2.1 Preliminaries and Main Results
The steady state solution S of free molecular flow under the Maxwell- type boundary condition (5) has been constructed explicitly, [13]:
$$\begin{aligned} S( \mathbf x , {\varvec{\zeta }}) =\,&\frac{ 1 }{ C_S } \alpha \sum \limits _{i=1}^{\infty }(1-\alpha )^{i-1} \left( \frac{2\pi }{RT( \mathbf x _{(i)})} \right) ^{\frac{1}{2}} M_{T( \mathbf x _{(i)})}( {\varvec{\zeta }}),\nonumber \\ C_S =\,&\frac{1}{|D|} \alpha \sum \limits _{i=1}^{\infty }(1-\alpha )^{i-1}\int \left( \frac{2\pi }{RT({ \mathbf x _{(i)}}_*)} \right) ^{\frac{1}{2}} M_{T({ \mathbf x _{(i)}}_*)}( {\varvec{\zeta }}_*) d \mathbf x _* d {\varvec{\zeta }}_*, \end{aligned}$$
(17)
where the boundary point obtained by tracing back from the given interior point \( \mathbf x \) along the direction \(-\frac{ {\varvec{\xi }}}{| {\varvec{\xi }}|}\):
$$\begin{aligned}&\mathbf y _B \left( \textstyle \mathbf x , \frac{ {\varvec{\xi }}}{| {\varvec{\xi }}|} \right) = \mathbf x - {\varvec{\xi }}\times \sup \left\{ s\ge 0: \mathbf x - {\varvec{\xi }}s' \in D, \text { for all } s' \in (0,s) \right\} ,\\&\mathbf x _{(1)} = \mathbf y _B \left( \textstyle \mathbf x , \frac{ {\varvec{\xi }}}{| {\varvec{\xi }}|} \right) ,\\&{\varvec{\xi }}^1 = {\varvec{\xi }}-2( {\varvec{\xi }}\cdot \mathbf n ( \mathbf x _{(1)})) \mathbf n ( \mathbf x _{(1)}),\\&\mathbf x _{(k+1)} = \mathbf y _B \left( \textstyle \mathbf x _{(k)}, \frac{ {\varvec{\xi }}^k}{| {\varvec{\xi }}^k|} \right) , \\&{\varvec{\xi }}^{k+1} = {\varvec{\xi }}^k -2( {\varvec{\xi }}^k\cdot \mathbf n ( \mathbf x _{(k+1)})) \mathbf n ( \mathbf x _{(k+1)}).\\ \end{aligned}$$
Since the domain D is symmetric, we have the following lemma:
Lemma 1
For \(( \mathbf x , {\varvec{\xi }})\in D\times \mathbb {R} ^d\) and each \(k\ge 1\),
$$\begin{aligned}&| {\varvec{\xi }}^{k}|=| {\varvec{\xi }}|,\\&| \mathbf x _{(k+1)}- \mathbf x _{(k)}|=| \mathbf x _{(k+2)}- \mathbf x _{(k+1)}|\ge | \mathbf x - \mathbf x _{(1)}|. \end{aligned}$$
From the explicit expression (17),
$$\begin{aligned} S( \mathbf x , {\varvec{\zeta }}) - M( {\varvec{\zeta }}) = O(1- T_* ) M( {\varvec{\zeta }}). \end{aligned}$$
(18)
We note that S has constant boundary flux \(1/C_S\):
$$\begin{aligned}&\int _{ {\varvec{\xi }}\cdot \mathbf n <0} - {\varvec{\xi }}\cdot \mathbf n S( \mathbf y , {\varvec{\zeta }}) d {\varvec{\zeta }}\nonumber \\&\quad \qquad = \frac{1}{C_S} \int _{ {\varvec{\xi }}\cdot \mathbf n <0} - {\varvec{\xi }}\cdot \mathbf n \alpha \sum \limits _{i=1}^{\infty }(1-\alpha )^{i-1} \left( \frac{2\pi }{RT( \mathbf y _{(i)})} \right) ^{\frac{1}{2}} M_{T\left( \mathbf y _{(i)}\right) }( {\varvec{\zeta }}) d {\varvec{\zeta }}\nonumber \\&\quad \qquad = \frac{1}{C_S} \alpha \sum \limits _{i=1}^{\infty }(1-\alpha )^{i-1} \left( 4\pi \right) ^\frac{1}{2} \int _{\hat{\varvec{\xi }}\cdot \mathbf n <0} -\hat{\varvec{\xi }}\cdot \mathbf n M\left( \hat{\varvec{\zeta }}\right) d\hat{\varvec{\zeta }}= \frac{1}{C_S}. \end{aligned}$$
(19)
Note that the molecular number is conserved under the evolutionary Eq. (4) and the boundary condition (3), and therefore the total molecular number \(\int g( \mathbf x , {\varvec{\zeta }},t) d \mathbf x d {\varvec{\zeta }}\) is a constant of time. Thus we may define the average total density as:
$$\begin{aligned} \rho _* \equiv \frac{1}{|D|} \int _{D\times \mathbb {R} ^3} g_{in}( \mathbf x , {\varvec{\zeta }}) d \mathbf x d {\varvec{\zeta }}= \frac{1}{|D|} \int _{D\times \mathbb {R} ^3} g( \mathbf x , {\varvec{\zeta }},t) d \mathbf x d {\varvec{\zeta }}, \end{aligned}$$
a constant associated with \(g_{in}\). Due to the equilibrating effect of the Maxwell-type boundary condition, one can expect the solution g to approach the steady state \(\rho _* S\). Namely, we expect the function \(g - \rho _* S( \mathbf x , {\varvec{\zeta }})\) to decay to zero. Moreover, \(g-\rho _*S\) satisfies the same evolutionary Eq. (4) and the boundary condition (3). Since the space dimension is d and \(d<3\), it is natural to integrate out the extra microscopic velocity degrees of freedom:
$$\begin{aligned} \bar{g} ( \mathbf x , {\varvec{\xi }},t) \equiv&\int _{ \mathbb {R} ^{3-d}} \left( g( \mathbf x , {\varvec{\zeta }},t) - \rho _* S( \mathbf x , {\varvec{\zeta }}) \right) d {\varvec{\eta }},\end{aligned}$$
(20)
$$\begin{aligned} \bar{g}_{in}( \mathbf x , {\varvec{\xi }}) \equiv&\int _{ \mathbb {R} ^{3-d}} \left( g_{in}( \mathbf x , {\varvec{\zeta }}) - \rho _* S( \mathbf x , {\varvec{\zeta }}) \right) d {\varvec{\eta }},\end{aligned}$$
(21)
$$\begin{aligned} \varvec{s}( \mathbf x , {\varvec{\xi }}) \equiv&\int _{ \mathbb {R} ^{3-d}} S( \mathbf x , {\varvec{\zeta }}) d {\varvec{\eta }}. \end{aligned}$$
(22)
Recall that \( {\varvec{\eta }}\) is the last \(3-d\) components of \( {\varvec{\zeta }}\), (2). Since S has constant boundary flux, the corresponding boundary flux becomes \(j_g-\rho _*/C_S\):
$$\begin{aligned} j( \mathbf y ,t) \equiv&\int \limits _{ {\varvec{\xi }}\cdot \mathbf n <0} - {\varvec{\xi }}\cdot \mathbf n \bar{g} ( \mathbf x , {\varvec{\xi }},t)d {\varvec{\xi }}\nonumber \\ =&\int \limits _{ {\varvec{\xi }}\cdot \mathbf n <0} - {\varvec{\xi }}\cdot \mathbf n \left( \int \limits _{ \mathbb {R} ^{3-d}} \left( g( \mathbf y , {\varvec{\zeta }},t) - \rho _* S( \mathbf y , {\varvec{\zeta }}) \right) d {\varvec{\eta }} \right) d {\varvec{\xi }}\nonumber \\ =&\int \limits _{ {\varvec{\xi }}\cdot \mathbf n <0} - {\varvec{\xi }}\cdot \mathbf n g( \mathbf y , {\varvec{\zeta }},t) d {\varvec{\zeta }}- \rho _*/C_S = j_{g}( \mathbf y ,t) - \frac{\rho _*}{C_S}\equiv j_{g}( \mathbf y ,t) -j_S, \end{aligned}$$
(23)
and the total molecular number becomes zero:
$$\begin{aligned} \int \limits _{D\times \mathbb {R} ^3} \left( g( \mathbf x , {\varvec{\zeta }},t) - \rho _* S( \mathbf x , {\varvec{\zeta }}) \right) d \mathbf x d {\varvec{\zeta }}= \int \limits _{D\times \mathbb {R} ^d} \bar{g}( \mathbf x , {\varvec{\xi }},t) d \mathbf x d {\varvec{\xi }}= 0. \end{aligned}$$
(24)
Moreover, the new functions \(\bar{g}( \mathbf x , {\varvec{\xi }},t), j( \mathbf y ,t)\) satisfy equations similar to that for the original functions:
$$\begin{aligned}&{\left\{ \begin{array}{ll} \frac{\partial \bar{g}}{\partial t} + \sum \limits _{i=1}^d \xi _i \frac{ \partial \bar{g} }{ \partial x_i } = 0, \quad \bar{g} = \bar{g}( \mathbf x , {\varvec{\xi }},t),\quad \mathbf x \in D \subset \mathbb {R} ^d, \quad {\varvec{\xi }}\in \mathbb {R} ^d, \quad t >0,\\ \bar{g}( \mathbf x , {\varvec{\xi }},0)= \bar{g}_{in}( \mathbf x , {\varvec{\xi }}), \end{array}\right. }\end{aligned}$$
(25a)
$$\begin{aligned}&{\left\{ \begin{array}{ll} \bar{g}( \mathbf y , {\varvec{\xi }},t) &{}= \alpha \left( \frac{2\pi }{RT( \mathbf y )} \right) ^{\frac{1}{2}}j( \mathbf y ,t) M_{T( \mathbf y )}( {\varvec{\xi }})\\ &{} \quad \,+\, (1-\alpha )\bar{g}( \mathbf y , {\varvec{\xi }}-2( {\varvec{\xi }}\cdot \mathbf n ) \mathbf n ,t), \quad \mathbf y \in \partial D, \quad {\varvec{\xi }}\cdot \mathbf n >0, \\ M_T ( {\varvec{\xi }}) &{}= \int \limits _{\mathbb {R}^{3-d}} M_T( {\varvec{\zeta }}) d {\varvec{\eta }}= \frac{ e^{ -\frac{ \left| \varvec{\xi } \right| ^2}{2RT} } }{ \left( 2\pi RT \right) ^{\frac{d}{2}} }, \end{array}\right. } \end{aligned}$$
(25b)
but with the additional zero total molecular number condition:
$$\begin{aligned} \int \limits _{D\times \mathbb {R} ^d} \bar{g}( \mathbf x , {\varvec{\xi }},t) d \mathbf x d {\varvec{\xi }}= 0, \ t\ge 0. \end{aligned}$$
(26)
Note that \(M_T( {\varvec{\zeta }})\), the Maxwellian, and \(M_T( {\varvec{\xi }})\), the reduced Maxwellian, are generally different as functions. To avoid confusion, we always refer to M as the abbreviation of \(M( {\varvec{\zeta }})\), not
\(M( {\varvec{\xi }})\).
