa <0$. Particularly, [20] found a velocity weight function containing an exponential factor $\exp \{c|v|^2/(1+t)^q\}$. We remark that this kind of weight could be useful for simultaneously dealing with the global existence and large-time behavior of solutions for the problem in the torus domain or even in the general bounded domain (for instance, [36]), since the typical function $\exp \{-\langle v\rangle ^\gamma t -c|v|^2/(1+t)^q\}$ induces a time-decay rate $\exp \{-\lambda t^{\beta '}\}$ with $\beta '=(2-q|\kappa |)/(2+|\kappa |)$. Therefore, the large-time behavior of solutions is gained by making the velocity weight in the solution space become lower and lower as time goes on. Indeed, this is also in the same spirit as in [41] on the basis of the velocity-time splitting technique.

By comparison with those results mentioned above, Theorems 1.2 or 1.3 implies that the large-time behavior of solutions to the initial-boundary value problem under consideration of this paper is established in the situation where solutions and initial data enjoy the same exponential velocity weight. In other words, to obtain the sub-exponential time-decay for soft potentials, it is no need to put any additional velocity weight on initial data. Roughly speaking, the main reason to realize this point is due to not only the boundedness of the domain but also the diffuse reflection boundary condition, which will be explained in more detail later on. We remark that the results are nontrivial to obtain even if the wall temperature is reduced to a constant implying that the stationary solution $F_*(x,v)$ is a global Maxwellian.

(b) Effect of non-isothermal boundary. The non-isothermal wall temperature provides an inhomogeneous source to force the Boltzmann solution to tend in large time to nontrivial stationary profiles. We review related works in the following two aspects which also involve the case of isothermal boundary. We mainly focus on general bounded domains. There exist also a number of papers in the setting of one-dimensional bounded intervals with different types of boundary conditions, cf. [37, 39]. Among them, we point out that Arkeryd, together with his collaborators, made great contributions in this direction, see for instance, [3, 4, 7] and references therein, where solutions are constructed mainly for large boundary data. The existence and dynamical stability of the stationary solution in a slab with diffuse reflection boundary was considered by Yu [46] in terms of a new probabilistic approach. Hydrodynamic limit to the compressible Navier-Stokes equations for the stationary Boltzmann equation in a slab was studied by Esposito–Lebowitz–Marra [23, 24]. For other related works on the effects of non-isothermal boundary, we also mention [14, 34, 35].

$\bullet$Time-dependent IBVP in general bounded domains. A first investigation of the IBVP was made by Hamdache [32] for a large-data existence theory in the sense of DiPerna–Lions [17]. Extensions of such result have been made in [2, 5, 12, 38] in several directions including the case of general diffuse reflection with variable wall temperature. The large-time behavior of weak solutions was studied in [6, 15, 16]. In the perturbation framework, via the idea of [45], Guo [29] developed a new approach to treat the global existence, uniqueness and continuity of bounded solutions with different types of boundary conditions. Further progress on high-order Sobolev regularity of solutions was recently made in [30]; see also references therein. For other related works on the study of the IBVP on the nonlinear Boltzmann equation, we would mention [9] for the general Maxwell boundary condition, [31] for the global existence of solutions with weakly inhomogeneous data in the case of specular reflection, [33] for the specular boundary condition in convex domains with $C^3$ smoothness, and [36] for a direct extension of [29] from hard potentials to soft potentials.

$\bullet$Steady problem in general bounded domains. There are much less known results on the mathematical analysis of the stationary Boltzmann equation in a general $3\hbox {D}$ bounded domain. First of all, it seems still open to establish a large-data DiPerna-Lions existence theory in the steady case; (see [8]), however, for an $L^1$ existence theorem with inflow data when the collision operator is truncated for small velocities. In Guiraud [26, 27], existence of stationary solutions was proved in convex bounded domains, but the positivity of obtained solutions remained unclear. Via the approach in [29], Esposito et al. [21] constructed the small-amplitude non-Maxwellian stationary solution for diffuse reflection when the space-dependent wall temperature has a small variation around a positive constant for hard potentials, and further obtained the positivity of stationary solutions as a consequence of the dynamical stability for the time-evolutionary Boltzmann equation. Indeed, [21] motivates us to study the steady Boltzmann equation for soft potentials, and we will explain the new mild formulation of solutions as well as new a priori estimates in more detail later on.

The hydrodynamic limit of the stationary Boltzmann equation on bounded domains in the incompressible setting was recently justified in [22]. Notice that such research topic was also discussed in [1] where the authors have particularly shown the non-existence of steady solutions for the Boltzmann equation with smooth divergence-free external forces in bounded domains with specular reflection.

1.5 Strategy of the Proof

In what follows, we briefly explain the key points in our proof of Theorems 1.1, 1.2 and 1.3 respectively.

(a) First, for the proof of Theorem 1.1, the key step is to establish a priori $L^\infty$-estimates on the steady solutions. The major difficulty comes from the degeneracy of collision frequency $\nu (v)\rightarrow 0$ as $|v|\rightarrow \infty$. Our strategy of overcoming this relies on introducing a new mild formulation of the steady Boltzmann equation along a speeded backward bi-characteristics on which the particles with large velocity move much faster than one along the classical characteristics. Precisely, we need to consider the solvability of the linearized steady Boltzmann equation with inhomogeneous source and boundary data

$$\begin{aligned} \left\{ \begin{aligned}&v\cdot \nabla _x f + L f=g,\\&f|_{\gamma _-}=P_\gamma f +r. \end{aligned}\right. \end{aligned}$$

(1.24)

See Lemma 3.8, particularly (3.72) for the $L^\infty$ bound of f in terms of g and r. Basing things on Lemma 3.8, Theorem 1.1 follows by showing the convergence of the iterative approximate solution sequence.

To show Lemma 3.8, we turn to study in Lemma 3.5 the solvability of the following approximate boundary-value problem:

$$\begin{aligned} \left\{ \begin{aligned}&{\mathcal {L}}_\lambda f:=\varepsilon f+v\cdot \nabla _x f + \nu (v) f-\lambda Kf=g,\\&f|_{\gamma _-}=\left( 1-\frac{1}{n}\right) P_\gamma f +r. \end{aligned}\right. \end{aligned}$$

(1.25)

Here, compared to the previous works [21, 22], we input an extra parameter $\lambda \in [0,1]$ in order to carry out a new strategy of the construction of solutions by making the interplay of $L^2$ and $L^\infty$ estimates. Specifically, we divide the proof by several steps as follows:

Step 1. To show the well-posedness of ${\mathcal {L}}_0^{-1}$ for $\lambda =0$. The reason why we start from the case of $\lambda =0$ is that there is no linear collision term Kf. In this case, we are able to directly construct the approximate solutions by solving the inflow problem, so the $L^\infty$ bound of approximate solutions is a consequence of $L^\infty$ bounds of the source term g as well as the corresponding boundary data. The uniform bound of solutions can be obtained in the same way as in the next Step 2.

Step 2. To obtain the a priori estimates of solutions in both $L^2$ and $L^\infty$ uniform in all parameters $\varepsilon$, n and $\lambda$. For the $L^2$ estimate, it is based on the fact that

$$\begin{aligned} \nu -\lambda K=(1-\lambda )\nu +\lambda L \end{aligned}$$

with $0\leqq \lambda \leqq 1$ is still nonnegative. For the velocity-weighted $L^\infty$ estimate on $h=wf$, we formally multiply the equation of (1.25) by $(1+|v|^2)^{|\kappa |/2}$ so as to get

$$\begin{aligned} {\hat{v}}\cdot \nabla _x h +(1+|v|^2)^{|\kappa |/2}[\varepsilon +\nu (v)] h=\lambda (1+|v|^2)^{|\kappa |/2} K_w h +(1+|v|^2)^{|\kappa |/2}w g, \end{aligned}$$

and then write it as the mild form along the backward bi-characteristic $[x-{\hat{v}}(t-s), v]$, where ${\hat{v}}=(1+|v|^2)^{|\kappa |/2} v$ is the transport velocity speeded up by comparison with the original velocity v. The advantage of such new mild formulation is that the corresponding new collision frequency

$$\begin{aligned} {\hat{\nu }}(v)=(1+|v|^2)^{|\kappa |/2}[\varepsilon +\nu (v)] \end{aligned}$$

has a uniform-in-$\varepsilon$ strictly positive lower bound independent of v. This is crucial for obtaining $L^\infty$-estimates for the steady problem. It should be pointed out that by using such new mild formulation, the $L^\infty$-estimates are also valid for the case that the domain is unbounded, so that our method in principle could be used to further study other physically important problems, such as exterior problems and shock wave theory.

Step 3. To prove the well-posedness of ${\mathcal {L}}_\lambda ^{-1}$ for any $\lambda \in [0,\lambda _*]$ with a constant $\lambda _*>0$ small enough. The main idea of showing the existence of solutions is based on the fixed point argument for the solution operator

$$\begin{aligned} {\mathcal {L}}_{\lambda }^{-1} f={\mathcal {L}}_{0}^{-1} (\lambda Kf +g). \end{aligned}$$

Note that the contraction property is essentially the consequence of the fact that we are restricted to $\lambda >0$ small enough. Once the existence of solutions is established, we also have the uniform estimates in $L^2$ and $L^\infty$ obtained in Step 2.

Step 4. To prove the well-posedness of ${\mathcal {L}}_{\lambda _*+\lambda }^{-1}$ for any $\lambda \in [0,\lambda _*]$ small enough. Formally, we have

$$\begin{aligned} {\mathcal {L}}_{\lambda _*+\lambda }^{-1}={\mathcal {L}}_{\lambda _*}^{-1}(\lambda Kf+g). \end{aligned}$$

Therefore, we may make use of the same arguments as in Step 3 and also obtain the corresponding uniform estimates. In the end, by repeating such procedure we can establish the solvability of ${\mathcal {L}}_{\lambda }^{-1}$ in the case of $\lambda =1$, and complete the construction of approximate solutions to the original boundary-value problem (1.24).

(b) Secondly, for the proof of Theorem 1.2, the key step is to study the time-decay structure of linearized IBVP problem around the steady solution provided by Theorem 1.1. In general, it is hard to obtain a satisfactory decay due to the degeneracy of $\nu (v)$ at large velocity. Unfortunately, our new mild formulation above no longer works for the time-dependent problem. Some new thought should be involved in. As mentioned before, so far there are basically two ways to get the decay of the Boltzmann solution for soft potentials. The first one, which was developed by Guo and Strain [43], is to first establish the global existence of the solution with an extra sufficiently strong velocity weight and then obtain the decay of the solution without weight by an interpolation technique. Following their idea, Liu and Yang [36] extend their work into the IBVP problem. The other one, which is developed by Duan et al. [20], is to introduce a velocity weight involving a time dependent factor $\exp \{c |v|^\zeta /(1+t)^q\}$ in order to compensate the degeneracy of collision frequency. One can see that there is an extra restriction in both theories that the initial datum must involve additional velocity weight. One of main contributions in the present work is to remove such a restriction.

More precisely, we are able to obtain simultaneously the global existence and the sub-exponential decay of the solution, without loss of any weight. The key observation used in our arguments is twofold. The first is to split the large velocity part and small velocity part in the following type estimate:

$$\begin{aligned} \int _{\max \{t-t_{\mathbf {b}},0\}}^te^{-\nu (v)(t-s)}\,\mathrm{d}s=\int _{\max \{t-t_{\mathbf {b}},0\}}^te^{-\nu (v)(t-s)}\{{\mathbf {1}}_{\{|v|\leqq d_\Omega \}}+{\mathbf {1}}_{\{|v|>d_\Omega \}}\}\,\mathrm{d}s, \end{aligned}$$

so it holds that

$$\begin{aligned} \int _{\max \{t-t_{\mathbf {b}},0\}}^te^{-\nu (v)(t-s)}\,\mathrm{d}s \leqq \int _{\max \{t-t_{\mathbf {b}},0\}}^te^{-{\bar{\nu }}_0(t-s)}{\mathbf {1}}_{\{|v|\leqq d_\Omega \}}+{\mathbf {1}}_{t-1\leqq s\leqq t}{\mathbf {1}}_{\{|v|>d_\Omega \}}\,\mathrm{d}s, \end{aligned}$$

(1.26)

which is essentially based on the elementary fact that the backward exit time

$$\begin{aligned} t_{\mathbf {b}}(x,v)\leqq \frac{\text {diam}(\Omega )}{|v|}. \end{aligned}$$

(1.27)

One can see that even for the soft potential, (1.26) still involves an exponential decay structure. The second is to notice that due to the diffusive reflection boundary condition, the boundary terms naturally exponentially decay in velocity. So, we can make use of the Caflish’s idea to obtain that

$$\begin{aligned} e^{-\nu (v)(t-t_1)}e^{-\frac{|v|^2}{16}}\leqq Ce^{-\lambda _1(t-t_1)^\alpha }, \end{aligned}$$

where $\lambda _1>0$ is obtained by taking the infimum of $\nu (v)(t-t_1)+|v|^2/16$ with respect to velocity. This reveals the decay structure for boundary terms. However, (1.26) and (1.27) only work when $\Omega$ is bounded and the solution satisfies the diffuse reflection boundary condition. So far we don’t know how to deal with the same problem for the specular reflection or even in the torus.

(c) Thirdly, the strategy of the proof of Theorem 1.3 is to use the linear decay theory provided by the second part to find a large time $T_0=T_0(M_0,{\varepsilon }_0)$, such that

$$\begin{aligned} \Vert wf(T_0)\Vert _{L^\infty }\leqq {\varepsilon }_0. \end{aligned}$$

(1.28)

Then we can extend our solution into $[T_0,\infty )$ by the previous small-amplitude theory. Since the initial data is allowed to have large oscillation around $F_*$, more efforts should be paid for treating the nonlinear term $w(v)\Gamma (f,f)$. The key point is to bound it pointwisely by a product of $L^\infty$-norm and $L^p$-norm with $\max \{\frac{3}{2},\frac{3}{3+\kappa }\}<p<\infty$ and then apply a nonlinear iteration. To show (1.28), we have to require that the following estimates holds:

$$\begin{aligned} C\left( \sup _{0\leqq t\leqq T_0}\Vert wf(t)\Vert _{L^\infty }\right) ^3\cdot |\theta -1|_{L^\infty (\partial \Omega )}\lesssim {\varepsilon }_0. \end{aligned}$$

Hence the restriction (1.21) on the amplitude of initial data is naturally required.

1.6 Plan of the Paper

In Section 2 we will provide some basic estimates on linear and nonlinear collision terms. In Section 3, we study the steady problem and give the proof of Theorem 1.1 for the existence of the stationary solution. In Section 4, we study the large-time asymptotic stability of the obtained stationary solution under small perturbations and give the proof of Theorem 1.2. In Section 5, we further extend the result to the situation where initial perturbation can have large amplitude with an extra restriction but be small in $L^p$ norm, and give the proof of Theorem 1.3. In Appendix, we give the proof of a technical lemma which has been used before, and also give the proof of the local-in-time existence of solutions for completeness.

1.7 Notations

Throughout this paper, C denotes a generic positive constant which may vary from line to line. $C_a,C_b,\ldots$ denote the generic positive constants depending on $a,~b,\ldots$, respectively, which also may vary from line to line. $A\lesssim B$ means that there exists a constant $C>0$ such that $A\leqq C B$. $\Vert \cdot \Vert _{L^2}$ denotes the standard $L^2(\Omega \times {\mathbb {R}}^3_v)$-norm and $\Vert \cdot \Vert _{L^\infty }$ denotes the $L^\infty (\Omega \times {\mathbb {R}}^3_v)$-norm. For the functions depending only in velocity v, we denote $\Vert \cdot \Vert _{L^p_v}$ as the $L^p({\mathbb {R}}^3_v)$-norm and $\langle \cdot ,\cdot \rangle$ as the $L^2(\Omega \times {\mathbb {R}}^3_v)$ inner product or $L^2({\mathbb {R}}^3_v)$ inner product. Moreover, we denote $\Vert \cdot \Vert _{\nu }:=\Vert \sqrt{\nu }\cdot \Vert _{L^2}$. For the phase boundary integration, we define $d\gamma \equiv |n(x)\cdot v| dS(x)dx$, where dS(x) is the surface measure and define $|f|_{L^p}^p=\int _{\gamma }|f(x,v)|^pd\gamma$ and the corresponding space is denoted as $L^p(\partial \Omega \times {\mathbb {R}}^3)=L^p(\partial \Omega \times {\mathbb {R}}^3;d\gamma )$. Furthermore, we denote $|f|_{L^p(\gamma _{\pm })}=|fI_{\gamma _{\pm }}|_{L^p}$ and $|f|_{L^\infty (\gamma _{\pm })}=|fI_{\gamma _{\pm }}|_{L^\infty }$. For simplicity, we denote $|f|_{L^\infty (\gamma )}=|f|_{L^\infty (\gamma _+)}+|f|_{L^\infty (\gamma _-)}$.

2 Preliminaries

2.1 Basic Properties of L

First of all, associated with the global Maxwellian $\mu =\mu (v)$ in (1.6), we introduce the linearized collision operator L around $\mu$ and the nonlinear collision operator $\Gamma (\cdot ,\cdot )$ respectively as

$$\begin{aligned} Lf =-\frac{1}{\sqrt{\mu }}\Big \{Q(\mu ,\sqrt{\mu }f)+Q(\sqrt{\mu }f,\mu )\Big \}, \end{aligned}$$

(2.1)

and

$$\begin{aligned} \Gamma (f,f)= & {} \frac{1}{\sqrt{\mu }}Q^+(\sqrt{\mu }f,\sqrt{\mu }f)-\frac{1}{\sqrt{\mu }}Q^-(\sqrt{\mu }f,\sqrt{\mu }f)\nonumber \\:= & {} \Gamma ^+(f,f)-\Gamma ^-(f,f), \end{aligned}$$

(2.2)

where $Q^{+}$ and $Q^{-}$ correspond to the gain part and loss part in Q in (1.2) respectively. As in [25], under the Grad’s angular cutoff assumption (1.3) and (1.4), L can be decomposed as $L=\nu -K$, where $\nu =\nu (v)$ is the velocity multiplication operator given by

$$\begin{aligned} \nu (v)=\int _{{\mathbb {R}}^3}\int _{{\mathbb {S}}^2}B(|v-u|,\omega )\mu (u)\,\mathrm{d}\omega \mathrm{d}u\sim (1+|v|)^{\kappa }, \end{aligned}$$

(2.3)

and $K=K_1-K_2$ is the integral operator in velocity given by

$$\begin{aligned} (K_1f)(v)&=\int _{{\mathbb {R}}^3}\int _{{\mathbb {S}}^2}B(|v-u|,\omega )\mu ^{\frac{1}{2}}(v)\mu ^{\frac{1}{2}}(u)f(u)\,\mathrm{d}\omega \mathrm{d}u, \end{aligned}$$

(2.4)

$$\begin{aligned} (K_2f)(v)&=\int _{{\mathbb {R}}^3}\int _{{\mathbb {S}}^2}B(|v-u|,\omega )\mu ^{\frac{1}{2}}(u)\left[ \mu ^{\frac{1}{2}}(u')f(v') +\mu ^{\frac{1}{2}}(v')f(u')\right] \,\mathrm{d}\omega \mathrm{d}u. \end{aligned}$$

(2.5)

It is well-known that L is a self-adjoint nonnegative-definite operator in $L^2_v$ space with the kernel

$$\begin{aligned} \text {Ker}\,L=\text {span}\,\{\phi _0,\ldots , \phi _4\}, \end{aligned}$$

where $\phi _{i}=\phi _{i}(v)$$(i=0,1,\ldots ,4)$ are the normal orthogonal basis of the null space $\text {Ker}\,L$ given by

$$\begin{aligned} \phi _0= & {} (2\pi )^{-\frac{1}{4}}\mu ^{\frac{1}{2}}(v),\\ \phi _i= & {} (2\pi )^{-\frac{1}{4}}v_i\mu ^{\frac{1}{2}}(v),\quad i=1,2,3,\\ \phi _4= & {} \frac{(2\pi )^{-\frac{1}{4}}}{\sqrt{6}}(|v|^2-3)\mu ^{\frac{1}{2}}(v). \end{aligned}$$

For each $f=f(v)\in L^2_v$, we denote the macroscopic part Pf as the projection of f onto $\text {Ker}\,L$, that is,

$$\begin{aligned} {P}f=\sum _{i=0}^{4}\langle f,\phi _i\rangle \,\phi _{i}, \end{aligned}$$

(2.6)

and further denote $(I-P)f=f-Pf$ to be the microscopic part of f. It is well-known (see [28] for instance) that there is a constant $c_0>0$ such that

$$\begin{aligned} \langle Lf,f \rangle \geqq c_0\int _{{\mathbb {R}}^3}\nu (v)| {(I-P)}f|^2\,\mathrm{d}v. \end{aligned}$$

(2.7)

Note that L has no spectral gap in case of soft potentials with $-3<\kappa <0$, particularly, the collision frequency $\nu (v)$ tends to zero as $|v|\rightarrow \infty$ due to (2.3).

2.2 Estimates on Collision Operators

Recall $K=K_1-K_2$ with $K_1$ and $K_2$ given in terms of (2.4) and (2.5). As in [28], it holds that

$$\begin{aligned} Kf(v)=\int _{{\mathbb {R}}^3}k(v,u)f(u)\,\mathrm{d}u, \end{aligned}$$

(2.8)

where the integral kernel k(v, u) is real and symmetric.

Lemma 2.1

([18]) The following estimate holds true:

$$\begin{aligned} |k(v,u)|\leqq C|v-u|^\kappa e^{-\frac{|v|^2}{4}}e^{-\frac{|u|^2}{4}} +\frac{C_\kappa }{|v-u|^{\frac{3-\kappa }{2}}}e^{-\frac{|v-u|^2}{8}}e^{-\frac{\left| |v|^2-|u|^2\right| ^2}{8|v-u|^2}}. \end{aligned}$$

(2.9)

Moreover, it holds that

$$\begin{aligned} \int _{{\mathbb {R}}^3}|k(v,u)|(1+|u|)^{-{\beta }}\,\mathrm{d}u\leqq C_{\kappa }(1+|v|)^{-1-{\beta }} \end{aligned}$$

for any ${\beta }\geqq 0$.

In order to deal with difficulties in the case of the soft potentials, as in [43] we introduce a smooth cutoff function $0\leqq \chi _m(s)\leqq 1$ for $s\geqq 0$ with $0<m\leqq 1$ such that

$$\begin{aligned} \chi _m(s)=1\ \text{ for }~0\leqq s\leqq m;\quad \chi _m(s)=0\ \text{ for }~s\geqq 2m. \end{aligned}$$

Then we define

$$\begin{aligned} (K^mf)(v)&=\int _{{\mathbb {R}}^3}\int _{{\mathbb {S}}^2}B(|v-u|,\omega )\chi _m(|v-u|)\mu ^{\frac{1}{2}}(u)\\&\quad \left[ \mu ^{\frac{1}{2}}(u')f(v')+\mu ^{\frac{1}{2}}(v')f(u')\right] \,\mathrm{d}\omega \mathrm{d}u\\&\quad -\int _{{\mathbb {R}}^3}\int _{{\mathbb {S}}^2}B(|v-u|,\omega )\chi _m(|v-u|)\mu ^{\frac{1}{2}}(v)\mu ^{\frac{1}{2}}(u)f(u)\,\mathrm{d}\omega \,\mathrm{d}u\\&:= (K_2^mf)(v)-(K^m_1f)(v), \end{aligned}$$

and

$$\begin{aligned} K^c=K-K^m. \end{aligned}$$

Similarly to (2.8), we denote

$$\begin{aligned} (K^mf)(v)=\int _{{\mathbb {R}}^3} k^m(v,u) f(u)\,\mathrm{d}u,\quad (K^cf)(v)=\int _{{\mathbb {R}}^3} k^c(v,u) f(u)\,\mathrm{d}u. \end{aligned}$$

(2.10)

In the following lemma we recall some basic estimates on $K^m$ and $K^c$, whose proof can be found in [43] and further refined in a recent work [18]:

Lemma 2.2

Let $-3<\kappa <0$. Then, for any $0<m\leqq 1$, it holds that

$$\begin{aligned} |(K^m f)(v)|\leqq Cm^{3+\kappa }e^{-\frac{|v|^2}{{6}}}\Vert f\Vert _{L^\infty _v}, \end{aligned}$$

(2.11)

where $C>0$ is independent of m. The kernels $k^m(v,u)$ and $k^c(v,u)$ in (2.10) satisfy

$$\begin{aligned} |k^m(v,u)|\leqq C_\kappa \Big \{|v-u|^\kappa +|v-u|^{-\frac{3-\kappa }{2}}\Big \}e^{-\frac{|v|^2+|u|^2}{16}}, \end{aligned}$$

and

$$\begin{aligned} |k^c(v,u)|&\leqq \frac{C_\kappa m^{a(\kappa -1)}}{|v-u|^{1+\frac{(1-a)}{2}(1-\kappa )}}\frac{1}{(1+|v|+|u|)^{a(1-\kappa )}}e^{-\frac{|v-u|^2}{10}}e^{-\frac{\left| |v|^2-|u|^2\right| ^2}{16|v-u|^2}}\nonumber \\&\quad +C|v-u|^\kappa [1-\chi _m(|v-u|)] e^{-\frac{|v|^2}{4}}e^{-\frac{|u|^2}{4}}, \end{aligned}$$

(2.12)

where $0\leqq a\leqq 1$ is an arbitrary constant, and $C_\kappa$ is a constant depending only on $\kappa$. It is worth to point out that $C_\kappa$ is independent of a and m. Moreover, by denoting

$$\begin{aligned} {k_w^c} (v,u)=k^{c}(v,u)\frac{w(v)}{w(u)}, \end{aligned}$$

it holds that

$$\begin{aligned} \int _{{\mathbb {R}}^3}|{k_w^c}(v,u)|{e^{\frac{|v-u|^2}{32}}}\,\mathrm{d}u \leqq C m^{\kappa -1}(1+|v|)^{\kappa -2}, \end{aligned}$$

(2.13)

and

$$\begin{aligned} \int _{{\mathbb {R}}^3}|{k_w^c}(v,u)|{e^{\frac{|v-u|^2}{32}}}\,\mathrm{d}u \leqq C (1+|v|)^{-1}, \end{aligned}$$

(2.14)

where $C>0$ is independent of m.

Furthermore, we need the following two lemmas for the later use:

Lemma 2.3

([43]) Assume (1.10), then for any $\eta >0$, it holds that

$$\begin{aligned} \left| \left\langle e^{\frac{\varpi }{2}|\cdot |^{ \zeta }}Kf, f\right\rangle \right| \leqq \left\| e^{\frac{\varpi }{4}|\cdot |^{ \zeta }}f\right\| _{\nu }\left( \eta \left\| e^{\frac{\varpi }{4}|\cdot |^{ \zeta }}f\right\| _{\nu }+C_{\eta }\left\| {\mathbf {1}}_{\{|\cdot |\leqq C_\eta \}}f\right\| _{L^{2}}\right) . \end{aligned}$$

(2.15)

Lemma 2.4

([36]) It holds that

$$\begin{aligned} \Vert \nu ^{-1}w\Gamma (f_1,f_2)\Vert _{L^\infty _v}\leqq C\Vert wf_1\Vert _{L^\infty _v}\cdot \Vert wf_2\Vert _{L^\infty _v}. \end{aligned}$$

(2.16)

In the end we conclude this subsection with the following $L^p$$(p>1)$ estimate on the nonlinear collision term, which will be used in Section 5.

Lemma 2.5

Let $1<p<\infty$ and $w_{\beta }:=(1+|v|^2)^{\frac{\beta }{2}}$ with $\beta p>3$. Then it holds that

$$\begin{aligned} \Vert \nu ^{-1/p'}\Gamma (f,g)\Vert _{L^p_{v}}\leqq C\min \left\{ \Vert w_\beta f\Vert _{L^\infty _v}\cdot \Vert \nu ^{1/p}g\Vert _{L^p_v},\,\Vert w_\beta g\Vert _{L^\infty _v}\cdot \Vert \nu ^{1/p}f\Vert _{L^p_v}\right\} , \end{aligned}$$

(2.17)

where $p'$ is the conjugate of p satisfying $1/p+1/p'=1.$

Proof

We first consider the loss part. By Hölder inequality, we have

$$\begin{aligned} \Vert \nu ^{-1/p'}\Gamma ^-(f,g)\Vert _{L^p_v}^p&\leqq \int _{{\mathbb {R}}^3}\nu (v)^{-(p-1)}|f(v)|^p\,\mathrm{d}v\nonumber \\&\qquad \times \bigg (\int _{{\mathbb {R}}^3}\int _{{\mathbb {S}}^2}B(v-u,\omega )\sqrt{\mu (u)}|g(u)|^p\,\mathrm{d}\omega \,\mathrm{d}u\bigg )\nonumber \\&\qquad \times \bigg (\int _{{\mathbb {R}}^3}\int _{{\mathbb {S}}^2}B(v-u,\omega )\sqrt{\mu (u)}\,\mathrm{d}\omega \mathrm{d}u\bigg )^{p-1}\nonumber \\&\leqq C\int _{{\mathbb {R}}^3}\int _{{\mathbb {R}}^3}\int _{{\mathbb {S}}^2}B(v-u,\omega )\sqrt{\mu (u)}|g(u)|^p|f(v)|^p\,\mathrm{d}\omega \mathrm{d}u \mathrm{d}v. \end{aligned}$$

(2.18)

Without loss of generality, we assume that

$$\begin{aligned} \Vert w_\beta f\Vert _{L^\infty _v}\cdot \Vert \nu ^{1/p}g\Vert _{L^p_v}\geqq \Vert w_\beta g\Vert _{L^\infty _v}\cdot \Vert \nu ^{1/p}f\Vert _{L^p_v}. \end{aligned}$$

Then from (2.18), it holds that

$$\begin{aligned} \Vert \nu ^{-1/p'}\Gamma ^-(f,g)\Vert _{L^p_v}^p&\leqq C\Vert w_\beta g\Vert ^p_{L^\infty _v}\cdot \int _{{\mathbb {R}}^3}|f(v)|^p\,\mathrm{d}v\nonumber \\&\quad \int _{{\mathbb {R}}^3}\int _{{\mathbb {S}}^2}B(v-u,\omega )(1+|u|^2)^{-\frac{\beta p}{2}}\,\mathrm{d}\omega \mathrm{d}u\nonumber \\&\leqq C\Vert w_\beta g\Vert _{L^\infty _v}^p\cdot \Vert \nu ^{1/p}f\Vert ^p_{L^p_v}\nonumber \\&\leqq C\min \left\{ \Vert w_\beta f\Vert _{L^\infty _v}\cdot \Vert \nu ^{1/p}g\Vert _{L^p_v},\Vert w_\beta g\Vert _{L^\infty _v}\cdot \Vert \nu ^{1/p}f\Vert _{L^p_v}\right\} ^p. \end{aligned}$$

(2.19)

Similarly, for the gain term $\Gamma ^+(f,g)$, by Hölder inequality, we have

$$\begin{aligned} \Vert \nu ^{-1/p'}\Gamma ^+(f,g)\Vert _{L^p_v}^p&\leqq \int _{{\mathbb {R}}^3}\nu (v)^{-(p-1)}\,\mathrm{d}v\\&\qquad \times \bigg (\int _{{\mathbb {R}}^3}\int _{{\mathbb {S}}^2}B(v-u,\omega )\sqrt{\mu (u)}|f(v')|^p|g(u')|^p\,\mathrm{d}\omega \mathrm{d}u\bigg )\\&\qquad \times \bigg (\int _{{\mathbb {R}}^3}\int _{{\mathbb {S}}^2}B(v-u,\omega )\sqrt{\mu (u)}\,\mathrm{d}\omega \mathrm{d}u\bigg )^{p-1}\\&\leqq C\int _{{\mathbb {R}}^3}\int _{{\mathbb {R}}^3}\int _{{\mathbb {S}}^2}B(v-u,\omega )\sqrt{\mu (u)}|g(u')|^p|f(v')|^p\,\mathrm{d}\omega \mathrm{d}u \mathrm{d}v. \end{aligned}$$

Making change of variable $(v,u)\rightarrow (v',u')$, it follows that

$$\begin{aligned} \Vert \nu ^{-1/p'}\Gamma ^+(f,g)\Vert _{L^p_v}^p\leqq C\int _{{\mathbb {R}}^3}\int _{{\mathbb {R}}^3}\int _{{\mathbb {S}}^2}B(v-u,\omega )|g(u)|^p|f(v)|^p\,\mathrm{d}\omega \mathrm{d}u \mathrm{d}v. \end{aligned}$$

By the same argument as in (2.19), one has

$$\begin{aligned} \Vert \nu ^{-1/p'}\Gamma ^+(f,g)\Vert _{L^p_v}^p\leqq C\min \left\{ \Vert w_\beta f\Vert _{L^\infty _v}\cdot \Vert \nu ^{1/p}g\Vert _{L^p_v},\Vert w_\beta g\Vert _{L^\infty _v}\cdot \Vert \nu ^{1/p}f\Vert _{L^p_v}\right\} ^p. \end{aligned}$$

From this and (2.19), we prove (2.17). Therefore, the proof of Lemma 2.5 is complete. $\quad \square$

2.3 Pointwise Weighted Estimates on Nonlinear Term $\Gamma (f,f)$

Recall (1.9), (1.10), and (2.2). We shall give the estimates on the upper bound of $|w(v)\Gamma ^{\pm }(f,f)(v)|$ for pointwise v in terms of the product of the weighted $L^\infty$ norm and the $L^p$ norm for a suitable $p>1$ which is finite. These estimates will play an essential role in the proof of Theorem 1.3 treating the initial data of large oscillations but with small $L^p$ perturbations.

Lemma 2.6

Let $-3<\kappa <0$, $\max \{\frac{3}{2}, \frac{3}{3+\kappa }\}<p<\infty$, $(\varpi ,\zeta )$ belong (1.10), and $\beta >4$. Then, for each $v\in {\mathbb {R}}^3$, it holds that

$$\begin{aligned}&|w\Gamma ^{-}(f,f)(v)|\leqq C\nu (v)\Vert wf\Vert _{L^\infty _v}\cdot \left( \int _{{\mathbb {R}}^{3}}|f(u)|^{p}\,\mathrm{d}u\right) ^{\frac{1}{p}}, \end{aligned}$$

(2.20)

$$\begin{aligned}&|w\Gamma ^{+}(f,f)(v)|\leqq C\nu (v)\Vert wf\Vert _{L^\infty _v}\cdot \left( \int _{{\mathbb {R}}^{3}}|wf(u)|^{p}(1+|u|)^{-p(\beta -4)-4}\,\mathrm{d}u\right) ^{\frac{1}{p}}, \end{aligned}$$

(2.21)

where the generic constant $C>0$ is independent of v.

Proof

First, we consider (2.20) regarding the estimate on the loss term. Indeed, it follows from Hölder inequality that

$$\begin{aligned} \begin{aligned} |w\Gamma ^{-}(f,f)(v)|&\leqq |wf(v)|\int _{{\mathbb {R}}^{3}}\int _{{\mathbb {S}}^{2}}B(|v-u|,\omega )\mu ^{\frac{1}{2}}(u)|f(u)|\,\mathrm{d}u \mathrm{d}\omega \\&\leqq C\Vert wf\Vert _{L^\infty _v}\cdot \left( \int _{{\mathbb {R}}^{3}}|f(u)|^{p}\,\mathrm{d}u\right) ^{\frac{1}{p}}\cdot \\&\quad \times \left( \int _{{\mathbb {R}}^{3}}|v-u|^{\frac{p\kappa }{p-1}}\mu ^{\frac{p}{2(p-1)}}(u)\,\mathrm{d}u\right) ^{\frac{p}{p-1}}\\&\leqq C \nu (v)\Vert wf\Vert _{L^\infty _v}\cdot {\left( \int _{{\mathbb {R}}^{3}}|f(u)|^{p}\,\mathrm{d}u\right) ^{\frac{1}{p}}}, \end{aligned} \end{aligned}$$

where we have used the fact that $\frac{p\kappa }{p-1}>-3$ due to $p>\frac{3}{3+\kappa }$. Then, (2.20) is proved.

To prove (2.21) for the gain term, we denote

$$\begin{aligned} {\hat{w}}(v):=e^{\varpi |v|^{\zeta }}, \end{aligned}$$

which is the pure exponential factor of w(v), and also set $g(v)={\hat{w}}(v)|f(v)|$. Since $|u|^{2}+|v|^{2}=|u'|^{2}+|v'|^{2}$, we have either $|u'|^{2}\geqq \frac{1}{2}\left( |u|^{2}+|v|^{2}\right)$ or $|v'|^{2}\geqq \frac{1}{2}\left( |u|^{2}+|v|^{2}\right)$. Without loss of generality, we may assume that $|v'|^{2}\geqq \frac{1}{2}\left( |u|^{2}+|v|^{2}\right)$, and then we have

$$\begin{aligned}&|w\Gamma ^{+}(f,f)(v)|\leqq C {\hat{w}}(v)\Vert wf\Vert _{L^\infty _v}\nonumber \\&\quad \cdot \int _{{\mathbb {R}}^{3}}\int _{{\mathbb {S}}^{2}}B(|v-u|,\omega )\mu ^{\frac{1}{2}}(u)|g(v')|{\hat{w}}(v')^{-1}{\hat{w}}(u')^{-1}\,\mathrm{d}u \mathrm{d}\omega . \end{aligned}$$

(2.22)

Note that in virtue of $0<\zeta \leqq 2$, it holds that

$$\begin{aligned} \begin{aligned} {\hat{w}}(v)&=e^{\varpi |v|^{\zeta }}\leqq e^{\varpi (|u|^{2}+|v|^{2})^{\frac{\zeta }{2}}}=e^{\varpi (|u'|^{2}+|v'|^{2})^{\frac{\zeta }{2}}}\leqq e^{\varpi (|u'|^{\zeta }+|v'|^{\zeta })}={\hat{w}}(v'){\hat{w}}(u'). \end{aligned} \end{aligned}$$

Using this, (2.22) gives that

$$\begin{aligned} |w\Gamma ^{+}(f,f)(v)|\leqq C \Vert wf\Vert _{L^\infty _v}\cdot \int _{{\mathbb {R}}^{3}}\int _{{\mathbb {S}}^{2}}B(|v-u|,\omega )\mu ^{\frac{1}{2}}(u)|g(v')|\,\mathrm{d}u \mathrm{d}\omega .\nonumber \\ \end{aligned}$$

(2.23)

To further estimate the integral term on the right-hand side of (2.23), we denote $z=u-v$, $z_{\parallel }=\{(u-v)\cdot \omega \}\omega$, and $z_{\perp }=z-z_{\parallel }$, then the collision kernel can be estimated as

$$\begin{aligned} B(|v-u|,\omega )\leqq {C|z_{\parallel }|\left( |z_{\parallel }|^{2}+|z_{\perp }|^{2}\right) ^{\frac{\kappa -1}{2}}.} \end{aligned}$$

(2.24)

Plugging (2.24) back to (2.23) and making change of variable $u\rightarrow z$, $|w\Gamma ^{+}(f,f)(v)|$ can be further bounded by

$$\begin{aligned} C\Vert wf\Vert _{L^\infty _v}\cdot \int _{{\mathbb {R}}^{3}}\int _{{\mathbb {S}}^{2}}|z_{\parallel }|\cdot |z|^{\kappa -1}g(v+z_{\parallel })\mu ^{\frac{1}{2}}(v+z)\,\mathrm{d}z \mathrm{d}\omega . \end{aligned}$$

By writing $\mathrm{d}z\mathrm{d}\omega =\frac{2}{|z_{\parallel }|^2} \mathrm{d}z_{\parallel }\mathrm{d}z_{\perp }$, we derive that the above term is bounded by

$$\begin{aligned} C\Vert wf\Vert _{L^\infty _v}\cdot \int _{{\mathbb {R}}^{3}}|z_{\parallel }|^{-1}g(v+z_{\parallel })\,\mathrm{d}z_{\parallel } \int _{\Pi _{\perp }}(|z_{\parallel }|^{2}+|z_{\perp }|^{2})^{\frac{\kappa -1}{2}}\mu ^{\frac{1}{2}}(v_{\perp }+z_{\perp })\,\mathrm{d}{z_{\perp }},\nonumber \\ \end{aligned}$$

(2.25)

where $\Pi _{\perp }=\left\{ z_{\perp }\in {\mathbb {R}}^{3}: z_{\parallel }\cdot z_{\perp }=0 \right\}$ and $v_{\perp }=\frac{ v \cdot z_{\perp }}{|v|\cdot |z_{\perp }|}z_{\perp }$ is the projection of v to $\Pi _{\perp }$. To estimate (2.25), we divide it by two cases.