For \( \mathbf x \in D\) and \( {\varvec{\xi }}\in \mathbb {R} ^d\), we define \(\tau _b = \tau _b( \mathbf x , {\varvec{\xi }})\) the backward exit time:
$$\begin{aligned} \tau _b( \mathbf x , {\varvec{\xi }}) \equiv \sup \left\{ s \ge 0 : \mathbf x - s' {\varvec{\xi }}\in D, \text { for all } s' \in (0,s) \right\} , \end{aligned}$$
(27)
and
$$\begin{aligned} t_1&=\tau _b( \mathbf x , {\varvec{\xi }})=\frac{| \mathbf x - \mathbf x _{(1)}|}{| {\varvec{\xi }}|},\\ t_{k+1}&=\tau _b\left( \mathbf x _{(k)}, {\varvec{\xi }}^{k}\right) =\frac{| \mathbf x _{(k)}- \mathbf x _{(k+1)}|}{| {\varvec{\xi }}^k|}. \end{aligned}$$
From Lemma 1, we have \(t_k=t_2\) for all \(k\ge 2\).
Suppose that the boundary flux j is given. Then the solutions of the transport equation (25) has explicit form by the characteristic method:
$$\begin{aligned} g( \mathbf x , {\varvec{\zeta }},t) - \rho _*S( \mathbf x , {\varvec{\zeta }}) = {\left\{ \begin{array}{ll} \alpha \sum \limits _{k=0}^{m-1}(1-\alpha )^{k} \Big (j_g \left( \textstyle \mathbf x _{(k+1)}, t-t_1-kt_2 \right) -j_S\Big ) \tilde{M}_{T\left( \mathbf x _{(k+1)}\right) }\\ \quad \qquad \,+\,\Big ((1-\alpha )^{m}g_{in}\left( \mathbf x _{(m)}- {\varvec{\xi }}^m(t-t_1-(m-1)t_2), {\varvec{\xi }}^m, {\varvec{\eta }}\right) \\ \quad \qquad \, -\,\alpha \sum \limits _{k=m}^{\infty }(1-\alpha )^{k} j_S \tilde{M}_{T\left( \mathbf x _{(k+1)}\right) }\Big ) \quad \text { for } \tau _b < t, \\ g_{in}( \mathbf x - {\varvec{\xi }}t, {\varvec{\zeta }})-\rho _*S( \mathbf x , {\varvec{\zeta }}) \quad \text { for } t < \tau _b, \end{array}\right. } \end{aligned}$$
(28)
$$\begin{aligned} \bar{g}( \mathbf x , {\varvec{\xi }},t) = {\left\{ \begin{array}{ll} \alpha \sum \limits _{k=0}^{m-1}(1-\alpha )^{k} j \left( \textstyle \mathbf x _{(k+1)}, t-t_1-kt_2 \right) \tilde{M}_{T\left( \mathbf x _{(k+1)}\right) }\left( {\varvec{\xi }}^k\right) \\ \quad \quad \,+\, (1-\alpha )^{m}\bar{g}_{in}\left( \mathbf x _{(m)}- {\varvec{\xi }}^m(t-t_1-(m-1)t_2), {\varvec{\xi }}^m\right) \quad \text { for } \tau _b < t,\\ \bar{g}_{in}( \mathbf x - {\varvec{\xi }}t, {\varvec{\xi }}) \quad \text { for } t < \tau _b, \end{array}\right. } \end{aligned}$$
(29)
where
$$\begin{aligned} m=\left\lfloor \frac{| {\varvec{\xi }}|t-| \mathbf x - \mathbf x _{(1)}|}{| \mathbf x _{(1)}- \mathbf x _{(2)}|}\right\rfloor +1, \end{aligned}$$
(30)
and for simplicity of notation we set
$$\begin{aligned} \tilde{M}_{T( \mathbf y )}\equiv \left( \frac{2\pi }{RT( \mathbf y )} \right) ^{\frac{1}{2}} M_{T( \mathbf y )}. \end{aligned}$$
The following are our main theorems for free molecular flow, which will be proven in the following four sections.
Theorem 5
(Global Existence for Boundary Flux) The solution of (4) and (25), with initial data \(g_{in} \in L^{\infty ,\mu }_{ \mathbf x , {\varvec{\zeta }}}\), exists and is unique for \(\mu >4\). Moreover, there exists \(C>0\) such that
$$\begin{aligned} j( \mathbf y ,t)=O(1) \left\| g_{in} \right\| _{\infty ,\mu }e^{C\frac{-\alpha \ln \alpha }{1-\alpha } t}, \end{aligned}$$
(31)
where C depends on \( T^* \) and \( T_* \).
Theorem 6
(Decay Rate for Boundary Flux) Suppose that \(g_{in}\in L^{\infty ,\mu }_{ \mathbf x , {\varvec{\zeta }}}\) for some constant \(\mu >4\). Then the boundary flux \(j( \mathbf y ,t)\), (23), satisfies
$$\begin{aligned} j( \mathbf y ,t)\le C \left\| g_{in} \right\| _{\infty ,\mu } \left( \frac{1}{(1+\alpha t)^d}+(1-\alpha )^{t^{\frac{1}{400}}} \right) \end{aligned}$$
for some constant C depending only on \(\mu \), \( T_* \) and \( T^* \).
From (30), for \(0<\epsilon <1\) we have \(m\ge t^{\epsilon }\) for \(| {\varvec{\xi }}|>\frac{2}{t^{1-\epsilon }}\). Therefore, Theorem 6 together with (28) yield immediately the pointwise convergence of the free molecular flow g, Theorem 1.
Remark 2
When \(\alpha =1\), the case of diffuse reflection boundary condition, we have the convergence rate \((1+t)^{-d}\) of the boundary flux [10, 11]. Roughly speaking, the equilibrating effect is mainly from sufficiently many collisions with the boundary of diffuse reflection condition when t is large. For Maxwell-type boundary condition, we have two possibilities after each collision with the boundary: one is diffuse reflection and another is specular reflection. This yields multiple scales in the convergence to the steady solution, and is one of the main causes of the analytical difficulty of this paper. Eventually, we have the convergence rate \((1+\alpha t)^{-d} +(1-\alpha )^{\frac{t^{\epsilon }}{2}}\) of the boundary flux, Theorem 6. \((1-\alpha )^{\frac{t^{\epsilon }}{2}}\) comes from the coefficient \((1-\alpha )\) of specular reflection although specular reflection condition itself has no equilibrating effect. \((1+\alpha t)^{-d}\) is from the diffuse reflection condition where the rate is essentially the same as before. However, we only have diffuse reflection for a multiple of \(\alpha \), and so the convergence to steady solution is slower than the case of the complete diffuse reflection. For instance, it starts to converge only when \(t>\frac{1}{\alpha }\); before that the solution is simply bounded.
2.2 The Global Exitence for Boundary Flux
In this subsection, we prove the global existence for the boundary flux function \(j( \mathbf y ,t)\). It should be noticed that we may associate the boundary flux with the backward flow of particles. Once particles collide with the boundary, both diffuse reflection and specular reflection occur for the Maxwell-type boundary condition. Note that diffuse reflection is stochastic and see [10] for more details. In contrast to diffuse reflection, specular reflection is deterministic and has no equilibrating effect. In the following discussion we first give a solution formula of boundary flux for general domains. Here we assume temporarily that the accommodation coefficient is variable, \(0<\alpha ( \mathbf y )<1\), for explaining what difficulties arise from this assumption.
Fix \( \mathbf y \in \partial D, t>0\), the boundary flux can be written as
$$\begin{aligned} j( \mathbf y ,t)&=\int \limits _{t<\frac{| \mathbf y - \mathbf y _{(1)}|}{| {\varvec{\xi }}_1|}} \left( - {\varvec{\xi }}_1\cdot \mathbf n ( \mathbf y ) \right) \bar{g}_{in}( \mathbf y - {\varvec{\xi }}_1t, {\varvec{\xi }}_1)d {\varvec{\xi }}_1\nonumber \\&\quad \,\,+\, \int \limits _{t>\frac{| \mathbf y - \mathbf y _{(1)}|}{| {\varvec{\xi }}_1|}} \left( - {\varvec{\xi }}_1\cdot \mathbf n ( \mathbf y ) \right) \alpha \left( \mathbf y _{(1)}\right) \tilde{M}_{T\left( \mathbf y _{(1)}\right) }( {\varvec{\xi }}_1)j \left( \mathbf y _{(1)},t-\frac{| \mathbf y - \mathbf y _{(1)}|}{| {\varvec{\xi }}_1|} \right) d {\varvec{\xi }}_1\nonumber \\&\quad \,\,+\, \int \limits _{t>\frac{| \mathbf y - \mathbf y _{(1)}|}{| {\varvec{\xi }}_1|}} \left( - {\varvec{\xi }}_1\cdot \mathbf n ( \mathbf y ) \right) \left( 1-\alpha \left( \mathbf y _{(1)}\right) \right) \bar{g} \left( \mathbf y _{(1)},t-\frac{| \mathbf y - \mathbf y _{(1)}|}{| {\varvec{\xi }}_1|}, {\varvec{\xi }}_1^1 \right) d {\varvec{\xi }}_1\nonumber \\&\equiv j_{in}^{(0)}( \mathbf y ,t)+D^{(1)}( \mathbf y ,t)+E^{(1)}( \mathbf y ,t), \end{aligned}$$
(32)
where \(j_{in}^{(0)}\) is a direct contribution of initial data, both \(D^{(1)}\) and \(E^{(1)}\) are the events that boundary collisions are more than once. More precisely, the first boundary collision takes place at \( \mathbf y _{(1)}\), \(D^{(1)}\) and \(E^{(1)}\) represent diffuse reflection and specular reflection of the backward flow respectively. We can continue to write down the formulas for \(D^{(1)}( \mathbf y ,t)\) and \(E^{(1)}( \mathbf y ,t)\):
$$\begin{aligned} D^{(1)}( \mathbf y ,t)&=D^{(1)}_{in}( \mathbf y ,t)+D^{(1)}_{dif}( \mathbf y ,t)+D^{(1)}_{spe}( \mathbf y ,t);\\ E^{(1)}( \mathbf y ,t)&=E^{(1)}_{in}( \mathbf y ,t)+E^{(1)}_{dif}( \mathbf y ,t)+E^{(1)}_{spe}( \mathbf y ,t), \end{aligned}$$
where
$$\begin{aligned} D^{(1)}_{in}( \mathbf y ,t)&=\int \limits _{0<t-\frac{| \mathbf y - \mathbf y _{(1)}|}{| {\varvec{\xi }}_1|}<\frac{| \mathbf y _{(1)}- \mathbf y _{(1,1)}|}{| {\varvec{\xi }}_2|}} \left( - {\varvec{\xi }}_1\cdot \mathbf n ( \mathbf y ) \right) \alpha ( \mathbf y _{(1)})\tilde{M}_{T\left( \mathbf y _{(1)}\right) }( {\varvec{\xi }}_1)\\&\quad \quad \left( - {\varvec{\xi }}_2\cdot \mathbf n \left( \mathbf y _{(1)}\right) \right) \bar{g}_{in}\left( \mathbf y _{(1)}- {\varvec{\xi }}_2(t-\frac{| \mathbf y - \mathbf y _{(1)}|}{| {\varvec{\xi }}_1|}), {\varvec{\xi }}_2\right) d {\varvec{\xi }}_2d {\varvec{\xi }}_1; \end{aligned}$$
$$\begin{aligned} D^{(1)}_{dif}( \mathbf y ,t)&=\int \limits _{t-\frac{| \mathbf y - \mathbf y _{(1)}|}{| {\varvec{\xi }}_1|}>\frac{| \mathbf y _{(1)}- \mathbf y _{(1,1)}|}{| {\varvec{\xi }}_2|}} \left( - {\varvec{\xi }}_1\cdot \mathbf n ( \mathbf y ) \right) \alpha ( \mathbf y _{(1)})\tilde{M}_{T( \mathbf y _{(1)})}( {\varvec{\xi }}_1) \left( - {\varvec{\xi }}_2\cdot \mathbf n \left( \mathbf y _{(1)}\right) \right) \\&\quad \quad \alpha \left( \mathbf y _{(1,1)}\right) \tilde{M}_{T\left( \mathbf y _{(1,1)}\right) }( {\varvec{\xi }}_2)j \left( \mathbf y _{(1,1)},t-\frac{| \mathbf y - \mathbf y _{(1)}|}{| {\varvec{\xi }}_1|}-\frac{| \mathbf y _{(1)}- \mathbf y _{(1,1)}|}{| {\varvec{\xi }}_2|} \right) d {\varvec{\xi }}_2d {\varvec{\xi }}_1; \end{aligned}$$
$$\begin{aligned} D^{(1)}_{spe}( \mathbf y ,t)&=\int \limits _{t-\frac{| \mathbf y - \mathbf y _{(1)}|}{| {\varvec{\xi }}_1|}>\frac{| \mathbf y _{(1)}- \mathbf y _{(1,1)}|}{| {\varvec{\xi }}_2|}} \left( - {\varvec{\xi }}_1\cdot \mathbf n ( \mathbf y ) \right) \alpha ( \mathbf y _{(1)})\tilde{M}_{T( \mathbf y _{(1)})}( {\varvec{\xi }}_1) \left( - {\varvec{\xi }}_2\cdot \mathbf n \left( \mathbf y _{(1)}\right) \right) \\&\quad \quad \left( 1-\alpha \left( \mathbf y _{(1,1)}\right) \right) \bar{g} \left( \mathbf y _{(1,1)},t-\frac{| \mathbf y - \mathbf y _{(1)}|}{| {\varvec{\xi }}_1|}-\frac{| \mathbf y _{(1)}- \mathbf y _{(1,1)}|}{| {\varvec{\xi }}_2|}, {\varvec{\xi }}_2^1 \right) d {\varvec{\xi }}_2d {\varvec{\xi }}_1; \end{aligned}$$
$$\begin{aligned} E^{(1)}_{in}( \mathbf y ,t)&=\int \limits _{0<t-\frac{| \mathbf y - \mathbf y _{(1)}|}{| {\varvec{\xi }}_1|}<\frac{| \mathbf y _{(1)}- \mathbf y _{(2)}|}{| {\varvec{\xi }}_1^1|}} \left( - {\varvec{\xi }}_1\cdot \mathbf n ( \mathbf y ) \right) (1-\alpha \left( \mathbf y _{(1)}\right) \\&\quad \quad \bar{g}_{in} \left( \mathbf y _{(1)}- {\varvec{\xi }}_1^1 \left( t-\frac{| \mathbf y - \mathbf y _{(1)}|}{| {\varvec{\xi }}_1|} \right) \right) d {\varvec{\xi }}_1;\\ \end{aligned}$$
$$\begin{aligned} E^{(1)}_{dif}( \mathbf y ,t)&=\int \limits _{t-\frac{| \mathbf y - \mathbf y _{(1)}|}{| {\varvec{\xi }}_1|}>\frac{| \mathbf y _{(1)}- \mathbf y _{(2)}|}{| {\varvec{\xi }}_1^1|}} \left( - {\varvec{\xi }}_1\cdot \mathbf n ( \mathbf y ) \right) (1-\alpha \left( \mathbf y _{(1)}\right) \\&\quad \quad \alpha ( \mathbf y _{(2)}) \tilde{M}_{T( \mathbf y _{(2)})}( {\varvec{\xi }}_1^1)j \left( \mathbf y _{(2)},t-\frac{| \mathbf y - \mathbf y _{(1)}|}{| {\varvec{\xi }}_1|}-\frac{| \mathbf y _{(1)}- \mathbf y _{(2)}|}{| {\varvec{\xi }}_1^1|} \right) d {\varvec{\xi }}_1; \end{aligned}$$
$$\begin{aligned} E^{(1)}_{spe}( \mathbf y ,t)&=\int \limits _{t-\frac{| \mathbf y - \mathbf y _{(1)}|}{| {\varvec{\xi }}_1|}>\frac{| \mathbf y _{(1)}- \mathbf y _{(2)}|}{| {\varvec{\xi }}_1^1|}} \left( - {\varvec{\xi }}_1\cdot \mathbf n ( \mathbf y ) \right) (1-\alpha \left( \mathbf y _{(1)}\right) \\&\quad \quad \left( 1-\alpha \left( \mathbf y _{(2)}\right) \right) \bar{g} \left( \mathbf y _{(2)},t-\frac{| \mathbf y - \mathbf y _{(1)}|}{| {\varvec{\xi }}_1|}-\frac{| \mathbf y _{(1)}- \mathbf y _{(2)}|}{| {\varvec{\xi }}_1^1|}, {\varvec{\xi }}_1^2 \right) d {\varvec{\xi }}_1. \end{aligned}$$
\(D^{(1)}_{in}\) is the event that the backward flow reaches initial state after diffuse reflection occurs once. \(D^{(1)}_{dif}\) is the event that diffuse reflection occurs again at \( \mathbf y _{(1,1)}\) after the first diffuse reflection at \( \mathbf y _{(1)}\), and \(D^{(1)}_{spe}\) is the event that specular reflection occurs at \( \mathbf y _{(1,1)}\) after the first diffuse reflection at \( \mathbf y _{(1)}\).
\(E^{(1)}_{in}\) is the event that the backward flow reaches initial state after specular reflection occurs once. \(E^{(1)}_{dif}\) is the event that diffuse reflection occurs at \( \mathbf y _{(2)}\) after the first specular reflection at \( \mathbf y _{(1)}\), and \(E^{(1)}_{spe}\) is the event that specular reflection occurs again at \( \mathbf y _{(2)}\) after the first specular reflection at \( \mathbf y _{(1)}\). Note that both \(D^{(1)}_{in}\) and \(E^{(1)}_{in}\) represent the contribution that the boundary collision takes place exactly once. More precisely, \(D^{(1)}_{in}\) and \(E^{(1)}_{in}\) represent exactly one diffuse collision and exactly one specular collision, respectively. If we want to compute the contribution that the boundary collision takes place exactly twice, we need to take \(D^{(1)}_{dif}, D^{(1)}_{spe},E^{(1)}_{dif}\) and \(E^{(1)}_{spe}\) into account. In other words, we must proceed to write down their formulas. Then there are four events arisen for exact two boundary collisions: (diffuse, diffuse), (diffuse, specular), (specular, diffuse) and (specular, specular). One can repeat this process inductively to compute the event that the boundary collision takes place exactly n times. In that case, we need to handle \(2^n\) possibilities. That would make the solution formula lengthy and complicated. That is one of the main causes of the analytical difficulty of this paper. Another tricky problem is the variable accommodation coefficient \(\alpha ( \mathbf y )\). In this paper we assume \(\alpha ( \mathbf y )=\alpha \) is a constant, this assumption not only makes the solution formula easier but also allows us to estimate all combinations of the events. We explain this by considering binomial expansion formally:
$$\begin{aligned} \left( \alpha \textit{diffuse}+(1-\alpha )\textit{specular} \right) ^n =\sum _{k=0}^n \left( \alpha \textit{diffuse} \right) ^k \left( (1-\alpha )\textit{specular} \right) ^{n-k}. \end{aligned}$$
(33)
Then all combinations of events, R.H.S. of (33), can be dominated by
$$\begin{aligned} \left( \alpha \textit{diffuse}+(1-\alpha )\textit{specular} \right) ^n =O(1)(\alpha +(1-\alpha ))^n=O(1), \end{aligned}$$
for example, if we can show each term of diffuse and specular is bounded. In other words, we can treat the effects caused by diffuse reflection and specular reflection independently when \(\alpha \) is constant. For the variable accommodation coefficient \(\alpha ( \mathbf y )\), the problem is more delicate and might involve different techniques. This will be our another research work in the future.
From now on we assume \(0<\alpha <1\) is constant. We define the following notations inductively:
$$\begin{aligned}&\mathbf y _{(0)}\equiv \mathbf y , \quad \mathbf y _{(k_1,\ldots ,k_l,0)}\equiv \mathbf y _{(k_1,\ldots ,k_l)}, \quad {\varvec{\xi }}^{0}_l \equiv {\varvec{\xi }}_l, \\&\mathbf y _{(k_1,\ldots ,k_{l-1},i)} = \mathbf y _B \left( \textstyle \mathbf y _{(k_1,\ldots ,k_{l-1},i-1)}, \frac{ {\varvec{\xi }}^{i-1}_{l}}{| {\varvec{\xi }}^{i-1}_{l}|} \right) ,\\&{\varvec{\xi }}^{i}_l = {\varvec{\xi }}^{i-1}_l -2\left( {\varvec{\xi }}^{i-1}_l\cdot \mathbf n ( \mathbf y _{(k_1,\ldots ,k_{l-1},i)})\right) \mathbf n ( \mathbf y _{(k_1,\ldots ,k_{l-1},i)}),\\ \end{aligned}$$
where \( \mathbf y _{(k_1,\ldots ,k_l)}\) indicates the location of particles via the backward flow process that:
$$\begin{aligned} (k_1-1)\text { specular }\rightarrow \text { diffuse }\rightarrow (k_2-1)\text { specular }\rightarrow \text { diffuse }\rightarrow \\ \cdots (k_{l-1}-1)\text { specular }\rightarrow \text { diffuse }\rightarrow (k_l-1)\text { specular }. \end{aligned}$$
According to the above discussion, we can find the solution formula of the boundary flux for general domains. Since the equation is quite lengthy, we omit it here and see [9] for more details.