Case 1:$-1<\kappa <0$. In this case, thanks to $-2<\kappa -1<-1$, we have

$$\begin{aligned} \int _{\Pi _{\perp }}\big (|z_{\parallel }|^{2}+|z_{\perp }|^{2}\big )^{\frac{\kappa -1}{2}}\mu ^{\frac{1}{2}}(v_{\perp }+z_{\perp })\,\mathrm{d}{z_{\perp }}\leqq \int _{\Pi _{\perp }}|z_{\perp }|^{\kappa -1}\mu ^{\frac{1}{2}}(v_{\perp }+z_{\perp })\,\mathrm{d}{z_{\perp }}\leqq C, \end{aligned}$$

for a finite constant $C>0$. Then it holds that for $p>\frac{3}{2}$,

$$\begin{aligned} |w\Gamma ^{+}(f,f)(v)|&\leqq C\int _{{\mathbb {R}}^3}|z_{\parallel }|^{-1}g(v+z_{\parallel })\,\mathrm{d}z_{\parallel }=C\int _{{\mathbb {R}}^3}|u-v|^{-1}g(u)\,\mathrm{d}u\nonumber \\&\leqq C\bigg (\int _{{\mathbb {R}}^3}\frac{(1+|u|)^{-4}}{|u-v|^{\frac{p}{p-1}}}\bigg )^{\frac{p-1}{p}}\bigg (\int _{{\mathbb {R}}^3}|g(u)|^p(1+|u|)^{4p-4}\,\mathrm{d}u\bigg )^{\frac{1}{p}}\nonumber \\&\leqq C(1+|v|)^{-1}\bigg (\int _{{\mathbb {R}}^3}|g(u)|^p(1+|u|)^{4p-4}\,\mathrm{d}u\bigg )^{\frac{1}{p}}. \end{aligned}$$

(2.26)

Case 2 :$-3<\kappa \leqq -1$. To avoid the higher singularity on the right hand side of (2.25), we choose $0<\varepsilon =\varepsilon (p,\kappa )<\min \{3+\frac{p\kappa }{p-1}, \frac{2p}{p-1}\}$ and bound (2.25) in terms of

$$\begin{aligned} |w\Gamma ^{+}(f,f)|&\leqq C\Vert wf\Vert _{L^\infty _v}\cdot \int _{{\mathbb {R}}^{3}}|z_{\parallel }|^{\kappa -\frac{(p-1)\varepsilon }{p}}g(v+z_{\parallel })\,\mathrm{d}z_{\parallel } \nonumber \\&\quad \int _{\Pi _{\perp }}|z_{\perp }|^{-2+\frac{(p-1)\varepsilon }{p}}\mu ^{\frac{1}{2}}(v_{\perp }+z_{\perp })\,\mathrm{d}{z_{\perp }}\nonumber \\&\leqq C\Vert wf\Vert _{L^\infty _v}\cdot \int _{{\mathbb {R}}^{3}}|z_{\parallel }|^{\kappa -\frac{(p-1)\varepsilon }{p}}g(v+z_{\parallel })\,\mathrm{d}z_{\parallel }, \end{aligned}$$

(2.27)

where in the second line we have used the fact that

$$\begin{aligned} \int _{\Pi _{\perp }}|z_{\perp }|^{-2+\frac{(p-1)\varepsilon }{p}}\mu ^{\frac{1}{2}}(v_{\perp }+z_{\perp })\,\mathrm{d}{z_{\perp }}<\infty . \end{aligned}$$

Making change of variable $z_{\parallel }+v\rightarrow u$ and using Hölder’s inequality, it holds that

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}|z_{\parallel }|^{\kappa -\frac{(p-1)\varepsilon }{p}}g(v+z_{\parallel })dz_{\parallel }= & {} \int _{{\mathbb {R}}^{3}}|u-v|^{\kappa -\frac{(p-1)\varepsilon }{p}}g(u)\,\mathrm{d}u\nonumber \\\leqq & {} \left[ \int _{{\mathbb {R}}^{3}}\frac{(1+|u|)^{-4}}{|u-v|^{\varepsilon +\frac{p|\kappa |}{p-1}}}\,\mathrm{d}u\right] ^{\frac{p-1}{p}}\nonumber \\&\times \left[ \int _{{\mathbb {R}}^{3}}|g(u)|^{p}(1+|u|)^{4p-4}\,\mathrm{d}u\right] ^{\frac{1}{p}}.\qquad \end{aligned}$$

(2.28)

Since $0<\varepsilon +\frac{p|\kappa |}{p-1}<3$, the right hand side of (2.28) can be further bounded by

$$\begin{aligned} C (1+|v|)^{\kappa -\frac{p-1}{p}\varepsilon }\left[ \int _{{\mathbb {R}}^{3}}|g(u)|^{p}(1+|u|)^{4p-4}\,\mathrm{d}u\right] ^{\frac{1}{p}} \end{aligned}$$

which combining with (2.27) and (2.26), immediately yields (2.21). Hence the proof of Lemma 2.6 is complete. $\quad \square$

3 Steady Problem

To construct the solution to the steady Boltzmann equation (1.8) and (1.5), we first consider the approximate linearized steady problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \varepsilon f+ v\cdot \nabla _xf+Lf=g,\\ f(x,v)|_{\gamma _{-}}=P_{\gamma }f+r, \end{array}\right. } \end{aligned}$$

(3.1)

where $P_{\gamma }f$ is defined as

$$\begin{aligned} P_{\gamma }f(x,v)=\mu ^{\frac{1}{2}}(v)\int _{v'\cdot n(x)>0} f(x,v') \mu ^{\frac{1}{2}}(v') \{v'\cdot n(x)\}\,\mathrm{d}v'. \end{aligned}$$

(3.2)

As in [21], the penalization term $\varepsilon f$ is used to guarantee the conservation of mass. Recall the weight function w(v) defined by (1.9) with (1.10). We also define

$$\begin{aligned} h(x,v):=w(v) f(x,v), \end{aligned}$$

then (3.1) can be rewritten as

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \varepsilon h+v\cdot \nabla _x h+\nu (v) h=K_{w} h+wg,\\ \displaystyle h(x,v)|_{\gamma _-}=\frac{1}{{\tilde{w}}(v)} \int _{v'\cdot n(x)>0} h(x,v') {\tilde{w}}(v') \,\mathrm{d}\sigma '+wr(x,v), \end{array}\right. } \end{aligned}$$

(3.3)

where

$$\begin{aligned} {\tilde{w}}(v)\equiv \frac{1}{w(v)\mu ^{\frac{1}{2}}(v)}, \quad K_wh=wK\left( \frac{h}{w}\right) . \end{aligned}$$

3.1 A Priori $L^\infty$ Estimate

For the approximate steady Boltzmann equation (3.3), the most difficult part is to obtain the $L^\infty$-bound due to the degeneration of frequency $\nu (v)$ as $|v|\rightarrow \infty$. To overcome this difficulty, the main idea is to introduce a new characteristic line.

Definition 3.1

Given (t, x, v), let $[{\hat{X}}(s),V(s)]$ be the backward bi-characteristics for the steady Boltzmann equation (1.8), which is determined by

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{d{\hat{X}}(s)}{ds}=(1+|V(s)|^2)^{\frac{|{\kappa }|}{2}} V(s):={\hat{V}}(s),\\ \displaystyle \frac{dV(s)}{ds}=0,\\ {[}{\hat{X}}(t),V(t)]=[x,v]. \end{array}\right. \end{aligned}$$

(3.4)

The solution is then given by

$$\begin{aligned}{}[{\hat{X}}(s;t,x,v),V(s;t,x,v)]=[x-{\hat{v}}(t-s),v],\quad {\hat{v}}:=(1+|v|^2)^{\frac{|\kappa |}{2}}v, \end{aligned}$$

(3.5)

which is called the speeded backward bi-characteristic for the problem (1.8).

We note that compared to the usual characteristic line as used in the time-evolutionary case, the particle along (3.4) or (3.5) with given (x, v) travels with the velocity ${\hat{v}}$ which has the much faster speed than |v| itself for |v| for large velocity. This is the key idea to overcome the difficulty of soft potentials in treating the steady problem on the Boltzmann equation.

In terms of the speeded backward bi-characteristic, we need to redefine the corresponding backward exit time etc.. Indeed, for each (x, v) with $x\in {\bar{\Omega }}$ and $v\ne 0,$ we define the backward exit time${\hat{t}}_{{\mathbf {b}}}(x,v)\geqq 0$ to be the last moment at which the back-time straight line $[{\hat{X}}({-\tau } ;0,x,v),V({-\tau } ;0,x,v)]$ remains in ${\bar{\Omega }}$:

$$\begin{aligned} {\hat{t}}_{{\mathbf {b}}}(x,v)={\sup \, \{s \geqq 0:x-{\hat{v}}\tau \in {\bar{\Omega }}\text { for }0\leqq \tau \leqq s\}.} \end{aligned}$$

We therefore have $x-{\hat{t}}_{{\mathbf {b}}}{\hat{v}}\in \partial \Omega$ and $\xi (x-{\hat{t}}_{{\mathbf {b}}}{\hat{v}})=0.$ We also define

$$\begin{aligned} {\hat{x}}_{{\mathbf {b}}}(x,v) =x-{\hat{t}}_{{\mathbf {b}}}{\hat{v}}\in \partial \Omega . \end{aligned}$$

Note that the fact that $v\cdot n({\hat{x}}_{{\mathbf {b}}})=v\cdot n({\hat{x}}_{{\mathbf {b}}}(x,v)) \leqq 0$ always holds true.

Let $x\in {\bar{\Omega }}$, $(x,v)\notin \gamma _{0}\cup \gamma _{-}$ and $(t_{0},x_{0},v_{0})=(t,x,v)$. For $v_{k+1}\in \hat{{\mathcal {V}}}_{k+1}:=\{v_{k+1}\cdot n({\hat{x}}_{k+1})>0\}$, the back-time cycle is defined as

$$\begin{aligned} \left\{ \begin{aligned} {\hat{X}}_{cl}(s;t,x,v)&=\sum _{k}{\mathbf {1}}_{[{\hat{t}}_{k+1},{\hat{t}}_{k})}(s)\{{\hat{x}}_{k}-{\hat{v}}_k({\hat{t}}_{k}-s)\},\\ V_{cl}(s;t,x,v)&=\sum _{k}{\mathbf {1}}_{[{\hat{t}}_{k+1},{\hat{t}}_{k})}(s)v_{k}, \end{aligned}\right. \end{aligned}$$

(3.6)

with

$$\begin{aligned} ({\hat{t}}_{k+1},{\hat{x}}_{k+1},v_{k+1}) =({\hat{t}}_{k}-{\hat{t}}_{{\mathbf {b}}}({\hat{x}}_{k},v_{k}), {\hat{x}}_{{\mathbf {b}}}({\hat{x}}_{k},v_{k}),v_{k+1}). \end{aligned}$$

We also define the iterated integral

$$\begin{aligned} \int _{\Pi _{j=1}^{k-1}{\hat{{\mathcal {V}}}}_j}\Pi _{j=1}^{k-1}\,\mathrm{d}{\hat{\sigma }}_j :=\int _{{\hat{{\mathcal {V}}}}_1}\cdots \left\{ \int _{{\hat{{\mathcal {V}}}}_{k-1}}\,\mathrm{d}{\hat{\sigma }}_{k-1}\right\} \cdots \,\mathrm{d}{\hat{\sigma }}_1, \end{aligned}$$

where

$$\begin{aligned} \mathrm{d}{\hat{\sigma }}_j:=\mu (v_j)\{n({\hat{x}}_j)\cdot v_j\},\quad j=1,\ldots , k-1 \end{aligned}$$

are probability measures.

Lemma 3.2

Let $(\eta ,\zeta )$ belong to

$$\begin{aligned} \{\zeta =2, 0\leqq \eta<1/2\}\cup \{0\leqq \zeta <2,\eta \geqq 0\}. \end{aligned}$$

For $T_0>0$ sufficiently large, there exist constants ${\hat{C}}_1$ and ${\hat{C}}_2$ independent of $T_0$ such that for $k={\hat{C}}_1T_0^{\frac{5}{4}}$ and $(t,x,v)\in [0,T_0]\times {\bar{\Omega }}\times {\mathbb {R}}^3$, it holds that

$$\begin{aligned} \int _{\Pi _{j=1}^{k-1}\hat{{\mathcal {V}}}_{j}} {\mathbf {1}}_{\{{\hat{t}}_k>0\}}~ \Pi _{j=1}^{k-1} {e^{\eta |v_j|^\zeta }} \,\mathrm{d}{\hat{\sigma }} _{j}\leqq \left( \frac{1}{2}\right) ^{{\hat{C}}_2T_0^{\frac{5}{4}}}. \end{aligned}$$

(3.7)

Proof

We take $\varepsilon >0$ small enough, and define the non-grazing sets

$$\begin{aligned} \hat{{\mathcal {V}}}_{j}^\varepsilon =\left\{ v_j\in \hat{{\mathcal {V}}}_{j}~:~ v_j\cdot n({\hat{x}}_j)\geqq \varepsilon ~\text{ and }~|v_j|\leqq \frac{1}{\varepsilon }\right\} ,\quad j\geqq 1. \end{aligned}$$

Then a direct calculation shows that

$$\begin{aligned} \int _{\hat{{\mathcal {V}}}_{j}\backslash \hat{{\mathcal {V}}}_{j}^{\varepsilon }} {e^{\eta |v_j|^\zeta }}\,\mathrm{d}{\hat{\sigma }}_j\leqq C\varepsilon , \end{aligned}$$

where the constant $C>0$ is independent of j. By similar arguments as in [29, Lemma 2], one can prove

$$\begin{aligned} {\hat{t}}_j-{\hat{t}}_{j+1}\geqq \frac{|v_j\cdot n({\hat{x}}_j)|}{C_\Omega |v_j|^2(1+|v_j|^2)^{\frac{|\kappa |}{2}}} \end{aligned}$$

with a positive constant $C_\Omega$ depending only on the domain. If $v_j\in \hat{{\mathcal {V}}}_j$, then we have ${\hat{t}}_j-{\hat{t}}_{j+1}\geqq \frac{\varepsilon ^{3+|\kappa |}}{C_\Omega }$. Therefore, if ${\hat{t}}_k={\hat{t}}_k(t,x,v,v_1, \cdots , v_{k-1})>0$, there can be at most $\left[ \frac{C_\Omega T_0}{\varepsilon ^{3+|\kappa |}}\right] +1$ number of $v_j\in \hat{{\mathcal {V}}}_j^\varepsilon$ for $1\leqq j\leqq k-1$. Hence we have

$$\begin{aligned}&\int _{\Pi _{j=1}^{k-1}\hat{{\mathcal {V}}}_{j}} {\mathbf {1}}_{\{{\hat{t}}_k>0\}}~ \Pi _{j=1}^{k-1} e^{\eta |v_j|^\zeta }\,\mathrm{d}{\hat{\sigma }} _{j}\nonumber \\&\quad \leqq \sum _{n=1}^{\left[ \frac{C_\Omega T_0}{\varepsilon ^{3+|\kappa |}}\right] +1} \int _{\{\text{ There } \text{ are } \text{ n } \text{ number } ~v_j\in \hat{{\mathcal {V}}}_j^{\varepsilon } ~\text{ for } \text{ some } ~1\leqq j\leqq k-1\}} \Pi _{j=1}^{k-1} e^{\eta |v_j|^\zeta }\mathrm{d}{\hat{\sigma }} _{j}\nonumber \\&\quad \leqq \sum _{n=1}^{\left[ \frac{C_\Omega T_0}{\varepsilon ^{3+|\kappa |}}\right] +1} \left( \begin{aligned} k-1\\ n \end{aligned} \right) \left| \sup _{j}\int _{\hat{{\mathcal {V}}}_j^\varepsilon }{e^{\eta |v_j|^\zeta }} \mathrm{d}{\hat{\sigma }}_j\right| ^n\cdot \left| \sup _{j}\int _{\hat{{\mathcal {V}}}_j\backslash \hat{{\mathcal {V}}}_j^\varepsilon }{e^{\eta |v_j|^\zeta }} \mathrm{d}{\hat{\sigma }}_j\right| ^{k-1-n}\nonumber \\&\quad \leqq \left( \left[ \frac{C_\Omega T_0}{\varepsilon ^{3+|\kappa |}}\right] +1\right) \cdot (k-1)^{\left[ \frac{C_\Omega T_0}{\varepsilon ^{3+|\kappa |}}\right] +1} (C\varepsilon )^{{k-1-2\left[ \frac{C_\Omega T_0}{\varepsilon ^{3+|\kappa |}}+1\right] }}. \end{aligned}$$

(3.8)

One can take $k-{1}=N\left( \left[ \frac{C_\Omega T_0}{\varepsilon ^{3+|\kappa |}}\right] +1\right)$ with $\left[ \frac{C_\Omega T_0}{\varepsilon ^{3+|\kappa |}}\right] \geqq 1$ and $N> 2(3+|\kappa |)$, so that (3.8) can be bounded as

$$\begin{aligned} \int _{\Pi _{j=1}^{k-1}\hat{{\mathcal {V}}}_{j}} ~{\mathbf {1}}_{\{{\hat{t}}_k>0\}}~ \Pi _{j=1}^{k-1}{e^{\eta |v_j|^\zeta }} \mathrm{d}{\hat{\sigma }} _{j}&\leqq \left\{ 2N\left( \left[ \frac{C_\Omega T_0}{\varepsilon ^{3+|\kappa |}}\right] +1\right) \right\} ^{\left[ \frac{C_\Omega T_0}{\varepsilon ^{3+|\kappa |}}\right] +{1}}\nonumber \\&\quad (C\varepsilon )^{\frac{N}{2}\left( {1}+\left[ \frac{C_\Omega T_0}{\varepsilon ^{3+|\kappa |}}\right] \right) }\\&\leqq \left\{ 4N\left[ \frac{C_\Omega T_0}{\varepsilon ^{3+|\kappa |}}\right] (C\varepsilon )^{\frac{N}{2}}\right\} ^{\left[ \frac{C_\Omega T_0}{\varepsilon ^{3+|\kappa |}}\right] +{1}} \\&\leqq \left\{ C_{\Omega ,N} \cdot T_0\cdot \varepsilon ^{\frac{N}{2}-3-|\kappa |}\right\} ^{\left[ \frac{C_\Omega T_0}{\varepsilon ^{3+|\kappa |}}\right] +{1}}. \end{aligned}$$

We choose

$$\begin{aligned} \varepsilon =\left( \frac{1}{2C_{\Omega ,N} \cdot T_0}\right) ^{\frac{1}{\frac{N}{2}-3-|\kappa |}} \end{aligned}$$

such that $C_{\Omega ,N} \cdot T_0\cdot \varepsilon ^{\frac{N}{2}-3-|\kappa |}=1/2$. Note that for large $T_0$, it holds that $\varepsilon >0$ is small, and

$$\begin{aligned} \left[ \frac{C_\Omega T_0}{\varepsilon ^{3+|\kappa |}}\right] +{1}\cong C_{\Omega ,N}T_0^{1+\frac{3+|\kappa |}{\frac{N}{2}-3-|\kappa |}}. \end{aligned}$$

Finally, we take $N=6(3+|\kappa |)$, so that $\left[ \frac{C_\Omega T_0}{\varepsilon ^{3+|\kappa |}}\right] +{1}=CT_0^{\frac{5}{4}}$ and

$$\begin{aligned} k=6(3+|\kappa |)\left\{ \left[ \frac{C_\Omega T_0}{\varepsilon ^{3+|\kappa |}}\right] +1\right\} +{1}=CT_0^{\frac{5}{4}}. \end{aligned}$$

Therefore, (3.7) follows. This completes the proof of Lemma 3.2. $\quad \square$

Along the back-time cycle (3.6), we can represent the solution of (3.3) in a mild formulation which enables us to get the $L^\infty$ bound of solutions in the steady case. Indeed, for later use, we consider the following iterative linear problems involving a parameter $\lambda \in [0,1]$:

$$\begin{aligned} {\left\{ \begin{array}{ll} \varepsilon h^{i+1}+v\cdot \nabla _x h^{i+1}+\nu (v) h^{i+1}=\lambda K_{ {w}}^m h^i+\lambda K^c_w h^i +wg,\\ \displaystyle h^{i+1}(x,v)|_{\gamma _-}=\frac{1}{{\tilde{w}}(v)} \int _{v'\cdot n(x)>0} h^i(x,v') {\tilde{w}}(v')\,\mathrm{d}\sigma '+w(v)r(x,v), \end{array}\right. } \end{aligned}$$

(3.9)

for $i=0,1, 2,\ldots$, where $h^0=h^0(x,v)$ is given. For the mild formulation of (3.9), we have the following lemma whose proof is omitted for brevity as it is similar to that in [29].

Lemma 3.3

Let $0\leqq \lambda \leqq 1$. Denote ${\hat{\nu }}(v):=(1+|v|^2)^{\frac{|\kappa |}{2}}[\varepsilon +\nu (v)]$. For each $t\in [0,T_0]$ and for each $(x,v)\in {\bar{\Omega }}\times {\mathbb {R}}^3\setminus (\gamma _0\cup \gamma _-)$, we have

$$\begin{aligned} h^{i+1}(x,v)=\sum _{n=1}^4J_n+\sum _{n=5}^{14}{\mathbf {1}}_{\{t_1>s\}}J_n, \end{aligned}$$

(3.10)

with

$$\begin{aligned} J_1&= {\mathbf {1}}_{\{{\hat{t}}_1\leqq 0\}} e^{-{\hat{\nu }}(v)t} h^{i+1}(x-{\hat{v}}t),\\ J_2+J_3+J_4&=\int _{\max \{{\hat{t}}_1,0\}}^t e^{-{\hat{\nu }}(v)(t-s)} (1+|v|^2)^\frac{|\kappa |}{2}\\&\qquad \Big [\lambda K_w^mh^{i}+\lambda K^c_wh^{i}+wg\Big ](x-{\hat{v}}(t-s),v)\,\mathrm{d}s,\\ J_5&=e^{-{\hat{\nu }}(v)(t-{\hat{t}}_1)}w(v) r({\hat{x}}_1,v),\\ J_6&=\frac{e^{-{\hat{\nu }}(v)(t-{\hat{t}}_1)}}{{\tilde{w}}(v)} \int _{\Pi _{j=1}^{k-1}\hat{{\mathcal {V}}}_{j}} \sum _{l=1}^{k-2} {\mathbf {1}}_{\{{\hat{t}}_{l+1}>0\}} w(v_l)r({\hat{x}}_{l+1},v_{l})\,\mathrm{d}{\hat{\Sigma }}_{l}({\hat{t}}_{l+1}),\\ J_7&=\frac{e^{-{\hat{\nu }}(v)(t-{\hat{t}}_1)}}{{\tilde{w}}(v)} \int _{\Pi _{j=1}^{k-1}\hat{{\mathcal {V}}}_{j}} \sum _{l=1}^{k-1} {\mathbf {1}}_{\{{\hat{t}}_{l+1}\leqq 0<{\hat{t}}_l\}} h^{i+1-l}({\hat{x}}_l-{\hat{v}}_l{\hat{t}}_l,v_l)\,\mathrm{d}{\hat{\Sigma }}_{l}(0),\\ J_8+J_9+J_{10}&=\frac{e^{-{\hat{\nu }}(v)(t-{\hat{t}}_1)}}{{\tilde{w}}(v)} \int _{\Pi _{j=1}^{k-1}\hat{{\mathcal {V}}}_{j}} \sum _{l=1}^{k-1}\int _0^{{\hat{t}}_l} {\mathbf {1}}_{\{{\hat{t}}_{l+1}\leqq 0<{\hat{t}}_l\}}\\&\qquad \times (1+|v_l|^2)^{\frac{|\kappa |}{2}}\Big [\lambda K_w^mh^{i-l}+\lambda K^c_wh^{i-l}+wg\Big ] \\&\quad ({\hat{x}}_l-{\hat{v}}_l({\hat{t}}_l-s),v_l) \,\mathrm{d}{\hat{\Sigma }}_l(s),\\ J_{11}+J_{12}+J_{13}&=\frac{e^{-{\hat{\nu }}(v)(t-{\hat{t}}_1)}}{{\tilde{w}}(v)} \int _{\Pi _{j=1}^{k-1}\hat{{\mathcal {V}}}_{j}} \sum _{l=1}^{k-1}\int _{{\hat{t}}_{l+1}}^{{\hat{t}}_l} {\mathbf {1}}_{\{{\hat{t}}_{l+1}\leqq 0<{\hat{t}}_l\}}\\&\qquad \times (1+|v_l|^2)^{\frac{|\kappa |}{2}}\Big [\lambda K_w^mh^{i-l}+\lambda K^c_wh^{i-l}+wg\Big ]\\&\qquad ({\hat{x}}_l-{\hat{v}}_l({\hat{t}}_l-s),v_l) \,\mathrm{d}{\hat{\Sigma }}_l(s),\\ J_{14}&=\frac{e^{-{\hat{\nu }}(v)(t-{\hat{t}}_1)}}{{\tilde{w}}(v)} \int _{\Pi _{j=1}^{k-1}\hat{{\mathcal {V}}}_{j}} I_{\{{\hat{t}}_{k}>0\}} h^{i+1-k}({\hat{x}}_k,v_{k-1}) \,\mathrm{d}{\hat{\Sigma }}_{k-1}({\hat{t}}_k), \end{aligned}$$

where we have denoted

$$\begin{aligned} \mathrm{d}{\hat{\Sigma }}_l(s) = \big \{\Pi _{j=l+1}^{k-1}\mathrm{d}{\hat{\sigma }}_j\big \}\cdot \big \{{\tilde{w}}(v_l) e^{-{\hat{\nu }}(v_l)({\hat{t}}_l-s)} \mathrm{d}{\hat{\sigma }}_l\big \}\cdot \big \{\Pi _{j=1}^{l-1} e^{-{\hat{\nu }}(v_j)({\hat{t}}_j-{\hat{t}}_{j+1})} \mathrm{d}{\hat{\sigma }}_j\big \}. \end{aligned}$$

Lemma 3.4

Let $\beta >3$. Let $h^i$, $i=0,1,2,\ldots$, be the solutions to (3.9), satisfying

$$\begin{aligned} \Vert h^i\Vert _{L^\infty }+|h^i|_{L^\infty (\gamma )}<\infty . \end{aligned}$$

Then there exists $T_0>0$ large enough such that for $i\geqq k:= {{\hat{C}}_1}T_0^{\frac{5}{4}}$, it holds that

$$\begin{aligned} \Vert h^{i+1}\Vert _{L^\infty }{+|h^{i+1}|_{L^\infty (\gamma )}}&\leqq \frac{1}{8} \sup _{0\leqq l\leqq k} \{\Vert h^{i-l}\Vert _{L^\infty }\}+C\Big \{\Vert \nu ^{-1}wg\Vert _{L^\infty }+|wr|_{{L^\infty (\gamma _-)}}\Big \}\nonumber \\&\quad +C \sup _{0\leqq l\leqq k}\left\{ \left\| \frac{\sqrt{\nu }h^{i-l}}{w}\right\| _{L^2}\right\} . \end{aligned}$$

(3.11)

Moreover, if $h^i\equiv h$ for $i=1,2,\ldots$, that is, h is a solution, then (3.11) is reduced to the following estimate

$$\begin{aligned} \Vert h\Vert _{L^\infty }{+|h|_{L^\infty (\gamma )}}&\leqq C\Big \{\Vert \nu ^{-1}wg\Vert _{L^\infty }+|wr|_{{L^\infty (\gamma _-)}}\Big \}+\left\| \frac{\sqrt{\nu }h}{w}\right\| _{L^2}. \end{aligned}$$

(3.12)

Here it is emphasized that the positive constant $C>0$ does not depend on $\lambda \in [0,1]$ and ${\varepsilon }>0$.

Proof

By the definition of ${\hat{\nu }}(v)$, we first note that

$$\begin{aligned} {\hat{\nu }}(v)\geqq (1+|v|^2)^{\frac{|\kappa |}{2}}\nu (v)\geqq {\hat{\nu }}_0>0, \end{aligned}$$

(3.13)

where ${\hat{\nu }}_0$ is a positive constant independent of $\varepsilon$ and $v\in {\mathbb {R}}^3$. For $J_1$, it follows from (3.13) that

$$\begin{aligned} |J_1|\leqq e^{-{\hat{\nu }}_0 t} \Vert h^{i+1}\Vert _{L^\infty }. \end{aligned}$$

(3.14)

For $J_2$, it follows from (2.11) that

$$\begin{aligned} |J_2|&\leqq Cm^{3+\kappa }\int _{\max \{{\hat{t}}_1,0\}}^t e^{-{\hat{\nu }}_0 (t-s)}(1+|v|^2)^{\frac{|\kappa |}{2}} e^{-\frac{|v|^2}{20}} \Vert h^{i}\Vert \,\mathrm{d}s\nonumber \\&\leqq Cm^{3+\kappa } e^{-\frac{|v|^2}{32}} \Vert h^i\Vert _{L^\infty }. \end{aligned}$$

(3.15)

For those terms involving the source g, we notice that

$$\begin{aligned} \frac{1}{{\tilde{w}}(v)}\leqq Cw(v)e^{-\frac{|v|^2}{4}}\leqq Ce^{-\frac{|v|^2}{8}}, \end{aligned}$$

(3.16)

which immediately yields that

$$\begin{aligned} \left\{ \begin{aligned}&\int _{\Pi _{j=1}^{k-1}\hat{{\mathcal {V}}}_{j}} {\tilde{w}}(v_l)[1+|v_l|^2]^{\frac{|\kappa |}{2}} \ \Pi _{j=1}^{k-1} \mathrm{d}{\hat{\sigma }}_j\leqq C<\infty ,\quad \text{ for }\quad 1\leqq l\leqq k-1,\\&\int _{\Pi _{j=1}^{k-1}\hat{{\mathcal {V}}}_{j}} \sum _{l=1}^{k-1} {\mathbf {1}}_{\{{\hat{t}}_{l+1}\leqq 0<{\hat{t}}_l\}} {\tilde{w}}(v_l)[1+|v_l|^2]^{\frac{|\kappa |}{2}} \ \Pi _{j=1}^{k-1} \mathrm{d}{\hat{\sigma }}_j\leqq Ck. \end{aligned}\right. \end{aligned}$$

(3.17)

Then it follows from (3.16) and (3.17) that

$$\begin{aligned}&|J_4|+|J_{10}|+|J_{13}|\leqq Ck \Vert \nu ^{-1}wg\Vert _{L^\infty }, \end{aligned}$$

(3.18)

$$\begin{aligned}&|J_5|+|J_6|\leqq Ck |wr|_{L^\infty {(\gamma _-)}}, \end{aligned}$$

(3.19)

and

$$\begin{aligned} |J_7|&\leqq C e^{-\frac{1}{8}|v|^2} e^{-{\hat{\nu }}_0 (t-t_1)} \int _{\Pi _{j=1}^{k-1}\hat{{\mathcal {V}}}_{j}} \sum _{l=1}^{k-1} {\mathbf {1}}_{\{{\hat{t}}_{l+1}\leqq 0<{\hat{t}}_l\}} \mathrm{d}{\hat{\Sigma }}_{l}(0)\cdot \sup _{1\leqq l\leqq k-1}\{\Vert h^{i+1-l}\Vert _{L^\infty }\}\nonumber \\&\leqq Ck e^{-{\hat{\nu }}_0 t} e^{-\frac{1}{8}|v|^2} \cdot \sup _{1\leqq l\leqq k-1}\{\Vert h^{i+1-l}\Vert _{L^\infty }\}. \end{aligned}$$

(3.20)

For the term $J_{14}$, it follows from (3.16) and Lemma 3.2 that

$$\begin{aligned} |J_{14}|\leqq C e^{-\frac{1}{8}|v|^2} \left( \frac{1}{2}\right) ^{{\hat{C}}_2 T_0^{\frac{5}{4}}}\cdot {|h^{i+1-k}|_{L^\infty (\gamma _-)}}, \end{aligned}$$

(3.21)

where we have taken $k= {{\hat{C}}_1}T_0^{\frac{5}{4}}$ and $T_0$ is a large constant to be chosen later. From the boundary condition given in the second equation of (3.9), it further holds that

$$\begin{aligned} |h^{i+1-k}|_{L^\infty (\gamma _-)}\leqq C|h^{i-k}|_{L^\infty (\gamma _+)}+|wr|_{L^\infty (\gamma _-)}. \end{aligned}$$

For $J_8$, using (2.11), (3.13), (3.16) and (3.17), one obtains that

$$\begin{aligned} {|J_8|}&\leqq C m^{3+\kappa } e^{-\frac{1}{8}|v|^2}\cdot \sup _{1\leqq l\leqq k-1}\{\Vert h^{i-l}\Vert _{L^\infty }\} \nonumber \\&\quad \times \int _{\Pi _{j=1}^{k-1}\hat{{\mathcal {V}}}_{j}} \sum _{l=1}^{k-1}I_{\{{\hat{t}}_{l+1}\leqq 0<{\hat{t}}_l\}}\int _0^{t_l} e^{-{\hat{\nu }}_0(t-s)} \,\mathrm{d}s \ \nu (v_l)^{-1} {\tilde{w}}(v_l)e^{-\frac{|v_l|^2}{8}}\Pi _{j=1}^{k-1} \,\mathrm{d}{\hat{\sigma }}_j\nonumber \\&\leqq C m^{3+\kappa } e^{-\frac{1}{8}|v|^2} \cdot \sup _{1\leqq l\leqq k-1}\{\Vert h^{i-l}\Vert _{L^\infty }\} \nonumber \\&\qquad \int _{\Pi _{j=1}^{k-1}\hat{{\mathcal {V}}}_{j}} \sum _{l=1}^{k-1}I_{\{{\hat{t}}_{l+1}\leqq 0<{\hat{t}}_l\}} \nu (v_l)^{-1} {\tilde{w}}(v_l) \Pi _{j=1}^{k-1} \,\mathrm{d}{\hat{\sigma }}_j\nonumber \\&\leqq Ck m^{3+\kappa } e^{-\frac{1}{8}|v|^2} \cdot \sup _{1\leqq l\leqq k-1}\Big \{\Vert h^{i-l}\Vert _{L^\infty }\Big \}. \end{aligned}$$

(3.22)

Here we remark that the factor $e^{-\frac{1}{8}|v|^2}$ on the right-hand side of (3.22) is very crucial for the later use of the Vidav’s iteration. For $J_9$, it holds that

$$\begin{aligned} |J_9|&\leqq C e^{-\frac{1}{8}|v|^2} \sum _{l=1}^{k-1}\int _{\Pi _{j=1}^{l-1}\hat{{\mathcal {V}}}_{j}} \mathrm{d}{\hat{\sigma }}_{l-1}\cdots \mathrm{d}{\hat{\sigma }}_1 \int _0^{{\hat{t}}_l} e^{-{\hat{\nu }}_0(t-s)} \,\mathrm{d}s\nonumber \\&\quad \times \int _{{\mathcal {V}}_l}\int _{{\mathbb {R}}^3} {\mathbf {1}}_{\{{\hat{t}}_{l+1}\leqq 0<{\hat{t}}_l\}} \nu (v_l)^{-1} {\tilde{w}}(v_l) |k^c_w(v_l,v') h^{i-l}({\hat{x}}_l-{\hat{v}}_{{l}}({\hat{t}}_l-s),v')|\,\mathrm{d}v' \mathrm{d}{\hat{\sigma }}_l \nonumber \\&=C e^{-\frac{1}{8}|v|^2} \sum _{l=1}^{k-1}\int _{\Pi _{j=1}^{l-1}\hat{{\mathcal {V}}}_{j}} \mathrm{d}{\hat{\sigma }}_{l-1}\cdots \mathrm{d}{\hat{\sigma }}_1 \int _0^{{\hat{t}}_l} e^{-{\hat{\nu }}_0(t-s)} \,\mathrm{d}s\int _{{\mathcal {V}}_l\cap \{|v_l|\geqq N\}}\int _{{\mathbb {R}}^3} (\cdots )\,\mathrm{d}v' \mathrm{d}{\hat{\sigma }}_l \nonumber \\&\quad +C e^{-\frac{1}{8}|v|^2} \sum _{l=1}^{k-1}\int _{\Pi _{j=1}^{l-1}\hat{{\mathcal {V}}}_{j}} \mathrm{d}{\hat{\sigma }}_{l-1}\cdots \mathrm{d}{\hat{\sigma }}_1 \int _0^{{\hat{t}}_l} e^{-{\hat{\nu }}_0(t-s)} \mathrm{d}s\int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{{\mathbb {R}}^3} (\cdots )\,\mathrm{d}v' \mathrm{d}{\hat{\sigma }}_l\nonumber \\&:=\sum _{l=1}^{k-1} \left( J_{91l}+J_{92l}\right) . \end{aligned}$$

(3.23)

We shall estimate the right-hand terms of (3.23) as follows. By using (2.14), we have

$$\begin{aligned} \sum _{l=1}^{k-1} J_{91l}&\leqq C e^{-\frac{1}{8}|v|^2} \sum _{l=1}^{k-1}\int _{\Pi _{j=1}^{l-1}\hat{{\mathcal {V}}}_{j}} \mathrm{d}{\hat{\sigma }}_{l-1}\cdots \mathrm{d}{\hat{\sigma }}_1 \int _0^{{\hat{t}}_l} e^{-{\hat{\nu }}_0(t-s)} \,\mathrm{d}s\nonumber \\&\quad \times \int _{{\mathcal {V}}_l\cap \{|v_l|\geqq N\}} e^{-\frac{1}{8}|v_l|^2} \mathrm{d}v_l \cdot \sup _{1\leqq l\leqq k-1}\{\Vert h^{i-l}\Vert _{L^\infty }\}\nonumber \\&\leqq Ck e^{-\frac{1}{8}|v|^2} e^{-\frac{1}{16}N^2} \cdot \sup _{1\leqq l\leqq k-1}\{\Vert h^{i-l}\Vert _{L^\infty }\}, \end{aligned}$$

(3.24)

and, for each term $J_{92l}$, we also have

$$\begin{aligned} J_{92l}&\leqq C e^{-\frac{1}{8}|v|^2} \int _{\Pi _{j=1}^{l-1}\hat{{\mathcal {V}}}_{j}} \mathrm{d}{\hat{\sigma }}_{l-1}\cdots \mathrm{d}{\hat{\sigma }}_1 \int _{{\hat{t}}_l-\frac{1}{N}}^{{\hat{t}}_l} e^{-{\hat{\nu }}_0(t-s)} \,\mathrm{d}s\int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{{\mathbb {R}}^3} (\cdots )\,\mathrm{d}v' \mathrm{d}{\hat{\sigma }}_l\nonumber \\&\quad +C e^{-\frac{1}{8}|v|^2} \int _{\Pi _{j=1}^{l-1}\hat{{\mathcal {V}}}_{j}} \mathrm{d}{\hat{\sigma }}_{l-1}\cdots \mathrm{d}{\hat{\sigma }}_1 \int _0^{{\hat{t}}_l-\frac{1}{N}} e^{-{\hat{\nu }}_0(t-s)} \,\mathrm{d}s\int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}} e^{-\frac{1}{8}|v_l|^2} \,\mathrm{d}v_l\nonumber \\&\quad \times \int _{|v'|\geqq 2N} |k_w^c(v_l,v')| e^{\frac{|v_l-v'|^2}{64}} \,\mathrm{d}v' e^{-\frac{N^2}{64}}\cdot \sup _{1\leqq l\leqq k-1}\{\Vert h^{i-l}\Vert _{L^\infty }\}\nonumber \\&\quad +C e^{-\frac{1}{8}|v|^2} \int _{\Pi _{j=1}^{l-1}\hat{{\mathcal {V}}}_{j}} \mathrm{d}{\hat{\sigma }}_{l-1}\cdots \mathrm{d}{\hat{\sigma }}_1 \int _0^{{\hat{t}}_l-\frac{1}{N}} e^{-{\hat{\nu }}_0(t-s)} \mathrm{d}s\int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{|v'|\leqq 2N} \nonumber \\&\quad \times {\mathbf {1}}_{\{{\hat{t}}_{l+1}\leqq 0<{\hat{t}}_l\}} e^{-\frac{1}{8}|v_l|^2} |k^c_w(v_l,v') h^{i-l}(x_l-{\hat{v}}_{{l}}({\hat{t}}_l-s),v')|\,\mathrm{d}v' \mathrm{d}v_l \nonumber \\&\leqq C e^{-\frac{1}{8}|v|^2} \int _{\Pi _{j=1}^{l-1}\hat{{\mathcal {V}}}_{j}} \mathrm{d}{\hat{\sigma }}_{l-1}\cdots \mathrm{d}{\hat{\sigma }}_1 \int _0^{{\hat{t}}_l-\frac{1}{N}} e^{-{\hat{\nu }}_0(t-s)} \mathrm{d}s\int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{|v'|\leqq 2N} \nonumber \\&\quad \times {\mathbf {1}}_{\{{\hat{t}}_{l+1}\leqq 0<{\hat{t}}_l\}} e^{-\frac{1}{8}|v_l|^2} |k^c_w(v_l,v') h^{i-l}(x_l-{\hat{v}}_{{l}}({\hat{t}}_l-s),v')|\,\mathrm{d}v' \mathrm{d}v_l\nonumber \\&\quad +\frac{C}{N}e^{-\frac{1}{8}|v|^2}\cdot \Vert h^{i-l}\Vert _{L^\infty }. \end{aligned}$$

(3.25)

To estimate the first term on the right-hand side of (3.25), it follows from (2.12) that

$$\begin{aligned}&\int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{|v'|\leqq 2N} {\mathbf {1}}_{\{{\hat{t}}_{l+1}\leqq 0<{\hat{t}}_l\}} e^{-\frac{1}{8}|v_l|^2} |k^c_w(v_l,v') h^{i-l}({\hat{x}}_l-{\hat{v}}_{{l}}({\hat{t}}_l-s),v')|\,\mathrm{d}v' \mathrm{d}v_l\nonumber \\&\quad \leqq C_N \left\{ \int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{|v'|\leqq 2N} e^{-\frac{1}{8}|v_l|^2} |k^c_w(v_l,v')|^2 \,\mathrm{d}v' \mathrm{d}v_l \right\} ^{\frac{1}{2}}\nonumber \\&\quad \times \left\{ \int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{|v'|\leqq 2N} {\mathbf {1}}_{\{{\hat{t}}_{l+1}\leqq 0<{\hat{t}}_l\}} \left| \frac{{\sqrt{\nu (v')}}h^{i-l}({\hat{x}}_l-{\hat{v}}_{{l}}({\hat{t}}_l-s),v')}{w(v')}\right| ^2 \,\mathrm{d}v' \mathrm{d}v_l \right\} ^{\frac{1}{2}}\nonumber \\&\quad \leqq C_N m^{\kappa -1}\left\{ \int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{|v'|\leqq 2N} {\mathbf {1}}_{\{{\hat{t}}_{l+1}\leqq 0<{\hat{t}}_l\}} \left| \frac{\sqrt{\nu (v')}h^{i-l}({\hat{x}}_l-{\hat{v}}_{{l}}({\hat{t}}_l-s),v')}{w(v')}\right| ^2 \,\mathrm{d}v' \mathrm{d}v_l \right\} ^{\frac{1}{2}}. \end{aligned}$$

(3.26)

Let $y_l={\hat{x}}_l-{\hat{v}}({\hat{t}}_l-s)\in \Omega$ for $s\in [0,{\hat{t}}_l-\frac{1}{N}]$. A direct computation shows that

$$\begin{aligned} \left| \frac{\partial y_l}{\partial v_l}\right| =|{\hat{t}}_l-s|\cdot \left| \frac{\partial {\hat{v}}}{\partial v}\right| \geqq \frac{(1+|v|^2)^{\frac{3|\kappa |}{2}-1}}{N^3}\cdot \{1+(1+|\kappa |)|v|^2\}\geqq \frac{1}{N^3}. \end{aligned}$$

(3.27)

Thus, by making change of variable ${\hat{v}}_l\rightarrow y_l$ and using (3.27), one obtains that

$$\begin{aligned}&\left\{ \int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{|v'|\leqq 2N} {\mathbf {1}}_{\{{\hat{t}}_{l+1}\leqq 0<{\hat{t}}_l\}} \left| \frac{\sqrt{\nu (v')}h^{i-l}({\hat{x}}_l-{\hat{v}}_{{l}}({\hat{t}}_l-s),v')}{w(v')}\right| ^2 \,\mathrm{d}v' \mathrm{d}v_l \right\} ^{\frac{1}{2}}\\&\quad \leqq C_{N}\left\{ \int _{\Omega }\int _{|v'|\leqq 2N}\left| \frac{\sqrt{\nu (v')}h^{i-l}(y_l,v')}{w(v')}\right| ^2 \,\mathrm{d}v' \mathrm{d}y_l \right\} ^{\frac{1}{2}}\leqq C_N \left\| \frac{\sqrt{\nu }h^{i-l}}{w}\right\| _{L^2}, \end{aligned}$$

which, together with (3.26) and (3.25), yields that

$$\begin{aligned} J_{92l}\leqq \frac{C}{N}e^{-\frac{1}{8}|v|^2}\cdot \sup _{1\leqq l\leqq k-1}\{\Vert h^{i-l}\Vert _{L^\infty }\}+C_N m^{\kappa -1} e^{-\frac{1}{8}|v|^2}\left\| \frac{\sqrt{\nu }h^{i-l}}{w}\right\| _{L^2}. \end{aligned}$$

(3.28)

Thus it follows from (3.28), (3.24) and (3.23) that

$$\begin{aligned} |J_9|&\leqq \frac{Ck}{N}e^{-\frac{1}{8}|v|^2}\cdot \sup _{1\leqq l\leqq k-1}\{\Vert h^{i-l}\Vert _{L^\infty }\}+C_N k m^{\kappa -1} e^{-\frac{1}{8}|v|^2}\cdot \sup _{1\leqq l\leqq k-1}\left\{ \left\| \frac{\sqrt{\nu }h^{i-l}}{w}\right\| _{L^2}\right\} . \end{aligned}$$

(3.29)

By similar arguments as to those in (3.22)–(3.29), one can obtain

$$\begin{aligned} |J_{11}|+|J_{12}|&\leqq Cke^{-\frac{1}{8}|v|^2}\left\{ m^{3+\kappa }+\frac{1}{N}\right\} \cdot \sup _{1\leqq l\leqq k-1}\{\Vert h^{i-l}\Vert _{L^\infty }\}\nonumber \\&\quad +C_N km^{\kappa -1} e^{-\frac{1}{8}|v|^2}\cdot \sup _{1\leqq l\leqq k-1}\left\{ \left\| \frac{\sqrt{\nu }h^{i-l}}{w}\right\| _{L^2}\right\} . \end{aligned}$$

(3.30)

Now substituting (3.30), (3.29), (3.22), (3.21), (3.20), (3.19), (3.18), (3.15) and (3.14) into (3.10), we get, for $t\in [0,T_0]$, that

$$\begin{aligned} |h^{i+1}(x,v)|&\leqq \int _{\max \{{\hat{t}}_1,0\}}^t e^{-{\hat{\nu }}_0(t-s)} (1+|v|^2)^{\frac{|\kappa |}{2}} \int _{{\mathbb {R}}^3}|k^c_w(v,v')h^{i}(x-{\hat{v}}(t-s),v')|\,\mathrm{d}v' \mathrm{d}s\nonumber \\&\quad + A_i(t,v), \end{aligned}$$

(3.31)

where we have denoted

$$\begin{aligned} A_i(t,v)&:=Ck e^{-\frac{1}{32}|v|^2}\left\{ m^{3+\kappa }+e^{-{\hat{\nu }}_0 t}+ \left( \frac{1}{2}\right) ^{{\hat{C}}_2T_0^{\frac{5}{4}}}+\frac{1}{N}\right\} \\&\quad \cdot \sup _{0\leqq l\leqq k-1}\{\Vert h^{i-l}\Vert _{L^\infty }{+|h^{i-l}|_{L^\infty (\gamma _+)}}\}\\&\quad +e^{-{\hat{\nu }}_0 t} \Vert h^{i+1}\Vert _{L^\infty }+Ck \Big \{\Vert \nu ^{-1}wg\Vert _{L^\infty }+|wr|_{{L^\infty (\gamma _-)}}\Big \}\\&\quad +C_{N,k,m}\ e^{-\frac{1}{8}|v|^2}\cdot \sup _{1\leqq l\leqq k-1}\left\{ \left\| \frac{\sqrt{\nu }h^{i-l}}{w}\right\| _{L^2}\right\} . \end{aligned}$$

We denote $x'=x-{\hat{v}}(t-s)\in \Omega$ and ${\hat{t}}_1'={\hat{t}}_1(s,x',v')$ for $s\in (\min \{t_1,0\},t)$. Using the Vidav’s iteration in (3.31), then we obtain that

$$\begin{aligned} |h^{i+1}(x,v)|&\leqq A_i(t,v)+\int _{0}^t e^{-{\hat{\nu }}_0(t-s)} (1+|v|^2)^{\frac{|\kappa |}{2}} \int _{{\mathbb {R}}^3}|k^c_w(v,v')| A_{i-1}(s,v')\,\mathrm{d}v' \mathrm{d}s \nonumber \\&\quad +\int _{0}^tds \int _0^s e^{-{\hat{\nu }}_0(t-\tau )} \,\mathrm{d}\tau \int _{{\mathbb {R}}^3}\int _{{\mathbb {R}}^3}|k^c_w(v,v') k^c_w(v',v'')|\nonumber \\&\quad \times {\mathbf {1}}_{\{\max \{{\hat{t}}_1,0\}<s<t\}} {\mathbf {1}}_{\{\max \{{\hat{t}}'_1,0\}<\tau <s\}} |h^{i-1}(x'-{\hat{v}}'(s-\tau ),v'')|\,\mathrm{d}v'' \mathrm{d}v'\nonumber \\&:=A_i(t,v)+B_1+B_2. \end{aligned}$$

(3.32)

For the term $B_1$, using (2.13) and (2.14), one has

$$\begin{aligned} B_1&\leqq Ck\left\{ m^{3+\kappa }+m^{\kappa -1}e^{-\frac{1}{2}{\hat{\nu }}_0t}+ \left( \frac{1}{2}\right) ^{{{\hat{C}}_2T_0^{\frac{5}{4}}}} +\frac{1}{N}\right\} \cdot \sup _{0\leqq l\leqq k}\{\Vert h^{i-l}\Vert _{L^\infty }+{|h^{i-l}|_{L^\infty (\gamma _+)}}\}\nonumber \\&\quad +C_{k,m}\Big \{\Vert \nu ^{-1}wg\Vert _{L^\infty }+|wr|_{{L^\infty (\gamma _-)}}\Big \}+C_{N,k,m}\ \sup _{0\leqq l\leqq k}\left\{ \left\| \frac{\sqrt{\nu }h^{i-l}}{w}\right\| _{L^2}\right\} . \end{aligned}$$

(3.33)

For the term $B_2$, we split the estimate by several cases.

Case 1. For $|v|\geqq N$, we have from (2.13) that

$$\begin{aligned} B_2\leqq Cm^{2(\kappa -1)} \Vert h^{i-1}\Vert _{L^\infty } (1+|v|)^{-4}\leqq \frac{Cm^{2(\kappa -1)} }{N^4} \Vert h^{i-1}\Vert _{L^\infty }. \end{aligned}$$

(3.34)

Case 2. For $|v|\leqq N, |v'|\geqq 2N$ or $|v'|\leqq 2N, |v''|\geqq 3N$. In this case, we note from (2.13) that

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \int _{|v|\leqq N, |v'|\geqq 2N} \Big |k_w^c(v,v') e^{\frac{|v-v'|^2}{32}}\Big |\,\mathrm{d}v' \leqq C m^{\kappa -1} (1+|v|)^{\kappa -2},\\ \displaystyle \int _{|v'|\leqq 2N, |v''|\geqq 3N} \Big |k_w^c(v',v'') e^{\frac{|v'-v''|^2}{32}}\Big |\,\mathrm{d}v'' \leqq C m^{\kappa -1} (1+|v'|)^{\kappa -2}. \end{array}\right. } \end{aligned}$$

This yields that

$$\begin{aligned}&\int _{0}^t\,\mathrm{d}s \int _0^s e^{-{\hat{\nu }}_0(t-\tau )} \,\mathrm{d}\tau \left\{ \int _{|v|\leqq N, |v'|\geqq 2N} +\int _{|v'|\leqq 2N, |v''|\geqq 3N} \right\} (\cdots )\,\mathrm{d}v'' \mathrm{d}v'\nonumber \\&\quad \leqq e^{-\frac{N^2}{32}}\Vert h^{i-1}\Vert _{L^\infty }\int _{|v|\leqq N, |v'|\geqq 2N} |k^c_w(v,v')e^{\frac{|v-v'|^2}{32}}| \cdot |k^c_w(v',v'')| \nu (v)^{-1}\nu (v')^{-1}\,\mathrm{d}v'' \mathrm{d}v'\nonumber \\&\qquad + e^{-\frac{N^2}{32}}\Vert h^{i-1}\Vert _{L^\infty } \int _{|v|\leqq N, |v'|\geqq 2N} |k^c_w(v,v')|\cdot |k^c_w(v',v'')e^{\frac{|v'-v''|^2}{32}}| \nu (v)^{-1}\nu (v')^{-1}\,\mathrm{d}v'' \mathrm{d}v'\nonumber \\&\quad \leqq C m^{2(\kappa -1)} e^{-\frac{N^2}{32}}\Vert h^{i-1}\Vert _{L^\infty }. \end{aligned}$$

(3.35)

Case 3. For $|v|\leqq N$, $|v'|\leqq 2N$, and $|v''|\leqq 3N$, we first note that

$$\begin{aligned}&\int _{0}^t\,\mathrm{d}s \int _0^s e^{-{\hat{\nu }}_0(t-\tau )} \,\mathrm{d}\tau \int _{|v'|\leqq 2N, |v''|\leqq 3N}(\cdots )\,\mathrm{d}v'' \mathrm{d}v'\nonumber \\&\quad \leqq \frac{C}{N} m^{2(\kappa -1)} \Vert h^{i-1}\Vert _{L^\infty }+\int _{0}^t\,\mathrm{d}s \int _0^{s-\frac{1}{N}} e^{-{\hat{\nu }}_0(t-\tau )} \,\mathrm{d}\tau \int _{{\mathbb {R}}^3}\int _{{\mathbb {R}}^3}(\cdots )\,\mathrm{d}v'' \mathrm{d}v'\nonumber \\&\quad \leqq \frac{C}{N} m^{2(\kappa -1)} \Vert h^{i-1}\Vert _{L^\infty }\nonumber \\&\quad \quad +C_{N,k}\int _{0}^t\,\mathrm{d}s \int _0^{s-\frac{1}{N}} e^{-{\hat{\nu }}_0(t-\tau )} \,\mathrm{d}\tau \left\{ \int _{|v'|\leqq 2N, |v''|\leqq 3N} |k^c(v,v') k^c(v',v'')|^2\,\mathrm{d}v'' \mathrm{d}v'\right\} ^{\frac{1}{2}}\nonumber \\&\quad \quad \times \left\{ \int _{|v'|\leqq 2N, |v''|\leqq 3N} {\mathbf {1}}_{\{\max \{{\hat{t}}_1,0\}<s<t\}} {\mathbf {1}}_{\{\max \{{\hat{t}}'_1,0\}<\tau<s\}} \Big |\frac{\sqrt{\nu (v'')}h^{i-1}(y',v'')}{w(v'')}\Big |^2\,\mathrm{d}v'' \mathrm{d}v'\right\} ^{\frac{1}{2}}\nonumber \\&\quad \leqq \frac{C}{N} m^{2(\kappa -1)} \Vert h^{i-1}\Vert _{L^\infty }+C_{N,k,m} \int _{0}^t\,\mathrm{d}s \int _0^{s-\frac{1}{N}} e^{-{\hat{\nu }}_0(t-\tau )} \,\mathrm{d}\tau \nonumber \\&\quad \quad \times \left\{ \int _{|v'|\leqq 2N, |v''|\leqq 3N} {\mathbf {1}}_{\{\max \{{\hat{t}}_1,0\}<s<t\}} {\mathbf {1}}_{\{\max \{{\hat{t}}'_1,0\}<\tau <s\}} \Big |\frac{\sqrt{\nu (v'')}h^{i-1}(y',v'')}{w(v'')}\Big |^2\,\mathrm{d}v'' \mathrm{d}v'\right\} ^{\frac{1}{2}}, \end{aligned}$$

(3.36)

where we have denoted $y'=x'-{\hat{v}}'(s-\tau )\in \Omega$ for $s\in (\max \{{\hat{t}}_1,0\}, s)$ and $\tau \in (\max \{{\hat{t}}'_1,0\}, s)$. Similar to (3.27), we make change of variable $v'\mapsto y'$, so that the second term on the right-hand side of (3.36) is bounded as

$$\begin{aligned} \int _{0}^t\,\mathrm{d}s \int _0^{s-\frac{1}{N}} e^{-{\hat{\nu }}_0(t-\tau )} \,\mathrm{d}\tau \left\{ \int _{|v'|\leqq 2N, |v''|\leqq 3N} (\cdots )\,\mathrm{d}v'' \mathrm{d}v'\right\} ^{\frac{1}{2}}\leqq CN^{\frac{3}{2}}\left\| \frac{\sqrt{\nu }h^{i-1}}{w}\right\| _{L^2}, \end{aligned}$$

which together with (3.36) yield that

$$\begin{aligned}&\int _{0}^t\,\mathrm{d}s \int _0^s e^{-{\hat{\nu }}_0(t-\tau )} \,\mathrm{d}\tau \int _{|v'|\leqq 2N, |v''|\leqq 3N}(\cdots )\,\mathrm{d}v'' \mathrm{d}v'\nonumber \\&\quad \leqq \frac{C}{N}m^{2(\kappa -1)} \Vert h^{i-1}\Vert _{L^\infty }+C_{N,k,m} \left\| \frac{\sqrt{\nu }h^{i-1}}{w}\right\| _{L^2}. \end{aligned}$$

(3.37)

Combining (3.34), (3.35) and (3.37), we have

$$\begin{aligned} B_2 \leqq \frac{C}{N}m^{2(\kappa -1)} \Vert h^{i-1}\Vert _{L^\infty }+C_{N,k,m} \left\| \frac{\sqrt{\nu }h^{i-1}}{w}\right\| _{L^2}. \end{aligned}$$

Hence, the above estimate together with (3.33) and (3.32) yields that for any $t\in [0,T_0]$,

$$\begin{aligned} |h^{i+1}(x,v)|&\leqq Ck\left\{ m^{3+\kappa }+m^{\kappa -1}e^{-\frac{1}{2}{\hat{\nu }}_0t}+\left( \frac{1}{2}\right) ^{{{\hat{C}}_2T_0^{\frac{5}{4}}}} +\frac{m^{2(\kappa -1)}}{N}\right\} \\&\quad \times \sup _{0\leqq l\leqq k}\{\Vert h^{i-l}\Vert _{L^\infty }{+|h^{i-l}|_{L^\infty (\gamma _+)}}\}\\&\quad +\,e^{-{\hat{\nu }}_0 t} \Vert h^{i+1}\Vert _{L^\infty }+C_{k,m}\Big \{\Vert \nu ^{-1}wg\Vert _{L^\infty }+|wr|_{{L^\infty (\gamma _-)}}\Big \}\\&\quad +\,C_{N,k,m}\ \sup _{0\leqq l\leqq k}\left\{ \left\| \frac{\sqrt{\nu }h^{i-l}}{w}\right\| _{L^2}\right\} . \end{aligned}$$

Now we take $k={\hat{C}}_1t^{\frac{5}{4}}={\hat{C}}_1T_0^{\frac{5}{4}}$ and choose $m=T_0^{-\frac{9}{4(3+\kappa )}}$. We first fix $t=T_0$ large enough, and then choose N large enough, so that one has $e^{-{\hat{\nu }}_0t}\leqq \frac{1}{2}$ and

$$\begin{aligned} Ck\left\{ m^{3+\kappa }+m^{\kappa -1}e^{-\frac{1}{2}{\hat{\nu }}_0t}+\left( \frac{1}{2}\right) ^{{{\hat{C}}_2T_0^{\frac{5}{4}}}} +\frac{m^{2(\kappa -1)}}{N}\right\} \leqq \frac{1}{16}. \end{aligned}$$

Therefore (3.11) follows. This completes the proof of Lemma 3.4. $\quad \square$

3.2 Approximate Sequence

Now we are in a position to construct solutions to (3.1) or equivalently (3.3). First of all, we consider the following approximate problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \varepsilon f^n+ v\cdot \nabla _xf^n+\nu (v) f^n-Kf^n=g,\\ \displaystyle f^n(x,v)|_{\gamma _{-}}=\left( 1-\frac{1}{n}\right) P_{\gamma }f^n+r, \end{array}\right. } \end{aligned}$$

(3.38)

where $\varepsilon \in (0,1]$ is arbitrary and $n>1$ is an integer. Recall $k={\hat{C}}_1T_0^{\frac{5}{4}}$ with $T_0$ large enough. To the end, we choose $n_0>1$ large enough such that

$$\begin{aligned} \frac{1}{8} \left( 1-\frac{2}{n}+\frac{3}{2n^2}\right) ^{-\frac{k+1}{2}}\leqq \frac{1}{2} \end{aligned}$$

for any $n\geqq n_0$.

Lemma 3.5

Let $\varepsilon >0$, $n\geqq n_0$, and $\beta >3$. Assume $\Vert \nu ^{-1}wg\Vert _{L^\infty }+|wr|_{L^\infty {(\gamma _-)}}<\infty$. Then there exists a unique solution $f^n$ to (3.38) satisfying

$$\begin{aligned} \Vert wf^{n}\Vert _{L^\infty }+|wf^n|_{L^\infty (\gamma )}\leqq C_{\varepsilon ,n}\Big ( |wr|_{L^\infty ({\gamma _-})}+\Vert \nu ^{-1}wg\Vert _{L^\infty } \Big ), \end{aligned}$$

where the positive constant $C_{\varepsilon ,n}>0$ depends only on $\varepsilon$ and n. Moreover, if $\Omega$ is a strictly convex domain, g is continuous in $\Omega \times {\mathbb {R}}^3$ and r is continuous in $\gamma _-$, then $f^n$ is continuous away from grazing set $\gamma _0$.

Proof

We consider the solvability of the following boundary value problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {L}}_\lambda f:=\varepsilon f+v\cdot \nabla _x f+\nu (v) f-\lambda Kf=g,\\ f(x,v)|_{\gamma _-}=\left( 1-\frac{1}{n}\right) P_{\gamma } f+r(x,v), \end{array}\right. } \end{aligned}$$

(3.39)

for $\lambda \in [0,1]$. For brevity we denote ${\mathcal {L}}_\lambda ^{-1}$ to be the solution operator associated with the problem, meaning that $f:={\mathcal {L}}_\lambda ^{-1} g$ is a solution to the BVP (3.39). Our idea is to prove the existence of ${\mathcal {L}}_0^{-1}$, and then extend to obtain the existence of ${\mathcal {L}}_1^{-1}$ in a continuous argument on $\lambda$. Since the proof is very long, we split it into several steps.

Step 1. In this step, we prove the existence of ${\mathcal {L}}_0^{-1}$. We consider the following approximate sequence

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {L}}_0 f^{i+1}=\varepsilon f^{i+1}+v\cdot \nabla _x f^{i+1} + \nu (v) f^{i+1}=g,\\ f^{i+1}(x,v)|_{\gamma _-}=\left( 1-\frac{1}{n}\right) P_{\gamma } f^{i}+r, \end{array}\right. } \end{aligned}$$

(3.40)

for $i=0,1,2,\ldots$, where we have set $f^0\equiv 0$. We will construct $L^\infty$ solutions to (3.40) for $i=0,1,2,\ldots$, and establish uniform $L^\infty$-estimates.

Firstly, we will solve inductively the linear equation (3.40) by the method of characteristics. Let $h^{i+1}(x,v)=w(v)f^{i+1}(x,v)$. For almost every $(x,v)\in {\bar{\Omega }}\times {\mathbb {R}}^3\backslash (\gamma _0\cup \gamma _-)$, one can write

$$\begin{aligned} h^{i+1}(x,v)&=e^{-(\varepsilon +\nu (v))t_{{\mathbf {b}}}} w(v)\left[ \left( 1-\frac{1}{n}\right) P_\gamma f^{i}+r\right] (x_{{\mathbf {b}}}(x,v),v) \nonumber \\&\quad +\int _{t-t_{\mathbf {b}}}^t e^{-(\varepsilon +\nu (v))(t-s)} (wg)(x-v(t-s),v) \,\mathrm{d}s, \end{aligned}$$

(3.41)

and for $(x,v)\in \gamma _-$, we write

$$\begin{aligned} h^{i+1}(x,v)=w(v)\left[ \left( 1-\frac{1}{n}\right) P_\gamma f^i+r\right] (x,v). \end{aligned}$$

(3.42)

Noting the definition of $P_{\gamma } f$, we have

$$\begin{aligned} |wP_{\gamma } f|_{L^\infty }\leqq C|f|_{L^\infty (\gamma _+)}. \end{aligned}$$

(3.43)

We consider (3.41) with $i=0$. Noting $h^0\equiv 0$, then it is straightforward to see that

$$\begin{aligned} \Vert h^{1}\Vert _{L^\infty }\leqq |wr|_{L^\infty (\gamma _-)}+ \frac{C}{\varepsilon } \Vert wg\Vert _{L^\infty }<\infty . \end{aligned}$$

Therefore we have obtained the solution to (3.40) with $i=0$. Assume that we have already solved (3.40) for $i\leqq l$ and obtained

$$\begin{aligned} \Vert h^{l+1}\Vert _{L^\infty }+|h^{l+1}|_{L^\infty (\gamma )}\leqq C_{\varepsilon ,n,l+1} \Big ( |wr|_{L^\infty {(\gamma _-)}}+ \Vert wg\Vert _{L^\infty }\Big )<\infty . \end{aligned}$$

(3.44)

We now consider (3.40) for $i=l+1$. Noting (3.44), then we can solve (3.40) by using (3.41) and (3.42) with $i=l+1$. We still need to prove $h^{l+2}\in L^\infty$. Indeed, it follows from (3.41), (3.42) and (3.43) that

$$\begin{aligned} \Vert h^{l+2}\Vert _{L^\infty }+{|h^{l+2}|_{L^\infty (\gamma )}}&\leqq C(|wr|_{L^\infty {(\gamma _-)}}+ |h^{l+1}|_{L^\infty {(\gamma _+)}} )+\frac{C}{\varepsilon } \Vert wg\Vert _{L^\infty }\\&\leqq C_{\varepsilon ,n,l+2} \Big ( |wr|_{L^\infty {{(}\gamma _-}{)}}+ \Vert wg\Vert _{L^\infty }\Big )<\infty . \end{aligned}$$

Therefore, inductively we have solved (3.40) for $i=0,1,2,\ldots$ and obtained

$$\begin{aligned} \Vert h^{i}\Vert _{L^\infty }+|h^{i}|_{L^\infty (\gamma )}\leqq C_{\varepsilon ,n,i}\Big ( |wr|_{L^\infty {(\gamma _-)}}+ \Vert wg\Vert _{L^\infty }\Big )<\infty \end{aligned}$$

(3.45)

for $i=0,1,2,\ldots$. The positive constant $C_{\varepsilon ,n,i}$ may increase to infinity as $i\rightarrow \infty$. Here, we emphasize that we first need to know the sequence $\{h^i\}_{i=0}^{\infty }$ is in $L^\infty$-space, otherwise one can not use Lemma 3.4 to get uniform $L^\infty$ estimates.

If $\Omega$ is a convex domain, let $(x,v)\in \Omega \times {\mathbb {R}}^3\backslash \gamma _0$, then it holds $v\cdot n(x_{{\mathbf {b}}}(x,v))<0$ which yields that $t_{{\mathbf {b}}}(x,v)$ and $x_{{\mathbf {b}}}(x,v)$ are smooth by Lemma 2 in [29]. Therefore if g and r are continuous, we have that $f^i(x,v)$ is continuous away from grazing set.

Secondly, in order to take the limit $i\rightarrow \infty$, one has to get some uniform estimates. Multiplying (3.40) by $f^{i+1}$ and integrating the resultant equality over $\Omega \times {\mathbb {R}}^3$, one obtains that

$$\begin{aligned}&\varepsilon \Vert f^{i+1}\Vert ^2_{L^2}+\frac{1}{2}|f^{i+1}|^2_{L^2(\gamma _+)}+\Vert f^{i+1}\Vert ^2_{\nu }\nonumber \\&\quad \leqq \frac{1}{2}\left( 1-\frac{2}{n}+\frac{3}{2n^2}\right) |f^{i}|^2_{L^2(\gamma _+)}+C_n |r|^2_{L^2(\gamma _-)}+\frac{C}{\varepsilon }\Vert g\Vert ^2_{L^2}+ \frac{\varepsilon }{4}\Vert f^{i+1}\Vert ^2_{L^2}, \end{aligned}$$

(3.46)

where we have used $|P_\gamma f^i|_{L^2(\gamma _-)}=|P_\gamma f^i|_{L^2(\gamma _+)}\leqq |f^i|_{L^2(\gamma _+)}$. Then, from (3.46), we have

$$\begin{aligned}&\frac{3}{2}\varepsilon \Vert f^{i+1}\Vert ^2_{L^2}+|f^{i+1}|^2_{L^2(\gamma _+)}+2\Vert f^{i+1}\Vert ^2_{\nu }\\&\quad \leqq \left( 1-\frac{2}{n}+\frac{3}{2n^2}\right) |f^{i}|^2_{L^2(\gamma _+)}+C_{\varepsilon ,n}\Big \{ |r|^2_{L^2(\gamma _-)}+\Vert g\Vert ^2_{L^2}\Big \}. \end{aligned}$$

Now we take the difference $f^{i+1}-f^i$ in (3.40), then by similar energy estimate as above, we obtain

$$\begin{aligned}&\frac{3}{2}\varepsilon \Vert f^{i+1}-f^i\Vert ^2_{L^2}+|f^{i+1}-f^i|^2_{L^2(\gamma _+)}+2\Vert f^{i+1}-f^i\Vert ^2_{\nu }\\&\quad \leqq \Big (1-\frac{2}{n}+\frac{3}{2n^2}\Big )|f^{i}-f^{i-1}|^2_{L^2(\gamma _+)}\\&\quad \leqq \cdots \\&\quad \leqq \Big (1-\frac{2}{n}+\frac{3}{2n^2}\Big )^{i} |f^1|^2_{L^2(\gamma )}\\&\quad \leqq C_\varepsilon \cdot \Big (1-\frac{2}{n}+\frac{3}{2n^2}\Big )^{i} \cdot \Big ( |wr|_{L^\infty {(\gamma _-)}}+ \Vert wg\Vert _{L^\infty }\Big )<\infty . \end{aligned}$$

Noting $1-\frac{2}{n}+\frac{3}{2n^2}<1$, thus $\{f^{i}\}_{i=0}^\infty$ is a Cauchy sequence in $L^2$, that is,

$$\begin{aligned} \Vert f^{i}-f^j\Vert ^2_{L^2}+|f^{i}-f^j|^2_{L^2(\gamma _+)}+\Vert {f^i-f^j}\Vert ^2_{\nu }\rightarrow 0,\quad \text{ as } \ i,j\rightarrow \infty . \end{aligned}$$

We also have, for $i=0,1,2,\ldots$, that

$$\begin{aligned} \Vert f^{i}\Vert ^2_{L^2}+|f^{i}|^2_{L^2(\gamma _+)}+\Vert f^{i}\Vert ^2_{\nu }\leqq C_{\varepsilon ,n}\Big \{ |r|^2_{L^2(\gamma _-)}+\Vert g\Vert ^2_{L^2}\Big \}, \end{aligned}$$

(3.47)

where $C_{\varepsilon ,n}>0$ is a positive constant which depends only on $\varepsilon$ and n.

Next we consider the uniform $L^\infty$ estimate. Here we point out that Lemma 3.4 still holds by replacing 1 with $1-\frac{1}{n}$ in the boundary condition, and the constants in Lemma 3.4 do not depend on $n\geqq 1$. Thus we apply Lemma 3.4 to obtain that

$$\begin{aligned} \Vert h^{i+1}\Vert _{L^\infty }&\leqq \frac{1}{8} \sup _{0\leqq l\leqq k} \{\Vert h^{i-l}\Vert _{L^\infty }\} +C\Big \{ |wr|_{L^\infty {(\gamma _-)}}+\Vert \nu ^{-1}wg\Vert _{L^\infty }\Big \}\\&\qquad +C\sup _{0\leqq l\leqq k} \Big \{ \Vert f^{i-l}\Vert _{\nu } \Big \}\\&\leqq \frac{1}{8} \sup _{0\leqq l\leqq k} \{\Vert h^{i-l}\Vert _{L^\infty }\} +C_{\varepsilon ,n}\Big \{ |wr|_{L^\infty {(\gamma _-)}}+\Vert \nu ^{-1}wg\Vert _{L^\infty }\Big \}, \end{aligned}$$

where we have used (3.47) in the second inequality. Now we apply Lemma 6.1 to obtain that for $i\geqq k+1$,

$$\begin{aligned} \Vert h^{i}\Vert _{L^\infty }&\leqq \frac{1}{8} \max _{0\leqq l\leqq k} \Big \{\Vert h^{1}\Vert _{L^\infty }, \ \Vert h^{2}\Vert _{L^\infty }, \cdots , \Vert h^{2k}\Vert _{L^\infty }\Big \}\nonumber \\&\quad +\frac{8+k}{7} C_{\varepsilon ,n}\Big \{ |wr|_{L^\infty {(\gamma _-)}}+\Vert \nu ^{-1}wg\Vert _{L^\infty }\Big \}\nonumber \\&\leqq C_{\varepsilon ,n,k}\Big \{ |wr|_{L^\infty {(\gamma _-)}}+\Vert \nu ^{-1}wg\Vert _{L^\infty }\Big \}, \end{aligned}$$

(3.48)

where we have used (3.45) in the second inequality. Hence it follows from (3.48) and (3.45) that

$$\begin{aligned} \Vert h^{i}\Vert _{L^\infty }&\leqq C_{\varepsilon ,n,k}\Big \{ |wr|_{L^\infty {(\gamma _-)}}+\Vert \nu ^{-1}wg\Vert _{L^\infty }\Big \}, \quad \text{ for } \ i\geqq 1. \end{aligned}$$

(3.49)

Taking the difference $h^{i+1}-h^i$ and then applying Lemma 3.4 to $h^{i+1}-h^i$, we have that for $i\geqq k$,

$$\begin{aligned}&\Vert h^{i+2}-h^{i+1}\Vert _{L^\infty }\nonumber \\&\quad \leqq \frac{1}{8} \max _{0\leqq l\leqq k} \Big \{\Vert h^{i+1-l}-h^{i-l}\Vert _{L^{\infty }}\Big \}+C \sup _{0\leqq l\leqq k} \Big \{\Vert f^{i+1-l}-f^{i-l}\Vert _{\nu }\Big \}\nonumber \\&\quad \leqq \frac{1}{8} \max _{0\leqq l\leqq k} \Big \{\Vert h^{i+1-l}-h^{i-l}\Vert _{L^{\infty }}\Big \}+C_{\varepsilon } \cdot \Big \{ |wr|_{L^\infty {(\gamma _-)}}+\Vert \nu ^{-1}wg\Vert _{L^\infty }\Big \} \cdot \eta _n^{i-k}\nonumber \\&\quad \leqq \frac{1}{8} \max _{0\leqq l\leqq k} \Big \{\Vert h^{i+1-l}-h^{i-l}\Vert _{L^{\infty }}\Big \}+C_{\varepsilon ,k} \Big \{ |wr|_{L^\infty {(\gamma _-)}}+\Vert \nu ^{-1}wg\Vert _{L^\infty }\Big \} \cdot \eta _n^{i+k+1}, \end{aligned}$$

(3.50)

where we have denoted $\eta _n:= (1-\frac{2}{n}+\frac{3}{2n^2})^{1/2} <1$. Here we choose n large enough so that $\frac{1}{8} \eta _n^{-k-1}\leqq \frac{1}{2}$, then it follows from (3.50) and Lemma 6.1 that

$$\begin{aligned} \Vert h^{i+2}-h^{i+1}\Vert _{L^\infty }&\leqq \left( \frac{1}{8}\right) ^{\left[ \frac{i}{k+1}\right] } \max _{0\leqq l\leqq 2k}\Big \{\Vert h^1\Vert _{L^\infty }, \Vert h^2\Vert _{L^\infty }, \cdots , \Vert h^{2k+1}\Vert _{L^\infty }\Big \}\nonumber \\&\quad +C_{\varepsilon , k} \Big \{ |wr|_{L^\infty {(\gamma _-)}}+\Vert \nu ^{-1}wg\Vert _{L^\infty }\Big \} \cdot \eta _n^{i}\nonumber \\&\leqq C_{\varepsilon ,n,k} \cdot \Big \{|wr|_{L^\infty {(\gamma _-)}}+\Vert \nu ^{-1}wg\Vert _{L^\infty }\Big \}\cdot \left\{ \left( \frac{1}{8}\right) ^{\left[ \frac{i}{k+1}\right] } +\eta _n^{i}\right\} , \end{aligned}$$

(3.51)

for $i\geqq k+1$. Then (3.51) implies immediately that $\{h^i\}_{i=0}^\infty$ is a Cauchy sequence in $L^\infty$, that is, there exists a limit function ${h}\in L^\infty$ so that $\Vert h^{i}- {h}\Vert _{L^\infty }\rightarrow 0$ as $i\rightarrow \infty$. Thus we obtained a function $f:=\frac{h}{w}$ solves

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {L}}_0 f=\varepsilon f+ v\cdot \nabla _xf+\nu (v) f=g,\\ \displaystyle f(x,v)|_{\gamma _{-}}=\left( 1-\frac{1}{n}\right) P_{\gamma }f+r, \end{array}\right. } \end{aligned}$$

with $n\geqq n_0$ large enough. Moreover, from (3.49), there exists a constant $C_{\varepsilon ,n,k}$ such that

$$\begin{aligned} \Vert h\Vert _{L^\infty }+|h|_{L^\infty (\gamma )}\leqq C_{\varepsilon ,n,k}\Big \{ |wr|_{L^\infty (\gamma _-)}+\Vert \nu ^{-1}wg\Vert _{L^\infty }\Big \}. \end{aligned}$$

Step 2.A priori estimates. For any given $\lambda \in [0,1]$, let $f^n$ be the solution of (3.39), that is,

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {L}}_\lambda f^n=\varepsilon f^{n}+v\cdot \nabla _x f^{n}+\nu (v) f^{n}-\lambda Kf^n=g,\\ f^{n}(x,v)|_{\gamma _-}=\left( 1-\frac{1}{n}\right) P_{\gamma }f^{n}+r(x,v). \end{array}\right. } \end{aligned}$$

(3.52)

Moreover we also assume that $\Vert wf^{n}\Vert _{L^\infty }+|wf^n|_{L^\infty (\gamma )}<\infty$. Firstly, we shall consider a priori$L^2$-estimates. Multiplying (3.52) by $f^{n}$, one has that

$$\begin{aligned}&\varepsilon \Vert f^{n}\Vert ^2_{L^2}+\frac{1}{2}|f^{n}|^2_{L^2(\gamma _+)}-\frac{1}{2}|\left( 1-\frac{1}{n}\right) P_{\gamma }f^{n}+r|^2_{L^2(\gamma _-)}+\Vert f^{n}\Vert ^2_{\nu }\nonumber \\&\quad \leqq \lambda \langle Kf^n, f^{n}\rangle +\frac{\varepsilon }{4} \Vert f^n\Vert _{L^2}+\frac{C}{\varepsilon }\Vert g\Vert ^2_{L^2}. \end{aligned}$$

(3.53)

We note that $\langle Lf^n, f^{n}\rangle \geqq 0$, which implies that

$$\begin{aligned} \lambda \langle Kf^n, f^{n}\rangle \leqq \lambda \Vert f^n\Vert _{\nu }. \end{aligned}$$

(3.54)

On the other hand, a direct computation shows that

$$\begin{aligned} \left| \left( 1-\frac{1}{n}\right) P_{\gamma }f^{n}+r\right| ^2_{L^2(\gamma _-)}\leqq \left( 1-\frac{2}{n}+\frac{3}{2n^2}\right) |f^n|_{L^2(\gamma _+)}+C_n |r|^2_{L^2(\gamma _-)}.\nonumber \\ \end{aligned}$$

(3.55)

Substituting (3.54) and (3.55) into (3.53), one has that

$$\begin{aligned} \Vert {\mathcal {L}}^{-1}_\lambda g\Vert ^2_{L^2}+|{\mathcal {L}}^{-1}_\lambda g|^2_{L^2(\gamma _+)}=\Vert f^{n}\Vert ^2_{L^2}+|f^{n}|^2_{L^2(\gamma _+)}\leqq C_{\varepsilon ,n} \Big [|r|^2_{L^2(\gamma _-)}+\Vert g\Vert ^2_{L^2}\Big ]. \end{aligned}$$

(3.56)

Let $h^n:=w f^n$. Then, by using (3.12) and (3.56), we obtain

$$\begin{aligned} \Vert w{\mathcal {L}}^{-1}_\lambda g\Vert _{L^\infty }=\Vert h^n\Vert _{L^\infty }\leqq C_{\varepsilon ,n} \Big \{ |wr|_{L^\infty (\gamma _-)}+\Vert \nu ^{-1}wg\Vert _{L^\infty } \Big \}. \end{aligned}$$

(3.57)

On the other hand, Let $\nu ^{-1}wg_1 \in L^\infty$ and $\nu ^{-1}wg_2 \in L^\infty$. Let $f^n_1={\mathcal {L}}^{-1}_\lambda g_1$ and $f^n_2={\mathcal {L}}^{-1}_\lambda g_2$ be the solutions to (3.52) with g replaced by $g_1$ and $g_2$, respectively. Then we have that

$$\begin{aligned} {\left\{ \begin{array}{ll} \varepsilon (f^{n}_2-f^n_1)+v\cdot \nabla _x (f^{n}_2-f^n_1)+\nu (v) (f^{n}_2-f^n_1)-\lambda K(f^{n}_2-f^n_1)=g_2-g_1,\\ (f^{n}_2-f^n_1)(x,v)|_{\gamma _-}=\left( 1-\frac{1}{n}\right) P_{\gamma }(f^{n}_2-f^n_1). \end{array}\right. } \end{aligned}$$

By similar arguments as to those in (3.52)–(3.57), we obtain

$$\begin{aligned} \Vert {\mathcal {L}}^{-1}_\lambda g_2-{\mathcal {L}}^{-1}_\lambda g_1\Vert ^2_{L^2}+|{\mathcal {L}}^{-1}_\lambda g_2-{\mathcal {L}}^{-1}_\lambda g_1|^2_{L^2(\gamma _+)}\leqq C_{\varepsilon ,n} \Vert g_2-g_1\Vert ^2_{L^2}, \end{aligned}$$

(3.58)

and

$$\begin{aligned} \Vert w({\mathcal {L}}^{-1}_\lambda g_2-{\mathcal {L}}^{-1}_\lambda g_1)\Vert _{L^\infty }\leqq C_{\varepsilon ,n} \Vert \nu ^{-1}w(g_2-g_1)\Vert _{L^\infty }. \end{aligned}$$

(3.59)

The uniqueness of solution to (3.52) also follows from (3.58). We point out that the constant $C_{\varepsilon ,n}$ in (3.56), (3.57), (3.58) and (3.59) does not depend on $\lambda \in [0,1]$. This property is crucial for us to extend ${\mathcal {L}}_0^{-1}$ to ${\mathcal {L}}_1^{-1}$ by a bootstrap argument.