In the present paper we assume spherical symmetric domains and therefore we can make use of this symmetric property to obtain more precise formulas for the boundary flux by using change of variables:
$$\begin{aligned} {\left\{ \begin{array}{ll} s_i=\frac{2}{k_i| {\varvec{\xi }}_i|} \quad \text { if } d=1,\\ {\left\{ \begin{array}{ll} s_i=\frac{| \mathbf y _{(k_1,\ldots ,k_{i-1},0)}- \mathbf y _{(k_1,\ldots ,k_{i-1},1)}|}{k_i| {\varvec{\xi }}_i|}\\ \cos \phi _i=- {\varvec{\xi }}_i\cdot \mathbf n ( \mathbf y _{(k_1,\ldots ,k_{i-1})}) \end{array}\right. } \quad \text { if } d=2. \end{array}\right. } \end{aligned}$$
We define
$$\begin{aligned} {\left\{ \begin{array}{ll} H(\sigma ) \equiv \left( \frac{2}{\sigma } \right) ^3 e^{- \left( \frac{2}{\sigma } \right) ^2} \quad \text { if } d=1,\\ G(\phi ,\sigma ) \equiv \frac{ 1 }{ \pi ^{\frac{1}{2}} } \left( \frac{ 2\cos \phi { } }{ \sigma _{} } \right) ^{4} e^{ - \left( \frac{ 2\cos \phi {} }{ \sigma _{} } \right) ^2 } \quad \text { if } d=2. \end{array}\right. } \end{aligned}$$
(34)
Here we give the formula of \(j( \mathbf y ,t)\) for \(d=2\). The formula for \(d=1\) is similar to the case of dimension 2 (replace G by H). We omit it and more details can be found in [9].
$$\begin{aligned} j( \mathbf y ,t)&=\sum \limits _{k=0}^n\left\{ \sum \limits _{l=1}^{k}\sum \limits _{k_1+k_2+ \cdots +k_l=l}^k \int (1-\alpha )^{k-k_1-\ldots -k_l} j_{in}^{(k-k_1-\ldots -k_l)}( \mathbf y _{(k_1,\ldots , k_l)},t-s_1-\cdots -s_l)\right. \\&\quad \prod \limits _{i=1}^{l}(1-\alpha )^{k_i-1}\alpha G \left( \phi _i,\frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots , k_i)})}s_i}{k_i} \right) \frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots , k_i)})}}{k_i}ds_i d\phi _i\\&\qquad \left. +\,(1-\alpha )^k j_{in}^{(k)}( \mathbf y ,t)\right\} \\&\qquad \, +\,\sum \limits _{l=1}^{n+1}\sum \limits _{k_1+k_2+ \cdots +k_l=l}^{n+1} \int (1-\alpha )^{n+1-k_1-\ldots -k_l}E^{(n+1-k_1-\cdots -k_l)}( \mathbf y _{(k_1,\ldots , k_l)},t-s_1-\cdots -s_l)\\&\quad \prod \limits _{i=1}^{l}(1-\alpha )^{k_i-1}\alpha G \left( \phi _i,\frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots , k_i)})}s_i}{k_i} \right) \frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots , k_i)})}}{k_i}ds_i d\phi _i\\&\qquad +\,(1-\alpha )^{n+1} E^{(n+1)}( \mathbf y ,t), \end{aligned}$$
where
$$\begin{aligned} j_{in}^{(k)}( \mathbf y ,t)&= \int \limits _{0<t-\sum \limits _{i=0}^{k-1}\frac{| \mathbf y _{(i)}- \mathbf y _{(i+1)}|}{| {\varvec{\xi }}_1^i|}<\frac{| \mathbf y _{(k)}- \mathbf y _{(k+1)}|}{| {\varvec{\xi }}_1^k|}} \left( - {\varvec{\xi }}_1\cdot \mathbf n ( \mathbf y ) \right) \nonumber \\&\quad \qquad \bar{g}_{in} \left( \mathbf y _{(k)}- {\varvec{\xi }}_1^k\left( t-\sum \limits _{i=0}^{k-1}\frac{| \mathbf y _{(i)}- \mathbf y _{(i+1)}|}{| {\varvec{\xi }}_1^i|}\right) , {\varvec{\xi }}_1^k \right) d {\varvec{\xi }}_1, \end{aligned}$$
(35)
$$\begin{aligned} E^{(n+1)}( \mathbf y ,t)&= \int \limits _{t>\sum \limits _{i=0}^{n}\frac{| \mathbf y _{(i)}- \mathbf y _{(i+1)}|}{| {\varvec{\xi }}_1^i|}} \left( - {\varvec{\xi }}_1\cdot \mathbf n ( \mathbf y ) \right) \bar{g} \left( \mathbf y _{(n+1)},t-\sum \limits _{i=0}^{n}\frac{| \mathbf y _{(i)}- \mathbf y _{(i+1)}|}{| {\varvec{\xi }}_1^i|}, {\varvec{\xi }}_1^{n+1} \right) d {\varvec{\xi }}_1,\nonumber \\ E^{(0)}( \mathbf y ,t)&=j( \mathbf y ,t). \end{aligned}$$
(36)
\(j^{(k)}_{in}\) represents the event that the backward flow reaches initial state after k times specular reflection. \(E^{(n+1)}\) represents the event that the boundary collisions are more than n times and the first \(n+1\) ones are precisely specular reflections. It should be notice that \(E^{(n+1)}\) is not the end. \(E^{(n+1)}\) itself involves an infinite series and we will use the coefficient \((1-\alpha )^{n+1}\) of \(E^{(n+1)}\) to get the decay for refined estimate later.
Now we are ready to prove the global existence of the boundary flux function.
Proof of Theorem 5
To compute the boundary flux \(j( \mathbf y ,t)\), we need to take all events into account. In other words, we have to sum up all events for each boundary collision. Hence, we have the following infinite series due to the above discussion.
The index k here means the exact number of boundary collision and we will show the convergence of the series. From (35) and noting that \(| {\varvec{\xi }}^i|=| {\varvec{\xi }}|\) for each i, we have
$$\begin{aligned}&\sum \limits _{k=0}^{\infty }(1-\alpha )^k j_{in}^{(k)}( \mathbf y ,t)\nonumber \\&\quad \quad \le \sum \limits _{k=0}^{\infty } \int _{\sum \limits _{i=0}^{k-1}\frac{| \mathbf y _{(i)}- \mathbf y _{(i+1)}|}{t}<| {\varvec{\xi }}| <\sum \limits _{i=0}^{k}\frac{| \mathbf y _{(i)}- \mathbf y _{(i+1)}|}{t}} \Big |- {\varvec{\xi }}\cdot \mathbf n ( \mathbf y )\Big |\int \frac{ \left\| g_{in} \right\| _{\infty ,\mu }}{(1+ {\varvec{\zeta }})^\mu }d {\varvec{\eta }}d {\varvec{\xi }}\nonumber \\&\quad \quad \le \left\| g_{in} \right\| _{\infty ,\mu }\int \frac{|- {\varvec{\xi }}\cdot \mathbf n ( \mathbf y )|}{(1+ {\varvec{\zeta }})^\mu }d {\varvec{\zeta }}\nonumber \\&\quad \quad =O(1) \left\| g_{in} \right\| _{\infty ,\mu } \end{aligned}$$
(38)
With (38), we rewrite (37) as
$$\begin{aligned} j( \mathbf y ,t)&=O(1) \left\| g_{in} \right\| _{\infty ,\mu }+\sum \limits _{k=0}^{\infty }\left\{ \sum \limits _{l=1}^{k}\sum \limits _{k_1+k_2+ \cdots +k_l=l}^k (1-\alpha )^{k-k_1-\ldots -k_l}\right. \\&\left. \quad \, \times \int \limits _{0<s_1+\ldots +s_l<t} j_{in}^{(k-k_1-\cdots -k_l)}( \mathbf y _{(k_1,\ldots , k_l)},t-s_1-\cdots -s_l)\right. \\&\left. \quad \, \times \left\{ \begin{array}{c@{,~}l} \prod \limits _{i=1}^{l}(1-\alpha )^{k_i-1}\alpha G \left( \phi _i,\frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots , k_i)})}s_i}{k_i} \right) \frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots , k_i)})}}{k_i}ds_i d\phi _i &{} d=2,\\ \prod \limits _{i=1}^{l}(1-\alpha )^{k_i-1}\alpha H \left( \frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots , k_i)})}s_i}{k_i} \right) \frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots , k_i)})}}{k_i}ds_i &{} d=1 \end{array} \right\} \right\} \end{aligned}$$
$$\begin{aligned}&=O(1) \left\| g_{in} \right\| _{\infty ,\mu }+\sum \limits _{l=1}^{\infty }\left( \frac{\alpha }{1-\alpha }\right) ^l \sum \limits _{k_1=1}^{\infty }\ldots \sum \limits _{k_l=1}^{\infty } \sum \limits _{k=k_1+\ldots +k_l}^{\infty } (1-\alpha )^{k-k_1-\ldots -k_l}\nonumber \\&\quad \,\times \int \limits _{0<s_1+\ldots +s_l<t} j_{in}^{(k-k_1-\cdots -k_l)}( \mathbf y _{(k_1,\ldots , k_l)},t-s_1-\cdots -s_l)\nonumber \\&\quad \, \times \left\{ \begin{array}{c@{,~}l} \prod \limits _{i=1}^{l}(1-\alpha )^{k_i} G \left( \phi _i,\frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots , k_i)})}s_i}{k_i} \right) \frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots , k_i)})}}{k_i}ds_i d\phi _i &{} d=2,\\ \prod \limits _{i=1}^{l}(1-\alpha )^{k_i} H \left( \frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots , k_i)})}s_i}{k_i} \right) \frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots , k_i)})}}{k_i}ds_i &{}d=1 \end{array} \right\} \end{aligned}$$
(39)
By plugging (38) into (39) and direct computations, we have