Step 3. In this step, we shall prove the existence of solution $f^n$ to (3.39) for sufficiently small $0<\lambda \ll 1$, that is, to prove the existence of operator ${\mathcal {L}}_{\lambda }^{-1}$. Firstly, we define the Banach space

$$\begin{aligned} {\mathbf {X}}:= & {} \Big \{f=f(x,v) : \ wf\in L^\infty (\Omega \times {\mathbb {R}}^3), \ wf\in L^\infty (\gamma ), \ \text{ and } \ f(x,v)|_{\gamma _-}\\= & {} \left( 1-\frac{1}{n}\right) P_\gamma f+r \Big \}. \end{aligned}$$

Now we define

$$\begin{aligned} T_\lambda f={\mathcal {L}}_0^{-1} \Big (\lambda K f+g\Big ). \end{aligned}$$

For any $f_1, f_2\in {\mathbf {X}}$, by using (3.59), we have that

$$\begin{aligned} \Vert w(T_\lambda f_1-T_\lambda f_2)\Vert _{L^\infty }&= \left\| w\{{\mathcal {L}}_0^{-1}(\lambda Kf_1+g)-{\mathcal {L}}_0^{-1}(\lambda Kf_2+g)\} \right\| _{L^\infty }\\&\leqq C_{\varepsilon ,n}\Vert \nu ^{-1} w\{(\lambda Kf_1+g)-(\lambda Kf_2+g)\}\Vert _{L^\infty }\\&\leqq \lambda \Vert \nu ^{-1} w (Kf_1-Kf_2)\Vert _{L^\infty } \\&\leqq \lambda C_{K,\varepsilon ,n} \Vert w(f_1-f_2)\Vert _{L^\infty }, \end{aligned}$$

where we have used (2.11) and (2.13) with $m=1$ in the last inequality. We take $\lambda _*>0$ sufficiently small such that $\lambda _*C_{K,\varepsilon ,n}\leqq 1/2$, then $T_\lambda : {\mathbf {X}}\rightarrow {\mathbf {X}}$ is a contraction mapping for $\lambda \in [0,\lambda _*]$. Thus $T_\lambda$ has a fixed point, that is, $\exists \, f^\lambda \in {\mathbf {X}}$ such that

$$\begin{aligned} f^\lambda =T_\lambda f^\lambda ={\mathcal {L}}_0^{-1} \Big (\lambda K f^\lambda +g\Big ), \end{aligned}$$

which immediately yields that

$$\begin{aligned} {\mathcal {L}}_\lambda f^\lambda =\varepsilon f^\lambda + v\cdot \nabla _x f^\lambda +\nu f^\lambda -\lambda Kf^{\lambda }=g. \end{aligned}$$

Hence, for any $\lambda \in [0,\lambda _*]$, we have solved (3.39) with $f^\lambda ={\mathcal {L}}_\lambda ^{-1}g\in {\mathbf {X}}$. Therefore we have obtained the existence of ${\mathcal {L}}_\lambda ^{-1}$ for $\lambda \in [0,\lambda _*]$. Moreover the operator ${\mathcal {L}}_\lambda ^{-1}$ has the properties (3.56), (3.57), (3.58) and (3.59).

Next we define

$$\begin{aligned} T_{\lambda _*+\lambda }f={\mathcal {L}}_{\lambda _*}^{-1}\Big (\lambda K f+g\Big ). \end{aligned}$$

Noting the estimates for ${\mathcal {L}}_{\lambda _*}^{-1}$ are independent of $\lambda _*$. By similar arguments, we can prove $T_{\lambda _*+\lambda } : {\mathbf {X}}\rightarrow {\mathbf {X}}$ is a contraction mapping for $\lambda \in [0,\lambda _*]$. Then we obtain the exitence of operator ${\mathcal {L}}_{\lambda _*+\lambda }^{-1}$, and (3.56), (3.57), (3.58) and (3.59). Step by step, we can finally obtain the existence of operator ${\mathcal {L}}_1^{-1}$, and ${\mathcal {L}}_1^{-1}$ satisfies the estimates in (3.56), (3.57), (3.58) and (3.59). The continuity is easy to obtain since the convergence of sequence under consideration is always in $L^\infty$. Therefore we complete the proof of Lemma 3.5. $\quad \square$

Lemma 3.6

Let $\varepsilon >0$ and $\beta >3$, and assume $\Vert \nu ^{-1}wg\Vert _{L^\infty }+|wr|_{L^\infty {(\gamma _-)}}<\infty$. Then there exists a unique solution $f^\varepsilon$ to solve the approximate linearized steady Boltzmann equation (3.1). Moreover, it satisfies

$$\begin{aligned} \Vert wf^\varepsilon \Vert _{L^\infty } +|wf^\varepsilon |_{L^\infty (\gamma )} \leqq C_{\varepsilon } \Big \{ |wr|_{L^\infty {(\gamma _-)}}+\Vert \nu ^{-1}wg\Vert _{L^\infty } \Big \}, \end{aligned}$$

(3.60)

where the positive constant $C_{\varepsilon }>0$ depends only on $\varepsilon$. Moreover, if $\Omega$ is a strictly convex domain, g is continuous in $\Omega \times {\mathbb {R}}^3$ and r is continuous in $\gamma _-$, then $f^\varepsilon$ is continuous away from the grazing set $\gamma _0$.

Proof

Let $f^n$ be the solution of (3.38) constructed in Lemma 3.5 for $n\geqq n_0$ with $n_0$ large enough. Multiplying (3.38) by $f^n$ and using the coercivity estimate (2.7), one obtains that

$$\begin{aligned} \varepsilon \Vert f^n\Vert ^2_{L^2}+|f^n|^2_{L^2(\gamma _+)}+2c_0 \Vert {(I-P)}f^n\Vert ^2_{\nu } \leqq C_{\varepsilon } \Vert g\Vert ^2_{L^2}+ \Big | \Big (1-\frac{1}{n}\Big )P_\gamma f^n+r \Big |^2_{L^2(\gamma _-)}. \end{aligned}$$

(3.61)

Here the projection P is defined by (2.6). A direct calculation shows that

$$\begin{aligned} \Big | \Big (1-\frac{1}{n}\Big )P_\gamma f^n+r \Big |^2_{L^2(\gamma _-)}&\leqq |P_\gamma f^n|^2_{L^2(\gamma _-)}+2|P_\gamma f^n|_{L^2(\gamma _-)}\cdot |r|_{L^2(\gamma _-)}+|r|^2_{L^2(\gamma _-)}\\&\leqq |P_\gamma f^n|^2_{L^2(\gamma _+)}+\eta |P_\gamma f^n|^2_{L^2(\gamma _+)}+C_\eta |r|^2_{L^2(\gamma _-)}, \end{aligned}$$

which, together with (3.61), yields that

$$\begin{aligned}&\varepsilon \Vert f^n\Vert ^2_{L^2}+|(I-P_\gamma )f^n|^2_{\gamma _+}+2c_0 \Vert {(I-P)}f^n\Vert ^2_{\nu } \leqq \eta |P_\gamma f^n|^2_{L^2(\gamma _+)}\nonumber \\&\quad +C_{\varepsilon ,\eta } (\Vert g\Vert ^2_{L^2} + |r|^2_{L^2(\gamma _-)}). \end{aligned}$$

(3.62)

where $\eta >0$ is a small constant to be chosen later.

We still need to bound the first term on the right-hand side of (3.62). Firstly, a direct calculation shows that

$$\begin{aligned} \frac{1}{2} |P_\gamma f^n|^2_{L^2(\gamma _+)}\leqq |P_\gamma f^n I_{\gamma _+\backslash \gamma _+^{\varepsilon '}}|^2_{L^2(\gamma _+)}, \end{aligned}$$

(3.63)

provided that $0<\varepsilon '\ll 1$. We note that

$$\begin{aligned} f^n(x,v)=(I-P_\gamma )f^n(x,v)+P_\gamma f^n(x,v),\quad \ \forall \, (x,v)\in \gamma _+, \end{aligned}$$

which yields that

$$\begin{aligned} |P_\gamma f^n I_{\gamma _+\backslash \gamma _+^{\varepsilon '}}|^2_{L^2(\gamma )}=2|f^n I_{\gamma _+\backslash \gamma _+^{\varepsilon '}}|^2_{L^2(\gamma )}+2|(I-P_\gamma )f^n I_{\gamma _+\backslash \gamma _+^{\varepsilon '}}|^2_{L^2(\gamma )}. \end{aligned}$$

(3.64)

On the other hand, it follows from (3.38) that

$$\begin{aligned} \frac{1}{2}v\cdot \nabla _x(|f^n|^2)=-\varepsilon |f^n|^2-f^n Lf^n+g f^n, \end{aligned}$$

which yields that

$$\begin{aligned} \Vert v\cdot \nabla _x(|f^n|^2)\Vert _{L^1}\leqq C\Big \{\Vert f^n\Vert ^2_{L^2}+\Vert {(I-P)}f^n\Vert ^2_{\nu }+\Vert g\Vert ^2_{L^2}\Big \}. \end{aligned}$$

(3.65)

It follows from (3.65) and (3.62) that

$$\begin{aligned} |f^n I_{\gamma _+\backslash \gamma _+^{\varepsilon '}}|^2_{L^2(\gamma )}&=|(f^n)^2 I_{\gamma _+\backslash \gamma _+^{\varepsilon '}}|_{L^1(\gamma )} \\&\leqq C_{\varepsilon '}\Big \{ \Vert f^n\Vert ^2_{L^2}+ \Vert v\cdot \nabla _x(|f^n|^2)\Vert _{L^1} \Big \}\\&\leqq C_{\varepsilon '} \Big \{\Vert f^n\Vert ^2_{L^2}+\Vert {(I-P)}f^n\Vert ^2_{\nu }+\Vert g\Vert ^2_{L^2}\Big \}\\&\leqq C_{\varepsilon ',\varepsilon } \eta |P_\gamma f^n|^2_{L^2(\gamma _+)}+C_{\varepsilon ',\varepsilon ,\eta } \Big \{\Vert g\Vert ^2_{L^2} + |r|^2_{L^2(\gamma _-)}\Big \}, \end{aligned}$$

where in the second line we have used (4.7) which will be given in Lemma 4.1 later on. The above estimate together with (3.63) and (3.64) yield that

$$\begin{aligned} |P_\gamma f^n|^2_{L^2(\gamma _+)}\leqq C_{\varepsilon ',\varepsilon } \eta |P_\gamma f^n|^2_{L^2(\gamma _+)}+C_{\varepsilon ',\varepsilon ,\eta } \Big \{\Vert g\Vert ^2_{L^2} + |r|^2_{L^2(\gamma _-)}\Big \}. \end{aligned}$$

Taking $\eta$ small so that $C_{\varepsilon ',v} \eta \leqq \frac{1}{2}$, one obtains that

(3.66)

Combining (3.66) and (3.62), one has

$$\begin{aligned} \Vert f^n\Vert ^2_{L^2}+|f^n|^2_{L^2(\gamma _+)}+2c_0 \Vert {(I-P)}f^n\Vert ^2_{\nu } \leqq C_{\varepsilon ',\varepsilon } (\Vert g\Vert ^2_{L^2} + |r|^2_{L^2(\gamma _-)}). \end{aligned}$$

(3.67)

We apply (3.12) and use (3.67) to obtain

$$\begin{aligned} \Vert wf^n\Vert _{L^\infty }{+|wf^n|_{L^\infty (\gamma )}}&\leqq C\Big \{\Vert \nu ^{-1}w g\Vert _{L^\infty }+|wr|_{L^\infty {(\gamma _-)}}+\Vert f^n\Vert _{L^2}\Big \}\\&\leqq C_{\varepsilon ',\varepsilon } \Big \{\Vert \nu ^{-1}w g\Vert _{L^\infty }+|wr|_{L^\infty {(\gamma _-)}}\Big \}. \end{aligned}$$

Taking the difference $f^{n_1}-f^{n_2}$ with $n_1,n_2\geqq n_0$, we know that

$$\begin{aligned} {\left\{ \begin{array}{ll} \varepsilon (f^{n_1}-f^{n_2})+ v\cdot \nabla _x (f^{n_1}-f^{n_2})+L (f^{n_1}-f^{n_2})=0,\\ (f^{n_1}-f^{n_2})(x,v)|_{\gamma _{-}}=(1-\frac{1}{n_1})P_{\gamma } (f^{n_1}-f^{n_2})+\left( \frac{1}{n_2}-\frac{1}{n_1}\right) P_\gamma f^{n_2}. \end{array}\right. } \end{aligned}$$

(3.68)

Multiplying (3.68) by $f^{n_1}-f^{n_2}$, and integrating it over $\Omega \times {\mathbb {R}}^3$, by similar arguments as in (3.61)–(3.67), we can obtain

$$\begin{aligned}&\Vert (f^{n_1}-f^{n_2})\Vert ^2_{L^2}+|(f^{n_1}-f^{n_2})|^2_{L^2(\gamma _+)}+2c_0 \Vert {(I-P)}(f^{n_1}-f^{n_2})\Vert ^2_{\nu }\nonumber \\&\quad \leqq C_{\varepsilon ',\varepsilon } \Big | \Big (\frac{1}{n_2}-\frac{1}{n_1}\Big ) P_\gamma f^{n_2}\Big |^2_{L^2(\gamma _-)} \leqq C_{\varepsilon ',\varepsilon } \cdot \Big \{\Big (\frac{1}{n_1}\Big )^2+\Big (\frac{1}{n_2}\Big )^2\Big \}\cdot |f^{n_2}|^2_{L^2(\gamma _+)}\nonumber \\&\quad \leqq C_{\varepsilon ',\varepsilon } \cdot (\Vert g\Vert ^2_{L^2} + |r|^2_{L^2(\gamma _-)})\cdot \Big \{\Big (\frac{1}{n_1}\Big )^2+\Big (\frac{1}{n_2}\Big )^2\Big \}\rightarrow 0, \end{aligned}$$

(3.69)

as $n_1$, $n_2 \rightarrow \infty$, where we have used the uniform estimate (3.67) in the last inequality. Applying (3.12) to $f^{n_1}-f^{n_2}$ and using (3.69), then one has

$$\begin{aligned}&\Vert w(f^{n_1}-f^{n_2})\Vert _{L^\infty }{+|w(f^{n_1}-f^{n_2})|_{L^\infty (\gamma )}}\\&\quad \leqq C \Big |w \Big (\frac{1}{n_2}-\frac{1}{n_1}\Big ) P_\gamma f^{n_2}\Big |_{L^\infty (\gamma _-)}+C\Vert f^{n_1}-f^{n_2}\Vert _{{L^2}}\\&\quad \leqq C_{\varepsilon ',\varepsilon } \cdot (\Vert wg\Vert _{L^\infty } + |wr|_{L^\infty (\gamma _-)})\cdot \Big \{\frac{1}{n_1}+\frac{1}{n_2}\Big \}\rightarrow 0, \end{aligned}$$

as $n_1,\ n_2 \rightarrow \infty$, which yields that $wf^n$ is a Cauchy sequence in $L^\infty$. We denote $f^\varepsilon =\lim _{n\rightarrow \infty } f^n$, then it is direct to check that $f^\varepsilon$ is a solution to (3.1), and (3.60) holds. The continuity of $f^\varepsilon$ is easy to obtain since the convergence of sequences is always in $L^\infty$ and $f^n$ is continuous away from the grazing set. Therefore we have completed the proof of Lemma 3.6. $\quad \square$

Now we assume that

$$\begin{aligned} \iint _{\Omega \times {\mathbb {R}}^3} g(x,v) \mu ^{\frac{1}{2}}(v) \,\mathrm{d}v\mathrm{d}x= \int _{\gamma _-} r(x,v) \mu ^{\frac{1}{2}}(v) \,\mathrm{d}\gamma =0. \end{aligned}$$

(3.70)

Lemma 3.7

Let f be a solution

$$\begin{aligned} v\cdot \nabla _x f+Lf=g,\quad f_{\gamma _-} = P_{\gamma }f+r \end{aligned}$$

in the weak sense of

$$\begin{aligned}&\int _{\gamma } \psi f \ \{v\cdot n(x)\} \,\mathrm{d}v \mathrm{d}S_x-\int _{\Omega \times {\mathbb {R}}^3} v\cdot \nabla _x\psi f \,\mathrm{d}v\mathrm{d}x\\&\quad = -\int _{\Omega \times {\mathbb {R}}^3} \psi \ Lf \,\mathrm{d}v\mathrm{d}x+\int _{\Omega \times {\mathbb {R}}^3} \psi g \mathrm{d}v \mathrm{d}x, \end{aligned}$$

where $\psi \in H^1(\Omega \times {\mathbb {R}}^3)$. Assume $\int _{\Omega \times {\mathbb {R}}^3}f \sqrt{\mu } \mathrm{d}v\mathrm{d}x=0$ and (3.70), then it holds that

$$\begin{aligned} \Vert {P}f\Vert ^2_{L^2}\leqq C\Big \{ \Vert {(I-P)}f\Vert ^2_{\nu }+\Vert g\Vert _{L^2} + |(I-P_\gamma )f|^2_{L^2(\gamma _+)} + |r|^2_{L^2(\gamma _-)}\Big \}. \end{aligned}$$

Proof

The proof is almost the same to Lemma 3.3 in [21], the details are omitted here for simplicity of presentation. $\quad \square$

Lemma 3.8

Assume (3.70). Let $\beta >3+|\kappa |$, and assume $\Vert \nu ^{-1}wg\Vert _{L^\infty }+|wr|_{L^\infty {(\gamma _-)}}<\infty$. Then there exists a unique solution $f=f(x,v)$ to the linearized steady Boltzmann equation

$$\begin{aligned} v\cdot \nabla _xf+Lf=g,\quad f(x,v)|_{\gamma _{-}}=P_{\gamma }f+r, \end{aligned}$$

(3.71)

such that $\int _{\Omega \times {\mathbb {R}}^3} f \sqrt{\mu } dvdx=0$ and

$$\begin{aligned} \Vert wf\Vert _{L^\infty } +|wf|_{L^\infty (\gamma )} \leqq C\Big \{ |wr|_{L^\infty {(\gamma _-)}}+\Vert \nu ^{-1}wg\Vert _{L^\infty } \Big \}. \end{aligned}$$

(3.72)

Moreover, if $\Omega$ is a strictly convex domain, g is continuous in $\Omega \times {\mathbb {R}}^3$ and r is continuous in $\gamma _-$, then f is continuous away from the grazing set $\gamma _0$.

Proof

Let $f^\varepsilon$ be the solution of (3.1) constructed in Lemma 3.6 for $\varepsilon >0$. Multiplying the first equation of (3.1) by $\sqrt{\mu }$, taking integration over $\Omega \times {\mathbb {R}}^3$, and noting (3.70), it is straightforward to see that

$$\begin{aligned} \int _{\Omega \times {\mathbb {R}}^3} f^\varepsilon (x,v)\mu ^{\frac{1}{2}}(v) \ \mathrm{d}v\mathrm{d}x=0 \end{aligned}$$

for any ${\varepsilon }>0$. Multiplying the first equation of (3.1) by $f^\varepsilon$ and integrating the resultant equation over $\Omega \times {\mathbb {R}}^3$, it follows from Cauchy inequality that

$$\begin{aligned}&\varepsilon \Vert f^\varepsilon \Vert ^2_{L^2}+\frac{1}{2}|(I-P_\gamma )f^\varepsilon |^2_{L^2(\gamma _+)}+c_0\Vert (I-P)f^\varepsilon \Vert ^2_{\nu }\nonumber \\&\quad \leqq \eta [\Vert f^\varepsilon \Vert ^2_{\nu }+|P_\gamma f^\varepsilon |^2_{L^2(\gamma _+)}]+C_\eta [\Vert \nu ^{-\frac{1}{2}}g\Vert ^2_{L^2}+|r|^2_{L^2(\gamma _-)}]. \end{aligned}$$

(3.73)

Applying Lemma 3.7 to $f^\varepsilon$, we obtain

$$\begin{aligned} \Vert {P}f^\varepsilon \Vert ^2_{L^2}\leqq C\Big \{ \Vert {(I-P)}f^\varepsilon \Vert ^2_{\nu }+\varepsilon \Vert f^\varepsilon \Vert ^2_{L^2}+ |(I-P_\gamma )f^\varepsilon |^2_{L^2(\gamma _+)} +\Vert g\Vert ^2_{L^2}+ |r|^2_{L^2(\gamma _-)}\Big \}, \end{aligned}$$

which, together with (3.73), implies that

$$\begin{aligned}&\varepsilon \Vert f^\varepsilon \Vert ^2_{L^2}+\Vert f^\varepsilon \Vert ^2_{\nu }+|(I-P_\gamma )f^\varepsilon |^2_{L^2(\gamma _+)}\nonumber \\&\quad \leqq C\eta [\Vert f^\varepsilon \Vert ^2_{\nu }+|P_\gamma f^\varepsilon |^2_{L^2(\gamma _+)}]+C_\eta [\Vert \nu ^{-\frac{1}{2}}g\Vert ^2_{L^2}+|r|^2_{L^2(\gamma _-)}], \end{aligned}$$

(3.74)

where $\eta >0$ is a small positive constant to be determined later.

To control the term $|P_\gamma f^\varepsilon |^2_{L^2(\gamma _+)}$ on the right-hand side of (3.74), we should be careful since we do not have the uniform bound on $\Vert f^\varepsilon \Vert _{L^2}$. Denote

$$\begin{aligned} z^\varepsilon _{\gamma }(x):=\int _{v'\cdot n(x)>0} f^\varepsilon (x,v') \sqrt{\mu (v')} \{v'\cdot n(x)\}\,\mathrm{d}v', \end{aligned}$$

then one has $P_{\gamma } f^\varepsilon =z^\varepsilon _{\gamma }(x) \mu ^{\frac{1}{2}}(v)$. A direct calculation shows that

$$\begin{aligned} \int _{v\cdot n(x)\geqq \varepsilon ', \ \varepsilon '\leqq |v|\leqq \frac{1}{\varepsilon '}} \nu (v) \mu (v) |v\cdot n(x)| \mathrm{d}v\geqq c_1>0, \end{aligned}$$

(3.75)

provided that $0<\varepsilon '\ll 1$, where $c_1>0$ is a positive constant independent of $\varepsilon '$. By using (3.75), we have that

$$\begin{aligned} |\sqrt{\nu }P_\gamma f^\varepsilon I_{\gamma _+\backslash \gamma _+^{\varepsilon '}}|^2_{L^2}&=\int _{\partial \Omega }|z_\gamma ^\varepsilon (x)|^2 {\mathrm{d}S_x}\cdot \int _{v\cdot n(x)\geqq \varepsilon ', \ \varepsilon '\leqq |v|\leqq \frac{1}{\varepsilon '}} \nu (v) \mu (v) |v\cdot n(x)| \mathrm{d}v\\&\geqq c_1 |P_\gamma f^\varepsilon |^2_{L^2(\gamma _+)}, \end{aligned}$$

which yields that

$$\begin{aligned} |P_\gamma f^\varepsilon |^2_{L^2(\gamma _+)}&\leqq C|\sqrt{\nu }P_\gamma f^\varepsilon I_{\gamma _+\backslash \gamma _+^{\varepsilon '}}|^2_{L^2}\nonumber \\&\leqq C \Big \{ |\sqrt{\nu } f^\varepsilon I_{\gamma _+\backslash \gamma _+^{\varepsilon '}}|^2_{L^2}+ |\sqrt{\nu }(I-P_\gamma ) f^\varepsilon I_{\gamma _+\backslash \gamma _+^{\varepsilon '}}|^2_{L^2}\Big \}\nonumber \\&\leqq C|\sqrt{\nu } f^\varepsilon I_{\gamma _+\backslash \gamma _+^{\varepsilon '}}|^2_{L^2}+C\eta [\Vert f^\varepsilon \Vert ^2_{\nu }+|P_\gamma f^\varepsilon |^2_{L^2(\gamma _+)}]\nonumber \\&\quad +C_\eta [\Vert \nu ^{-\frac{1}{2}}g\Vert ^2_{L^2}+|r|^2_{L^2(\gamma _-)}]. \end{aligned}$$

(3.76)

It follows from (3.1) that

$$\begin{aligned} {\frac{1}{2}}v\cdot \nabla _x(\nu |f^\varepsilon |^2)=-\varepsilon \nu |f^\varepsilon |^2-\nu f^\varepsilon Lf^\varepsilon +\nu f^\varepsilon g, \end{aligned}$$

which implies that

$$\begin{aligned} \Vert v\cdot \nabla _x(\nu |f^\varepsilon |^2)\Vert _{L^1}\leqq C\{\Vert {f^{\varepsilon }}\Vert ^2_{\nu }+\Vert g\Vert ^2_{\nu }\}. \end{aligned}$$

(3.77)

It follows from (3.77) and (4.7) that

$$\begin{aligned} \frac{1}{2}|\sqrt{\nu } f^\varepsilon I_{\gamma _+\backslash \gamma _+^{\varepsilon '}}|^2_{L^2(\gamma _+)}&={\frac{1}{2}}|\nu (f^\varepsilon )^2 I_{\gamma _+\backslash \gamma _+^{\varepsilon '}}|_{L^1(\gamma _+)}\\&\quad \leqq C\Big \{ \Vert \nu \ (f^\varepsilon )^2\Vert _{L^1} +\Vert v\cdot \nabla _x(\nu |f^\varepsilon |^2)\Vert _{L^1}\Big \}\\&\leqq C\{\Vert f^\varepsilon \Vert ^2_{\nu }+\Vert g\Vert ^2_{\nu }\}, \end{aligned}$$

which, together with (3.76), and by taking $\eta >0$ suitably small, yields that

$$\begin{aligned} |P_\gamma f^\varepsilon |^2_{L^2(\gamma _+)}\leqq C\Vert f^\varepsilon \Vert ^2_{\nu }+C [\Vert \nu ^{-\frac{1}{2}}g\Vert ^2_{L^2}+|r|^2_{L^2(\gamma _-)}]. \end{aligned}$$

(3.78)

Combining (3.78) and (3.74), then taking $\eta >0$ small, one has that

$$\begin{aligned} \Vert f^\varepsilon \Vert ^2_{\nu }+|f^\varepsilon |^2_{L^2(\gamma _+)}\leqq C [\Vert \nu ^{-\frac{1}{2}}g\Vert ^2_{L^2}+|r|^2_{L^2(\gamma _-)}]. \end{aligned}$$

(3.79)

Applying (3.12) to $f^\varepsilon$ and using (3.79), then we obtain

$$\begin{aligned} \Vert wf^\varepsilon \Vert _{L^\infty }{+|wf^{\varepsilon }|_{L^\infty (\gamma )}}\leqq C [\Vert \nu ^{-\frac{1}{2}}wg\Vert _{L^\infty }+|wr|_{L^\infty (\gamma _-)}]. \end{aligned}$$

(3.80)

Next we consider the convergence of $f^\varepsilon$ as $\varepsilon \rightarrow 0+$. For any $\varepsilon _1,\varepsilon _2>0$, we consider the difference $f^{\varepsilon _2}-f^{\varepsilon _1}$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} v\cdot \nabla _x (f^{\varepsilon _2}-f^{\varepsilon _1})+ L(f^{\varepsilon _2}-f^{\varepsilon _1})=-\varepsilon _2 f^{\varepsilon _2}+\varepsilon _1 f^{\varepsilon _1},\\ (f^{\varepsilon _2}-f^{\varepsilon _1})|_{\gamma _-}=P_\gamma (f^{\varepsilon _2}-f^{\varepsilon _1}). \end{array}\right. } \end{aligned}$$

(3.81)

Multiplying (3.81) by $f^{\varepsilon _2}-f^{\varepsilon _1}$, integrating the resultant equation and by similar arguments as in (3.73)–(3.79), one gets

$$\begin{aligned}&\Vert f^{\varepsilon _2}-f^{\varepsilon _1}\Vert ^2_{\nu }+|f^{\varepsilon _2}-f^{\varepsilon _1}|^2_{L^2(\gamma _+)}\nonumber \\&\quad \leqq C \Vert \nu ^{-\frac{1}{2}}(\varepsilon _2 f^{\varepsilon _2}-\varepsilon _1 f^{\varepsilon _1})\Vert ^2_{L^2} \leqq C(\varepsilon _1^2+\varepsilon _2^2) [\Vert w f^{\varepsilon _1}\Vert ^2_{L^\infty } + \Vert w f^{\varepsilon _2}\Vert ^2_{L^\infty } ]\nonumber \\&\quad \leqq C(\varepsilon _1^2+\varepsilon _2^2)\cdot [\Vert \nu ^{-1}wg\Vert _{L^\infty }+|wr|_{L^\infty (\gamma _-)}]^2, \end{aligned}$$

(3.82)

as $\varepsilon _1$, $\varepsilon _2\rightarrow 0+$, where we have used (3.80) in the last inequality. Finally, applying (3.12) to $f^{\varepsilon _2}-f^{\varepsilon _1}$, then we obtain

$$\begin{aligned}&\Vert \nu w(f^{\varepsilon _2}-f^{\varepsilon _1})\Vert _{L^\infty }{+|\nu w(f^{\varepsilon _2}-f^{\varepsilon _1})|_{L^\infty (\gamma )}}\nonumber \\&\quad \leqq C\Big \{ \Vert w (\varepsilon _2 f^{\varepsilon _2}-\varepsilon _1 f^{\varepsilon _1})\Vert _{L^\infty } +\Vert f^{\varepsilon _2}-f^{\varepsilon _1}\Vert _{\nu } \Big \}\nonumber \\&\quad \leqq C(\varepsilon _1+\varepsilon _2)\cdot [\Vert \nu ^{-1}wg\Vert _{L^\infty }+|wr|_{L^\infty (\gamma _-)}], \end{aligned}$$

(3.83)

as $\varepsilon _1$, $\varepsilon _2\rightarrow 0+$, where we have used (3.82) and (3.80) above. Here we have to demand $\beta >3+|\kappa |$ so that we can apply (3.12). We also point out here that we can only obtain the convergence in a weak norm $L^\infty _{\nu w}$ but not $L^\infty _w$. The main reason is that for soft potentials, in order to get the uniform $L^\infty _w$ estimate, one has to demand g to has the more velocity weight. With (3.83), we know that there exists a function f so that $\Vert \nu w(f^{\varepsilon }-f)\Vert _{L^\infty }\rightarrow 0$ as $\varepsilon \rightarrow 0+$. And it is direct to see that f solves (3.71). Also, (3.72) follows immediately from (3.80). If $\Omega$ is convex, the continuity of f directly follows from the $L^\infty$-convergence. Therefore the proof of Lemma 3.8 is complete. $\quad \square$

3.3 Proof of Theorem 1.1.

We consider the following iterative sequence

$$\begin{aligned} {\left\{ \begin{array}{ll} v\cdot \nabla _x f^{j+1}+Lf^{j+1}=\Gamma (f^j,f^j),\\ f^{j+1}|_{\gamma _-}=P_\gamma f^{j+1}+\frac{{\mu _\theta }-\mu }{\sqrt{\mu }} +\frac{{\mu _\theta }-\mu }{\sqrt{\mu }}\int _{v'\cdot n(x)>0} f^j(x,v')\sqrt{\mu (v')} \{v'\cdot n(x)\}\, \mathrm{d}v', \end{array}\right. } \end{aligned}$$

(3.84)

for $j=0,1,2\cdots$ with $f^0\equiv 0$. A direct computation shows that

$$\begin{aligned} \int _{\Omega \times {\mathbb {R}}^3} \Gamma (f^j,f^j)\mu ^{\frac{1}{2}}(v)\,\mathrm{d}v\mathrm{d}x=0,\quad \int _{v\cdot n(x)<0} [\mu _{{\theta }} (v)-\mu (v)] \{v\cdot n(x)\}\,\mathrm{d}v =0, \end{aligned}$$

(3.85)

which yields that

$$\begin{aligned} \int _{\gamma _-} \left\{ \frac{{\mu _\theta }-\mu }{\sqrt{\mu }} +\frac{{\mu _\theta }-\mu }{\sqrt{\mu }}\int _{v'\cdot n(x)>0} f^j(x,v')\sqrt{\mu (v')} \{v'\cdot n(x)\}\,\mathrm{d}v' \right\} \mu ^{\frac{1}{2}}(v)\,\mathrm{d}\gamma \equiv 0. \end{aligned}$$

We note that

$$\begin{aligned} \Vert \nu ^{-1} w \Gamma (f^j,f^j)\Vert _{L^\infty }\leqq C \Vert wf^j\Vert ^2_{L^\infty }, \end{aligned}$$

(3.86)

and

$$\begin{aligned}&\left| w\left\{ \frac{{\mu _\theta }-\mu }{\sqrt{\mu }} +\frac{{\mu _\theta }-\mu }{\sqrt{\mu }}\int _{v'\cdot n(x)>0} f^j(x,v')\sqrt{\mu (v')} \{v'\cdot n(x)\}\,\mathrm{d}v'\right\} \right| _{L^\infty {(\gamma _-)}}\nonumber \\&\quad \leqq C\delta +C\delta |f^j|_{L^\infty {(\gamma _+)}}. \end{aligned}$$

(3.87)

Noting (3.85)–(3.87), and using Lemma 3.8, we can solve (3.84) inductively for $j=0,1,2,\ldots$. Moreover, it follows from (3.72), (3.86) and (3.87) that

$$\begin{aligned} \Vert wf^{j+1}\Vert _{L^\infty }+|wf^{j+1}|_{L^\infty (\gamma )}\leqq C_1\delta +C_1\delta |f^j|_{L^\infty {(\gamma _+)}}+C_1 \Vert wf^j\Vert ^2_{L^\infty }. \end{aligned}$$

(3.88)

By induction, we shall prove that

$$\begin{aligned} \Vert wf^{j}\Vert _{L^\infty }+|wf^{j}|_{L^\infty (\gamma )}\leqq 2C_1\delta ,\quad \text{ for } \ j=1,2,\ldots . \end{aligned}$$

(3.89)

Indeed, for $j=0$, it follows from $f^0\equiv 0$ and (3.88) that

$$\begin{aligned} \Vert wf^{1}\Vert _{L^\infty }+|wf^{1}|_{L^\infty (\gamma )}\leqq C_1\delta . \end{aligned}$$

Now we assume that (3.89) holds for $j=1,2\cdots , l$, then we consider the case for $j=l+1$. Indeed it follows from (3.88) that

$$\begin{aligned} \Vert wf^{l+1}\Vert _{L^\infty }+|wf^{l+1}|_{L^\infty (\gamma )}&\leqq C_1\delta +C_1\delta |f^l|_{L^\infty {(\gamma )}}+C_1 \Vert wf^l\Vert ^2_{L^\infty }\\&\leqq C_1\delta (1+2C_1\delta +4C_1^2\delta )\leqq \frac{3}{2}C_1\delta , \end{aligned}$$

where we have used (3.89) with $j=l$, and chosen $\delta >0$ small enough such that $2C_1\delta +4C_1^2 \delta \leqq 1/2$. Therefore we have proved (3.89) by induction.

Finally we consider the convergence of sequence $f^j$. For the difference $f^{j+1}-f^j$, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle v\cdot \nabla _x (f^{j+1}-f^j)+L(f^{j+1}-f^j)=\Gamma (f^{j}-f^{j-1},f^{j})+\Gamma (f^{j-1},f^{j}-f^{j-1}),\\[2mm] \displaystyle (f^{j+1}-f^j)|_{\gamma _-}=P_\gamma (f^{j+1}-f^j)\\ \displaystyle \quad +\frac{{\mu _\theta }-\mu }{\sqrt{\mu }}\int _{v'\cdot n(x)>0} [f^j-f^{j-1}](x,v')\sqrt{\mu (v')} \{v'\cdot n(x)\}\,\mathrm{d}v'. \end{array}\right. } \end{aligned}$$

(3.90)

Applying (3.72) to (3.90), we have that

$$\begin{aligned}&\Vert w\{f^{j+1}-f^j\}\Vert _{L^\infty }+|w\{f^{j+1}-f^j\}|_{L^\infty (\gamma )}\nonumber \\&\quad \leqq C\Big \{ \Vert \nu ^{-1}w\Gamma (f^{j}-f^{j-1},f^{j})\Vert _{L^\infty } +\Vert \nu ^{-1}w\Gamma (f^{j-1},f^{j}-f^{j-1})\Vert _{L^\infty }\Big \}\nonumber \\&\qquad +C\Big |w\Big \{\frac{{\mu _\theta }-\mu }{\sqrt{\mu }}\int _{v'\cdot n(x)>0} [f^j-f^{j-1}](x,v')\sqrt{\mu (v')} \{v'\cdot n(x)\} \mathrm{d}v'\Big \}\Big |_{L^\infty {(\gamma _-)}}\nonumber \\&\quad \leqq C[\delta +\Vert wf^{j}\Vert _{L^\infty }+\Vert wf^{j-1}\Vert _{L^\infty }]\cdot \Big \{\Vert w(f^{j}-f^{j-1})\Vert _{L^\infty }+|w(f^j-f^{j-1})|_{L^\infty {(\gamma _+)}}\Big \}\nonumber \\&\quad \leqq C\delta \Big \{\Vert w(f^{j}-f^{j-1})\Vert _{L^\infty }+|w(f^j-f^{j-1})|_{L^\infty {(\gamma _+)}}\Big \}\nonumber \\&\quad \leqq \frac{1}{2} \Big \{\Vert w(f^{j}-f^{j-1})\Vert _{L^\infty }+|w(f^j-f^{j-1})|_{L^\infty {(\gamma _+)}}\Big \}, \end{aligned}$$

(3.91)

where we have used (3.89) and taken $\delta >0$ small such that $C\delta \leqq 1/2$. Hence $f^j$ is a Cauchy sequence in $L^\infty$, then we obtain the solution by taking the limit $f_*=\lim _{j\rightarrow \infty } f^j$. The uniqueness can also be obtained by using the inequality as (3.91).

If $\Omega$ is convex, the continuity of $f_*$ is a direct consequence of $L^\infty$-convergence. The positivity of $F_*:=\mu +\sqrt{\mu } f_*$ will be proved in Section 4. Therefore we complete the proof of Theorem 1.1. $\quad \square$

4 Dynamical Stability Under Small Perturbations

In this section, we are concerned with the large-time asymptotic stability of the steady solution $F_*$ obtained in Theorem 1.1. For this purpose, we introduce the perturbation

$$\begin{aligned} f(t,x,v):=\frac{F(t,x,v)-F_*(x,v)}{\sqrt{\mu }}. \end{aligned}$$

Then the initial-boundary value problem on f(t, x, v) reads as

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t}f+v\cdot \nabla _{x}f+Lf=-L_{\sqrt{\mu }f_{*}}f+\Gamma (f,f),\\&f(t,x,v)|_{t=0}=f_{0}(x,v):=\frac{F(0,x,v)-F_*(x,v)}{\mu ^{\frac{1}{2}}(v)},\\&f(t,x,v)|_{\gamma _{-}}=P_{\gamma }f+\frac{\mu _{\theta }-\mu }{\sqrt{\mu }}\int _{n(x)\cdot u>0}f(t,x,u)\sqrt{\mu (u)}|n(x)\cdot u|\,\mathrm{d}u. \end{aligned} \right. \end{aligned}$$

(4.1)

Here $P_{\gamma }f$ is defined in (3.2), the linearized collision operator L is defined in (2.1), the nonlinear term $\Gamma (f,f)$ is defined in (2.2) and

$$\begin{aligned} L_{\sqrt{\mu }f_{*}}f:=-\frac{1}{\sqrt{\mu }}\left[ Q(\sqrt{\mu }f_{*},\sqrt{\mu }f)+Q(\sqrt{\mu }f,\sqrt{\mu }f_{*})\right] . \end{aligned}$$

Recall (1.9). Let

$$\begin{aligned} h(t,x,v):=w(v)f(t,x,v). \end{aligned}$$

(4.2)

Then one can reformulate (4.1) as

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t}h+v\cdot \nabla _{x}h+\nu (v) h-K_{w}h=-wL_{\sqrt{\mu }f_*}f+w\Gamma (f,f),\\&h|_{t=0}=wf_0:=h_0,\\&h|_{\gamma _{-}}=\frac{1}{{\tilde{w}}(v)}\int _{n(x)\cdot u>0}h(u){\tilde{w}}(u)\,\mathrm{d}\sigma (x)+w(v)\frac{\mu _{\theta }-\mu }{\sqrt{\mu }}\int _{n(x)\cdot u>0}h(u){\tilde{w}}(u)\,\mathrm{d}\sigma (x), \end{aligned}\right. \end{aligned}$$

(4.3)

where ${\tilde{w}}$ and $K_wh$ are the same as ones defined before, and for each $x\in \partial \Omega$, $\mathrm{d}\sigma (x)$ denotes the probability measure

$$\begin{aligned} \mathrm{d}\sigma (x)=\{n(x)\cdot u\}\,\mathrm{d}u \end{aligned}$$

in the velocity space ${\mathcal {V}}(x):=\{u\in {\mathbb {R}}^3:n(x)\cdot u>0\}$. For simplicity, to the end we denote

$$\begin{aligned} r[h](t,x,v):=\frac{\mu _{\theta }-\mu }{\sqrt{\mu }}\int _{n(x)\cdot u>0}h(t,x,v){\tilde{w}}(u)\,\mathrm{d}\sigma (x). \end{aligned}$$

(4.4)

4.1 Characteristics for Time-Dependent Problem

Given (t, x, v),  let [X(s), V(s)] be the backward bi-characteristics associated with the initial-boundary value problem (4.1) on the Boltzmann equation, which is determined by

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \frac{{\mathrm{d}}X(s)}{{\mathrm{d}}s}=V(s), \frac{{\mathrm{d}}V(s)}{{\mathrm{d}}s}=0,\\ {[}X(t),V(t)]=[x,v]. \end{array}\right. } \end{aligned}$$

The solution is then given by

$$\begin{aligned}{}[X(s),V(s)]=[X(s;t,x,v),V(s;t,x,v)]=[x-(t-s)v,v]. \end{aligned}$$

Similarly as to the steady case, for each (x, v) with $x\in {\bar{\Omega }}$ and $v\ne 0,$ we define its backward exit time $t_{{\mathbf {b}}}(x,v)\geqq 0$ to be the last moment at which the back-time straight line $[X({-\tau } ;0,x,v),V({-\tau } ;0,x,v)]$ remains in ${\bar{\Omega }}$:

$$\begin{aligned} t_{{\mathbf {b}}}(x,v)={\sup \{s \geqq 0:x-\tau v\in {\bar{\Omega }}\text { for }0\leqq \tau \leqq s\}.} \end{aligned}$$

We also define

$$\begin{aligned} x_{{\mathbf {b}}}(x,v)=x(t_{{\mathbf {b}}})=x-t_{{\mathbf {b}}}v\in \partial \Omega . \end{aligned}$$

Let $x\in {\bar{\Omega }}$, $(x,v)\notin \gamma _{0}\cup \gamma _{-}$ and $(t_{0},x_{0},v_{0})=(t,x,v)$. For $v_{k+1}\in {\mathcal {V}}_{k+1}:=\{v_{k+1}\cdot n(x_{k+1})>0\}$, the back-time cycle is defined as

$$\begin{aligned} \left\{ \begin{aligned} X_{cl}(s;t,x,v)&=\sum _{k}{\mathbf {1}}_{[t_{k+1},t_{k})}(s)\{x_{k}-v_k(t_{k}-s)\},\\ V_{cl}(s;t,x,v)&=\sum _{k}{\mathbf {1}}_{[t_{k+1},t_{k})}(s)v_{k}, \end{aligned}\right. \end{aligned}$$

(4.5)

with

$$\begin{aligned} (t_{k+1},x_{k+1},v_{k+1})=(t_{k}-t_{{\mathbf {b}}}(x_{k},v_{k}),x_{{\mathbf {b}}}(x_{k},v_{k}),v_{k+1}). \end{aligned}$$

For $k\geqq 2$, the iterated integral means that

$$\begin{aligned} \int _{\Pi _{l=1}^{k-1}{\mathcal {V}}_{l}}\Pi _{l=1}^{k-1}\mathrm{d}\sigma _{l}:= \int _{{\mathcal {V}}_{1}}...\left\{ \int _{{\mathcal {V}}_{k-1}}\,\mathrm{d}\sigma _{k-1}\right\} \cdots \mathrm{d}\sigma _{1}. \end{aligned}$$

where

$$\begin{aligned} {\mathrm{d}\sigma _i:=\mu (v_i)\{n(x_i)\cdot v_i\}\mathrm{d}v_i.} \end{aligned}$$

Note that all $v_{l}$ ($l=1,2,\ldots$) are independent variables, and $t_k$, $x_k$ depend on $t_l$, $x_l$, $v_l$ for $l\leqq k-1$, and the velocity space ${\mathcal {V}}_{l}$ implicitly depends on $(t,x,v,v_{1},v_{2},\ldots ,v_{l-1}).$

Define the near-grazing set of $\gamma _{+}$ and $\gamma _{-}$ as

$$\begin{aligned} \gamma ^\varepsilon _{\pm }=\left\{ (x,v)\in \gamma _{\pm }~:~ |v\cdot n(x)|<\varepsilon ~\text{ or }~|v|\leqq \varepsilon ~\text{ or }~|v|\geqq \frac{1}{\varepsilon }\right\} . \end{aligned}$$

(4.6)

Then we have

Lemma 4.1

([13, 29]) Let $\varepsilon >0$ be a small positive constant, then it holds that

$$\begin{aligned} |f{\mathbf {1}}_{\gamma _{\pm }\setminus \gamma _{\pm }^\varepsilon }|_{L^1(\gamma )} \leqq C_{\varepsilon ,\Omega } \Big \{\Vert f\Vert _{L^1} + \Vert v\cdot \nabla _x f\Vert _{L^1}\Big \}, \end{aligned}$$

(4.7)

and

$$\begin{aligned}&\int _0^t |f(\tau ){\mathbf {1}}_{\gamma _+\setminus \gamma _{+}^\varepsilon }|_{L^1(\gamma )}\,\mathrm{d}\tau \nonumber \\&\quad \leqq C_{\varepsilon ,\Omega } \bigg \{\Vert f(0)\Vert _{L^1}+\int _0^t \Big [\Vert f(\tau )\Vert _{L^1}+\Vert [\partial _{\tau }+v\cdot \nabla _x] f(\tau )\Vert _{L^1}\Big ]\,\mathrm{d}s\bigg \}, \end{aligned}$$

(4.8)

where the positive constant $C_{\varepsilon }>0$ depends only on $\varepsilon$.

4.2 Linear Problem

In this part, we study the following linear inhomogeneous problem:

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t}f+v\cdot \nabla _{x}f+Lf=g, \\&f|_{\gamma _{-}}=P_{\gamma }f+\frac{\mu _{\theta }-\mu }{\sqrt{\mu }}\int _{n(x)\cdot u>0}f\sqrt{\mu }|n(x)\cdot u|\,\mathrm{d}u,\\&f(t,x,v)|_{t=0}=f_0(x,v), \end{aligned} \right. \end{aligned}$$

(4.9)

where g is a given function. Recall that h(t, x, v) defined in (4.2). Then the equation of h as well as the boundary condition reads

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t}h+v\cdot \nabla _{x}h+\nu (v)h-K_wh=wg, \\&h|_{\gamma _{-}}=\frac{1}{{\tilde{w}}(v)}\int _{n(x)\cdot u>0}h(u){\tilde{w}}(u)\mathrm{d}\sigma (x)+w(v)\frac{\mu _{\theta }-\mu }{\sqrt{\mu }}\int _{n(x)\cdot u>0}h(u){\tilde{w}}(u)\,\mathrm{d}\sigma (x).\\ \end{aligned} \right. \end{aligned}$$

(4.10)

Recall the stochastic cycle defined in (4.5). For any $0\leqq s\leqq t$, we define

$$\begin{aligned} \displaystyle I(t,s):=e^{-\int _s^t\nu (V_{cl}(\tau ))\,\mathrm{d}\tau }. \end{aligned}$$

The following Lemma is to establish the mild formulation for (4.10). (Since its proof is almost the same as for [29, Lemma 24], we omit details for brevity):

Lemma 4.2

Let $k\geqq 1$ be an integer and $h(t,x,v) \in L^{\infty }$ satisfy (4.10). For any $t>0$, for almost every $(x,v)\in {\bar{\Omega }}\times {\mathbb {R}}^3\setminus \gamma _0\cup \gamma _-$ and for any $0\leqq s\leqq t$, it holds that

$$\begin{aligned} h(t,x,v)=\sum _{n=1}^4{\mathcal {I}}_n+\sum _{n=5}^{14}{\mathbf {1}}_{\{t_1>s\}}{\mathcal {I}}_{n}, \end{aligned}$$

(4.11)

where

$$\begin{aligned}&{\mathcal {I}}_1={\mathbf {1}}_{\{t_1\leqq s\}}I(t,s)h(s,x-(t-s)v,v)\\&\quad {\mathcal {I}}_2+{\mathcal {I}}_3+{\mathcal {I}}_4=\int _{\max \{t_{1},s\}}^{t}I(t,\tau )[K_w^mh+K_w^ch+g](\tau ,x-(t-\tau )v,v)\,\mathrm{d}\tau \\&\quad {\mathcal {I}}_5=\frac{I(t,t_{1})}{{\tilde{w}}(v)}\int _{\prod _{j=1}^{k-1}{\mathcal {V}}_{j}}\sum ^{k-1}_{l=1}{\mathbf {1}}_{\{t_{l+1}\leqq s<t_{l}\}}h(s,x_{l}-(t_{l}-s)v_{l},v_{l})\,\mathrm{d}\Sigma _{l}(s)\\&\quad {\mathcal {I}}_6+{\mathcal {I}}_7+{\mathcal {I}}_8=\frac{I(t,t_{1})}{{\tilde{w}}(v)}\int _{\prod _{j=1}^{k-1}{\mathcal {V}}_{j}}\int _{s}^{t_{l}}\sum ^{k-1}_{l=1}{\mathbf {1}}_{\{t_{l+1}\leqq s<t_{l}\}}\\&\quad \times [K_w^mh+K^c_wh+wg](\tau ,x_{l}-(t_{l}-\tau )v_{l},v_{l})\,\mathrm{d}\Sigma _{l}(\tau )\,\mathrm{d}\tau \\&\quad {\mathcal {I}}_9+{\mathcal {I}}_{10}+{\mathcal {I}}_{11}=\frac{I(t,t_{1})}{{\tilde{w}}(v)}\int _{\prod _{j=1}^{k-1}{\mathcal {V}}_{j}}\int _{t_{l+1}}^{t_{l}}\sum ^{k-1}_{l=1}{\mathbf {1}}_{\{t_{l+1}> s\}}\\&\quad \times [K_w^mh+K^c_wh+wg](\tau ,x_{l}-(t_{l}-\tau )v_{l},v_{l})\,\mathrm{d}\Sigma _{l}(\tau )\,\mathrm{d}\tau \\&{\mathcal {I}}_{12}=I(t,t_{1})wr(t_{1},x_{1},v),\\&{\mathcal {I}}_{13}=\frac{I(t,t_{1})}{{\tilde{w}}(v)}\int _{\prod _{j=1}^{k-1}{\mathcal {V}}_{j}}\sum ^{k-2}_{l=1}{\mathbf {1}}_{\{t_{l+1}> s\}}wr(t_{l+1},x_{l+1},v_{l})\,\mathrm{d}\Sigma _{l}(t_{l+1})\\&\quad {\mathcal {I}}_{14}=\frac{I(t,t_{1})}{{\tilde{w}}(v)}\int _{\prod _{j=1}^{k-1}{\mathcal {V}}_{j}}{\mathbf {1}}_{\{t_{k}>s\} }h(t_{k},x_{k},v_{k-1})\,\mathrm{d}\Sigma _{k-1}(t_{k}), \end{aligned}$$

where the measure $\mathrm{d}\Sigma _{l}(\tau )$ is defined by

$$\begin{aligned} d\Sigma _{l}(\tau ):=\left\{ \prod _{j=l+1}^{k-1}\mathrm{d}\sigma _{j}\right\} \left\{ {\tilde{w}}(v_{l})I(t_{l},\tau )\,\mathrm{d}\sigma _{l}\right\} \left\{ \prod _{j=1}^{l-1}I(t_{j},t_{j+1})\,\mathrm{d}\sigma _{j}\right\} . \end{aligned}$$

The same as for Lemma 3.2, we have

Lemma 4.3

Let $(\eta ,\zeta )$ belong to

$$\begin{aligned} \{\zeta =2, 0\leqq \eta<1/2\}\cup \{0\leqq \zeta <2,\eta \geqq 0\}. \end{aligned}$$

For $T_0$ sufficiently large, there exist constants ${\hat{C}}_3$ and ${\hat{C}}_4$ independent of $T_0$ such that for $k={\hat{C}}_3T_0^{\frac{5}{4}}$ and $(t,x,v)\in [s,s+T_0]\times {\bar{\Omega }}\times {\mathbb {R}}^3$, it holds that

$$\begin{aligned} \int _{\Pi _{j=1}^{k-1}\hat{{\mathcal {V}}}_{j}} {\mathbf {1}}_{\{{t}_k>s\}}~ \Pi _{j=1}^{k-1} e^{\eta |v_j|^\zeta } \mathrm{d}{\sigma } _{j}\leqq \left( \frac{1}{2}\right) ^{{\hat{C}}_4T_0^{\frac{5}{4}}}. \end{aligned}$$

(4.12)

The following proposition is to clarify the solvability of the linear problem (4.9):

Proposition 4.4

Let $-3<\gamma <0$, $\beta >3+|\kappa |,$ and $(\varpi ,\zeta )$ satisfy (1.10). Assume that

$$\begin{aligned} \int _{\Omega }\int _{{\mathbb {R}}^3} f_0(x,v)\mu ^{\frac{1}{2}}(v)\,\mathrm{d}x\mathrm{d}v=\int _{\Omega } \int _{{\mathbb {R}}^3} g(t,x,v)\mu ^{\frac{1}{2}}(v)\,\mathrm{d}x\mathrm{d}v=0, \end{aligned}$$

and also assume that

$$\begin{aligned} \Vert wf_0\Vert _{L^{\infty }}+\sup _{s\geqq 0}e^{\lambda _0 s^\alpha }\Vert \nu ^{-1}wg(s)\Vert _{L^{\infty }}<\infty , \end{aligned}$$

where $\lambda _0>0$ is a small constant to be chosen in the proof. Then the linear IBVP problem (4.9) admits a unique solution f(t, x, v) satisfying

$$\begin{aligned} \sup _{0\leqq s\leqq t}e^{\lambda _0 s^\alpha }\{\Vert wf(s)\Vert _{L^{\infty }}+|wf(s)|_{L^{\infty }(\gamma )}\}\leqq C\Vert wf_0\Vert _{L^{\infty }}+C\sup _{0\leqq s\leqq t}e^{\lambda _0 s^\alpha }\Vert \nu ^{-1}wg(s)\Vert _{L^{\infty }} \end{aligned}$$

(4.13)

for any $t\geqq 0$. Moreover, if $\Omega$ is convex, $f_0(x,v)$ is continuous except on $\gamma _0$, g is continuous in $(0,\infty )\times \Omega \times {\mathbb {R}}^3$,

$$\begin{aligned} f_0|_{\gamma _-}=P_{\gamma }f_0+\frac{\mu _{\theta }-\mu }{\sqrt{\mu }}\int _{\{n(x)\cdot v'>0\}}f_0\sqrt{\mu }|n(x)\cdot v'|\,\mathrm{d}v', \end{aligned}$$

and $\theta (x)$ is continuous on $\partial \Omega$, then f(t, x, v) is continuous over $[0,\infty )\times \{{\bar{\Omega }}\times {\mathbb {R}}^{3}\setminus \gamma _0\}$.

The proof of Proposition 4.4 will be given after we prepare two lemmas.

Lemma 4.5

Let h(t, x, v) be the $L^\infty$-solution of the linear problem (4.10). Then for any $s\geqq 0$, for any $s\leqq t \leqq s+T_0$ with $T_0>0$ is sufficiently large, and for almost everywhere $(x,v)\in {\bar{\Omega }}\times {\mathbb {R}}^3\setminus \gamma _0$, it holds that

$$\begin{aligned} |h(t,x,v)|\leqq \,&CT_0^{5/2}e^{-\lambda _1(t-s)^{\alpha }}\Vert h(s)\Vert _{L^\infty }\nonumber \\&+\,CT_{0}^{5/2}e^{-{\tilde{\lambda }}t^{\alpha }} \left\{ m^{3+\kappa }+\delta +2^{-T_0}+\frac{1}{N}\right\} \cdot \sup _{s\leqq \tau \leqq t}e^{{\tilde{\lambda }}\tau ^\alpha } \left\{ \Vert h(\tau )\Vert _{L^\infty }+|h(\tau )|_{L^\infty (\gamma )}\right\} \nonumber \\&+\,C_{N, T_0,m}e^{-{\tilde{\lambda }}t^{\alpha }}\sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}\tau ^{\alpha }} f(\tau )\Vert _{L^2}+CT_0^{5/2}e^{-{\tilde{\lambda }}t^\alpha }\sup _{s\leqq \tau \leqq t}\Vert \nu ^{-1}e^{{\tilde{\lambda }}\tau ^\alpha }wg(\tau )\Vert _{L^\infty }, \end{aligned}$$

(4.14)

where $\lambda _1>0$ is a generic constant given in (4.21), and $0< {\tilde{\lambda }}<\lambda _1$ is an arbitrary constant to be chosen later. Here $m>0$ can be chosen arbitrarily small and N can be chosen arbitrarily large.

Proof

We take $T_0$ sufficiently large and $k={\hat{C}}_3T_0^{5/4}$ such that (4.12) holds for $\eta =\frac{5}{16}$. We first estimate ${\mathcal {I}}_1$. On one hand, if $s\leqq t\leqq s+1$, it obviously holds that

$$\begin{aligned} |{\mathcal {I}}_1|\leqq \Vert h(s)\Vert _{L^\infty }\leqq e^{{\tilde{\lambda }}}e^{-{\tilde{\lambda }}(t-s)}\Vert h(s)\Vert _{L^\infty }, \end{aligned}$$

(4.15)

for any ${\tilde{\lambda }}>0$. On the other hand, if $t>s+1$, we note that

$$\begin{aligned} 0\leqq t_{{\mathbf {b}}}(x,v)\leqq \frac{d_\Omega }{|v|}, \end{aligned}$$

where $d_\Omega :=\sup _{x,y\in \Omega }|x-y|$ is the diameter of the bounded domain $\Omega$. Then for $|v|>d_{\Omega }$, it holds that

$$\begin{aligned} t_1-s=t-t_{{\mathbf {b}}}(x,v)-s >0. \end{aligned}$$

In other words, ${\mathcal {I}}_1$ appears only when the particle velocity |v| is not greater than $d_\Omega$, so that we have

$$\begin{aligned} |{\mathcal {I}}_1|\leqq Ce^{-\nu (v)(t-s)}{\mathbf {1}}_{\{|v|\leqq d_\Omega \}}\Vert h(s)\Vert _{L^\infty }\leqq C_{\Omega }e^{-{\bar{\nu }}_0(t-s)}\Vert h(s)\Vert _{L^\infty }, \end{aligned}$$

(4.16)

where

$$\begin{aligned} {\bar{\nu }}_0:=\inf _{|v|\leqq d_{\Omega }}\nu (v)>0 \end{aligned}$$

depends only on $d_\Omega$. Collecting the estimates (4.15) and (4.16) on these two cases, we have

$$\begin{aligned} |{\mathcal {I}}_1|\leqq C_{\Omega }e^{-{\bar{\nu }}_0(t-s)}\Vert h(s)\Vert _{L^\infty }. \end{aligned}$$

(4.17)

For ${\mathcal {I}}_4$, we split the velocity to estimate it as

$$\begin{aligned} |{\mathcal {I}}_4|\leqq \int _{\max \{t_1,s\}}^te^{-\nu (v)(t-\tau )}\nu (v)\cdot \{{\mathbf {1}}_{\{|v|\leqq d_\Omega \}}+{\mathbf {1}}_{\{|v|> d_{\Omega }\}}\}\cdot \Vert \nu ^{-1}wg(\tau )\Vert _{L^\infty }\,\mathrm{d}\tau . \end{aligned}$$

For $|v|>d_\Omega$, we have $\max \{t_1,s\}\geqq t_1\geqq t-t_{{\mathbf {b}}}(x,v)\geqq t-1$, which implies, for any ${\tilde{\lambda }}>0$. that

$$\begin{aligned} \begin{aligned}&\int _{\max \{t_1,s\}}^te^{-\nu (v)(t-\tau )}\nu (v){\mathbf {1}}_{\{|v|> d_\Omega \}}\Vert \nu ^{-1}wg(\tau )\Vert _{L^\infty }\\&\quad \leqq C\int _{\max \{t_1,s\}}^t{\mathbf {1}}_{\{t-1\leqq \tau \leqq t\}}\Vert \nu ^{-1}wg(\tau )\Vert _{L^\infty }\\&\quad \leqq e^{{\tilde{\lambda }}}\int _{\max \{t_1,s\}}^te^{-{\tilde{\lambda }}(t-\tau )}{\mathbf {1}}_{\{t-1\leqq \tau \leqq t\}}\Vert \nu ^{-1}wg(\tau )\Vert _{L^\infty }\,\mathrm{d}\tau \\&\quad \leqq C_{{\tilde{\lambda }},\alpha }e^{-{\tilde{\lambda }}t^\alpha }\cdot \sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}\tau ^\alpha }\nu ^{-1}wg(\tau )\Vert _{L^\infty }. \end{aligned} \end{aligned}$$

For $|v|\leqq d_\Omega$, we have

$$\begin{aligned}&\int _{\max \{t_1,s\}}^te^{-\nu (v)(t-\tau )}\nu (v){\mathbf {1}}_{\{|v|\leqq d_{\Omega }\}}\Vert \nu ^{-1}wg(\tau )\Vert _{L^\infty }\,\mathrm{d}\tau \\&\quad \leqq C_{\Omega }\int _{\max \{t_1,s\}}^t e^{-{\bar{\nu }}_0(t-\tau )}\Vert \nu ^{-1}wg(\tau )\Vert _{L^\infty }\,\mathrm{d}\tau \\&\quad \leqq C_{\Omega ,{\tilde{\lambda }},\alpha }e^{-{\tilde{\lambda }}t^\alpha }\cdot \sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}\tau ^\alpha }\nu ^{-1}wg(\tau )\Vert _{L^\infty }. \end{aligned}$$

Conbining these two estimates, we have

$$\begin{aligned} |{\mathcal {I}}_4|\leqq Ce^{-{\tilde{\lambda }}t^\alpha }\cdot \sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}\tau ^\alpha }\nu ^{-1}wg(\tau )\Vert _{L^\infty }. \end{aligned}$$

(4.18)

Similarly, by using (2.11), it holds that

$$\begin{aligned} |{\mathcal {I}}_2|\leqq Cm^{3+\kappa }e^{-{\tilde{\lambda }}t^\alpha }e^{-\frac{|v|^2}{16}}\sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}\tau ^\alpha }h(\tau )\Vert _{L^\infty }. \end{aligned}$$

(4.19)

For ${\mathcal {I}}_{5}$, we borrow an idea from [36]. Take $|v_m|=\max \{|v_1|,|v_2|,\ldots ,|v_l|\}$. By a direct computation, we have, for some positive constant $c>0$, that

$$\begin{aligned}&\frac{I(t,t_{1})}{{\tilde{w}}(v)}\bigg [\prod _{j=1}^{l-1}I(t_{j},t_{j+1})\bigg ]\times {\tilde{w}}(v_{l})I(t_{l},s)\nonumber \\&\quad \leqq \frac{e^{-c\langle v\rangle ^{\kappa }(t-t_1)}}{{\tilde{w}}(v)}\prod _{j=1}^{l-1}e^{-c\langle v_j\rangle ^{\kappa }(t_j-t_{j+1})}\cdot e^{-c\langle v_l\rangle ^{\kappa }(t_l-s)}{\tilde{w}}(v_l)\nonumber \\&\quad \leqq \frac{Ce^{-c\langle v\rangle ^{\kappa }(t-t_1)}}{{\tilde{w}}(v)} e^{-c\langle v_m\rangle ^{\kappa }(t_1-s)}e^{\frac{|v_m|^2}{4}}. \end{aligned}$$

(4.20)

Here we have denoted $\langle v\rangle :=(1+|v|^2)^{1/2}$. We note the fact from Young’s inequality that

$$\begin{aligned} c\langle v\rangle ^{\kappa }(\tau _1-\tau _2)+\frac{|v|^{\zeta }}{16}\geqq \lambda _1(\tau _1-\tau _2)^{\alpha }, \end{aligned}$$

(4.21)

for a generic constant $\lambda _1>0$ depending only on $\zeta$. Then the right-hand side of (4.20) is further bounded as

$$\begin{aligned} \begin{aligned} C \frac{e^{-c\langle v\rangle ^{\kappa }(t-t_1)}e^{-\frac{|v|^\zeta }{16}}}{{\tilde{w}}(v)e^{-\frac{|v|^\zeta }{16}}} e^{-c\langle v_m\rangle ^{\kappa }(t_1-s)-\frac{|v_m|^\zeta }{16}}e^{\frac{5|v_m|^2}{16}}\leqq Ce^{-\frac{|v|^2}{16}}e^{-\lambda _1(t-s)^\alpha }e^{\frac{5|v_m|^2}{16}}. \end{aligned} \end{aligned}$$

Here we have used the elementary fact that $x^{\alpha }+y^{\alpha }\geqq (x+y)^{\alpha }$ for $x,y\geqq 0$ and $0\leqq \alpha \leqq 1$. Therefore, it holds that

$$\begin{aligned} |{\mathcal {I}}_5|\leqq \,&\sum _{l=1}^{k-1}\sum _{m=1}^{l}Ce^{-\frac{|v|^2}{16}}e^{-\lambda _1(t-s)^{\alpha }}\Vert h(s)\Vert _{L^\infty } \nonumber \\&\times \bigg \{\int _{\prod _{j=1}^{l}{\mathcal {V}}_{j}}{\mathbf {1}}_{\{t_{l+1}\leqq s<t_{l}\}}\times {\mathbf {1}}_{\big \{|v_{m}|= \max \big [|v_{1}|,|v_{2}|,...|v_{l}|\big ]\big \}}e^{\frac{5|v_{m}|^{2}}{16}} \prod _{j=1}^{l}\mathrm{d}\sigma _{j}\bigg \}\nonumber \\ \leqq&Ck^2e^{-\frac{|v|^2}{16}}e^{-\lambda _1(t-s)^{\alpha }}\Vert h(s)\Vert _{L^\infty }\cdot \sup _{j}\left| \int _{{\mathcal {V}}_j}e^{\frac{5|v_{j}|^{2}}{16}}\mathrm{d}\sigma _j\right| \nonumber \\ \leqq \,&Ck^2e^{-\frac{|v|^2}{16}}e^{-\lambda _1(t-s)^{\alpha }}\Vert h(s)\Vert _{L^\infty }. \end{aligned}$$

(4.22)

Similarly, we have

$$\begin{aligned} |{\mathcal {I}}_{8}|+|{\mathcal {I}}_{11}|\leqq Ck^2e^{-\frac{|v|^2}{16}}e^{-{\tilde{\lambda }}t^{\alpha }}\cdot \sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}\tau ^\alpha }\nu ^{-1}wg(\tau )\Vert _{L^\infty }. \end{aligned}$$

(4.23)

Recall r defined in (4.4). Similar for obtaining (4.22), we have

$$\begin{aligned} |{\mathcal {I}}_{12}|+|{\mathcal {I}}_{13}|\leqq C\delta k^2e^{-\frac{|v|^2}{16}}e^{-{\tilde{\lambda }}t^{\alpha }}\cdot \sup _{s\leqq \tau \leqq t}|e^{{\tilde{\lambda }}\tau ^\alpha }h(\tau )|_{L^\infty (\gamma _+)}. \end{aligned}$$

(4.24)

By (2.11), it holds that

$$\begin{aligned} |{\mathcal {I}}_{6}|+|{\mathcal {I}}_{9}|\leqq Ck^2m^{3+\kappa }e^{-\frac{|v|^2}{16}}e^{-{\tilde{\lambda }}t^{\alpha }}\cdot \sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}\tau ^\alpha }h(\tau )\Vert _{L^\infty }. \end{aligned}$$

(4.25)

For ${\mathcal {I}}_{14}$, we have, from (4.12), that

$$\begin{aligned} |{\mathcal {I}}_{14}|\leqq&\,Ce^{-\frac{|v|^2}{16}}e^{-{\tilde{\lambda }}t^{\alpha }}\cdot \sup _{s\leqq \tau \leqq t}|e^{{\tilde{\lambda }}\tau ^\alpha }h(\tau )|_{L^\infty (\gamma _-)}\cdot \int _{\Pi _{j=1}^{k-1}\hat{{\mathcal {V}}}_{j}} {\mathbf {1}}_{\{{t}_k>s\}}~ \Pi _{j=1}^{k-1} e^{\frac{5|v_j|^2}{16}}\,\mathrm{d}{\sigma } _{j}\nonumber \\ \leqq&\,Ce^{-\frac{|v|^2}{16}}e^{-{\tilde{\lambda }}t^{\alpha }}\cdot \sup _{s\leqq \tau \leqq t}|e^{{\tilde{\lambda }}\tau ^\alpha }h(\tau )|_{L^\infty (\gamma _-)}\cdot \left( \frac{1}{2}\right) ^{{\hat{C}}_4T_0^{\frac{5}{4}}}. \end{aligned}$$

(4.26)

Now we consider the terms involving $K^c_{w}$. Similar as in (4.22), we have

$$\begin{aligned} |{\mathcal {I}}_7|&\leqq C e^{-\frac{|v|^2}{16}} \sum _{l=1}^{k-1}\sum _{m=1}^{l}\int _{\Pi _{j=1}^{l-1}{{\mathcal {V}}}_{j}}\mathrm{d}{\sigma }_{l-1}\cdots \mathrm{d}{\sigma }_1 \\&\quad \times \int _{{\mathcal {V}}_l}\int _{{\mathbb {R}}^3} \int _s^{{t}_l} e^{-\lambda _1(t-\tau )^{\alpha }} {\mathbf {1}}_{\{{t}_{l+1}\leqq s<{t}_l\}} e^{\frac{5|v_m|^2}{16}} |k^c_w(v_l,v')\\&\quad h(\tau ,{x}_l-{v}_l({t}_l-\tau ),v')|\,\mathrm{d}\tau \,\mathrm{d}v' \mathrm{d}{\sigma }_l. \end{aligned}$$

Then, by splitting the integral domain, we further have

$$\begin{aligned} |{\mathcal {I}}_7|&\leqq C e^{-\frac{|v|^2}{16}} \sum _{l=1}^{k-1}\sum _{m=1}^{l-1}\int _{\Pi _{j=1}^{l-1}{{\mathcal {V}}}_{j}}e^{\frac{5|v_m|^2}{16}} \mathrm{d}{\sigma }_{l-1}\cdots \mathrm{d}{\sigma }_1 \int _{{\mathcal {V}}_l\cap \{|v_l|\geqq N\}}\int _{{\mathbb {R}}^3}\int _s^{{t}_l} \Delta \,\mathrm{d}\tau \mathrm{d}v' \mathrm{d}{\sigma }_l \nonumber \\&\quad +C e^{-\frac{|v|^2}{16}} \sum _{l=1}^{k-1}\sum _{m=1}^{l-1}\int _{\Pi _{j=1}^{l-1}{{\mathcal {V}}}_{j}}e^{\frac{5|v_m|^2}{16}} \mathrm{d}{\sigma }_{l-1}\cdots \mathrm{d}{\sigma }_1 \int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{{\mathbb {R}}^3} \int _{t_l-\frac{1}{N}}^{{t}_l} \Delta \,\mathrm{d}\tau \mathrm{d}v' \mathrm{d}{\sigma }_l\nonumber \\&\quad + C e^{-\frac{|v|^2}{16}} \sum _{l=1}^{k-1}\sum _{m=1}^{l-1}\int _{\Pi _{j=1}^{l-1}{{\mathcal {V}}}_{j}} e^{\frac{5|v_m|^2}{16}} \mathrm{d}{\sigma }_{l-1}\cdots \mathrm{d}{\sigma }_1 \int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{\{|v'|\geqq 2N\}} \int _s^{{t}_l-\frac{1}{N}} \Delta \,\mathrm{d}\tau \mathrm{d}v' \mathrm{d}{\sigma }_l\nonumber \\&\quad + C e^{-\frac{|v|^2}{16}} \sum _{l=1}^{k-1}\sum _{m=1}^{l-1}\int _{\Pi _{j=1}^{l-1}{{\mathcal {V}}}_{j}} e^{\frac{5|v_m|^2}{16}} \mathrm{d}{\sigma }_{l-1}\cdots \mathrm{d}{\sigma }_1 \int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{\{|v'|\leqq 2N\}} \int _s^{{t}_l-\frac{1}{N}} \Delta \mathrm{d}\tau \,\mathrm{d}v' \mathrm{d}{\sigma }_l\nonumber \\&\quad :=Ce^{-\frac{|v|^2}{16}}\sum _{l=1}^{k-1}\bigg \{{\mathcal {I}}_{71l}+{\mathcal {I}}_{72l}+{\mathcal {I}}_{73l}+{\mathcal {I}}_{74l}\bigg \}, \end{aligned}$$

(4.27)

where we have denoted that

$$\begin{aligned} \Delta :=e^{-\lambda _1(t-\tau )^{\alpha }} {\mathbf {1}}_{\{{t}_{l+1}\leqq s<{t}_l\}}e^{\frac{5|v_l|^2}{16}}|k^c_w(v_l,v') h(\tau ,{x}_l-{v}_l({t}_l-\tau ),v')|. \end{aligned}$$

For ${\mathcal {I}}_{71l}$, we use (2.14) to obtain, for any $0<{\tilde{\lambda }}<\lambda _1$, that

$$\begin{aligned} {\mathcal {I}}_{71l}&\leqq Ck e^{-{\tilde{\lambda }}t^{\alpha }}\cdot \sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}\tau ^\alpha }h(\tau )\Vert _{L^\infty }\cdot \sup _{j=0,\ldots ,l-1}\left| \int _{{\mathcal {V}}_j}e^{\frac{5|v_j|^2}{16}}\mathrm{d}\sigma _j\right| \cdot \left| \int _{|v_l|\geqq N}e^{-\frac{|v_l|^2}{16}}\mathrm{d}v_l\right| \nonumber \\&\leqq Cke^{-\frac{N^2}{32}} e^{-{\tilde{\lambda }}t^{\alpha }}\cdot \sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}\tau ^\alpha }h(\tau )\Vert _{L^\infty }. \end{aligned}$$

(4.28)

For ${\mathcal {I}}_{72l}$, it is straightforward to see that

$$\begin{aligned} {\mathcal {I}}_{72l}\leqq \frac{Ck}{N} e^{-{\tilde{\lambda }}t^{\alpha }}\cdot \sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}\tau ^\alpha }h(\tau )\Vert _{L^\infty }. \end{aligned}$$

(4.29)

For ${\mathcal {I}}_{73l}$, since $|v'-v_l|\geqq N$, then by (2.14), it holds that

$$\begin{aligned} |{\mathcal {I}}_{73l}|\leqq Cke^{-\frac{N^2}{32}} e^{-{\tilde{\lambda }}t^{\alpha }}\cdot \sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}\tau ^\alpha }h(\tau )\Vert _{L^\infty }. \end{aligned}$$

(4.30)

For ${\mathcal {I}}_{74l}$, by Hölder’s inequality, it holds that

$$\begin{aligned}&\int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{\{|v'|\leqq 2N\}} \int _s^{{t}_l-\frac{1}{N}} \Delta \mathrm{d}\tau \,\mathrm{d}v' \mathrm{d}{\sigma }_l\\&\quad \leqq C \int _s^{t_l-\frac{1}{N}}e^{-\lambda _1(t-\tau )^{\alpha }}\,\mathrm{d}\tau \bigg \{\int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{|v'|\leqq 2N} e^{-\frac{1}{8}|v_l|^2} |k^c_w(v_l,v')|^2 \,\mathrm{d}v' \mathrm{d}v_l \bigg \}^{1/2}\\&\qquad \times \bigg \{\int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{|v'|\leqq 2N} {\mathbf {1}}_{\{{t}_{l+1}\leqq s<{t}_l\}} \left| h(\tau ,{x}_l-{v_l}({t}_l-\tau ),v')\right| ^2 \,\mathrm{d}v' \mathrm{d}v_l \bigg \}^{1/2}\\&\quad \leqq C_{N} m^{\kappa -1}\int _s^{t_l-\frac{1}{N}}e^{-\lambda _1(t-\tau )^{\alpha }}\,\mathrm{d}\tau \\&\qquad \times \bigg \{\int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{|v'|\leqq 2N} {\mathbf {1}}_{\{{t}_{l+1}\leqq s<{t}_l\}} \left| f(\tau ,{x}_l-{v_l}({t}_l-\tau ),v')\right| ^2 \,\mathrm{d}v' \mathrm{d}v_l\bigg \}^{1/2}. \end{aligned}$$

Here we have used (2.12) for $\alpha =1$ in the last inequality. Note that $y_l:={x}_l-{v}({t}_l-\tau )\in \Omega$ for $s\leqq \tau \leqq t_l-\frac{1}{N}$. Making change of variable $v_l\rightarrow y_l$, we obtain that

$$\begin{aligned} \int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{{\mathbb {R}}^3} \int _s^{{t}_l-\frac{1}{N}} \Delta \mathrm{d}\tau \,\mathrm{d}v' \mathrm{d}{\sigma }_l\leqq C_Nm^{\kappa -1}e^{-{\tilde{\lambda }}t^\alpha }\cdot \sup _{s\leqq \tau \leqq t}e^{{\tilde{\lambda }}{\tau }^\alpha }\Vert f(\tau )\Vert _{L^2}. \end{aligned}$$

Combing this with (4.27), (4.28), (4.29), (4.30), we have

$$\begin{aligned} |{\mathcal {I}}_7|\leqq&\frac{Ck}{N}e^{-\frac{|v|^2}{16}}e^{-{\tilde{\lambda }}t^\alpha }\cdot \sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}\tau ^\alpha }h(\tau )\Vert _{L^\infty }+C_Nm^{\kappa -1}e^{-\frac{|v|^2}{16}}e^{-{\tilde{\lambda }}t^\alpha }\nonumber \\&\quad \cdot \sup _{s\leqq \tau \leqq t}e^{{\tilde{\lambda }}{\tau }^\alpha }\Vert f(\tau )\Vert _{L^2}. \end{aligned}$$

(4.31)

Similarly,

$$\begin{aligned} |{\mathcal {I}}_{10}|&\leqq \frac{Ck}{N}e^{-\frac{|v|^2}{16}}e^{-{\tilde{\lambda }}t^\alpha }\cdot \sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}\tau ^\alpha }h(\tau )\Vert _{L^\infty }+C_Nm^{\kappa -1}e^{-\frac{|v|^2}{16}}e^{-{\tilde{\lambda }}t^\alpha }\nonumber \\&\quad \cdot \sup _{s\leqq \tau \leqq t}e^{{\tilde{\lambda }}{\tau }^\alpha }\Vert f(\tau )\Vert _{L^2}. \end{aligned}$$

(4.32)

Substituting (4.17), (4.18), (4.19), (4.22), (4.24), (4.23), (4.25), (4.26), (4.31), (4.32) into (4.11), we have

$$\begin{aligned} |h(t,x,v)|\leqq \int _{\max \{t_1,s\}}^t\int _{{\mathbb {R}}^3}I(t,\tau )\left| k_w^c(v,u)h(\tau ,x-(t-\tau )v,u)\right| \,\mathrm{d}u\mathrm{d}\tau +{\hat{A}}(t,v), \end{aligned}$$