$$\begin{aligned} j( \mathbf y ,t)&=O(1) \left\| g_{in} \right\| _{\infty ,\mu }\left\{ 1+ \sum \limits _{l=1}^{\infty }\left( \frac{\alpha }{1-\alpha }\right) ^l \sum \limits _{k_1=1}^{\infty }\ldots \sum \limits _{k_l=1}^{\infty } \int \limits _{0<s_1+\ldots +s_l<t}\right. \\&\left. \quad \, \times \left\{ \begin{array}{c@{,~}l} \prod \limits _{i=1}^{l}(1-\alpha )^{k_i} G \left( \phi _i,\frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots , k_i)})}s_i}{k_i} \right) \frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots , k_i)})}}{k_i}ds_i d\phi _i &{} d=2,\\ \prod \limits _{i=1}^{l}(1-\alpha )^{k_i} H \left( \frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots , k_i)})}s_i}{k_i} \right) \frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots , k_i)})}}{k_i}ds_i &{} d=1 \end{array} \right\} \right\} \end{aligned}$$
$$\begin{aligned} =O(1) \left\| g_{in} \right\| _{\infty ,\mu }\left\{ 1+ \sum \limits _{l=1}^{\infty }\left( \frac{\alpha }{1-\alpha }\right) ^l \sum \limits _{k_1=1}^{\infty }\frac{(1-\alpha )^{k_1}}{k_1}\ldots \sum \limits _{k_l=1}^{\infty } \frac{(1-\alpha )^{k_l}}{k_l}\right. \\ \left. \int \limits _{0<s_1+\ldots +s_l<t} \times \prod \limits _{i=1}^{l} \left\{ \begin{array}{c@{,~}l} \left( \pi \left\| G \right\| _{L^\infty }\sqrt{2R T^* } \right) ds_i &{} d=2,\\ \left( \left\| H \right\| _{L^\infty }\sqrt{2R T^* } \right) ds_i &{} d=1, \end{array} \right\} \right\} \end{aligned}$$
$$\begin{aligned} \qquad \qquad \quad \,\,=O(1) \left\| g_{in} \right\| _{\infty ,\mu }\left\{ 1+ \sum \limits _{l=1}^{\infty }\left( \frac{\alpha }{1-\alpha }\right) ^l \Bigg (\sum \limits _{k_1=1}^{\infty }\frac{(1-\alpha )^{k_1}}{k_1}\Bigg )^l \left\{ \begin{array}{c@{,~}l} \frac{ \left( \pi \left\| G \right\| _{L^\infty }\sqrt{2R T^* }t \right) ^l}{l!} &{} d=2,\\ \frac{ \left( \left\| H \right\| _{L^\infty }\sqrt{2R T^* }t \right) ^l}{l!} &{} d=1, \end{array} \right\} \!\right\} \!. \end{aligned}$$
It is easy to check that
$$\begin{aligned} \sum \limits _{k_1=1}^{\infty }\frac{(1-\alpha )^{k_1}}{k_1}=-\ln \alpha , \end{aligned}$$
and therefore we have
$$\begin{aligned} j( \mathbf y ,t)=O(1) \left\| g_{in} \right\| _{ \infty , \mu } \left\{ \begin{array}{c@{,~}l} e^{\frac{-\alpha \ln \alpha }{1-\alpha }\pi \left\| G \right\| _{L^\infty }\sqrt{2R T^* }t} &{}\quad d=2,\\ e^{\frac{-\alpha \ln \alpha }{1-\alpha } \left\| H \right\| _{L^\infty }\sqrt{2R T^* }t} &{}\quad d=1 \end{array} \right\} . \end{aligned}$$
\(\square \)
2.3 Preliminary Estimates
The discussion of this subsection applies to all space dimension. To avoid complication in notations, we treat only the 2d case. And we will use the following Law of Large Numbers to get a refined estimate. It can be proved by the similar argument as in [10]. Therefore we omit it.
Theorem 7
(Law of Large Numbers) There exists some constant \(C>0\) such that, for any \(\gamma \) and m with \(\gamma /(mn)^{\frac{1}{d+1}} > C\),
$$\begin{aligned} \int \limits _{ \frac{\gamma }{m} < |\sigma -n{{\mathrm{E}}}(X_1)| } H_n(\sigma ) d\sigma ={{\mathrm{P}}}\left\{ \frac{\gamma }{m} < |X_1+\ldots +X_n-n{{\mathrm{E}}}(X_1)| \right\} = O(1) \frac{ m^{d+1}n^d \log (\gamma +1) }{ \gamma ^{d+1} }. \end{aligned}$$
where \(X_1,X_2,\ldots ,\) are i.i.d. with the probability density function
$$\begin{aligned} H(\sigma )= {\left\{ \begin{array}{ll} \left( \frac{2}{\sigma } \right) ^{3} e^{ - \left( \frac{2}{\sigma } \right) ^2 }&{}\quad \text {if } d=1,\\ \int G(\phi ,\sigma )d\phi &{}\quad \text {if } d=2, \end{array}\right. } \end{aligned}$$
and \(H_n\) is the probability density function of the sum of i.i.d. random variables, \(X_1 +X_2 + \cdots + X_n\). Notice that \(H_n\) is a convolution of H itself, \(H_n=\underbrace{ \left( H*\cdots *H \right) }_{n\text { times }}\).
In order to get a refined estimate, we start with the specular reflection:
$$\begin{aligned} j( \mathbf y ,t)&=\sum \limits _{k=0}^{m-1}(1-\alpha )^k j_{in}^{(k)}( \mathbf y ,t)\nonumber \\&\quad \,+ \, \alpha \sum \limits _{k=1}^{m}(1-\alpha )^{k-1} \int \limits _{-\pi /2}^{\pi /2}\int \limits _{0}^{t} j( \mathbf y _{(k)},t-s)G\left( \phi ,\frac{s\sqrt{2RT( \mathbf y _{(k)})}}{k}\right) \frac{\sqrt{2RT( \mathbf y _{(k)})}}{k} ds d\phi \nonumber \\&\quad \,+\, (1-\alpha )^{m}E^{(m)}( \mathbf y ,t) \quad \text {when } d=2, \end{aligned}$$
(40)
$$\begin{aligned} j(\pm 1,t)&=\sum \limits _{k=0}^{m-1}(1-\alpha )^k j_{in}^{(k)}(\pm 1,t)\\&\quad \,+\, \alpha \sum \limits _{k=1}^{m}(1-\alpha )^{k-1} \int \limits _{0}^{t} j(\pm (-1)^k,t-s)H\left( \frac{s\sqrt{2RT(\pm (-1)^k)}}{k}\right) \frac{\sqrt{2RT(\pm (-1)^k)}}{k} ds \\&\quad \,+\,(1-\alpha )^{m}E^{(m)}(\pm 1,t) \quad \text {when } d=1. \end{aligned}$$
Define
$$\begin{aligned} J_{in}^{(m)}( \mathbf y ,t)=\sum \limits _{k=0}^{m-1}(1-\alpha )^k j_{in}^{(k)}( \mathbf y ,t), \end{aligned}$$
and rewrite (40) as:
$$\begin{aligned} j( \mathbf y ,t)&=J_{in}^{(m)}( \mathbf y ,t)+(1-\alpha )^{m}E^{(m)}( \mathbf y ,t)\nonumber \\&\quad \,+\, \alpha \sum \limits _{k=1}^{m}(1-\alpha )^{k-1} \int \limits _{-\pi /2}^{\pi /2}\int \limits _{0}^{t} j( \mathbf y _{(k)},t-s)G\left( \phi ,\frac{s\sqrt{2RT( \mathbf y _{(k)})}}{k}\right) \frac{\sqrt{2RT( \mathbf y _{(k)})}}{k} ds d\phi \end{aligned}$$
(41)
By iterating (41) n times, we have
$$\begin{aligned} j( \mathbf y ,t)&=J_{in}^{(m)}( \mathbf y ,t)+(1-\alpha )^{m}E^{(m)}( \mathbf y ,t)\nonumber \\&\quad \,+\,\sum \limits _{i=1}^{n-1}\alpha ^i\sum \limits _{k_1=1}^{m}(1-\alpha )^{k_1-1}\cdots \sum \limits _{k_i=1}^{m}(1-\alpha )^{k_i-1} \int \limits _{0<s_1+s_2+\cdots +s_i<t}\nonumber \\&\quad \quad \prod \limits _{j=1}^{i} G\left( \phi _j,\frac{s_j\sqrt{2RT( \mathbf y _{(k_1,\ldots ,k_j)})}}{k_j}\right) \frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots ,k_j)})}}{k_j}\nonumber \\&\left\{ J_{in}^{(m)}( \mathbf y _{(k_1,\ldots , k_i)},t-s_1-\cdots -s_i)+(1-\alpha )^{m}E^{(m)}( \mathbf y _{(k_1,\ldots , k_i)},t-s_1-\cdots -s_i) \right\} \nonumber \\&\quad ds_i d\phi _i \cdots ds_1 d\phi _1\nonumber \\&\quad \,+\, \alpha ^n\sum \limits _{k_1=1}^{m}(1-\alpha )^{k_1-1}\cdots \sum \limits _{k_n=1}^{m}(1-\alpha )^{k_n-1}\nonumber \\&\quad \int \limits _{0<s_1+s_2+\cdots +s_n<t}j( \mathbf y _{(k_1,k_2,\ldots , k_n)},t-s_1-s_2-\cdots -s_n)\nonumber \\&\quad \quad \prod \limits _{i=1}^{n} G\left( \phi _i,\frac{s_i\sqrt{2RT( \mathbf y _{(k_1,\ldots ,k_i)})}}{k_i}\right) \frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots ,k_i)})}}{k_i} ds_n d\phi _n \cdots ds_1 d\phi _1. \end{aligned}$$
(42)
To estimate (42), we define
$$\begin{aligned} j_{(k_1,\ldots ,k_i)}^{(i,m)}( \mathbf y ,t)&= \int \limits _{0<s_1+s_2+\cdots +s_i<t} \prod \limits _{j=1}^{i} G\left( \phi _j,\frac{s_j\sqrt{2RT( \mathbf y _{(k_1,\ldots ,k_j)})}}{k_j}\right) \frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots ,k_j)})}}{k_j}\\&\quad \left\{ J_{in}^{(m)}( \mathbf y _{(k_1,\ldots , k_i)},t-s_1-\cdots -s_i)+(1-\alpha )^{m}E^{(m)}\right. \\&\left. ( \mathbf y _{(k_1,\ldots , k_i)},t-s_1-\cdots -s_i) \right\} ds_i d\phi _i \cdots ds_1 d\phi _1\\ \end{aligned}$$
and
$$\begin{aligned} J_{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ,t)&= \int \limits _{0<s_1+s_2+\cdots +s_n<t}j( \mathbf y _{(k_1,k_2,\ldots , k_n)},t-s_1-s_2-\cdots -s_n)\\&\qquad \prod \limits _{i=1}^{n} G\left( \phi _i,\frac{s_i\sqrt{2RT( \mathbf y _{(k_1,\ldots ,k_i)})}}{k_i}\right) \frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots ,k_i)})}}{k_i} ds_n d\phi _n \cdots ds_1 d\phi _1. \end{aligned}$$