(4.33)

where we have denoted

$$\begin{aligned} {\hat{A}}(t,v):=\,&Ck^2e^{-\lambda _1(t-s)^{\alpha }}\Vert h(s)\Vert _{L^\infty }\\&+Ck^2e^{-{\tilde{\lambda }}t^\alpha }e^{-\frac{|v|^2}{16}} \left\{ m^{3+\kappa }+\delta +2^{-T_0}+\frac{1}{N}\right\} \\&\quad \cdot \sup _{s\leqq \tau \leqq t}e^{{\tilde{\lambda }}\tau ^\alpha }\{\Vert h(\tau )\Vert _{L^\infty }+|h(\tau )|_{L^\infty (\gamma )}\}\\&+C_{N,T_0,m}e^{-\frac{|v|^2}{16}}e^{-{\tilde{\lambda }}t^\alpha }\cdot \sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}{\tau }^\alpha }f(\tau )\Vert _{L^2}\\&+Ck^2e^{-{\tilde{\lambda }}t^\alpha }\sup _{s\leqq \tau \leqq t}\Vert \nu ^{-1}e^{{\tilde{\lambda }}\tau ^\alpha }wg(\tau )\Vert _{L^\infty }. \end{aligned}$$

Denote $x':=x-(t-\tau )v$ and $t_{1}':=t_1(\tau ,x',u)$. Now we use (4.33) for $h(\tau ,x-(t-\tau )v,u)$ to evaluate

$$\begin{aligned} |h(t,x,v)|&\leqq {\hat{A}}(t,v)+\int _{\max \{t_1,s\}}^t\int _{{\mathbb {R}}^3}I(t,\tau )\left| k_w^c(v,u){\hat{A}}(\tau ,u)\right| \,\mathrm{d}u\mathrm{d}\tau \nonumber \\&\quad +\int _{s}^t\int _{s}^\tau \int _{{\mathbb {R}}^3}\int _{{\mathbb {R}}^3}{\mathbf {1}}_{\{\max \{t_1,s\}\leqq \tau \leqq t\}}{\mathbf {1}}_{\{\max \{t_1',s\}\leqq \tau '\leqq \tau \}}\nonumber \\&\quad \times I(t,\tau ')\left| k_w^c(v,u)k_w^c(u,u')h(\tau ',x'-(\tau -\tau ')u,u')\right| \,\mathrm{d}u\,\mathrm{d}u'\mathrm{d}\tau '\mathrm{d}\tau \nonumber \\&:={\hat{A}}(t,v)+{\hat{B}}_1+{\hat{B}}_2. \end{aligned}$$

(4.34)

Similar for obtaining (4.18), we have

$$\begin{aligned} |{\hat{B}}_1|\leqq&\int _{\max \{t_1,s\}}^t\int _{{\mathbb {R}}^3}\left\{ {\mathbf {1}}_{\{|v|\leqq d_\Omega \}}+{\mathbf {1}}_{\{|v|>d_{\Omega }\}}\right\} I(t,\tau )|k_w(v,u)A(\tau ,u)|\,\mathrm{d}u\mathrm{d}\tau \nonumber \\ \leqq \,&C_{\Omega }\int _{\max \{t_1,s\}}^t\int _{{\mathbb {R}}^3}\left\{ e^{-{\bar{\nu }}_0(t-\tau )}{\mathbf {1}}_{\{t-1\leqq \tau \leqq t\}}+e^{-{\bar{\nu }}_0(t-\tau )}{\mathbf {1}}_{\{|v|\leqq d_\Omega \}}\right\} |k_w(v,u)A(\tau ,u)|\,\mathrm{d}u \mathrm{d}\tau \nonumber \\ \leqq \,&Ck^2e^{-\lambda _1(t-s)^{\alpha }}\Vert h(s)\Vert _{L^\infty }\nonumber \\&+Ck^2e^{-{\tilde{\lambda }}t^\alpha } \left\{ m^{3+\kappa }+\delta +2^{-T_0}+\frac{1}{N}\right\} \cdot \sup _{s\leqq \tau \leqq t}e^{{\tilde{\lambda }}\tau ^\alpha }\{\Vert h(\tau )\Vert _{L^\infty }+|h(\tau )|_{L^\infty (\gamma )}\}\nonumber \\&+C_{N,T_0,m}e^{-{\tilde{\lambda }}t^\alpha }\cdot \sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}{\tau }^\alpha }f(\tau )\Vert _{L^2}+Ck^2e^{-{\tilde{\lambda }}t^\alpha }\sup _{s\leqq \tau \leqq t}\Vert \nu ^{-1}e^{{\tilde{\lambda }}\tau ^\alpha }wg(\tau )\Vert _{L^\infty }. \end{aligned}$$

(4.35)

Finally, we estimate ${\hat{B}}_2$. If $|v|>N$, we have, from (2.14), that

$$\begin{aligned} |{\hat{B}}_2|&\leqq C(1+|v|)^{-2}e^{-{\tilde{\lambda }}t^\alpha }\sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}\tau ^\alpha }h(\tau )\Vert _{L^\infty }\nonumber \\&\leqq \frac{C}{N^2}e^{-{\tilde{\lambda }}t^\alpha }\sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}\tau ^\alpha }h(\tau )\Vert _{L^\infty }. \end{aligned}$$

(4.36)

If $|v|\leqq N$, we denote the integrand of $B_2$ as $U(\tau ',v',v'';\tau ,v)$, and split the integral domain with respect to $\mathrm{d}\tau '\,\mathrm{d}v''\mathrm{d}v'$ into the following four parts:

$$\begin{aligned} \cup _{i=1}^4{\mathcal {O}}_i:=\,&\{|v'|\geqq 2N\}\\&\cup \{|v'|\leqq 2N, |v''|>3N\}\\&\cup \left\{ |v'|\leqq 2N, |v''|\leqq 3N, \tau -\frac{1}{N}\leqq \tau '\leqq \tau \right\} \\&\cup \left\{ |v'|\leqq 2N, |v''|\leqq 3N, s\leqq \tau '\leqq \tau -\frac{1}{N}\right\} . \end{aligned}$$

Over ${\mathcal {O}}_1$ and ${\mathcal {O}}_2$, we have either $|v-v'|\geqq N$ or $|v'-v''|\geqq N$, so that one of the following is valid:

$$\begin{aligned} \left\{ \begin{aligned}&|k^c_w(v,v')|\leqq e^{-\frac{N^2}{32}}e^{\frac{|v-v'|^2}{32}}|k^{c}_w(v,v')|,\\&|k^c_w(v',v'')|\leqq e^{-\frac{N^2}{32}}e^{\frac{|v'-v''|^2}{32}}|k^{c}_w(v,v')|. \end{aligned}\right. \end{aligned}$$

By (2.14), one has

$$\begin{aligned} \int _{s}^t\int _{{\mathcal {O}}_1\cup {\mathcal {O}}_2}U(\tau ',v',v'';\tau ,v)\,\mathrm{d}v''\mathrm{d}\tau '\,\mathrm{d}v'\mathrm{d}\tau \leqq Ce^{-\frac{N^2}{32}}e^{-{\tilde{\lambda }}t^\alpha }\sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}\tau ^\alpha }h(\tau )\Vert _{L^{\infty }}.\nonumber \\ \end{aligned}$$

(4.37)

Over ${\mathcal {O}}_3$, it is direct to obtain

$$\begin{aligned} \int _{s}^t\int _{{\mathcal {O}}_3}U(\tau ',v',v'';\tau ,v)\,\mathrm{d}v''\mathrm{d}\tau '\,\mathrm{d}v'\mathrm{d}\tau \leqq \frac{C}{N}e^{-{\tilde{\lambda }}t^\alpha }\sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}\tau ^\alpha }h(\tau )\Vert _{L^{\infty }}.\nonumber \\ \end{aligned}$$

(4.38)

For ${\mathcal {O}}_4$, it holds from Holder’s inequality that

$$\begin{aligned}&\int _{{\mathcal {O}}_4}U(\tau ',v',v'';\tau ,v)\,\mathrm{d}v''\mathrm{d}\tau '\mathrm{d}v'\nonumber \\&\quad \leqq C_Ne^{-{\bar{\nu }}_0(t-\tau )}\left( \int _{{\mathcal {O}}_4}e^{-{\bar{\nu }}_0(\tau -\tau ')}|k_w^c(v,u)k_{w}^c(u,u')|^2\,\mathrm{d}\tau '\mathrm{d}u\mathrm{d}u'\right) ^{1/2}\nonumber \\&\qquad \times \left( \int _{{\mathcal {O}}_4}e^{-{\bar{\nu }}_0(\tau -\tau ')}{\mathbf {1}}_{\{y'\in \Omega \}}|h(\tau ',y',u')|^2\,\mathrm{d}\tau '\mathrm{d}u\mathrm{d}u'\right) ^{1/2}\nonumber \\&\quad \leqq C_{N}m^{2(\kappa -1)}e^{-{\bar{\nu }}_0(t-\tau )}\left( \int _{{\mathcal {O}}_4}e^{-{\bar{\nu }}_0(\tau -\tau ')}{\mathbf {1}}_{\{y'\in \Omega \}}|f(\tau ',y',u')|^2\,\mathrm{d}\tau '\mathrm{d}u\mathrm{d}u'\right) ^{1/2}, \end{aligned}$$

(4.39)

where we have denoted $y':=x'-(\tau -\tau ')u.$ Making change of variable $u\rightarrow y'$ and noting that the Jacobian $\left| \frac{\mathrm{d}y'}{\mathrm{d}u}\right| \geqq \frac{1}{N^3}>0$ for $s\leqq \tau '<\tau -\frac{1}{N}$, the right-hand side of (4.39) is bounded by

$$\begin{aligned} C_{N}m^{2(\kappa -1)}e^{-{\tilde{\lambda }}t^\alpha }e^{-\frac{{\bar{\nu }}_0(t-\tau )}{2}}\cdot \sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}{\tau }^\alpha }f(\tau )\Vert _{L^2}, \end{aligned}$$

which implies that

$$\begin{aligned} \int _s^t\int _{{\mathcal {O}}_4}U(\tau ',v',v'';\tau ,v)\,\mathrm{d}v''\mathrm{d}\tau '\,\mathrm{d}v'\mathrm{d}\tau \leqq C_{N}m^{2(\kappa -1)}e^{-{\tilde{\lambda }}t^\alpha }\cdot \sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}{\tau }^\alpha }f(\tau )\Vert _{L^2}. \end{aligned}$$

Combining this with (4.36), (4.37), (4.38) yields that

$$\begin{aligned} |{\hat{B}}_2|\leqq \frac{C}{N}e^{-{\tilde{\lambda }}t^\alpha }\sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}\tau ^\alpha }h(\tau )\Vert _{L^\infty }+C_{N}m^{2(\kappa -1)}e^{-{\tilde{\lambda }}t^\alpha }\cdot \sup _{s\leqq \tau \leqq t}\Vert e^{{\tilde{\lambda }}{\tau }^\alpha }f(\tau )\Vert _{L^2}. \end{aligned}$$

(4.40)

Substituting (4.35) and (4.40) into (4.34), one has (4.14). The proof of Lemma 4.5 is complete. $\quad \square$

The following lemma gives the $L^2$-decay of the solution:

Lemma 4.6

If

$$\begin{aligned} \int _{\Omega }\int _{{\mathbb {R}}^3}f_0(x,v)\mu ^{\frac{1}{2}}(v)\,\mathrm{d}x\mathrm{d}v=\int _{\Omega }\int _{{\mathbb {R}}^3}g(t,x,v)\mu ^{\frac{1}{2}}(v)\,\mathrm{d}x\mathrm{d}v=0, \end{aligned}$$

(4.41)

and $|\theta -1|_{L^\infty (\partial \Omega )}$ is sufficiently small, then there exists a constant $\lambda _2>0$ such that for any $t\geqq 0$,

$$\begin{aligned} \Vert f(t)\Vert _{L^2}\leqq Ce^{-\lambda _2t^{\alpha }}\Vert wf_0\Vert _{L^\infty }+\int _0^te^{-\lambda _2(t-s)^{\alpha }}\Vert \nu ^{-1}wg(s)\Vert _{L^\infty }\,\mathrm{d}s. \end{aligned}$$

(4.42)

Proof

We first consider the case $g\equiv 0$. Multiplying both sides of (4.9) by f, we have

$$\begin{aligned}&\frac{1}{2}\Vert f(t)\Vert _{L^2}^2+\frac{1}{2}\int _0^t|f(s)|_{L^2(\gamma _+)}^2\,\mathrm{d}s+\int _0^t\langle Lf(s),f(s)\rangle \,\mathrm{d}s \nonumber \\&\quad =\frac{1}{2}\Vert f_0\Vert _{L^2}^2+\frac{1}{2}\int _0^t|P_{\gamma }f+r|_{L^2(\gamma _-)}^2\,\mathrm{d}s, \end{aligned}$$

(4.43)

where r is defined in (4.4). By the coercivity estimate (2.7), it holds that

$$\begin{aligned} \int _0^t\langle Lf(s),f(s)\rangle \mathrm{d}s\geqq c_0\int _0^t\Vert {\nu ^{1/2}}(I-P)f(s)\Vert _{L^2}^2\,\mathrm{d}s. \end{aligned}$$

(4.44)

Notice that $P_{\gamma }r\equiv 0$. Therefore, it follows that

$$\begin{aligned} \frac{1}{2}\int _0^t|P_{\gamma }f+r|_{L^2(\gamma _-)}^2\,\mathrm{d}s&= \frac{1}{2}\int _{0}^t|P_{\gamma }f(s)|_{L^2(\gamma _+)}^2\,\mathrm{d}s+\frac{1}{2}\int _{0}^t|r(s)|_{L^2(\gamma _-)}^2\,\mathrm{d}s\nonumber \\&\leqq \frac{1}{2}\int _{0}^t|P_{\gamma }f(s)|_{L^2(\gamma _+)}^2\,\mathrm{d}s+C\delta \int _{0}^t|f(s)|_{L^2(\gamma _+)}^2\,\mathrm{d}s. \end{aligned}$$

(4.45)

To estimate $|P_{\gamma }f|_{L^2(\gamma _+)}$, recall the cutoff function ${\mathbf {1}}_{\gamma _+^{{\varepsilon }}}$ with respect to the near grazing set $\gamma _{+}^{\varepsilon }$ defined in (4.6). Then we have

$$\begin{aligned} |P_{\gamma }f(s)|_{L^2(\gamma _+)}^2&=\int _{\gamma _-}\mu (v)|n(x)\cdot v|\,\mathrm{d}\gamma \nonumber \\&\quad \times \left( \int _{n(x)\cdot v'>0}f(s)\{{\mathbf {1}}_{\gamma _{+}^{{\varepsilon }}}+{\mathbf {1}}_{\gamma _+^{\varepsilon }\setminus \gamma _{+}^{{\varepsilon }}}\}\sqrt{\mu }|n(x)\cdot v'|\,\mathrm{d}v'\right) ^2\nonumber \\&\leqq C{\varepsilon }|f(s)|_{L^2(\gamma _+)}^2+C|e^{-\frac{|v|^2}{16}}f(s){\mathbf {1}}_{\gamma _{+}\setminus \gamma _+^{\varepsilon }}|_{L^2(\gamma _+)}^2. \end{aligned}$$

(4.46)

Notice that

$$\begin{aligned} {\frac{1}{2}}(\partial _t+v\cdot \nabla _x)e^{-\frac{|v|^2}{8}}f^2=e^{-\frac{|v|^2}{8}}fLf, \end{aligned}$$

which implies that

$$\begin{aligned} \int _0^t\Vert (\partial _t+v\cdot \nabla _x)e^{-\frac{|v|^2}{8}}f^2(s)\Vert _{L^1}\mathrm{d}s\leqq C\int _0^t\Vert e^{-\frac{|v|^2}{16}}f(s)\Vert _{L^2}^2\,\mathrm{d}s. \end{aligned}$$

Therefore, by the trace estimate (4.8), we have

$$\begin{aligned} \int _0^t|e^{-\frac{|v|^2}{16}}f(s){\mathbf {1}}_{\gamma _{+}\setminus \gamma _+^{\varepsilon }}|_{L^2(\gamma _+)}^2\,\mathrm{d}s&=\int _0^t|e^{-\frac{|v|^2}{8}}f^2(s){\mathbf {1}}_{\gamma _{+}\setminus \gamma _+^{\varepsilon }}|_{L^1(\gamma _+)}\mathrm{d}s\\&\leqq C_{\varepsilon }\Vert e^{-\frac{|v|^2}{16}}f_0\Vert _{L^2}^2+C_{\varepsilon }\int _0^t\Vert e^{-\frac{|v|^2}{16}}f(s)\Vert _{L^2}^2\,\mathrm{d}s \\&\leqq C_{{\varepsilon }}\Vert f_0\Vert _{L^2}^2+C_{{\varepsilon }}\int _0^t\Vert \nu ^{1/2}f(s)\Vert _{L^2}^2\,\mathrm{d}s. \end{aligned}$$

Combining this with (4.46), we have

$$\begin{aligned}&\int _0^t|f(s)|_{L^{2}(\gamma _+)}^2\,\mathrm{d}s\nonumber \\&\quad =\int _0^t|P_{\gamma }f(s)|_{L^{2}(\gamma _+)}^2\,\mathrm{d}s+\int _0^t|(I-P_{\gamma })f(s)|_{L^{2}(\gamma _+)}^2\,\mathrm{d}s\nonumber \\&\quad \leqq C\int _0^t|(I-P_{\gamma })f(s)|_{L^{2}(\gamma _+)}^2\,\mathrm{d}s+C{\varepsilon }\int _0^t|f(s)|_{L^{2}(\gamma _+)}^2\,\mathrm{d}s\nonumber \\&\quad +C_{\varepsilon }\Vert f_0\Vert _{L^2}^2+C_{{\varepsilon }}\int _0^t\Vert \nu ^{1/2}f(s)\Vert _{L^2}^2\,\mathrm{d}s\nonumber \\&\quad \leqq C\int _0^t|(I-P_{\gamma })f(s)|_{L^{2}(\gamma _+)}^2\,\mathrm{d}s+C\Vert f_0\Vert _{L^2}^2+C\int _0^t\Vert \nu ^{1/2}f(s)\Vert _{L^2}^2\,\mathrm{d}s. \end{aligned}$$

(4.47)

Here we have taken ${\varepsilon }>0$ suitably small in the last inequality. For the macroscopic part Pf, we multiply $\sqrt{\mu }$ to both sides of the first equation in (4.9) and use (4.41) to get

$$\begin{aligned} \int _{\Omega }\int _{{\mathbb {R}}^3}f(t,x,v)\mu ^{\frac{1}{2}}(v)\,\mathrm{d}x\mathrm{d}v=\int _{\Omega }\int _{{\mathbb {R}}^3}f_0(x,v)\mu ^{\frac{1}{2}}(v)\,\mathrm{d}x\mathrm{d}v=0. \end{aligned}$$

Then is a fashion similar to [21, Lemma 6.1], there exists a functional ${\mathfrak {e}}_{f}(t)$ with $|{\mathfrak {e}}_{f}(t)|\lesssim \Vert f(t)\Vert _{L^2}^2$ such that

$$\begin{aligned} \int _0^t\Vert \nu ^{1/2}Pf(s)\Vert _{L^2}^2\,\mathrm{d}s&\lesssim \bigg ({\mathfrak {e}}_{f}(t)-{\mathfrak {e}}_{f}(0)\bigg )+\int _0^t\Vert \nu ^{1/2}(I-P)f(s)\Vert _{L^2}^2\,\mathrm{d}s\\&\quad +\int _0^t|r(s)|^2_{L^2(\gamma _-)}\mathrm{d}s+\int _0^t|(I-P_{\gamma })f(s)|_{L^2(\gamma _+)}^2\,\mathrm{d}s.\\&\lesssim \bigg ({\mathfrak {e}}_{f}(t)-{\mathfrak {e}}_{f}(0)\bigg )+\int _0^t\Vert \nu ^{1/2}(I-P)f(s)\Vert _{L^2}^2\,\mathrm{d}s\\&\quad +\delta \int _0^t|f(s)|^2_{L^2(\gamma _+)}\mathrm{d}s+\int _0^t|(I-P_{\gamma })f(s)|_{L^2(\gamma _+)}^2\,\mathrm{d}s. \end{aligned}$$

Suitably combining the estimate above with (4.43), (4.44), (4.45) and (4.47) and taking $\delta >0$ suitably small, we have

$$\begin{aligned} \Vert f(t)\Vert _{L^2}^2+\int _{0}^t\Vert \nu ^{1/2}f(s)\Vert _{L^2}^2\,\mathrm{d}s+\int _{0}^t|f(s)|_{L^2(\gamma _+)}^2\,\mathrm{d}s\leqq C\Vert f_0\Vert _{L^2}^2. \end{aligned}$$

(4.48)

Next we need to obtain the weighted $L^2$ estimate in order to obtain $L^2$ decay of f. Multiplying to both sides of the first equation in (4.9), we have

$$\begin{aligned}&\frac{1}{2}\Vert e^{\frac{\varpi |\cdot |^\zeta }{4}}f(t)\Vert _{L^2}^2+\frac{1}{2}\int _0^t|e^{\frac{\varpi |\cdot |^\zeta }{4}}f(s)|_{L^2(\gamma _+)}^2\,\mathrm{d}s +\int _0^t\Vert \nu ^{1/2}e^{\frac{\varpi |\cdot |^\zeta }{4}}f(s)\Vert _{L^2}^2\,\mathrm{d}s\nonumber \\&\quad =\frac{1}{2}\Vert e^{\frac{\varpi |\cdot |^\zeta }{4}}f_0\Vert _{L^2}^2+\frac{1}{2}\int _0^t|e^{\frac{\varpi |\cdot |^\zeta }{4}}f(s)|_{L^2(\gamma _-)}^2 +\int _0^t\langle Kf(s), e^{\frac{\varpi |\cdot |^\zeta }{2}}f(s)\rangle \mathrm{d}s. \end{aligned}$$

(4.49)

A direct computation shows that

$$\begin{aligned} \int _0^t|e^{\frac{\varpi |\cdot |^\zeta }{4}}f(s)|_{L^2(\gamma _-)}^2&\leqq C\int _0^t|e^{\frac{\varpi |\cdot |^\zeta }{4}}P_{\gamma }f(s)|_{L^2(\gamma _-)}^2\,\mathrm{d}s+C\int _0^t|e^{\frac{\varpi |\cdot |^\zeta }{4}}r(s)|_{L^2(\gamma _-)}^2\,\mathrm{d}s\nonumber \\&\leqq C\int _0^t|f(s)|_{L^2(\gamma _+)}^2\,\mathrm{d}s. \end{aligned}$$

(4.50)

As for the last term on the right-hand side of (4.49), we use (2.15) to obtain

$$\begin{aligned} \int _0^t\left| \langle Kf(s), e^{\frac{\varpi |\cdot |^\zeta }{2}}f(s)\rangle \right| \,\mathrm{d}s\leqq \eta \int _0^t\Vert \nu ^{1/2}e^{\frac{\varpi |\cdot |^\zeta }{4}}f(s)\Vert _{L^2}^2\,\mathrm{d}s+C_\eta \int _0^t\Vert \nu ^{1/2}f(s)\Vert _{L^2}^2\,\mathrm{d}s. \end{aligned}$$

(4.51)

Therefore, suitably combining (4.48), (4.49), (4.50) and (4.51) and taking $\eta >0$ suitably small, we have

$$\begin{aligned} \Vert e^{\frac{\varpi |\cdot |^\zeta }{4}}f(t)\Vert _{L^2}^2+\int _{0}^t\Vert e^{\frac{\varpi |\cdot |^\zeta }{4}}\nu ^{1/2}f(s)\Vert _{L^2}^2\,\mathrm{d}s+\int _{0}^t|e^{\frac{\varpi |\cdot |^\zeta }{4}}f(s)|_{L^2(\gamma _+)}^2\,\mathrm{d}s\leqq C\Vert e^{\frac{\varpi |\cdot |^\zeta }{4}}f_0\Vert _{L^2}^2. \end{aligned}$$

Now we are ready for obtaining $L^2$ decay of f in terms of the idea in [43]. Let ${\tilde{f}}=e^{\lambda ' (1+t)^{\alpha }}f$, with $\lambda '>0$ is a suitably small constant to be determined later. Then applying the same energy estimate for obtaining (4.48), we have

$$\begin{aligned} \Vert {\tilde{f}}(t)\Vert _{L^2}^2+\int _0^t\Vert \nu ^{1/2}{\tilde{f}}(s)\Vert _{L^2}^2\,\mathrm{d}s\leqq C\Vert f_0\Vert _{L^2}^2+C\lambda '\int _0^t(s+1)^{\alpha -1}\Vert {\tilde{f}}(s)\Vert _{L^2}^2\,\mathrm{d}s. \end{aligned}$$

(4.52)

To estimate the last term, we split the v-integration domain into

$$\begin{aligned} {\mathbb {R}}^3_v=\{1+|v|\geqq (1+t)^{\frac{1}{\zeta +|\kappa |}}\}\cup \{1+|v|< (1+t)^{\frac{1}{\zeta +|\kappa |}}\}={\mathcal {M}}(t)\cup {\mathcal {M}}^c(t). \end{aligned}$$

One one hand, we have

$$\begin{aligned} \begin{aligned}\int _0^t(s+1)^{\alpha -1}\Vert {\tilde{f}}{\mathbf {1}}_{{\mathcal {M}}(s)}(s)\Vert _{L^2}^2\,\mathrm{d}s&\leqq C\int _0^t\left\| e^{\lambda '(1+|\cdot |)^{\zeta }}f(s)\right\| _{L^2}^2\,\mathrm{d}s\\&\leqq C\int _0^t\Vert \nu ^{1/2}e^{\frac{\varpi |\cdot |^\zeta }{4}}f(s)\Vert _{L^2}^2\,\mathrm{d}s\\&\leqq C\Vert e^{\frac{\varpi |\cdot |^\zeta }{4}}f_0\Vert _{L^2}^2, \end{aligned} \end{aligned}$$

by taking $\lambda '>0$ suitably small. On the other hand, in ${\mathcal {M}}^c(s)$, we have

$$\begin{aligned} (1+s)^{\alpha -1}\leqq {(1+|v|)^{(\alpha -1)\cdot (\zeta +|\kappa |)}{\mathbf {1}}_{{\mathcal {M}}^c(s)}}\leqq C\nu (v){\mathbf {1}}_{{\mathcal {M}}^c(s)}. \end{aligned}$$

Therefore, we have

$$\begin{aligned} \int _0^t(s+1)^{\alpha -1}\Vert {\tilde{f}}{\mathbf {1}}_{{\mathcal {M}}^c(s)}(s)\Vert _{L^2}^2\,\mathrm{d}s\leqq C\int _0^t\Vert \nu ^{1/2}{\tilde{f}}(s)\Vert _{L^2}^2\,\mathrm{d}s. \end{aligned}$$

Combining these estimates with (4.52) and taking $\lambda '>0$ suitably small, we have

$$\begin{aligned} \Vert {\tilde{f}}(t)\Vert _{L^2}^2+\int _0^t\Vert \nu ^{1/2}{\tilde{f}}(s)\Vert _{L^2}^2\,\mathrm{d}s&\leqq C\Vert e^{\frac{\varpi |\cdot |^\zeta }{4}}f_0\Vert _{L^2}^2+C\lambda '\int _0^t\Vert \nu ^{1/2}{\tilde{f}}(s)\Vert _{L^2}^2\,\mathrm{d}s\nonumber \\&\leqq C\Vert e^{\frac{\varpi |\cdot |^\zeta }{4}}f_0\Vert _{L^2}^2. \end{aligned}$$

(4.53)

Then (4.42) for $g\equiv 0$ naturally follows. We denote G(t) as the solution operator to the linear homogeneous problem (4.9) with $g\equiv 0$. Then for non-trivial g, from Duhamel’s formula, it holds that

$$\begin{aligned} \Vert f(t)\Vert _{L^2}&\leqq \Vert G(t)f_0\Vert _{L^2}+\int _0^t\Vert G(t-s)g(s)\Vert _{L^2}\mathrm{d}s\nonumber \\&\leqq Ce^{-\lambda 't^{\alpha }}\Vert e^{\frac{\varpi |\cdot |^\zeta }{4}}f_0\Vert _{L^2}+C\int _0^te^{-\lambda '(t-s)^{\alpha }}\Vert e^{\frac{\varpi |\cdot |^\zeta }{4}}g(s)\Vert _{L^2}\mathrm{d}s\nonumber \\&\leqq Ce^{-\lambda 't^{\alpha }}\Vert wf_0\Vert _{L^\infty }+\int _0^te^{-\lambda '(t-s)^{\alpha }}\Vert \nu ^{-1}wg(s)\Vert _{L^\infty }\mathrm{d}s. \end{aligned}$$

(4.54)

Here we have used (4.53) in the second inequality. Then (4.42) follows from (4.54) by taking $\lambda _2=\lambda '$. The proof of Lemma 4.6 is complete. $\quad \square$

Proof of Proposition 4.4

The local existence and uniqueness of solutions to the linear inhomogeneous problem (4.10) can be obtained in a similar way as in Proposition 6.2. We omit the details for brevity. In what follows we will show the decay estimate (4.13). Recall the finite-time estimate (4.14). We define $\lambda _0=\min \{\frac{\lambda _1}{4},\frac{\lambda _2}{4}\}$, and

$$\begin{aligned} m=\left( \frac{\eta }{2C}\right) ^{\frac{1}{3+\kappa }}T_0^{-\frac{5}{2(3+\kappa )}} \end{aligned}$$

with $\eta >0$ suitably small to be determined later. Then we choose $T_0$ suitably large and $\delta >0$ suitably small, and also take N suitably large, such that

$$\begin{aligned} CT_0^{5/2}\left\{ m^{3+\kappa }+\delta +2^{-T_0}+\frac{1}{N}\right\} \leqq \eta ,\quad CT_0^{5/2}e^{-\frac{\lambda _1}{2}T_0^\alpha }\leqq 1. \end{aligned}$$

Then it holds from (4.14) with the choice of ${\tilde{\lambda }}=\lambda _0$ that for any $s\geqq 0$ and any $t\in [s,s+T_0]$,

$$\begin{aligned} \Vert h(t)\Vert _{L^\infty }+|h(t)|_{L^\infty (\gamma )}\leqq e^{-\frac{\lambda _1}{2}(t-s)^\alpha }\Vert h(s)\Vert _{L^\infty }+e^{-{\lambda _0}t^\alpha }{\mathcal {D}}(t,s), \end{aligned}$$

(4.55)

where we have defined

$$\begin{aligned} {\mathcal {D}}(t,s)&=\eta \sup _{s\leqq \tau \leqq t}e^{{\lambda _0}\tau ^\alpha }\{\Vert h(\tau )\Vert _{L^\infty }+|h(\tau )|_{L^\infty (\gamma )}\}\nonumber \\&\quad +C\sup _{s\leqq \tau \leqq t}\Vert e^{{\lambda _0}\tau ^\alpha }f(\tau )\Vert _{L^2} +C\sup _{s\leqq \tau \leqq t}\Vert \nu ^{-1}e^{{\lambda _0}\tau ^\alpha }wg(\tau )\Vert _{L^\infty }. \end{aligned}$$

(4.56)

For any $t>0$, there exists a positive integer $n\geqq 1$, such that $nT_0\leqq t<(n+1)T_0$. Then applying (4.55) to $[0,T_0], [T_0,2T_0], \cdots , [(n-1)T_0,nT_0]$ inductively, we have

$$\begin{aligned} \Vert h(nT_0)\Vert _{L^\infty }&\leqq e^{-\frac{\lambda _1}{2}T_0^\alpha }\Vert h(n-1)T_0\Vert _{L^\infty }+e^{-{\lambda _0}(nT_0)^\alpha }D(nT_0,[n-1]T_0)\nonumber \\&\leqq e^{-\lambda _1T_0^\alpha }\Vert h(n-2)T_0\Vert _{L^\infty }+e^{-{\lambda _0}(nT_0)^\alpha }D(nT_0,[n-1]T_0)\nonumber \\&\quad +e^{-\frac{\lambda _1T_0^\alpha }{2}-{\lambda _0}([n-1]T_0)^\alpha } D([n-1]T_0,[n-2]T_0)\nonumber \\&\leqq e^{-\lambda _1T_0^\alpha }\Vert h(n-2)T_0\Vert _{L^\infty }+e^{-{\lambda _0}(nT_0)^\alpha } \left\{ 1+e^{-\frac{\lambda _1T_0^\alpha }{4}}\right\} D(nT_0,[n-2]T_0)\nonumber \\&\leqq \cdots \nonumber \\&\leqq e^{-\frac{\lambda _1nT_0^\alpha }{2}}\Vert h_0\Vert _{L^\infty }+e^{-{\lambda _0}(nT_0)^\alpha }\left\{ 1+e^{-\frac{\lambda _1T_0^\alpha }{4}}+ \cdots +e^{-\frac{(n-1)\lambda _1T_0^\alpha }{4}}\right\} {\mathcal {D}}(nT_0,0)\nonumber \\&\leqq Ce^{-\frac{\lambda _1(nT_0)^\alpha }{2}}\Vert h_0\Vert _{L^\infty }+Ce^{-{\lambda _0}(nT_0)^\alpha }{\mathcal {D}}(nT_0,0), \end{aligned}$$

(4.57)

where in the third inequality we have used $0< {\lambda _0}\leqq \frac{\lambda _1}{4}$. Here we also have used the elementary fact that $x^{\alpha }+y^\alpha \geqq (x+y)^\alpha$ for x, $y\geqq 0$ and $0\leqq \alpha \leqq 1$. Finally applying (4.55) in $[nT_0, (n+1)T_0]$ and using (4.57), we have

$$\begin{aligned}&\Vert h(t)\Vert _{L^\infty }+|h(t)|_{L^\infty (\gamma )}\nonumber \\&\quad \leqq e^{-\frac{\lambda _1(t-nT_0)^\alpha }{2}}\Vert h(nT_0)\Vert _{L^\infty }+e^{-{\lambda _0}t^\alpha }{\mathcal {D}}(t,nT_0)\nonumber \\&\quad \leqq e^{-{\lambda _0}t^\alpha }{\mathcal {D}}(t,nT_0)+ e^{-\frac{\lambda _1(t-nT_0)^\alpha }{2}}\left\{ Ce^{-\frac{\lambda _1(nT_0)^\alpha }{2}}\Vert h_0\Vert _{L^\infty }+Ce^{-{\lambda _0}(nT_0)^\alpha }D(nT_0,0)\right\} \nonumber \\&\quad \leqq Ce^{-\frac{\lambda _1t^\alpha }{2}}\Vert h_0\Vert _{L^\infty }+Ce^{-{\lambda _0}t^\alpha }{\mathcal {D}}(t,0). \end{aligned}$$

(4.58)

Recall (4.56). Let $\eta >0$ be suitably small, then (4.13) follows from (4.58) and (4.42). Therefore, the proof of Proposition 4.4 is complete. $\quad \square$

4.3 Proof of Theorem 1.2.

The local existence and uniqueness of the solution to nonlinear problem (4.1) is provided in Proposition 6.2. In what follows we will show (1.17). Notice that for any $t>0$, it holds that

$$\begin{aligned} \int _{\Omega }\int _{{\mathbb {R}}^3}\sqrt{\mu }L_{\sqrt{\mu }f_*}f\mathrm{d}x\mathrm{d}v=\int _{\Omega }\int _{{\mathbb {R}}^3}\sqrt{\mu }\Gamma (f,f)\,\mathrm{d}x\mathrm{d}v\equiv 0. \end{aligned}$$

Then applying the linear theory Proposition 4.4 to f, we have

$$\begin{aligned}&\sup _{0\leqq s\leqq t}e^{\lambda _0 s^\alpha }\{\Vert wf(s)\Vert _{L^\infty }+|wf(s)|_{L^\infty (\gamma )}\}\nonumber \\&\quad \leqq C\Vert wf_0\Vert _{L^\infty }+C\sup _{0\leqq s\leqq t}e^{\lambda _0 s^\alpha }\{\Vert \nu ^{-1}wL_{\sqrt{\mu }f_*}f(s)\Vert _{L^\infty }+\Vert \nu ^{-1}w\Gamma (f,f)(s)\Vert _{L^\infty }\}\nonumber \\&\quad \leqq C\Vert wf_0\Vert _{L^\infty }+C\{\delta +\sup _{0\leqq s\leqq t}\Vert wf(s)\Vert _{L^\infty }\}\cdot \sup _{0\leqq s\leqq t}e^{\lambda _0 s^\alpha }\Vert wf(s)\Vert _{L^\infty }, \end{aligned}$$

(4.59)

where we have used the nonlinear estimate (2.16). Now we make the a priori assumption that

$$\begin{aligned} \sup _{0\leqq s\leqq t}e^{\lambda _0 s^\alpha }\Vert wf(s)\Vert _{L^\infty }\leqq 2C\Vert wf_0\Vert _{L^\infty }. \end{aligned}$$

Then from (4.59), we have

$$\begin{aligned}&\sup _{0\leqq s\leqq t}e^{\lambda _0 s^\alpha }\{\Vert wf(s)\Vert _{L^\infty }+|wf(s)|_{L^\infty (\gamma )}\}\\&\quad \leqq C\Vert wf_0\Vert _{L^\infty }+2C^2\Vert wf_0\Vert _{L^\infty }\cdot \{\delta +2C\Vert wf_0\Vert _{L^\infty }\} \leqq \frac{3C}{2}\Vert wf_0\Vert , \end{aligned}$$

provided that both $\delta >0$ and $\Vert wf_0\Vert _{L^\infty }$ are suitably small. This justifies that the a priori assumption can be closed. Then from a standard continuity argument, the global existence together with the estimate (1.17) follow. Therefore, the proof of Theorem 1.2 is complete. Note that since the obtained time-independent solution F(t, x, v) is nonnegative for all $t\geqq 0$ and converges to the stationary solution $F_*(x,v)$ in large time, one then has the non-negativity of $F_*(x,v)$.$\quad \square$

5 Dynamical Stability Under a Class of Large Perturbations

The section is devoted to proving Theorem 1.3. Recall $h(t,x,v):=wf(t,x,v)$. In what follows, we make the following a priori assumption:

$$\begin{aligned} \sup _{0\leqq s\leqq T}\Vert h(s)\Vert _{L^\infty }+\sup _{0\leqq s\leqq T}|h(s)|_{ {L^\infty (\gamma )}}\leqq {\bar{M}}, \end{aligned}$$

(5.1)

where $T>0$ is an arbitrary constant and ${\bar{M}}$ is a positive constant depending only on $M_0$ as given in (1.21). We emphasize here that $M_0$ is not necessarily small and will be determined at the end of the proof.

5.1 $L^p_{x,v}$ Estimates

First of all, we have

Lemma 5.1

([30]) Let $1<p<\infty$. Assume that f, $\partial _t+v\cdot \nabla _xf\in L^p([0,T]; L^p)$ and $f{\mathbf {1}}_{\gamma _-}\in L^p([0,T]; L^p(\gamma ))$. Then $f\in C^{0}([0,T];L^p)$ and $f{\mathbf {1}}_{\gamma _+}\in L^p([0,T]; L^p(\gamma ))$ and for almost every $t\in [0,T]$:

$$\begin{aligned}&\Vert f(t)\Vert _{L^p}^p+\int _0^t|f(s)|_{L^p(\gamma _+)}^p=\Vert f_0\Vert _{L^p}^p\nonumber \\&\quad +\int _0^t|f(s)|_{L^p(\gamma _-)}^p+p\int _0^t\int _{\Omega }\int _{{\mathbb {R}}^3}\{\partial _tf+v\cdot \partial _xf\}|f|^{p-2}f. \end{aligned}$$

(5.2)

Moreover, we prove that the $L^p$ bound of solutions grows in time exponentially related to ${\bar{M}}$. Note that ${\bar{M}}$ is to be chosen depending only on $M_0$, so that within a finite time interval, the solution can be uniformly small in $L^p$ if it is so initially.

Lemma 5.2

Let $1<p<\infty$. Under the assumption (5.1), it holds that

$$\begin{aligned} \Vert f(t)\Vert _{L^p}\leqq e^{C_3{\bar{M}}t}\Vert f_0\Vert _{L^p}, \end{aligned}$$

(5.3)

for any $t\in [0,T]$. Here $C_3>1$ is a generic constant depending only on $\kappa$ and p.

Proof

By Green’s identity (5.2), one has

$$\begin{aligned}&\Vert f(t)\Vert _{L^p}^p+\int _0^t|f(s)|_{L^p(\gamma _+)}^p\mathrm{d}s+p\int _0^t\Vert \nu ^{1/p}f(s)\Vert _{L^p}^p\mathrm{d}s\nonumber \\&\quad =\Vert f_0\Vert _{L^p}^p+\int _0^t|f(s)|_{L^p(\gamma _-)}^p+p\int _0^t\langle |f(s)|^{p-2}f(s),Kf(s)\rangle \mathrm{d}s\nonumber \\&\quad +p\int _0^t\langle |f(s)|^{p-2}f(s),-L_{\sqrt{\mu } f_*}f(s)+\Gamma (f,f)(s)\rangle \mathrm{d}s. \end{aligned}$$

(5.4)

It is straightforward to see from (2.9) that K is bounded from $L^p$ to $L^p$. Therefore, one has

$$\begin{aligned} \left| \int _0^t\langle |f(s)|^{p-2}f(s),Kf(s)\rangle \mathrm{d}s\right|&\leqq C\int _0^t\Vert f(s)\Vert _{L^p}^{p-1}\cdot \Vert Kf(s)\Vert _{L^p}\mathrm{d}s\nonumber \\&\leqq C\int _0^t\Vert f(s)\Vert _{L^p}^{p}\mathrm{d}s. \end{aligned}$$

(5.5)

As for the last term on the right-hand side of (5.4), it holds from (2.17) that

$$\begin{aligned} \left| \langle |f|^{p-2}f,-L_{\sqrt{\mu } f_*}f+\Gamma (f,f)\rangle \right|&\leqq \Vert \nu ^{1/p}f\Vert _{L^p}^{p-1}\Vert \nu ^{-1/p'}\left| -L_{\sqrt{\mu } f_*}f+\Gamma (f,f)\right| \Vert _{L^p}\nonumber \\&\leqq C\{\Vert wf_*\Vert _{L^\infty }+\sup _{0\leqq s\leqq t}\Vert wf(s)\Vert _{L^\infty }\}\cdot \Vert \nu ^{1/p}f\Vert _{L^p}^p\nonumber \\&\leqq C({\bar{M}}+1)\Vert \nu ^{1/p}f\Vert _{L^p}^p. \end{aligned}$$

(5.6)

To treat the boundary term $|f|_{L^p(\gamma _-)}$, the same as before, we introduce the cutoff function ${\mathbf {1}}_{\gamma _+^{\varepsilon }}$ near the grazing set $\gamma _+^{\varepsilon }$ defined in (4.6). Then by a direct computation, we have

$$\begin{aligned} |f|_{L^p(\gamma _-)}^p&\leqq C|P_\gamma f|_{L^p(\gamma _-)}^p+C|r|_{L^p(\gamma _-)}^p\leqq C|P_\gamma f|_{L^p(\gamma _-)}^p+C\delta ^p|f|_{L^p(\gamma _+)}^p\nonumber \\&\leqq \int _{\gamma _-}\mu ^{p/2}(v)|n(x)\cdot v|\,\mathrm{d}\gamma \left( \int _{{\mathcal {V}}(x)}\sqrt{\mu (u)}\{f{\mathbf {1}}_{\gamma _+^{\varepsilon }} +f{\mathbf {1}}_{\gamma _+\setminus \gamma _+^{\varepsilon }}\}|n(x)\cdot u|\,\mathrm{d}u\right) ^p\nonumber \\&\quad +C\delta ^{{p}}|f|_{L^p(\gamma _+)}^p\nonumber \\&\leqq C\{{\varepsilon }^p+C\delta ^p\}\cdot |f|_{L^p(\gamma _+)}^p+C|f{\mathbf {1}}_{\gamma _+\setminus \gamma _+^{\varepsilon }}|_{L^p(\gamma _+)}^p. \end{aligned}$$

(5.7)

From the trace estimate (4.8), it holds that

$$\begin{aligned}&\int _0^t|f{\mathbf {1}}_{\gamma _+\setminus \gamma _+^{\varepsilon }}(s)|_{L^p(\gamma _+)}^p\mathrm{d}s=\int _0^t\left| |f|^p{\mathbf {1}}_{\gamma _+\setminus \gamma _+^{\varepsilon }}(s)\right| _{L^1(\gamma _+)}\mathrm{d}s \nonumber \\&\quad \leqq C_{{\varepsilon },\Omega }\left\{ \Vert f_0\Vert _{L^p}^p+\int _0^t\Vert f(s)\Vert _{L^p}^p\mathrm{d}s+\int _0^t\left\| [\partial _t+v\cdot \nabla _x]|f(s)|^p\right\| _{L^1}\mathrm{d}s\right\} .\nonumber \\ \end{aligned}$$

(5.8)

Notice that

$$\begin{aligned}&[\partial _t+v\cdot \nabla _x]|f|^p=p|f|^{p-2}f\{\partial _tf+v\cdot \nabla _xf\}\\&\quad =p|f|^{p-2}f\left\{ -\nu f+Kf+\Gamma (f,f)-L_{\sqrt{\mu }f_*}f\right\} . \end{aligned}$$

Then from (5.5) and (5.6), it holds that

$$\begin{aligned} \int _0^t\left\| [\partial _t+v\cdot \nabla _x]|f(s)|^p\right\| _{L^1}\mathrm{d}s\leqq C({\bar{M}}+1)\int _0^t\Vert {f(s)}\Vert _{L^p}^p\mathrm{d}s. \end{aligned}$$

Combining this with (5.7) and (5.8), we obtain that

$$\begin{aligned}&\int _0^t|f(s)|_{L^p(\gamma _-)}^p\mathrm{d}s\leqq C\{{\varepsilon }^{p}+\delta ^p\}\cdot \int _0^t|f(s)|_{L^p(\gamma _+)}^p\mathrm{d}s\nonumber \\&\quad +\,C_{{\varepsilon }}\Vert f_0\Vert _{L^p}^p+C_{{\varepsilon }}({\bar{M}}+1)\int _0^t\Vert f(s)\Vert _{L^p}^p\mathrm{d}s. \end{aligned}$$

(5.9)

Substituting (5.5), (5.6) and (5.9) into (5.4), and then taking both ${\varepsilon }>0$ as well as $\delta >0$ suitably small, we have

$$\begin{aligned}&\Vert f(t)\Vert _{L^p}^p+\int _0^t|f(s)|_{L^p}^p\mathrm{d}s+\int _0^t\Vert \nu ^{1/p}f(s)\Vert _{L^p}^p\mathrm{d}s\\&\quad \leqq C\Vert f_0\Vert _{L^p}^p+C({\bar{M}}+1)\int _0^t\Vert f(s)\Vert _{L^p}^p\mathrm{d}s. \end{aligned}$$

Then (5.3) follows from Gronwall’s inequality. Therefore, the proof of Lemma 5.2 is complete. $\quad \square$

5.2 $L^\infty _{x,v}$-Estimate

Lemma 5.3

Under the a priori assumption (5.1), for $t\in [0,\min \{T,T_0\}]$ and for almost every $(x,v)\in {\bar{\Omega }}\times {\mathbb {R}}^3\setminus \gamma _0$, it holds that:

$$\begin{aligned} |h(t,x,v)|\leqq&S(t)+ \int _{\max \{t_1,0\}}^tI(t,s)\left| w\Gamma (f,f)(s,x-(t-s)v,v)\right| \,\mathrm{d}s, \end{aligned}$$

(5.10)

where

$$\begin{aligned} \begin{aligned}S(t)=\,&Ce^{-\lambda t^\alpha }M_0+CT_0^{5/2}\big \{m^{3+\kappa }+\delta +2^{-{\hat{C}}_4T_0^{5/4}}+\frac{1}{N}+\frac{1}{N^{\beta -4}}\big \}\cdot \{{\bar{M}}+{\bar{M}}^2\}\\&+C_{N,T_0,m}\left\{ \sup _{0\leqq s\leqq t}\Vert f(s)\Vert _{L^p}+\sup _{0\leqq s\leqq t}\Vert f(s)\Vert _{L^p}^2\right\} . \end{aligned} \end{aligned}$$

Here the positive constants $T_0$ and N can be taken arbitrarily large and m can be taken arbitrarily small.

Proof

We denote ${\tilde{G}}(t)$ as the solution operator of (4.10) provided by Proposition 4.4. Then the solution h of (4.3) is given in terms of Duhamel’s formula as

$$\begin{aligned} h(t,x,v)&=\left( {\tilde{G}}(t)h_0\right) (t,x,v)\nonumber \\&\quad +\int _0^t \left( {\tilde{G}}(t-s)[-wL_{\sqrt{\mu }f_*}f(s)+w\Gamma (f,f)(s)]\mathrm{d}s\right) (t,x,v). \end{aligned}$$

(5.11)

Using (4.13) and (2.16), we have

$$\begin{aligned} \Vert {\tilde{G}}(t)h_0\Vert _{L^\infty }\leqq Ce^{-\lambda _0 t^\alpha }\Vert h_0\Vert _{L^\infty }, \end{aligned}$$

(5.12)

and

$$\begin{aligned}&\left| \int _0^t {\tilde{G}}(t-s)wL_{\sqrt{\mu }f_*}f(s)\,\mathrm{d}s\right| \leqq C\Vert wf_*\Vert _{L^\infty }\cdot \int _0^te^{-\lambda _0(t-s)^\alpha }\Vert h(s)\Vert _{L^\infty }\mathrm{d}s\nonumber \\&\quad \leqq C\delta \sup _{0\leqq s\leqq t}\Vert h(s)\Vert _{L^\infty }. \end{aligned}$$

(5.13)

To estimate the last term on the right-hand side of (5.11), denoting

$$\begin{aligned} Z(t,x,v):=\bigg ({\tilde{G}}(t-s)w\Gamma (f,f)(s)\bigg )(t,x,v) \end{aligned}$$

and then applying the mild formulation (4.11) to Z(t, x, v), we obtain that

$$\begin{aligned} \left( {\tilde{G}}(t-s)w\Gamma (f,f)\right) (t,x,v)=\sum _{i=1}^3H_i+{\mathbf {1}}_{\{t_1>s\}}\sum _{i=4}^{11}H_i, \end{aligned}$$

(5.14)

where

$$\begin{aligned} H_1&={\mathbf {1}}_{\{t_1\leqq s\}}I(t,s)w\Gamma (f,f)(s,x-(t-s)v,v)\\ H_2+H_3&=\int _{\max \{t_{1},s\}}^{t}I(t,\tau )[K_w^m+K_w^c]Z(\tau ,x-(t-\tau )v,v)\,\mathrm{d}\tau ,\quad H_4=I(t,t_{1})wr[Z](t_{1},x_{1},v)\\ H_5&=\frac{I(t,t_{1})}{{\tilde{w}}(v)}\int _{\prod _{j=1}^{k-1}{\mathcal {V}}_{j}}\sum ^{k-1}_{l=1}{\mathbf {1}}_{\{t_{l+1}\leqq s<t_{l}\}}w\Gamma (f,f)(s,x_{l}-(t_{l}-s)v_{l},v_{l})\,\mathrm{d}\Sigma _{l}(s)\\ H_6+H_7&=\frac{I(t,t_{1})}{{\tilde{w}}(v)}\int _{\prod _{j=1}^{k-1}{\mathcal {V}}_{j}}\int _{s}^{t_{l}}\sum ^{k-1}_{l=1}{\mathbf {1}}_{\{t_{l+1}\leqq s<t_{l}\}}[K_w^m+K^c_w]Z(\tau ,x_{l}-(t_{l}-\tau )v_{l},v_{l})\,\mathrm{d}\Sigma _{l}(\tau )\,\mathrm{d}\tau \\ H_8+H_9&=\frac{I(t,t_{1})}{{\tilde{w}}(v)}\int _{\prod _{j=1}^{k-1}{\mathcal {V}}_{j}}\int _{t_{l+1}}^{t_{l}}\sum ^{k-1}_{l=1}{\mathbf {1}}_{\{t_{l+1}> s\}}[K_w^m+K^c_w]Z(\tau ,x_{l}-(t_{l}-\tau )v_{l},v_{l})\,\mathrm{d}\Sigma _{l}(\tau )\,\mathrm{d}\tau \\ H_{10}&=\frac{I(t,t_{1})}{{\tilde{w}}(v)}\int _{\prod _{j=1}^{k-1}{\mathcal {V}}_{j}}\sum ^{k-2}_{l=1}{\mathbf {1}}_{\{t_{l+1}> s\}}wr[Z](t_{l+1},x_{l+1},v_{l})\,\mathrm{d}\Sigma _{l}(t_{l+1})\\ H_{11}&=\frac{I(t,t_{1})}{{\tilde{w}}(v)}\int _{\prod _{j=1}^{k-1}{\mathcal {V}}_{j}}{\mathbf {1}}_{\{t_{k}>s\} }Z(t_{k},x_{k},v_{k-1})\,\mathrm{d}\Sigma _{k-1}(t_{k}), \end{aligned}$$

and r is defined in (4.4). Here the same as before, $k={\hat{C}}_3T_0^{5/4}$ such that (4.12) holds for $\eta =\frac{5}{16}$. We first consider terms $H_2$, $H_5$ and $H_7$ involving $K_w^m$. On one hand, similar for obtaining (4.19), we have

$$\begin{aligned} |H_2|&\leqq \int _{\max \{t_{1},s\}}^{t}I(t,\tau )\left| K_w^mZ(\tau ,x-(t-\tau )v,v)\right| \,\mathrm{d}\tau \nonumber \\&\leqq \int _{\max \{t_{1},s\}}^{t}I(t,\tau )\{{\mathbf {1}}_{\{|v|\leqq d_\Omega \}}+{\mathbf {1}}_{\{|v|>d_\Omega \}}\}\left| K_w^mZ(\tau ,x-(t-\tau )v,v)\right| \,\mathrm{d}\tau \nonumber \\&\leqq Cm^{3+\kappa }\int _{\max \{t_1,s\}}^te^{-{\bar{\nu }}_0(t-\tau )}\Vert {\tilde{G}}(\tau -s)w\Gamma (f,f)(s)\Vert _{L^\infty }\,\mathrm{d}\tau \nonumber \\&\leqq Cm^{3+\kappa }\Vert h(s)\Vert _{L^\infty }^{{2}}\cdot \int _{\max \{t_1,s\}}^te^{-{\bar{\nu }}_0(t-\tau )-\lambda _0(\tau -s)^\alpha }\,\mathrm{d}\tau \nonumber \\&\leqq Cm^{3+\kappa }e^{-\lambda _0(t-s)^\alpha }\Vert h(s)\Vert ^2_{L^\infty }. \end{aligned}$$

(5.15)

On the other hand, similar for obtaining (4.22), we have

$$\begin{aligned} |H_6|\leqq \,&\sum _{l=1}^{k-1}\sum _{m=1}^{l}C\bigg \{\int _{\prod _{j=1}^{l}{\mathcal {V}}_{j}}{\mathbf {1}}_{\{t_{l+1}\leqq s<t_{l}\}}\times {\mathbf {1}}_{\big \{|v_{m}|= \max \big [|v_{1}|,|v_{2}|,...|v_{l}|\big ]\big \}}e^{\frac{5|v_{m}|^{2}}{16}} \prod _{j=1}^{l}\mathrm{d}\sigma _{j}\nonumber \\&\times \int _s^{t_l}e^{-\lambda _1(t-\tau )^\alpha }\Vert K_w^mZ(\tau )\Vert _{L^\infty }\,\mathrm{d}\tau \bigg \}\nonumber \\ \leqq \,&Ck^2m^{3+\kappa }e^{-\lambda _0(t-s)^{\alpha }}\Vert h(s)\Vert ^2_{L^\infty }\cdot \sup _{j}\left| \int _{{\mathcal {V}}_j}e^{\frac{5|v_{j}|^{2}}{16}}\mathrm{d}\sigma _j\right| \nonumber \\ \leqq \,&Ck^2m^{3+\kappa }e^{-\lambda _0(t-s)^{\alpha }}\Vert h(s)\Vert ^2_{L^\infty }. \end{aligned}$$

(5.16)

Similarly, it holds that

$$\begin{aligned} |H_8|\leqq Ck^2m^{3+\kappa }e^{-\lambda _0(t-s)^{\alpha }}\Vert h(s)\Vert ^2_{L^\infty }. \end{aligned}$$

(5.17)

For the terms $H_4$ and $H_{10}$ involving r, we see from (4.13) and (4.4) that

$$\begin{aligned} \begin{aligned} |r[Z](\tau )|_{L^\infty (\gamma _-)}\leqq C\delta e^{-\lambda _0(\tau -s)^\alpha }|h(s)|^2_{L^\infty (\gamma _+)}. \end{aligned} \end{aligned}$$

Therefore, similar for obtaining (5.16), we have

$$\begin{aligned} |H_{10}|\leqq \,&\sum _{l=1}^{k-1}\sum _{m=1}^{l}C\bigg \{\int _{\prod _{j=1}^{l}{\mathcal {V}}_{j}}{\mathbf {1}}_{\{t_{l+1}>s\}}\nonumber \\&\times {\mathbf {1}}_{\big \{|v_{m}|= \max \big [|v_{1}|,|v_{2}|,...|v_{l}|\big ]\big \}}e^{\frac{5|v_{m}|^{2}}{16}}e^{-\lambda _1(t-t_{l+1})^\alpha }|r[Z](t_{l+1})|_{L^\infty (\gamma _-)} \prod _{j=1}^{l}\mathrm{d}\sigma _{j}\bigg \}\nonumber \\ \leqq \,&Ck^2\delta e^{-\lambda _0(t-s)^{\alpha }}|h(s)|^2_{L^\infty (\gamma _+)}\cdot \sup _{j}\left| \int _{{\mathcal {V}}_j}e^{\frac{5|v_{j}|^{2}}{16}}\mathrm{d}\sigma _j\right| \nonumber \\ \leqq \,&Ck^2\delta e^{-\lambda _0(t-s)^{\alpha }}|h(s)|^2_{L^\infty (\gamma _+)}, \end{aligned}$$

(5.18)

and

$$\begin{aligned} |H_4|&\leqq C{\mathbf {1}}_{\{t_1>s\}}I(t,t_{1})\{{\mathbf {1}}_{\{|v|\leqq d_\Omega \}}+{\mathbf {1}}_{\{|v|>d_\Omega \}}\}\left| wr[Z](t_{1},x_{1},v)\right| \nonumber \\&\leqq C e^{-{\bar{\nu }}_0(t-t_1)}\left| wr[Z](t_{1},x_{1},v)\right| \leqq C\delta e^{-\lambda _0(t-s)^\alpha }|h(s)|^2_{L^\infty (\gamma _+)}. \end{aligned}$$

(5.19)

For $H_{11}$, we note that

$$\begin{aligned} |Z(t_{k},x_k,v_{k-1})|\leqq \left| {\tilde{G}}(t_{k}-s)w\Gamma (f,f)(s)\right| _{L^\infty {(\gamma _-)} }\leqq e^{-\lambda _0(t_k-s)^\alpha }\Vert h(s)\Vert ^2_{L^\infty }. \end{aligned}$$

Then by (4.12), we have

$$\begin{aligned} |H_{11}|\leqq Ce^{-\lambda _0(t-s)^\alpha }\Vert h(s)\Vert ^2_{L^\infty }\cdot \left( \frac{1}{2}\right) ^{{\hat{C}}_4T_0^{5/4}}. \end{aligned}$$

(5.20)

For the terms $H_7$ and $H_9$ involving $K_w^c$, similar for obtaining (4.27), we have

$$\begin{aligned} |H_8|&\leqq C \sum _{l=1}^{k-1}\sum _{m=1}^{{l} }\int _{\Pi _{j=1}^{l-1}{{\mathcal {V}}}_{j}}\mathrm{d}{\sigma }_{l-1}\cdots \mathrm{d}{\sigma }_1 \\&\quad \times \int _{{\mathcal {V}}_l}\int _{{\mathbb {R}}^3} \int _s^{{t}_l} e^{-\lambda _1(t-\tau )^{\alpha }} {\mathbf {1}}_{\{{t}_{l+1}\leqq s<{t}_l\}} e^{\frac{5|v_m|^2}{16}} |k^c_w(v_l,v')\\&\qquad Z(\tau ,{x}_l-{v}_l({t}_l-\tau ),v')|\,\mathrm{d}\tau \,\mathrm{d}v' \mathrm{d}{\sigma }_l, \end{aligned}$$

and further split it as

$$\begin{aligned} |H_8|&\leqq C \sum _{l=1}^{k-1}\sum _{m=1}^{l-1}\int _{\Pi _{j=1}^{l-1}{{\mathcal {V}}}_{j}}e^{\frac{5|v_m|^2}{16}} \mathrm{d}{\sigma }_{l-1}\cdots \mathrm{d}{\sigma }_1 \int _{{\mathcal {V}}_l\cap \{|v_l|\geqq N\}}\int _{{\mathbb {R}}^3}\int _s^{{t}_l} {\bar{\Delta }}\mathrm{d}\tau \mathrm{d}v' \mathrm{d}{\sigma }_l \\&\quad +C \sum _{l=1}^{k-1}\sum _{m=1}^{l-1}\int _{\Pi _{j=1}^{l-1}{{\mathcal {V}}}_{j}}e^{\frac{5|v_m|^2}{16}} \mathrm{d}{\sigma }_{l-1}\cdots \mathrm{d}{\sigma }_1 \int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{{\mathbb {R}}^3} \int _{t_l-\frac{1}{N}}^{{t}_l} {\bar{\Delta }}\mathrm{d}\tau \mathrm{d}v' \mathrm{d}{\sigma }_l\\&\quad + C \sum _{l=1}^{k-1}\sum _{m=1}^{l-1}\int _{\Pi _{j=1}^{l-1}{{\mathcal {V}}}_{j}} e^{\frac{5|v_m|^2}{16}} \mathrm{d}{\sigma }_{l-1}\cdots \mathrm{d}{\sigma }_1 \int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{\{|v'|\geqq 2N\}} \int _s^{{t}_l-\frac{1}{N}} {\bar{\Delta }} \mathrm{d}\tau \mathrm{d}v' \mathrm{d}{\sigma }_l\\&\quad + C \sum _{l=1}^{k-1}\sum _{m=1}^{l-1}\int _{\Pi _{j=1}^{l-1}{{\mathcal {V}}}_{j}} e^{\frac{5|v_m|^2}{16}} \mathrm{d}{\sigma }_{l-1}\cdots \mathrm{d}{\sigma }_1 \int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{\{|v'|\leqq 2N\}} \int _s^{{t}_l-\frac{1}{N}} {\bar{\Delta }} \mathrm{d}\tau \mathrm{d}v' \mathrm{d}{\sigma }_l, \end{aligned}$$

where

$$\begin{aligned} {\bar{\Delta }}:=e^{-\lambda _1(t-\tau )^{\alpha }} {\mathbf {1}}_{\{{t}_{l+1}\leqq s<{t}_l\}}e^{\frac{5|v_l|^2}{16}}|k^c_w(v_l,v') Z(\tau ,{x}_l-{v}_l({t}_l-\tau ),v')|. \end{aligned}$$

Then it follows that

$$\begin{aligned} |H_8|&\leqq \frac{Ck^2e^{-\lambda _0(t-s)^\alpha }}{N}\sup _{s\leqq \tau \leqq t}\Vert h(\tau )\Vert ^2_{L^\infty }\nonumber \\&\quad +\,C \sum _{l=1}^{k-1}\sum _{m=1}^{l-1}\int _{\Pi _{j=1}^{l-1}{{\mathcal {V}}}_{j}} e^{\frac{5|v_m|^2}{16}} \mathrm{d}{\sigma }_{l-1}\cdots \mathrm{d}{\sigma }_1 \int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{\{|v'|\leqq 2N\}} \int _s^{{t}_l-\frac{1}{N}} {\bar{\Delta }} \mathrm{d}\tau \mathrm{d}v' \mathrm{d}{\sigma }_l, \end{aligned}$$

(5.21)

By Hölder inequality, it holds that

$$\begin{aligned}&\int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{\{|v'|\leqq 2N\}} \int _s^{{t}_l-\frac{1}{N}} {\bar{\Delta }} \mathrm{d}\tau \mathrm{d}v' \mathrm{d}{\sigma }_l \nonumber \\&\quad \leqq C\int _s^{t_l-\frac{1}{N}}e^{-\lambda _1(t-\tau )^\alpha }\mathrm{d}\tau \left( \int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{\{|v'|\leqq 2N\}}e^{-\frac{|v_l|^2}{8}}|k_w^c(v_l,v')|^{p'}\mathrm{d}v'\mathrm{d}v_l\right) ^{\frac{1}{p'}}\nonumber \\&\qquad \times \left( \int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{\{|v'|\leqq 2N\}}{\mathbf {1}}_{{\{}t_{l+1}\leqq s<t_l{\}}}|Z(\tau ,x_{l}-(t_l-\tau )v_l,v')|^p\mathrm{d}v'\mathrm{d}v_l\right) ^{\frac{1}{p}}.\nonumber \\ \end{aligned}$$

(5.22)

Since $1<p'=\frac{p}{p-1}<3$, then by (2.12) with $a=1$, it holds that

$$\begin{aligned} \int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{\{|v'|\leqq 2N\}}e^{-\frac{|v_l|^2}{8}}|k_w^c(v_l,v')|^{p'}\mathrm{d}v'\mathrm{d}v_l\leqq Cm^{p'(\kappa -1)}. \end{aligned}$$

Therefore, it holds that

$$\begin{aligned} \begin{aligned}&\text {Right-hand side of }(5.22) \leqq \, C_Nm^{{\kappa } -1}\int _s^{t_l-\frac{1}{N}}e^{-\lambda _1(t-\tau )^\alpha }\,\mathrm{d}\tau \\&\quad \times \left( \int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{\{|v'|\leqq 2N\}}{\mathbf {1}}_{{\{}t_{l+1}\leqq s<t_l{\}}}\left| \frac{Z}{w}(\tau ,x_{l}-(t_l-\tau )v_l,v')\right| ^p\mathrm{d}v'\mathrm{d}v_l\right) ^{\frac{1}{p}}. \end{aligned} \end{aligned}$$

Note that $y_l:=x_l-(t_l-\tau )v_l\in \Omega$ for $s\leqq \tau \leqq t_l-\frac{1}{N}$. Then making chang of variable $v_l\rightarrow y_l$, we obtain that

$$\begin{aligned} \int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{\{|v'|\leqq 2N\}} \int _s^{{t}_l-\frac{1}{N}} {\bar{\Delta }} \mathrm{d}\tau \mathrm{d}v' \mathrm{d}{\sigma }_l\leqq C_{N,m}\int _{s}^{t_l-\frac{1}{N}}e^{-\lambda _1(t-\tau )^\alpha }\left\| \frac{Z(\tau )}{w}\right\| _{L^p}\,\mathrm{d}\tau . \end{aligned}$$

Similarly, for obtaining (5.3), we have

$$\begin{aligned} \left\| \frac{Z(\tau )}{w}\right\| _{L^p}&=\left\| \frac{{\tilde{G}}(\tau -s)w\Gamma (f,f)(s)}{w}\right\| _{L^p}\leqq Ce^{C (\tau -s)}\Vert \Gamma (f,f)(s)\Vert _{L^p}\nonumber \\&\leqq C_{T_0}e^{-\lambda _0 (\tau -s)^\alpha }\Vert \Gamma (f,f)(s)\Vert _{L^p}\leqq C_{T_0}e^{-\lambda _0 (\tau -s)^\alpha }\Vert h(s)\Vert _{L^\infty }\cdot \Vert f(s)\Vert _{L^p}\nonumber \\&\leqq \frac{Ce^{-\lambda _0 (\tau -s)^\alpha }}{N}\Vert h(s)\Vert ^2_{L^\infty }+C_{N,T_0}e^{-\lambda _0 (\tau -s)^\alpha }\Vert f(s)\Vert _{L^p}^2, \end{aligned}$$

(5.23)

which implies that

$$\begin{aligned}&\int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{\{|v'|\leqq 2N\}} \int _s^{{t}_l-\frac{1}{N}} {\bar{\Delta }} \mathrm{d}\tau \mathrm{d}v' \mathrm{d}{\sigma }_l\\&\quad \leqq \frac{Ce^{-\lambda _0 (t-s)^\alpha }}{N}\Vert h(s)\Vert ^2_{L^\infty }+C_{N,T_0,m}e^{-\lambda _0 (t-s)^\alpha }\Vert f(s)\Vert _{L^p}^2. \end{aligned}$$

Substituting this into (5.21), we have

$$\begin{aligned} |H_8|\leqq \frac{Ck^2e^{-\lambda _0(t-s)^\alpha }}{N}\sup _{0\leqq s\leqq t}\Vert h(s)\Vert ^2_{L^\infty }+C_{N,T_0,m}e^{-\lambda _0(t-s)^\alpha }\sup _{0\leqq s\leqq t}\Vert f(s)\Vert ^2_{L^p}. \end{aligned}$$

(5.24)

Similarly, we have

$$\begin{aligned} |H_{10}|\leqq \frac{Ck^2e^{-\lambda _0(t-s)^\alpha }}{N}\sup _{0\leqq s\leqq t}\Vert h(s)\Vert ^2_{L^\infty }+C_{N,T_0,m}e^{-\lambda _0(t-s)^\alpha }\sup _{0\leqq s\leqq t}\Vert f(s)\Vert ^2_{L^p}. \end{aligned}$$

(5.25)

For $H_5$, similar as (4.22), we have

$$\begin{aligned} \int _0^t|H_{5}| \mathrm{d}s&\leqq C\int _0^t e^{-\lambda _1(t-s)^{\alpha }}\mathrm{d}s\sum _{l=1}^{k-1}\sum _{m=1}^{l}\int _{\Pi _{j=1}^{l-1}{{\mathcal {V}}}_{j}}\mathrm{d}{\sigma }_{l-1}\cdots \mathrm{d}{\sigma }_1 \\&\quad \times \int _{{\mathcal {V}}_l}\int _{{\mathbb {R}}^3} {\mathbf {1}}_{\{{t}_{l+1}\leqq s<{t}_l\}} e^{\frac{5|v_m|^2}{16}} \left| w\Gamma (f,f)(s,x_l-(t_l-s)v_l,v_l)\right| \mathrm{d}\tau \mathrm{d}v' \mathrm{d}{\sigma }_l. \end{aligned}$$

Then it follows that

$$\begin{aligned} \int _0^t|H_{5}| \mathrm{d}s&\leqq C\int _0^{t}e^{-\lambda _1(t-s)^\alpha }\Vert h(s)\Vert _{L^\infty }\mathrm{d}s \sum _{l=1}^{k-1}\sum _{m=1}^{l-1}\int _{\Pi _{j=1}^{l-1}{{\mathcal {V}}}_{j}}e^{\frac{5|v_m|^2}{16}} \mathrm{d}{\sigma }_{l-1}\cdots \mathrm{d}{\sigma }_1\nonumber \\&\quad \times \int _{{\mathcal {V}}_l}e^{-\frac{|v_l|^2}{8}}{\mathbf {1}}_{\{t_{l+1}\leqq s<t_l\}}\mathrm{d}v_l\left\{ \int _{{\mathbb {R}}^3} \left| h(s,x_{l}-(t_l-s)v_l,v')\right| ^p\langle v'\rangle ^{-4-p(\beta -4)}\,\mathrm{d}v'\right\} ^{1/p}\nonumber \\&\leqq C\int _0^{t}e^{-\lambda _1(t-s)^\alpha }\Vert h(s)\Vert _{L^\infty }\mathrm{d}s\nonumber \\&\quad \times \bigg (\sum _{l=1}^{k-1}\sum _{m=1}^{l-1}\int _{\Pi _{j=1}^{l-1}{{\mathcal {V}}}_{j}}e^{\frac{5|v_m|^2}{16}} \mathrm{d}{\sigma }_{l-1}\cdots \mathrm{d}{\sigma }_1\left\{ \int _{{\mathcal {V}}_l}\int _{{\mathbb {R}}^3}{\tilde{\Delta }}\mathrm{d}v_l\,\mathrm{d}v'\right\} ^{1/p}\bigg ), \end{aligned}$$

(5.26)

where we have used (2.20) and (2.21) in the first inequality, and also denoted that

$$\begin{aligned} {\tilde{\Delta }}:=e^{-\frac{|v_l|^2}{8}}{\mathbf {1}}_{\{t_{l+1}\leqq s<t_l\}}\left| h(s,x_{l}-(t_l-s)v_l,v')\right| ^p\langle v'\rangle ^{-4-p(\beta -4)}. \end{aligned}$$

Now, we consider the integral in (5.26) over either $\{|v_l|\geqq N\}$ or $\{|v_l|\leqq N, |v'|\geqq N\}$ or $\{|v_l|\leqq N, |v'|\leqq N, t_l-1/N\leqq s \leqq t_l\}$ or $\{|v_l|\leqq N, |v'|\leqq N, 0\leqq s \leqq t_l-1/N\}$. Over $\{|v_l|\geqq N\}$ or $\{|v_l|\leqq N, |v'|\geqq N\}$ or $\{|v_l|\leqq N, |v'|\leqq N, t_l-1/N\leqq s \leqq t_l\}$, it is bounded by

$$\begin{aligned} Ck^2\left( \frac{1}{N}+\frac{1}{N^{\beta -4}}\right) {\bar{M}}^2. \end{aligned}$$

Over $\{|v_l|\leqq N, |v'|\leqq N, 0\leqq s \leqq t_l-1/N\}$, it is bounded by

$$\begin{aligned}&C_N\sum _{l=1}^{k-1}\sum _{m=1}^{l-1}\int _0^{t}e^{-\lambda _1(t-s)^\alpha }\Vert h(s)\Vert _{L^\infty }\mathrm{d}s \int _{\Pi _{j=1}^{l-1}{{\mathcal {V}}}_{j}}e^{\frac{5|v_m|^2}{16}} \mathrm{d}{\sigma }_{l-1}\cdots \mathrm{d}{\sigma }_1\\&\qquad \times \left\{ \int _{{\mathcal {V}}_l\cap \{|v_l|\leqq N\}}\int _{\{|v'|\leqq N\}}e^{-\frac{|v_l|^2}{8}}{\mathbf {1}}_{\{t_{l+1}\leqq s<t_l-\frac{1}{N}\}}\left| f(s,x_{l}-(t_l-s)v_l,v')\right| ^p\mathrm{d}v_l\mathrm{d}v'\right\} ^{1/p}\\&\quad \leqq C_Nk^2\int _0^{t}e^{-\lambda _1(t-s)^\alpha }\Vert h(s)\Vert _{L^\infty }\cdot \Vert f(s)\Vert _{L^p}\mathrm{d}s\leqq \frac{Ck^2}{N}{\bar{M}}^2+C_Nk^2\sup _{0\leqq s\leqq t}\Vert f(s)\Vert _{L^p}^2, \end{aligned}$$

where we have used the change of variable $v_l\rightarrow y_l:=x_l-(t_l-s)v_l$ above. Therefore, for $H_5$, it holds that

$$\begin{aligned} \int _0^t|H_5|\,\mathrm{d}s\leqq Ck^2\left( \frac{1}{N}+\frac{1}{N^{\beta -4}}\right) {\bar{M}}^2+C_Nk^2\sup _{0\leqq s\leqq t}\Vert f(s)\Vert _{L^p}^2. \end{aligned}$$

(5.27)

Substituting (5.15), (5.16), (5.17), (5.18), (5.19), (5.20), (5.24) (5.25) and (5.27) into (5.14), we have

$$\begin{aligned}&\int _0^t\left| \left( {\tilde{G}}(t-s)w\Gamma (f,f)(s)\right) (t,x,v)\right| \,\mathrm{d}s\nonumber \\&\quad \leqq \int _0^t{\mathbf {1}}_{\{t_1\leqq s\}}I(t,s)w\Gamma (f,f)(s,x-(t-s)v,v)\,\mathrm{d}s\nonumber \\&\qquad +\int _0^t\mathrm{d}s\int _{\max \{t_{1},s\}}^{t}I(t,\tau )\,\mathrm{d}\tau \int _{{\mathbb {R}}^3}\left| k_w^c(v,u)\left( {\tilde{G}}(\tau -s)w\Gamma (f,f)(s)\right) (\tau ,x-(t-\tau )v,u)\right| \,\mathrm{d}u\nonumber \\&\qquad +CT_0^{5/2}{\bar{M}}^2\big \{m^{3+\kappa }+\delta +2^{-{\hat{C}}_4T_0^{5/4}}+\frac{1}{N}+\frac{1}{N^{\beta -4}}\big \}+C_{N,T_0,m}\sup _{0\leqq s\leqq t}\Vert f(s)\Vert _{L^p}^2\nonumber \\&\quad :=H_{12}+H_{13}+H_{14}. \end{aligned}$$

(5.28)

To further estimate $H_{13}$, we denote $x'=x-(t-\tau )v$ and $\tau _1'=t_1(\tau ,x',u).$ Then by Fubini Theorem and (5.28), it holds that

$$\begin{aligned} |H_{13}|&=\left| \int _{\max \{t_1,0\}}^t\int _{{\mathbb {R}}^3}I(t,\tau )k_w^c(v,u)\,\mathrm{d}u\mathrm{d}\tau \int _{0}^\tau \left( {\tilde{G}}(\tau -s)w\Gamma (f,f)(s)\right) (\tau ,x',u)\,\mathrm{d}s\right| \nonumber \\&\leqq H_{13,1}+H_{13,2}+H_{13,3}, \end{aligned}$$

(5.29)

where

$$\begin{aligned} H_{13,1}&= \int _{\max \{t_1,0\}}^t\int _0^\tau \int _{{\mathbb {R}}^3}|k_w^c(v,u)|{\mathbf {1}}_{\{t_1'\leqq s\}}I(t,s)\left| w\Gamma (f,f)(s,x'-(\tau -s)u,u)\right| \,\mathrm{d}u\mathrm{d}s\mathrm{d}\tau ,\\ H_{13,2}&=\int _{\max \{t_1,0\}}^t\int _0^\tau \int _{{\mathbb {R}}^3}\int _{\max \{t_1',s\}}^\tau I(t,\tau ')|k_w^c(v,u)|\,\mathrm{d}\tau '\mathrm{d}u\mathrm{d}s\mathrm{d}\tau \\&\quad \times \int _{{\mathbb {R}}^3} \left| k_{w}^c(u,u')\left( {\tilde{G}}(\tau '-s)w\Gamma (f,f)(s)\right) (\tau ',x'-(\tau -\tau ')u,u')\right| \,\mathrm{d}u',\\ H_{13,3}&=|H_{14}|\cdot \int _{\max \{t_1,0\}}^t\int _{{\mathbb {R}}^3}I(t,\tau )|k_w^c(v,u)|\,\mathrm{d}u\mathrm{d}\tau . \end{aligned}$$

Similarly to before, we have

$$\begin{aligned} |H_{13,3}|&\leqq |H_{14}|\cdot \int _{\max \{t_1,0\}}^tI(t,\tau )\{{\mathbf {1}}_{\{|v|\leqq d_\Omega \}}+{\mathbf {1}}_{\{|v|>d_\Omega \}}\}\,\mathrm{d}\tau \int _{{\mathbb {R}}^3}|k_w^c(v,u)|\,\mathrm{d}u\nonumber \\&\leqq C|H_{14}|\cdot \int _{\max \{t_1,0\}}^te^{-{\bar{\nu }}_0(t-\tau )}\,\mathrm{d}\tau \leqq CH_{14}. \end{aligned}$$

(5.30)

For $H_{13,1}$, we have from (2.20) and (2.21) that

Now we divide the estimates by the following cases:

Case 1.$|v|\geqq N$. We have from (2.14) that

$$\begin{aligned} |H_{13,1}|\leqq \frac{C}{N}\sup _{0\leqq s\leqq t}\Vert h(s)\Vert _{L^\infty }^2. \end{aligned}$$

Case 2.$|v|\leqq N, |u|\geqq 2N$. In this case, we have $|v-u|\geqq N$, so that by (2.14),

$$\begin{aligned} \int _{\{|u|\leqq 2N\}}|k_{w}^c(v,u)|\,\mathrm{d}u\leqq e^{-\frac{N^2}{32}}\int _{\{|u|\leqq 2N\}}|k_{w}^c(v,u)e^{\frac{|v-u|^2}{32}}|\,\mathrm{d}u\leqq Ce^{-\frac{N^2}{32}}. \end{aligned}$$

It then follows that

$$\begin{aligned} |H_{13,1}|\leqq Ce^{-\frac{N^2}{32}}\sup _{0\leqq s\leqq t}\Vert h(s)\Vert _{L^\infty }^2. \end{aligned}$$

Case 3.$|v|\leqq N, |u|\leqq 2N, |u'|>N$. By $\beta >4$, it holds that

$$\begin{aligned} |H_{13,1}|\leqq \frac{C}{N^{\beta -4}}\sup _{0\leqq s\leqq t}\Vert h(s)\Vert _{L^\infty }^2. \end{aligned}$$

Case 4.$|v|\leqq N, |u|\leqq 2N, |u'|\leqq N, \tau -1/N<s\leqq \tau$. It is straightforward to see that

$$\begin{aligned} |H_{13,1}|\leqq \frac{C}{N}\sup _{0\leqq s\leqq t}\Vert h(s)\Vert _{L^\infty }^2. \end{aligned}$$