Recall
$$\begin{aligned} J_{in}^{(m)}( \mathbf y ,t)&=\sum \limits _{k=0}^{m-1}(1-\alpha )^k j_{in}^{(k)}( \mathbf y ,t)\\&=O(1)\sum \limits _{k=0}^{m-1}(1-\alpha )^k \left\| g_{in} \right\| _{\infty ,\mu }\\&=O(1)m \left\| g_{in} \right\| _{\infty ,\mu }. \end{aligned}$$
Moreover, we have for \(t>1\),
$$\begin{aligned} j_{in}^{(k)}( \mathbf y ,t)&= \int \limits _{\frac{k| \mathbf y - \mathbf y _{(1)}|}{t}<| {\varvec{\xi }}|<\frac{(k+1)| \mathbf y - \mathbf y _{(1)}|}{t}} - {\varvec{\xi }}\cdot \mathbf n ( \mathbf y )\bar{g}_{in}( \mathbf y _{(k)}- {\varvec{\xi }}^k(t-kt_1), {\varvec{\xi }}^k)d {\varvec{\xi }}\\&=O(1) \left\| g_{in} \right\| _{\infty ,\mu } \int \limits _{\frac{k| \mathbf y - \mathbf y _{(1)}|}{t}<| {\varvec{\xi }}|<\frac{(k+1)| \mathbf y - \mathbf y _{(1)}|}{t}} | {\varvec{\xi }}|^d d| {\varvec{\xi }}|\\&=O(1) \left\| g_{in} \right\| _{\infty ,\mu } \frac{ \left( (k+1)| \mathbf y - \mathbf y _{(1)}| \right) ^{d+1}- \left( k| \mathbf y - \mathbf y _{(1)}| \right) ^{d+1}}{t^{d+1}}\\&=O(1) \left\| g_{in} \right\| _{\infty ,\mu } {\left\{ \begin{array}{ll} \frac{2k+1}{t^2} \quad \text { for } d=1,\\ \frac{3k^2+3k+1}{t^3} \quad \text { for } d=2. \end{array}\right. } \end{aligned}$$
Thus we have
$$\begin{aligned} J^{(m)}_{in}( \mathbf y ,t)&=O(1) \left\| g_{in} \right\| _{\infty ,\mu } \sum \limits _{k=0}^{m-1} (1-\alpha )^k {\left\{ \begin{array}{ll} \frac{2k+1}{t^2} \quad \text { for } d=1,\\ \frac{3k^2+3k+1}{t^3} \quad \text { for } d=2, \end{array}\right. }\nonumber \\&=O(1) \left\| g_{in} \right\| _{\infty ,\mu }\frac{1+(1-\alpha )m^{d+1}}{t^{d+1}}. \end{aligned}$$
(43)
In order to estimate the remainder terms, we define a priori bound of boundary flux \(j( \mathbf y ,t)\).
Definition 1
Define the a priori bound \( {\mathcal J} \) by
$$\begin{aligned} {\mathcal J} (t) \equiv \sup _{0\le s\le t} {\left\{ \begin{array}{ll} \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s) &{} \text { for } d=2, \\ \Big ( \left| j_+(s) \right| + \left| j_-(s) \right| \Big ) &{} \text { for } d=1, \end{array}\right. } \end{aligned}$$
where \(j_\pm (t) \equiv j(\pm 1,t)\).
From (29) and (36), it is easy to show
$$\begin{aligned} E^{(m)}( \mathbf y ,t)=O(1) \left( {\mathcal J} (t)+ \left\| g_{in} \right\| _{\infty ,\mu } \right) . \end{aligned}$$
For \(i<n\), we divide \(j^{(i,m)}_{(k_1,\ldots ,k_i)}( \mathbf y ,t)\) into the following two parts:
$$\begin{aligned} j^{(i,m)}_{(k_1,\ldots ,k_i)}( \mathbf y ,t)&= \left( \int _{\mathscr {A}_1} + \int _{ \mathscr {A}_2} \right) \left\{ J_{in}^{(m)} \left( \textstyle \mathbf y _{(k_1,\ldots ,k_i)}, t - \frac{k_1\sigma _1}{\sqrt{2RT_{(1)}}} - \ldots - \frac{k_i\sigma _i}{\sqrt{2RT_{(k_1,\ldots ,k_i)}}} \right) \right. \\&\quad \left. +\,(1-\alpha )^{m}E^{(m)} \left( \textstyle \mathbf y _{(k_1,\ldots ,k_i)}, t - \frac{k_1\sigma _1}{\sqrt{2RT_{(1)}}} - \ldots - \frac{k_i\sigma _i}{\sqrt{2RT_{(k_1,\ldots ,k_i)}}} \right) \right\} \\&\quad \times \prod _{l=1}^i \left( G(\phi _l,\sigma _l) d\sigma _l d\phi _l \right) \\&\equiv j^{(i,m)slow}_{(k_1,\ldots ,k_i)}( \mathbf y ,t) + j^{(i,m)rare}_{(k_1,\ldots ,k_i)}( \mathbf y ,t), \end{aligned}$$
where
$$\begin{aligned} \mathscr {A}_1 \equiv&\left\{ 0 < \frac{ k_1\sigma _1 }{ \sqrt{2RT_{(k_1)} }} + \ldots + \frac{ k_i\sigma _i }{ \sqrt{2RT_{(k_1,\ldots ,k_i)} }} < \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} \right\} ,\\ \mathscr {A}_2 \equiv&\left\{ \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} < \frac{ k_1\sigma _1 }{ \sqrt{2RT_{(k_1)}} } + \ldots + \frac{ k_i\sigma _i }{ \sqrt{2RT_{(k_1,\ldots ,k_i)}} } < t \right\} . \end{aligned}$$
For \(j^{(i,m)slow}_{(k_1,\ldots ,k_i)}( \mathbf y ,t)\), the time needed to trace back to an interior point is at least t / 2:
$$\begin{aligned}&\frac{ k_1\sigma _1 }{ \sqrt{2RT_{(k_1)}} } + \ldots + \frac{ k_i\sigma _i }{ \sqrt{2RT_{(k_1,\ldots ,k_i)} }} < \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2}\\&\Longrightarrow \quad t - \left( \frac{ k_1\sigma _1 }{ \sqrt{2RT_{(k_1)} }} + \ldots + \frac{ k_i\sigma _i }{ \sqrt{2RT_{(k1,\ldots ,k_i)}} } \right) > t - \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} \ge \frac{t}{2}. \end{aligned}$$
Thus
$$\begin{aligned}&j^{(i,m)slow}_{(k_1,\ldots ,k_i)}( \mathbf y ,t)\nonumber \\ \le&\int \limits _{\mathscr {A}_1} \sup _{\frac{t}{2}<s<t} \left( \left\| J_{in}^{(m)} \right\| _{L^\infty _ \mathbf y }+(1-\alpha )^m \left\| E^{(m)} \right\| _{L^\infty _ \mathbf y } \right) (s) \times \int \prod _{l=1}^i G(\phi _l,\sigma _l)d\phi _l d\sigma _l\nonumber \\ =\,&O(1) \left( \left\| g_{in} \right\| _{\infty ,\mu }\frac{1+(1-\alpha )m^{d+1}}{t^{d+1}} +(1-\alpha )^m \left( \left\| g_{in} \right\| _{\infty ,\mu }+ {\mathcal J} (t) \right) \right) . \end{aligned}$$
(44)
Note that the estimate (44) relies merely on the smallness of the speed.
For \(j^{(i,m)rare}_{(k_1,\ldots ,k_i)}( \mathbf y ,t)\), the time consumed to trace back is at least \( \sqrt{\frac{ T_* }{ T^* }} t/2\). Therefore,
$$\begin{aligned} \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2m}\sqrt{2R T_* } < \sigma _1 + \ldots + \sigma _i, \end{aligned}$$
$$\begin{aligned} j^{(i,m)rare}_{(k_1,\ldots ,k_i)}( \mathbf y ,t)&\le O(1) \left( m \left\| g_{in} \right\| _{\infty ,\mu } +(1-\alpha )^m \left( \left\| g_{in} \right\| _{\infty ,\mu }+ {\mathcal J} (t) \right) \right) \nonumber \\&\quad \,\times \int \limits _{ \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2m} \sqrt{2R T_* } < \sigma _1 + \ldots + \sigma _i } \prod _{l=1}^i G(\phi _l,\sigma _l)d\phi _l d\sigma _l\nonumber \\&= O(1) \left( m \left\| g_{in} \right\| _{\infty ,\mu } + {\mathcal J} (t) \right) \times \int _{ \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2m}\sqrt{2R T_* } }^{\infty } H_i(\sigma ) d\sigma \nonumber \\&= O(1) \left( m \left\| g_{in} \right\| _{\infty ,\mu } + {\mathcal J} (t) \right) \times \Pr \left\{ X_1+\ldots +X_i > \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2m}\sqrt{2R T_* } \right\} . \end{aligned}$$
(45)
Note that n, m, the index of \(J^{(n,m)}_{(k_1,\ldots ,k_i)}( \mathbf y ,t)\), are variables at our disposal. Throughout this paper we assume
\(t/mn \gg 1\). Recall that our final choice of n and m is \(mn=\lfloor t^r \rfloor \), \(r\in (0,(d+1)^{-1})\). Thus \(r<1\) and so, for large t, \(mn\ll t.\) Since \({{\mathrm{E}}}(X_1+\ldots +X_i)=i{{\mathrm{E}}}(X_1) \sim i \le n \ll \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2m}\sqrt{2R T_* }\), (45) represents the probability of a rare event. We now apply the law of large numbers, Theorem 7, to estimate (45): choose the truncation variable \(\gamma \) to be \(\sqrt{2R T_* } \frac{t}{3}\). Under the assumption \(t/mi > t/mn \gg 1\), we have
$$\begin{aligned} \left\{ \sqrt{\frac{ T_* }{ T^* }} \sqrt{2R T_* }\frac{t}{2m} < \sigma \right\} \subset \{\gamma <|\sigma -i{{\mathrm{E}}}(X_1)| \}, \quad \frac{ \gamma }{ (mn)^{\frac{1}{d+1}} } \sim \frac{ t }{ (mn)^{\frac{1}{d+1}} } \gg 1. \end{aligned}$$
Therefore, we can apply Theorem 7 to obtain
$$\begin{aligned} j^{(i,m)rare}_{(k_1,\ldots ,k_i)}( \mathbf y ,t) = O(1) \left( m \left\| g_{in} \right\| _{\infty ,\mu } + {\mathcal J} (t) \right) \frac{ m^{d+1}i^d \log (t+1) }{ t^{d+1} }. \end{aligned}$$
(46)
Note that the estimate (46) relies merely on the law of large numbers.
For \(J^{(n,m)}_{(k_1,\ldots ,k_n)}\), we conduct a similar decomposition:
$$\begin{aligned} J^{(n,m)}_{(k_1,\ldots ,k_n)}( \mathbf y ,t)&= \left( \int _{\mathscr {B}_1} + \int _{ \mathscr {B}_2} \right) j \left( \textstyle \mathbf y ^{(k_1,\ldots ,k_n)}, t - \frac{k_1\sigma _1}{\sqrt{2RT_{(k_1)}}} - \ldots - \frac{k_n\sigma _n}{\sqrt{2RT_{(k_1,\ldots ,k_n)}}} \right) \\&\quad \,\times \prod _{l=1}^n G(\phi _l,\sigma _l)d\phi _l d\sigma _l\\&\equiv \Lambda _{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ,t) + J^{(n,m)rare}_{(k_1,\ldots ,k_i)}( \mathbf y ,t), \end{aligned}$$
where
$$\begin{aligned} \mathscr {B}_1 \equiv&\left\{ 0 < \frac{ k_1\sigma _1 }{ \sqrt{2RT_{(k_1)} }} + \ldots + \frac{ k_n\sigma _n }{ \sqrt{2RT_{(k_1,\ldots ,k_n)}} } < \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} \right\} \\ \mathscr {B}_2 \equiv&\left\{ \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} < \frac{ k_1\sigma _1 }{ \sqrt{2RT_{(k_1)}} } + \ldots + \frac{ k_1\sigma _n }{ \sqrt{2RT_{(k_1,\ldots ,k_n)}} } < t \right\} . \end{aligned}$$
We can apply the same argument of \(j^{(n,m)rare}_{(k_1,\ldots ,k_i)}\) to \(J^{(n,m)rare}_{(k_1,\ldots ,k_i)}\) to obtain:
$$\begin{aligned} J^{(n,m)rare}_{(k_1,\ldots ,k_i)}( \mathbf y ,t) = O(1) {\mathcal J} (t) \frac{ m^{d+1} n^d \log (t+1) }{ t^{d+1} }, \text { whenever } t/mn \gg 1. \end{aligned}$$
(47)
We omit the details.
Lemma 2
$$\begin{aligned}&\sum \limits _{k_1=0}^{m-1}\cdots \sum \limits _{k_i=0}^{m-1}(1-\alpha )^{k_1+\ldots +k_i} = \left( \frac{1-(1-\alpha )^{m}}{\alpha } \right) ^i\\&\sum \limits _{i=1}^{n-1}\alpha ^i\sum \limits _{k_1=0}^{m-1}\cdots \sum \limits _{k_i=0}^{m-1}(1-\alpha )^{k_1+\ldots +k_i} =\sum \limits _{i=1}^{n-1} \left( 1-(1-\alpha )^{m} \right) ^i=O(1)\alpha n \end{aligned}$$
Since
$$\begin{aligned} j( \mathbf y ,t)&= J^{(m)}_{in}( \mathbf y ,t)+(1-\alpha )^m E^{(m)}( \mathbf y ,t)\\&\quad \,+\, \sum _{i=1}^{n-1}\alpha ^i \sum _{k_1=1}^m (1-\alpha )^{k_1-1}\cdots \sum _{k_i=1}^m (1-\alpha )^{k_i-1} j^{(i,m)}_{(k_1,\ldots ,k_i)}( \mathbf y ,t)\\&\quad \,+\, \alpha ^n \sum _{k_1=1}^m (1-\alpha )^{k_1-1}\cdots \sum _{k_n=1}^m (1-\alpha )^{k_n-1} J^{(n,m)}_{(k_1,\ldots ,k_n)}( \mathbf y ,t)\\ \end{aligned}$$
putting (44), (46), and (47) together we have:
Theorem 8
For \(t/mn \gg 1\),
$$\begin{aligned} j( \mathbf y ,t)&= O(1) \left( \frac{ m(mn)^{d+1}\log (t+1) }{ t^{d+1} }+(1-\alpha )^m \right) \left\| g_{in} \right\| _{\infty ,\mu } \\&\quad \,+\, O(1) \left( \frac{ (mn)^{d+1}\log (t+1) }{ t^{d+1} }+(1-\alpha )^m \right) {\mathcal J} (t)\\&\quad \,+\,\alpha ^n \sum _{k_1=1}^m (1-\alpha )^{k_1-1}\cdots \sum _{k_n=1}^m (1-\alpha )^{k_n-1} \Lambda _{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ,t), \end{aligned}$$
where
$$\begin{aligned} \Lambda _{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ,t)&\equiv \int \limits _{ 0 < s_1 + \ldots + s_n < \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} } j \left( \mathbf y _{(k_1,\ldots ,k_n)}, t-s_1-\ldots -s_n \right) \nonumber \\&\quad \, \times \left( \prod _{l=1}^n G \left( \textstyle \phi _l, \frac{s_l \sqrt{2RT( \mathbf y _{(k_1,\ldots ,k_l)})}}{k_l} \right) \frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots ,k_l)})}}{k_l}d\phi _l d s_l \right) \quad \text { when } d=2, \end{aligned}$$
(48a)
$$\begin{aligned} \Lambda _{(k_1,\ldots ,k_n)}^{(n,m)}(\pm 1,t)&\equiv \int \limits _{ 0 < s_1 + \ldots + s_n < \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} } j \left( \pm 1_{(k_1,\ldots ,k_n)}, t-s_1-\ldots -s_n \right) \nonumber \\&\quad \,\times \left( \prod _{l=1}^n H \left( \textstyle \frac{s_l \sqrt{2RT(\pm 1_{(k_1,\ldots ,k_l)})}}{k_l} \right) \frac{\sqrt{2RT(\pm 1_{(k_1,\ldots ,k_l)})}}{k_l} d s_l \right) \quad \text { when } d=1, \end{aligned}$$
(48b)
With the aid of the law of large numbers, we have estimated \(j_{(k_1,\ldots ,k_n)}^{(n,m)}\) and \(J_{(k_1,\ldots ,k_n)}^{(n,m)rare}\). The remaining term \(\Lambda _{(k_1,\ldots ,k_n)}^{(n,m)}\) consists of the main event, which requires more effort to estimate.
To show the convergence of boundary flux j, we need to use the crucial conservation of molecular number (24) and (28), as
$$\begin{aligned} j( \mathbf y ,t)&= j( \mathbf y ,t) - \frac{1}{C_S |D|} \int _{D\times \mathbb {R} ^d } \bar{g}( \mathbf x , {\varvec{\xi }},t) ~ d \mathbf x d {\varvec{\xi }}\nonumber \\&= \frac{1}{|D|} \int _{D\times \mathbb {R} ^d} \Big (j( \mathbf y ,t)\varvec{s}( \mathbf x , {\varvec{\xi }}) - \frac{1}{C_S}\bar{g}( \mathbf x , {\varvec{\xi }},t) \Big ) d \mathbf x d {\varvec{\xi }}\nonumber \\&= \frac{1}{|D|} \int _{| {\varvec{\xi }}|<\frac{| \mathbf x - \mathbf x _{(1)}|}{t}} \Big (j( \mathbf y ,t)\varvec{s}( \mathbf x , {\varvec{\xi }}) - \frac{1}{C_S}\bar{g}( \mathbf x , {\varvec{\xi }},t) \Big ) d \mathbf x d {\varvec{\xi }}\nonumber \\&\quad \,+\, \frac{1}{|D|} \int _{\frac{| \mathbf x - \mathbf x _{(1)}|}{t}<| {\varvec{\xi }}|<\frac{| \mathbf x _{(1)}- \mathbf x _{(2)}|}{\log (t+1)}} \Big (j( \mathbf y ,t)\varvec{s}( \mathbf x , {\varvec{\xi }}) - \frac{1}{C_S}\bar{g}( \mathbf x , {\varvec{\xi }},t) \Big ) d \mathbf x d {\varvec{\xi }}\nonumber \\&\quad \,+\, \frac{1}{|D|} \int _{| {\varvec{\xi }}|>\frac{| \mathbf x _{(1)}- \mathbf x _{(2)}|}{\log (t+1)}} \Big (j( \mathbf y ,t)\varvec{s}( \mathbf x , {\varvec{\xi }}) - \frac{1}{C_S}\bar{g}( \mathbf x , {\varvec{\xi }},t) \Big ) d \mathbf x d {\varvec{\xi }}\nonumber \\&\equiv j_{in}( \mathbf y ,t) + j_{mid}( \mathbf y ,t) + j_{fl}( \mathbf y ,t) . \end{aligned}$$
(49)
It is easy to see that if we choose \(K=K( \mathbf x , {\varvec{\xi }},t)\) such that
$$\begin{aligned} K-1<\frac{t}{\log (t+1)}-\frac{| \mathbf x - \mathbf x _{(1)}|}{| \mathbf x _{(1)}- \mathbf x _{(2)}|}<K, \end{aligned}$$
then we have
$$\begin{aligned} \frac{| \mathbf x - \mathbf x _{(1)}|+(K-1)| \mathbf x _{(1)}- \mathbf x _{(2)}|}{t}<\frac{| \mathbf x _{(1)}- \mathbf x _{(2)}|}{\log (t+1)}<\frac{| \mathbf x - \mathbf x _{(1)}|+K| \mathbf x _{(1)}- \mathbf x _{(2)}|}{t}. \end{aligned}$$
(50)
Since the domain is spherically symmetric, it is easy to show that
$$\begin{aligned} \frac{| \mathbf x - \mathbf x _{(1)}|}{| \mathbf x _{(1)}- \mathbf x _{(2)}|}\le 1, \end{aligned}$$
and therefore \(K\approx \frac{t}{\log (t+1)}\). With (50), (29), and (30), we have
$$\begin{aligned} j_{in}( \mathbf y ,t)&= \frac{1}{C_S|D|} \int _{| {\varvec{\xi }}|<\frac{| \mathbf x - \mathbf x _{(1)}|}{t}} \Bigg \{ \alpha \sum \limits _{i=1}^{\infty }(1-\alpha )^{i-1}j( \mathbf y ,t) \left( \frac{2\pi }{RT( \mathbf x _{(i)})} \right) ^{\frac{1}{2}} M_{T( \mathbf x _{(i)})}( {\varvec{\xi }}) \nonumber \\&\qquad \qquad \qquad \qquad \,-\, \bar{g}_{in}( \mathbf x - {\varvec{\xi }}t, {\varvec{\xi }})\Bigg \} d \mathbf x d {\varvec{\xi }}, \end{aligned}$$
(51)
$$\begin{aligned} j_{mid}( \mathbf y ,t)&\le \frac{1}{C_S|D|} \sum \limits _{k=1}^{K} \int _{A_k} \left\{ \alpha \sum \limits _{i=1}^{k}(1-\alpha )^{i-1} \left[ j( \mathbf y ,t)-j( \mathbf x _{(i)},t-t_1-...-t_i) \right] \right. \nonumber \\&\left. \quad \,\times \left( \frac{2\pi }{RT( \mathbf x _{(i)})} \right) ^{\frac{1}{2}}M_{T( \mathbf x _{(i)})}( {\varvec{\xi }}) +(1-\alpha )^k\left( \alpha \sum \limits _{i=1}^{\infty }(1-\alpha )^{i-1}j( \mathbf y ,t) \left( \frac{2\pi }{RT( \mathbf x _{(k+i)})} \right) ^{\frac{1}{2}}\right. \right. \nonumber \\&\left. \left. \quad \,\times \, M_{T( \mathbf x _{(k+i)})}( {\varvec{\xi }}) - \bar{g}_{in}\left( \mathbf x _{(k)}- {\varvec{\xi }}^k(t-t_1-...-t_k), {\varvec{\xi }}^k\right) \right) \right\} d \mathbf x d {\varvec{\xi }}, \end{aligned}$$