Case 5.$|v|\leqq N, |u|\leqq 2N, |u'\leqq N, 0\leqq s\leqq \tau -1/N.$ By Hölder’s inequality, we have

$$\begin{aligned} \begin{aligned} |H_{13,1}|&\leqq C_N\sup _{0\leqq s\leqq t}\Vert h(s)\Vert _{L^\infty }\cdot \int _{\max \{t_1,0\}}^t\int _0^{\tau -\frac{1}{N}}e^{-{\bar{\nu }}_0(t-s)}\mathrm{d}s\mathrm{d}\tau \left( \int _{\{|u|\leqq 2N\}}|k_w^c(v,u)|^{p'}\mathrm{d}u\right) ^{1/p'}\\&\quad \cdot \left( \int _{\{|u|\leqq 2N\}}\int _{\{|u'|\leqq N\}}{\mathbf {1}}_{\{t_1'\leqq s\}}\left| f(s,x'-(\tau -s)u,u')\right| ^p\mathrm{d}u\mathrm{d}u'\right) ^{1/p}\\&\leqq C_{N,m}\sup _{0\leqq s\leqq t}\Vert h(s)\Vert _{L^\infty }\cdot \sup _{0\leqq s\leqq t}\Vert f(s)\Vert _{L^p}\leqq \frac{C}{N}\sup _{0\leqq s\leqq t}\Vert h(s)\Vert ^2_{L^\infty }+C_{N,m}\sup _{0\leqq s\leqq t}\Vert f(s)\Vert _{L^p}^2. \end{aligned} \end{aligned}$$

Collecting the estimates for these cases, we have

$$\begin{aligned} |H_{13,1}|\leqq C\left( \frac{1}{N}+\frac{1}{N^{\beta -4}}\right) \sup _{0\leqq s\leqq t}\Vert h(s)\Vert ^2_{L^\infty }+C_{N,m}\sup _{0\leqq s\leqq t}\Vert f(s)\Vert _{L^p}^2. \end{aligned}$$

(5.31)

Similarly, for $H_{13,2}$, we have

$$\begin{aligned} |H_{13,2}|\leqq&\int _{\max \{t_1,0\}}^t\int _{0}^\tau \mathrm{d}s\mathrm{d}\tau \int _{s}^\tau e^{-{\bar{\nu }}_0(t-\tau ')}\,\mathrm{d}\tau '\nonumber \\&\times \int _{{\mathbb {R}}^3}\int _{{\mathbb {R}}^3}\left| k_{w}^c(v,u)k_{w}^c(u,u'){\mathbf {1}}_{\{\max \{t_1', s\}\leqq \tau '\leqq \tau \}}Z(\tau ',x'-(\tau -\tau ')u,u')\right| \,\mathrm{d}u\mathrm{d}u'\nonumber \\ \leqq&\frac{C}{N}\sup _{0\leqq s\leqq t}\Vert h(s)\Vert _{L^\infty }^2+C_N\int _{\max \{t_1,0\}}^t\int _{0}^\tau \mathrm{d}s\mathrm{d}\tau \int _{s}^{\tau -\frac{1}{N}} e^{-{\bar{\nu }}_0(t-\tau ')}\,\mathrm{d}\tau '\nonumber \\&\times \left( \int _{\{|u|\leqq 2N\}}\int _{\{|u'|\leqq 3N\}}\left| k_{w}^c(v,u)k_{w}^c(u,u')\right| ^{p'}\mathrm{d}u\mathrm{d}u'\right) ^{1/p'}\nonumber \\&\times \left( \int _{\{|u|\leqq 2N\}}\int _{\{|u'|\leqq 3N\}} {\mathbf {1}}_{\{\max \{t_1', s\}\leqq \tau '\leqq \tau \}}\left| \frac{Z}{w}(\tau ',x'-(\tau -\tau ')u,u')\right| ^p\mathrm{d}u\mathrm{d}u'\right) ^{1/p}\nonumber \\ \leqq \,&\frac{C}{N}\sup _{0\leqq s\leqq t}\Vert h(s)\Vert _{L^\infty }^2+C_{N,m}\int _{\max \{t_1,0\}}^t\int _{0}^\tau \mathrm{d}s\mathrm{d}\tau \int _{s}^{\tau -\frac{1}{N}} e^{-{\bar{\nu }}_0(t-\tau ')}\left\| \frac{Z(\tau ')}{w}\right\| _{L^p}\,\mathrm{d}\tau '\nonumber \\ \leqq \,&\frac{C}{N}\sup _{0\leqq s\leqq t}\Vert h(s)\Vert _{L^\infty }^2+C_{N,{T_0,}m}\sup _{0\leqq s\leqq t}\Vert f(s)\Vert _{L^p}^2. \end{aligned}$$

(5.32)

Here we have used (5.23) in the last inequality. Substituting (5.30), (5.31) and (5.32) into (5.29), we have

$$\begin{aligned}&\int _0^t\left| \left( {\tilde{G}}(t-s)w\Gamma (f,f)(s)\right) (t,x,v)\right| \,\mathrm{d}s\nonumber \\&\quad \leqq \int _0^t{\mathbf {1}}_{\{t_1\leqq s\}}I(t,s)w\Gamma (f,f)(s,x-(t-s)v,v)\,\mathrm{d}s\nonumber \\&\quad + CT_0^{5/2}\big \{m^{3+\kappa }+\delta +2^{-{\hat{C}}_4T_0^{5/4}}+\frac{1}{N}+\frac{1}{N^{\beta -4}}\big \}{\bar{M}}^2+C_{N,T_0,m}\sup _{0\leqq s\leqq t}\Vert f(s)\Vert _{L^p}^2. \end{aligned}$$

(5.33)

Then (5.10) naturally follows from (5.12), (5.13) and (5.33). Therefore, the proof of Lemma 5.3 is complete. $\quad \square$

Lemma 5.4

Under the assumption (5.1), there exists a constant $C>0$ independent of t, such that, for any $0\leqq t\leqq \min \{T,T_0\}$, it holds that

$$\begin{aligned} \Vert h(t)\Vert _{L^\infty }+|h(t)|_{L^\infty {(\gamma )}}\leqq \,&CM_0e^{-\lambda _0t^\alpha }\left( 1+\int _{0}^t\Vert h(\tau )\Vert _{L^\infty }\,\mathrm{d}\tau \right) \nonumber \\&+CT_0^{5/2}\left\{ m^{3+\kappa }+\delta +2^{-{\hat{C}}_4T_0^{5/4}}+\frac{1}{N} +\frac{1}{N^{\beta -4}}\right\} \cdot \{{\bar{M}}+{\bar{M}}^3\}\nonumber \\&+C_{N,T_0,m}\left\{ \sup _{0\leqq s\leqq t}\Vert f(s)\Vert _{L^p}+\sup _{0\leqq s\leqq t}\Vert f(s)\Vert _{L^p}^3\right\} . \end{aligned}$$

(5.34)

Here the positive constants $T_0$ and N can be chosen arbitrarily large and m can be chosen arbitrarily small.

Proof

We denote $x':=x-(t-s)v$ and $t_1':=t_{1}(s,x',u)$. It suffices to consider the last term on the right-hand side of (5.10). By (2.20) and (2.21), it holds that

$$\begin{aligned}&\int _{\max \{t_1,0\}}^tI(t,s)\left| w\Gamma (f,f)(s,x-(t-s)v,u)\right| \,\mathrm{d}u\mathrm{d}s\\&\quad \leqq \int _{\max \{t_1,0\}}^te^{-{\bar{\nu }}_0(t-s)}\Vert wf(s)\Vert _{L^\infty }\mathrm{d}s\left( \int _{{\mathbb {R}}^3}\left| h(s,x-(t-s)v,u)\right| ^p\langle u\rangle ^{-4-p(\beta -4)}\right) ^{1/p} \\&\quad \leqq \frac{C}{N^{\beta -4}}\sup _{0\leqq s\leqq t}\Vert h(s)\Vert _{L^\infty }^2\\&\qquad +C\int _{\max \{t_1,0\}}^te^{-{\bar{\nu }}_0(t-s)}\Vert wf(s)\Vert _{L^\infty }\mathrm{d}s\times \left( \int _{\{|u|\leqq N\}}\left| h(s,x',u)\right| ^p\langle u\rangle ^{-4-p(\beta -4)}\right) ^{1/p}. \end{aligned}$$

Notice that $x':=x-(t-s)v\in \Omega$. Then applying (5.10) to $h(s,x',u)$, we have

$$\begin{aligned}&\int _{\max \{t_1,0\}}^tI(t,s)\left| w\Gamma (f,f)(s,x-(t-s)v,u)\right| \,\mathrm{d}u\mathrm{d}\tau \nonumber \\&\quad \leqq \frac{C}{N^{\beta -4}}\sup _{0\leqq s\leqq t}\Vert h(s)\Vert _{L^\infty }^2+R_1+R_2, \end{aligned}$$

(5.35)

where we have defined

$$\begin{aligned} R_1=C \int _{\max \{t_1,0\}}^te^{-{\bar{\nu }}_0(t-s)}\Vert wf(s)\Vert _{L^\infty }S(s)\,\mathrm{d}s, \end{aligned}$$

and

$$\begin{aligned} R_2&=C \int _{\max \{t_1,0\}}^te^{-{\bar{\nu }}_0(t-s)}\Vert wf(s)\Vert _{L^\infty }\mathrm{d}s\times \bigg (\int _{\{|u|\leqq N\}}\langle u \rangle ^{-4-p(\beta -4)}\mathrm{d}u\\&\quad \times \bigg \{\int _{\max \{t_1',0\}}^s e^{-{\bar{\nu }}_0(s-s')}|w\Gamma (f,f)(s',x'-(s-s')u,u)|\,\mathrm{d}\tau '\bigg \}^p\bigg )^{1/p}. \end{aligned}$$

A direct computation shows that

$$\begin{aligned} |R_1|\leqq&\,C\int _{\max \{t_1,0\}}^te^{-{\bar{\nu }}_0(t-s)}\Vert wf(s)\Vert _{L^\infty }e^{-\lambda _0 s^\alpha }\Vert h_0\Vert _{L^\infty }\mathrm{d}s\nonumber \\&+CT_0^{5/2}\sup _{0\leqq s\leqq t}\Vert h(s)\Vert _{L^\infty }\times \big \{m^{3+\kappa }+\delta +2^{-T_0}+\frac{1}{N}+\frac{1}{N^{\beta -4}}\big \}\cdot \{{\bar{M}}+{\bar{M}}^2\}\nonumber \\&+C_{N,T_0,m}\sup _{0\leqq s\leqq t}\Vert h(s)\Vert _{L^\infty }\cdot \big \{\sup _{0\leqq s\leqq t}\Vert f(s)\Vert _{L^p}+\sup _{0\leqq s\leqq t}\Vert f(s)\Vert _{L^p}^2\big \}\nonumber \\ \leqq&\,CM_0e^{-\lambda _0 t^\alpha }\int _{0}^t\Vert h(s)\Vert _{L^\infty }\mathrm{d}s+CT_0^{5/2}\left\{ m^{3+\kappa }+\delta +2^{-T_0}+\frac{1}{N}+\frac{1}{N^{\beta -4}}\right\} \cdot \big \{{\bar{M}}+{\bar{M}}^3\big \}\nonumber \\&+C_{N,T_0,m}\big \{\sup _{0\leqq s\leqq t}\Vert f(s)\Vert _{L^p}+\sup _{0\leqq s\leqq t}\Vert f(s)\Vert _{L^p}^3\big \}. \end{aligned}$$

(5.36)

For $R_2$, using (2.20) and (2.21) again, we have

$$\begin{aligned} |R_2|\leqq \,&C\sup _{0\leqq s\leqq t}\Vert h(s)\Vert _{L^\infty }^2\cdot \int _{\max \{t_1,0\}}^te^{-{\bar{\nu }}_0(t-s)}\mathrm{d}s\nonumber \\&\times \bigg (\int _{|u|\leqq N}\int _{\max \{t_1',0\}}^\tau \int _{{\mathbb {R}}^3}e^{-{\bar{\nu }}_0(s-s')}|h(s',x'-(s-s')u,u')|^p\big [\langle u\rangle \langle u'\rangle \big ]^{-4-p(\beta -4)}\bigg )^{1/p}\nonumber \\ \leqq&\,C\left( \frac{1}{N}+\frac{1}{N^{\beta -4}}\right) \sup _{0\leqq s\leqq t}\Vert h(s)\Vert _{L^\infty }^3+C_N\sup _{0\leqq s\leqq t}\Vert h(s)\Vert _{L^\infty }^2\cdot \int _{\max \{t_1,0\}}^te^{-{\bar{\nu }}_0(t-s)}\mathrm{d}s\nonumber \\&\times \bigg (\int _{0}^{s-\frac{1}{N}}e^{-{\bar{\nu }}_0(s-s')}\mathrm{d}s'\int _{|u|\leqq N}\int _{|u'|\leqq N}{\mathbf {1}}_{\max \{t_1',0\}\leqq s'\leqq s}|f(s',x'-(s-s')u,u')|^p\mathrm{d}u\mathrm{d}u'\bigg )^{1/p}\nonumber \\ \leqq&\, C\left( \frac{1}{N}+\frac{1}{N^{\beta -4}}\right) \sup _{0\leqq s\leqq t}\Vert h(s)\Vert _{L^\infty }^3+C_{N}\sup _{0\leqq s\leqq t}\Vert f(s)\Vert _{L^p}^3. \end{aligned}$$

(5.37)

Here we have used the change of variable $u\rightarrow y':=x'-(s-s')u$. Then from combining (5.10), (5.35), (5.36) and (5.37), (5.34) follows. Therefore, the proof of Lemma 5.4 is complete. $\quad \square$

5.3 Proof of Theorem 1.3.

Let

$$\begin{aligned} {\mathcal {E}}(t):=1+\int _0^t\left( \Vert h(s)\Vert _{L^\infty }{+|h(s)|_{L^\infty (\gamma )}}\right) \,\mathrm{d}s. \end{aligned}$$

Then it holds from (5.34) that

$$\begin{aligned} \Vert h(t)\Vert _{L^\infty }+|h(t)|_{L^\infty (\gamma )}={\mathcal {E}}'(t)\leqq CM_0e^{-\lambda _0 t^\alpha }{\mathcal {E}}(t)+{\mathcal {D}}, \end{aligned}$$

(5.38)

where

$$\begin{aligned}&{\mathcal {D}}:=CT_0^{5/2}\left\{ m^{3+\kappa }+\delta +\frac{1}{N}+\frac{1}{N^{\beta -4}}\right\} {\bar{M}}^3+2^{-T_0}{\bar{M}}^3\\&\quad +C_{N,T_0,m}\left\{ e^{C_3{\bar{M}}T_0}\Vert f_0\Vert _{L^p}+ \left( e^{C_3{\bar{M}}T_0}\Vert f_0\Vert _{L^p}\right) ^3\right\} . \end{aligned}$$

From (5.38), we have

$$\begin{aligned} {\mathcal {E}}(t)\leqq {\mathcal {E}}(0)e^{CM_0\int _{0}^te^{-\lambda _0 s^\alpha }\mathrm{d}s}+{\mathcal {D}}\cdot \int _0^te^{CM_0\int _s^te^{-\lambda _0\tau ^\alpha }\,\mathrm{d}\tau }\mathrm{d}s\leqq (1+{\mathcal {D}}t)e^{CM_0}. \end{aligned}$$

(5.39)

Substituting (5.39) into (5.38), we have

$$\begin{aligned} \Vert h(t)\Vert _{L^\infty }+|h(t)|_{L^\infty (\gamma )}\leqq CM_0e^{CM_0}(1+{\mathcal {D}}t)e^{-\lambda _0 t^\alpha }+{\mathcal {D}}\leqq e^{CM_0}(1+{\mathcal {D}})e^{-\frac{\lambda _0 t^\alpha }{2}}+{\mathcal {D}}. \end{aligned}$$

(5.40)

Take ${\bar{M}}=2e^{CM_0}$, ${\tilde{{\varepsilon }}}=\min \{{\varepsilon }_0,(2C_0)^{-1}\}$ where $C_0$ and ${\varepsilon }_0$ are the same as ones in Theorem 1.2, and

$$\begin{aligned} T_0:=\max \left\{ 3\left( \log _2{\bar{M}}+1\right) +|\log _2{\tilde{{\varepsilon }}}|, \left( \frac{2\left( \log 4{\bar{M}}+|\log {\tilde{{\varepsilon }}}|\right) }{\lambda _0}\right) ^{1/\alpha }\right\} , \end{aligned}$$

such that $2^{-T_0}{\bar{M}}^3\leqq \frac{{\tilde{{\varepsilon }}}}{8}$ and ${\bar{M}}e^{-\frac{\lambda _0 T_0^\alpha }{2}}\leqq \frac{{\tilde{{\varepsilon }}}}{4}$. Then it holds that

$$\begin{aligned} CT_0^{5/2}\delta {\bar{M}}^3\leqq C_4\left[ |\log {\tilde{{\varepsilon }}}|^{\frac{5}{2\alpha }}+1\right] e^{C_4M_0}\delta , \end{aligned}$$

for some universal constant $C_4>1$. Let

$$\begin{aligned} 0<\delta \leqq \delta _0<\left( \frac{{\tilde{{\varepsilon }}}}{16C_4\left[ |\log {\tilde{{\varepsilon }}}|^{\frac{5}{2\alpha }}+1\right] }\right) ^{2}, \end{aligned}$$

and $0<M_0\leqq \frac{|\log \delta |}{2C_4}.$ Then it is straightforward to see that $CT_0^{5/2}\delta {\bar{M}}^3\leqq \frac{{\tilde{{\varepsilon }}}}{16}$. Now we take $0<m<1$ suitably small and N suitably large, and finally take $\Vert f_0\Vert _{L^p}\leqq {\varepsilon }_{1}$, with ${\varepsilon }_1>0$ sufficiently small, such that

$$\begin{aligned} CT_0^{5/2}\left\{ m^{3+\kappa }+\frac{1}{N}+\frac{1}{N^{\beta -4}}\right\} {\bar{M}}^3+C_{N,T_0,m}\left\{ e^{C_3{\bar{M}}T_0}\Vert f_0\Vert _{L^p}+ \left( e^{C_3{\bar{M}}T_0}\Vert f_0\Vert _{L^p}\right) ^3\right\} \leqq \frac{{\tilde{{\varepsilon }}}}{16}. \end{aligned}$$

Therefore, we have ${\mathcal {D}}\leqq \frac{{\tilde{{\varepsilon }}}}{4}.$ From (5.40), it holds, for any $t\in [0,T_0]$, that

$$\begin{aligned} \Vert h(t)\Vert _{L^\infty }+|h(t)|_{L^\infty (\gamma )}\leqq e^{CM_0}(1+{\mathcal {D}})+{\mathcal {D}}\leqq \left( 1+\frac{{\tilde{{\varepsilon }}}}{4}\right) e^{CM_0}+\frac{{\tilde{{\varepsilon }}}}{4}\leqq \frac{3{\bar{M}}}{4}. \end{aligned}$$

(5.41)

Notice that at $t=T_0$, we have from (5.40) that

$$\begin{aligned} \Vert h(T_0)\Vert _{L^\infty }\leqq e^{CM_0}(1+{\mathcal {D}})e^{-\frac{\lambda _0 T_0^{\alpha }}{2}}+{\mathcal {D}}\leqq \frac{{\tilde{{\varepsilon }}}}{2}. \end{aligned}$$

Then from (1.17), we have, for $t>T_0$,

$$\begin{aligned} \Vert h(t)\Vert _{L^\infty }+|h(t)|_{L^\infty (\gamma )}\leqq C_{{0}}\Vert h(T_0)\Vert _{{L^\infty }}\leqq C_0{\tilde{{\varepsilon }}}\leqq \frac{3{\bar{M}}}{4}. \end{aligned}$$

(5.42)

A combination of (5.41) and (5.42) justifies that the a priori assumption (5.1) can be closed by our choice. Notice that the local existence has been established in Proposition 6.2. Then the global existence of the solution follows from a standard continuity argument. For large time behavior, it holds, for $t\in [0,T_0]$, that

$$\begin{aligned} \Vert h(t)\Vert _{L^\infty }+|h(t)|_{L^\infty (\gamma )}\leqq {\bar{M}}\leqq e^{2CM_0}e^{\lambda _0 T_0^{\alpha } }e^{-\lambda _0 t^\alpha } \leqq C_5e^{C_5M_0}e^{-\lambda _0 t^\alpha }, \end{aligned}$$

(5.43)

for some constant $C_5>1$. For $t>T_0$, it holds from (1.17) that

$$\begin{aligned} \Vert h(t)\Vert _{L^\infty }+|h(t)|_{L^\infty (\gamma )}\leqq C_0e^{-\lambda _0(t-T_0)^\alpha }\Vert h(T_0)\Vert _{L^\infty }\leqq C_0C_5e^{C_5M_0}e^{-\lambda _0 t^\alpha }. \end{aligned}$$

(5.44)

By taking $C_2=C_0C_5$, (1.23) follows from (5.43) and (5.44). Therefore, the proof of Theorem 1.3 is complete. $\quad \square$

Appendix

1.1 An Iteration Lemma

Lemma 6.1

Consider a sequence $\{a_i\}_{i=0}^\infty$ with each $a_i\geqq 0$. For any fixed $k\in {\mathbb {N}}_+$, we denote

$$\begin{aligned} A_i^k=\max \{a_i, a_{i+1},\ldots , a_{i+k}\}. \end{aligned}$$

(1) Assume $D\geqq 0$. If $a_{i+1+k}\leqq \frac{1}{8} A_i^{k}+D$ for $i=0,1,\ldots$, then it holds that

$$\begin{aligned} A_i^k\leqq \left( \frac{1}{8}\right) ^{\left[ \frac{i}{k+1}\right] }\cdot \max \{A_0^k, \ A_1^k, \cdots , \ A_k^k \}+\frac{8+k}{7} D,\quad \text{ for }\quad i\geqq k+1. \end{aligned}$$

(6.1)

(2) Let $0\leqq \eta <1$ with $\eta ^{k+1}\geqq \frac{1}{4}$. If $a_{i+1+k}\leqq \frac{1}{8} A_i^{k}+C_k \cdot \eta ^{i+k+1}$ for $i=0,1,\ldots$, then it holds that

$$\begin{aligned} A_i^k\leqq \left( \frac{1}{8}\right) ^{\left[ \frac{i}{k+1}\right] }\cdot \max \{A_0^k, \ A_1^k, \ldots , \ A_k^k \}+2C_k\frac{8+k}{7} \eta ^{i+k},\quad \text{ for }\quad i\geqq k+1. \end{aligned}$$

(6.2)

Proof

We first show (6.1). By iteration in $i\geqq 0$, we obtain that

$$\begin{aligned} a_{i+1+2k}&\leqq \frac{1}{8} A_{i+k}^k+D=\frac{1}{8} \max \{a_{i+2k}, a_{i+2k-1},\ldots ,a_{i+k}\}+D\nonumber \\&\leqq \frac{1}{8}\max \left\{ a_{i+2k}, \ A_{i+k-1}^k\right\} +D\leqq \frac{1}{8}\max \left\{ \frac{1}{8} A_{i+k-1}^k+D, \ A_{i+k-1}^k\right\} +D\nonumber \\&\leqq \frac{1}{8} A_{i+k-1}^k+\left( 1+\frac{1}{8}\right) D \cdots \leqq \frac{1}{8} A_{i}^k+\left( 1+\frac{k}{8}\right) D. \end{aligned}$$

(6.3)

Similarly, for all $j=0,1,\ldots ,k-1$, we also have

$$\begin{aligned} a_{i+1+j+k}\leqq \frac{1}{8} A_{i}^k+\frac{8+k}{8}D. \end{aligned}$$

(6.4)

Therefore, for $1\leqq \frac{i}{k+1}\in {\mathbb {N}}_+$, it follows from (6.3) and (6.4) that

$$\begin{aligned} A_{i+1+k}^k&=\max \big \{a_{i+1+2k}, \ a_{i+2k},\ \cdots ,\ a_{i+1+k}\big \}\nonumber \\&\leqq \frac{1}{8} A_{i}^k+\frac{8+k}{8}D\leqq \left( \frac{1}{8}\right) ^2A_{i-2(k+1)}^k+\frac{1}{8} \frac{8+k}{8}D+\frac{8+k}{8}D\nonumber \\&=\left( \frac{1}{8}\right) ^2A_{i-2(k+1)}^k+\frac{8+k}{8}\left( 1+\frac{1}{8}\right) D=\cdots \nonumber \\&\leqq \left( \frac{1}{8}\right) ^{\frac{i}{k+1}}A_0^k+\frac{8+k}{8}\left( 1+\frac{1}{8}+\big (\frac{1}{8}\big )^2+\cdots \right) D\nonumber \\&\leqq \left( \frac{1}{8}\right) ^{\frac{i}{k+1}}A_0^k+\frac{8+k}{7}D. \end{aligned}$$

(6.5)

If $\frac{i}{k+1}\notin {\mathbb {N}}_+$ and $i=(k+1) \left[ \frac{i}{k+1}\right] +j$ for some $1\leqq j\leqq k$, then by similar arguments we have

$$\begin{aligned} A_{i+1+k}^k&\leqq \left( \frac{1}{8}\right) ^{\left[ \frac{i}{k+1}\right] +1} A_j^k+\frac{8+k}{7}D. \end{aligned}$$

(6.6)

Hence, from (6.5) and (6.6), we complete the proof of (6.1).

It remains to show (6.2). Noting $\eta <1$ and by similar arguments as in (6.3) and (6.4), we can get

$$\begin{aligned} a_{i+j+k+1}\leqq \frac{1}{8} A_{i}^k +C_k \left( 1+\frac{k}{8}\right) \eta ^{i+k+1},\ \text{ for } \ 0\leqq j\leqq k. \end{aligned}$$

(6.7)

Hence, for $1\leqq \frac{i}{k+1}\in {\mathbb {N}}_+$, noting $\frac{1}{8}\cdot \eta ^{-k-1}\leqq \frac{1}{2}$ and using (6.7), then we have

$$\begin{aligned}&A^k_{i+k+1}=\max \{a_{i+2k+1}, \cdots , a_{i+k+1} \}\leqq \frac{1}{8} A_{i}^k +C_k \left( 1+\frac{k}{8}\right) \eta ^{i+k+1}\nonumber \\&\quad \leqq \cdots \leqq \left( \frac{1}{8}\right) ^{\frac{i}{k+1}} A_0^k+ C_k \left( 1+\frac{k}{8}\right) \eta ^{i+k+1}\cdot \left\{ 1+\frac{1}{8} \eta ^{-k-1}+\left( \frac{1}{8} \eta ^{-k-1}\right) ^2+\cdots \right\} \nonumber \\&\quad \leqq \left( \frac{1}{8}\right) ^{\frac{i}{k+1}} A_0^k+2C_k \left( 1+\frac{k}{8}\right) \eta ^{i+k+1}. \end{aligned}$$

(6.8)

If $\frac{i}{k+1}\notin {\mathbb {N}}_+$ and $i=(k+1) \left[ \frac{i}{k+1}\right] +j$ for some $1\leqq j\leqq k$, then by similar arguments as above, we have

$$\begin{aligned} A^k_{i+k+1} \leqq \left( \frac{1}{8}\right) ^{\left[ \frac{i}{k+1}\right] +1} A_j^k+2C_k \left( 1+\frac{k}{8}\right) \eta ^{i+k+1}. \end{aligned}$$

(6.9)

Thus we prove (6.2) from (6.8) and (6.9). Therefore the proof of lemma 6.1 is complete. $\quad \square$

1.2 Local-in-Time Existence

Proposition 6.2

Let w(v) be the weight function defined in (1.9). Assume

$$\begin{aligned} |\theta -\theta _0|_{L^{\infty }(\partial \Omega )}=\delta \ll 1,\quad F_0(x,v)=F_*(x,v)+\mu ^{\frac{1}{2}}(v)f_0(x,v)\geqq 0 \end{aligned}$$

and $\Vert wf_0\Vert _{L^\infty }:=M_{0}<\infty .$ Then there exists a positive time

$$\begin{aligned} {\hat{t}}:=\bigg [{\hat{C}}\bigg (1+M_{0}\bigg )\bigg ]^{-1} \end{aligned}$$

such that the IBVP (1.1), (1.5) and (1.13) has a unique nonnegative solution

$$\begin{aligned} F(t,x,v)=F_*(x,v)+\mu ^{\frac{1}{2}}(v)f(t,x,v)\geqq 0 \end{aligned}$$

in $[0,{\hat{t}}]$ satisfying

$$\begin{aligned} \sup _{0\leqq t\leqq {\hat{t}}}\bigg \{\Vert wf(t)\Vert _{L^\infty }+|wf(t)|_{L^{\infty }(\gamma )}\bigg \}\leqq 2{\tilde{C}}(M_0+1). \end{aligned}$$

Here ${\tilde{C}}>0$ and ${\hat{C}}>1$ are generic constants independent of $M_0$. Moreover, if the domain $\Omega$ is strictly convex, $\theta (x)$ is continuous over $\partial \Omega$, the initial data $F_0(x,v)$ is continuous except on $\gamma _0$ and satisfies (1.19) then the solution F(t, x, v) is continuous in $[0,{\hat{t}}]\times \{\Omega \times {\mathbb {R}}^3\setminus \gamma _0\}$.

Proof

We consider the following iteration scheme:

$$\begin{aligned} \left\{ \begin{aligned}&\partial _tF^{n+1}+v\cdot \nabla _x F^{n+1}+F^{n+1}\cdot R(F^{n})=Q^+(F^n,F^n),\\&F^{n+1}(t,x,v)\Big |_{t=0}=F_0(x,v)\geqq 0,\\&F^{n+1}(t,x,v)|_{\gamma _-}=\mu _{\theta }(x,v)\int _{n(x)\cdot u>0} F^{n+1}(t,x,u)\{u \cdot n(x)\}\,\mathrm{d}u,\\&F^0(t,x,v)=\mu (v), \end{aligned} \right. \end{aligned}$$

(6.10)

where

$$\begin{aligned} {R}(F^n)(t,x,v)=\int _{{\mathbb {R}}^{3}\times {\mathbb {S}}^{2}}{B}(v-u,\omega )F^n(t,x,u)\,\mathrm{d}u\mathrm{d}\omega . \end{aligned}$$

Let

$$\begin{aligned} f^{n+1}(t,x,v)=\frac{F^{n+1}(t,x,v)-F_{*}(x,v)}{\mu ^{\frac{1}{2}}(v)},\quad h^{n+1}(t,x,v)=wf^{n+1}(t,x,v). \end{aligned}$$

Then the equation of $h^{n+1}$ reads as

$$\begin{aligned} \left\{ \begin{aligned}&\partial _th^{n+1}+v\cdot \nabla _x h^{n+1}+h^{n+1}\cdot {R}(F^{n})=wK_{*}f^n+w\Gamma ^+(f^n,f^n),\\&h^{n+1}(t,x,v)\Big |_{t=0}=h_0(x,v),\\&h^{n+1}\Big |_{\gamma _-}=\frac{1}{{\tilde{w}}(v)}\int _{\{n(x)\cdot u>0\}} h^{n+1} {\tilde{w}}(u)\,\mathrm{d}\sigma (x)+w(v)\frac{\mu _{\theta }-\mu }{\sqrt{\mu }}\int _{n(x)\cdot u>0}h^{n+1}{\tilde{w}}(u)\,\mathrm{d}\sigma (x),\\&h^{0}(t,x,v)=-wf_*(x,v), \end{aligned} \right. \end{aligned}$$

(6.11)

where we have denoted

$$\begin{aligned} K_{*}f^n:=-\sqrt{\mu }^{-1}\left\{ R(\sqrt{\mu }f^n)F_*+\Gamma ^{+}(\sqrt{\mu }f^{n},F_*)+\Gamma ^+(F_*,\sqrt{\mu }f^n)\right\} . \end{aligned}$$

Now we shall use the induction on $n=0,1,\ldots$ to show that there exists a positive time ${\hat{t}}_1>0$, independent of n, such that (6.10) or equivalently (6.11) admits a unique mild solution on the time interval $[0,{\hat{t}}_1]$, and the following uniform bound and positivity hold true:

$$\begin{aligned} \Vert h^n(t)\Vert _{L^\infty }+|h^n(t)|_{L^{\infty }(\gamma )}\leqq 2{\tilde{C}}[\Vert h_0\Vert _{L^\infty }+1], \end{aligned}$$

(6.12)

and

$$\begin{aligned} F^n(t,x,v)\geqq 0, \end{aligned}$$

(6.13)

for $0\leqq t\leqq {\hat{t}}=\left( {\hat{C}}\{1+\Vert h_0\Vert _{L^\infty }\}\right) ^{-1}$ and suitably chosen constants ${\tilde{C}}>0$ and ${\hat{C}}$ independent of t. Thanks to the fact that

$$\begin{aligned} \Vert h^0(t)\Vert _{L^\infty }+|h^0(t)|_{L^\infty (\gamma )}\leqq \Vert wf_*\Vert _{L^\infty }+|wf_*|_{L^\infty (\gamma )}\leqq C\delta , \end{aligned}$$

we see that (6.12) is obviously true for $n=0$. To proceed, we assume that (6.12) holds true up to $n\geqq 0$. Since $F^n\geqq 0$, it holds that ${R}(F^n)\geqq 0$. Then by using a similar argument as in [19, Lemma 3.4], one can construct the solution operator $G^n(t)$ to the following linear problem:

$$\begin{aligned} \left\{ \begin{aligned}&\partial _th+v\cdot \nabla _x h+{R}(F^n)h=0, \quad t>0,\quad x\in \Omega , \quad v\in {\mathbb {R}}^3,\\&h(t,x,v)|_{t=0}=h_0(x,v),\\&h(t,x,v)\Big |_{\gamma _-}=\frac{1}{{\tilde{w}}(v)}\int _{n(x)\cdot u>0} h(t,x,u) {\tilde{w}}(u)\,\mathrm{d}\sigma (x)+w(v)\frac{\mu _{\theta }-\mu }{\sqrt{\mu }}\int _{n(x)\cdot u>0}h(t,x,u){\tilde{w}}(u)\,\mathrm{d}\sigma (x) \end{aligned} \right. \end{aligned}$$

over $(0,\rho )$ for some universal constant $\rho >0$ independent of n, provided that $|\theta -1|_{L^\infty (\partial \Omega )}$ is sufficiently small. Moreover, $G^n(t)$ satisfies the estimate

$$\begin{aligned} \Vert G^{n}(t)h_0\Vert _{L^{\infty }}+|G^n(t)h_0|_{L^\infty (\gamma )}\leqq C_{\rho }\Vert h_0\Vert _{L^{\infty }}. \end{aligned}$$

(6.14)

Here the constant $C_\rho >0$ is independent of n. Then applying Duhamel’s formula to (6.11), we have, for $0<t<\rho ,$ that

$$\begin{aligned} h^{n+1}(t)= {G}^{n}(t,0)h_0+\int _0^t {G}^{n}(t,s)[w K_*f^n(s)+w\Gamma ^+(f^n,f^n)(s)]\mathrm{d}s. \end{aligned}$$

(6.15)

Taking $L^\infty$-norm on the both sides of (6.15) and using (2.16) and (6.14), we have

$$\begin{aligned}&\Vert h^{n+1}(t)\Vert _{L^\infty }+|h^{n+1}(t)|_{L^\infty {(\gamma )}}\nonumber \\&\quad \leqq C_\rho \Vert h_0\Vert _{L^\infty }+C_\rho \int _0^t\Vert wK_{*}f^n(s)\Vert _{L^\infty }+\Vert w\Gamma ^+(f^n,f^n)(s)\Vert _{L^\infty }\mathrm{d}s\nonumber \\&\quad \leqq {C} \Vert h_0\Vert _{L^\infty }+C\int _0^t\Vert h^n(s)\Vert _{L^\infty }+\Vert h^n(s)\Vert ^2_{L^\infty }\mathrm{d}s\nonumber \\&\quad \leqq C_{{3}}\Vert h_0\Vert _{L^{\infty }}+C_{{3}}t\cdot \{\sup _{0\leqq s\leqq t}\Vert h^n(s)\Vert _{L^{\infty }}+\sup _{0\leqq s\leqq t}\Vert h^n(s)\Vert _{L^\infty }^2\} \end{aligned}$$

(6.16)

for some constants $C_3>1$. Now we take ${\tilde{C}}=C_3$ and ${\hat{C}}=8C_3^2$. Then by the induction hypothesis (6.12), for any $0<t<{\hat{t}}$, it follows from (6.16) that

$$\begin{aligned} \begin{aligned} \Vert h^{n+1}(t)\Vert _{L^\infty }+|h^{n+1}(t)|_{L^\infty (\gamma )}&\leqq C_{3}\{\Vert h_0\Vert _{L^{\infty }}+1\}\cdot \{1+2C_3t[1+2C_3]\\&\quad \cdot [1+\Vert h_0\Vert _{L^\infty }]\}\\&\leqq 2C_3\{\Vert h_0\Vert _{L^\infty }+1\}. \end{aligned} \end{aligned}$$

This then proves (6.12) for $n+1$. Next we show the non-negativity (6.13) for $n+1$. We denote that

$$\begin{aligned} I^{n}(t,s):=\exp \left\{ -\int _s^t[R(F^n)](\tau ,X_{cl}(\tau ),V_{cl}(\tau ))\,\mathrm{d}\tau \right\} \end{aligned}$$

and

$$\begin{aligned} \mathrm{d}\Sigma _{l}^n(\tau ):= & {} \left\{ \prod _{j=l+1}^{k-1}\mathrm{d}\sigma _{j}\right\} \cdot I^n(t_l,\tau )[v_l\cdot n(v_l)]\mathrm{d}v_l\\&\quad \cdot \prod _{j=1}^{l-1}\left\{ I^n(t_{j},t_{j+1})\mu _{\theta }(x_{j+1},v_{j})[v_j\cdot n(x_j)]\mathrm{d}v_{j}\right\} . \end{aligned}$$

Then we have the following mild formulation for $F^{n+1}$:

(6.17)

for $t>0$, $x\in {\bar{\Omega }}\times {\mathbb {R}}^3\setminus \gamma _0\cup \gamma _-$ and integer $k\geqq 1$. From (6.12), it holds that

$$\begin{aligned} |F^{n+1}(t,x,v)|=\left| F_{*}(x,v)+\mu ^{\frac{1}{2}}(v)\frac{h^{n+1}(t,x,v)}{w(v)}\right| \leqq C(1+\Vert h_0\Vert _{L^{\infty }})\mu ^{\frac{1}{2}}(v) \end{aligned}$$

(6.18)

for some constant $C>0$. Furthermore, a direct computation shows that

$$\begin{aligned} 0<\mu _{\theta }(x_{j+1},v_{j})\leqq \frac{1}{(1-\delta )^2}\exp \left\{ \frac{\delta |v_j|^2}{2(1-\delta )}\right\} \mu (v_j). \end{aligned}$$

Then similarly as for (4.12), we have, for sufficiently large $T_0>0$ and for $k={\hat{C}}_5T_0^{5/4}$, that

$$\begin{aligned} \int _{\prod _{j=1}^{k-2}{\mathcal {V}}_j}{\mathbf {1}}_{\{t_{k-1}>0\}}\prod _{j=1}^{k-2}\mu _{\theta }(x_{j+1},v_j)\{n(x_j)\cdot v_j\}\mathrm{d}v_j\leqq \left( \frac{1}{2}\right) ^{{\hat{C}}_6T_0^{5/4}} \end{aligned}$$

for some generic constant ${\hat{C}}_5>0$ and ${\hat{C}}_6>0$. Then, by (6.17) and (6.18), we have

$$\begin{aligned} \begin{aligned} F^{n+1}&\geqq -C\mu _\theta (x_{1},v)\{1+\Vert h_0\Vert _{L^\infty }\} \int _{\prod _{j=1}^{k-2}{\mathcal {V}}_j}{\mathbf {1}}_{\{t_{k-1}>0\}}\\&\quad \prod _{j=1}^{k-2}\mu _{\theta }(x_{j+1},v_j)\{n(x_j)\cdot v_j\}\mathrm{d}v_j\\&\geqq -C\mu _\theta (x_{1},v)\{1+\Vert h_0\Vert _{L^\infty }\}\cdot \left( \frac{1}{2}\right) ^{{\hat{C}}_6T_0^{5/4}}. \end{aligned} \end{aligned}$$

(6.19)

Since $T_0>0$ can be taken arbitrarily large, we have $F^{n+1}\geqq 0.$ This then proves (6.12) and (6.13). Finally, with the uniform estimates (6.12) in hand, we can use a similar argument as one in [19, Theorem 4.1] to show that $h^n,$$n=0,1,2\cdots ,$ is a Cauchy sequence in $L^\infty$. We omit here for brevity. The solution is obtained by taking the limit $n\rightarrow \infty$. If $\Omega$ is convex and the compatibility condition (1.19) holds, the continuity is a direct consequence of the $L^\infty$-convergence. The uniqueness is standard. The proof of Proposition 6.2 is complete. $\quad \square$