(52)
$$\begin{aligned}&j_{fl}( \mathbf y ,t)= \frac{1}{C_S|D|} \int _{| {\varvec{\xi }}|>\frac{| \mathbf x _{(1)}- \mathbf x _{(2)}|}{\log (t+1)}} \left\{ \alpha \sum \limits _{i=1}^{K}(1-\alpha )^{i-1} \left[ j( \mathbf y ,t)-j( \mathbf x _{(i)},t-t_1-...-t_i) \right] \right. \nonumber \\&\left. \left( \frac{2\pi }{RT( \mathbf x _{(i)})} \right) ^{\frac{1}{2}}M_{T( \mathbf x _{(i)})}( {\varvec{\xi }}) +(1-\alpha )^K\Bigg (\alpha \sum \limits _{i=1}^{\infty }(1-\alpha )^{i-1}j( \mathbf y ,t) \left( \frac{2\pi }{RT( \mathbf x _{(K+i)})} \right) ^{\frac{1}{2}}\right. \nonumber \\&\left. M_{T( \mathbf x _{(K+i)})}( {\varvec{\xi }}) - \bar{g}\left( \mathbf x _{(K)}- {\varvec{\xi }}^K(t-t_1-...-t_K), {\varvec{\xi }}^K\right) \Bigg )\right\} d \mathbf x d {\varvec{\xi }}, \end{aligned}$$
(53)
where
$$\begin{aligned} A_k= \left\{ \frac{| \mathbf x - \mathbf x _{(1)}|+(k-1)| \mathbf x _{(1)}- \mathbf x _{(2)}|}{t}<| {\varvec{\xi }}|<\frac{| \mathbf x - \mathbf x _{(1)}|+k| \mathbf x _{(1)}- \mathbf x _{(2)}|}{t} \right\} . \end{aligned}$$
Each component of j can be estimated in terms of \( {\mathcal J} (t)\) and the fluctuation of j. Therefore, it suffices to consider the fluctuation of j. For \(t'<t\), \(mn/t'\ll 1\), Theorem 8 yields
$$\begin{aligned} j( \mathbf y ,t) - j( \mathbf y ',t')&= O(1) \left( \frac{ m(mn)^{d+1}\log (t'+1) }{ t'^{d+1} }+(1-\alpha )^m \right) \left\| g_{in} \right\| _{\infty ,\mu }\nonumber \\&\quad \,+\,O(1) \left( \frac{ (mn)^{d+1}\log (t'+1) }{ t'^{d+1} }+(1-\alpha )^m \right) {\mathcal J} (t) \nonumber \\&\quad \,+\, \alpha ^n \sum _{k_1=1}^m (1-\alpha )^{k_1-1}\cdots \sum _{k_n=1}^m (1-\alpha )^{k_n-1}\nonumber \\&\quad \times \Big ( \Lambda _{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ,t) - \Lambda _{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ',t') \Big ). \end{aligned}$$
(54)
And note that \(\alpha ^n \sum _{k_1=1}^m (1-\alpha )^{k_1-1}\cdots \sum _{k_n=1}^m (1-\alpha )^{k_n-1}\le 1\). Therefore, we need only to estimate the fluctuation of \(\Lambda _{(k_1,\ldots ,k_n)}^{(n,m)}\) and show that they are uniform for each \((k_1,\ldots ,k_n)\). Since
$$\begin{aligned} \Lambda _{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ,t) - \Lambda _{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ',t')&= \Big ( \Lambda _{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ,t) - \Lambda _{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ,t') \Big ) \nonumber \\&\quad \,+\, \Big ( \Lambda _{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ,t') - \Lambda _{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ',t') \Big ), \end{aligned}$$
(55)
we may consider temporal and spacial fluctuation separately. We study the temporal fluctuation in Sect. 2.4 and the spacial fluctuation in Sect. 2.5.
Remark 3
For the Maxwell-type boundary condition, which is a convex combination of the specular reflection condition and the diffuse reflection condition, the intricate dependence on the accommodation coefficient \(\alpha \) yields serious analytical difficulties beyond those in [10] and [11]. One needs to consider all the events of particle colliding with the boundary many times (37). In that case we need to deal with the problem that the diffuse reflections are coupled with specular reflections. Roughly speaking, for the events that specular reflections are more than diffuse reflections, we need to obtain the decay rate even if the specular reflection itself doesn’t have the equilibrating effect. We achieve the aim through (43) and (44). For the events that diffusion reflections are more than specular reflections, we may modify the analysis from the previous works [10] and [11] to get the decay rate (47) and the upcoming fluctuation estimates. However, the appearance of specular reflection will slow down the decay rate. Finally, we succeed in combining all events via Theorem 8.
2.4 Temporal Fluctuation Estimate
In this subsection we consider the temporal fluctuation. Recall
$$\begin{aligned} \Lambda _{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ,t)&= \int \limits _{0<s_1+s_2+\cdots +s_n< \sqrt{\frac{ T_* }{ T^* }} t/2}j( \mathbf y _{(k_1,k_2,\ldots , k_n)},t-s_1-s_2-\cdots -s_n)\\&\quad \qquad \prod \limits _{i=1}^{n} G\left( \phi _i,\frac{s_i\sqrt{2RT( \mathbf y _{(k_1,\ldots ,k_i)})}}{k_i}\right) \frac{\sqrt{2RT( \mathbf y _{(k_1,\ldots ,k_i)})}}{k_i} ds_n d\phi _n \cdots ds_1 d\phi _1\\&= \int \limits _{0<\frac{k_1\sigma _1}{2RT_{(k_1)}}+\frac{k_2\sigma _2}{2RT_{(k_1,k_2)}}+\cdots +\frac{k_n\sigma _n}{2RT_{(k_1,\ldots ,k_n)}}<t/2}\\&\quad \quad j \left( \mathbf y _{(k_1,k_2,\ldots , k_n)},t-\frac{k_1\sigma _1}{2RT_{(k_1)}}-\ldots -\frac{k_n\sigma _n}{2RT_{(k_1,\ldots ,k_n)}} \right) \\&\quad \quad \prod \limits _{i=1}^{n} G(\phi _i,\sigma _i) d\sigma _n d\phi _n \cdots d\sigma _1 d\phi _1 \quad \text {when } d=2, \end{aligned}$$
$$\begin{aligned} \Lambda _{(k_1,\ldots ,k_n)}^{(n,m)}(\pm 1,t)&= \int \limits _{0<s_1+s_2+\cdots +s_n< \sqrt{\frac{ T_* }{ T^* }} t/2}j(\pm 1_{(k_1,k_2,\ldots , k_n)},t-s_1-s_2-\cdots -s_n)\\&\quad \quad \prod \limits _{i=1}^{n} H\left( \frac{s_i\sqrt{2RT(\pm 1_{(k_1,\ldots ,k_i)})}}{k_i}\right) \frac{\sqrt{2RT(\pm 1_{(k_1,\ldots ,k_i)})}}{k_i} ds_n \cdots ds_1\\&= \int \limits _{0<\frac{k_1\sigma _1}{2RT_{(k_1)}}+\frac{k_2\sigma _2}{2RT_{(k_1,k_2)}}+\cdots +\frac{k_n\sigma _n}{2RT_{(k_1,\ldots ,k_n)}}<t/2}\\&\quad \quad j \left( \pm 1_{(k_1,k_2,\ldots , k_n)},t-\frac{k_1\sigma _1}{2RT_{(k_1)}}-\ldots -\frac{k_n\sigma _n}{2RT_{(k_1,\ldots ,k_n)}} \right) \\&\quad \quad \prod \limits _{i=1}^{n} H(\sigma _i) d\sigma _n \cdots d\sigma _1 \quad \text {when } d=1. \end{aligned}$$
We note that the kernel \(H(\sigma )\) and \(G(\phi ,\sigma )\) are smooth in \(\sigma \), and hence we may differentiate \(\Lambda _{(k_1,\ldots ,k_n)}^{(n,m)}\) with respect to t directly to obtain an explicit expression.
Lemma 3
Let n be any positive integer. \(\Lambda _{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ,t)\) is \(C^1\) with respect to t. Their derivatives has two parts:
$$\begin{aligned} \frac{\partial \Lambda _{(k_1,\ldots ,k_n)}^{(n,m)}}{ \partial t} ( \mathbf y ,t) =&\mathcal B_{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ,t) + \mathcal V_{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ,t), \end{aligned}$$
The first term \(\mathcal B_{(k_1,\ldots ,k_n)}^{(n,m)}\) is the boundary term:
$$\begin{aligned} \mathcal B_{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ,t)&= - \left( 1- \sqrt{\frac{ T_* }{ T^* }} \frac{1}{2} \right) \int \limits _{ s_1+\ldots +s_n = \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} } \prod _{l=1}^n G \left( \textstyle \phi _l, \frac{s_l \sqrt{2RT_{(k_1,\ldots ,k_l)} }}{k_l} \right) \frac{\sqrt{2RT_{(k_1,\ldots ,k_l)}}}{k_l}\nonumber \\&\quad \, \times j \left( \mathbf y _{(k_1,\ldots ,k_n)}, t- \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} \right) ds_1 \cdots ds_{n-1}d^n\phi \quad \text {when } d=2, \end{aligned}$$
(56)
$$\begin{aligned} \mathcal B_{(k_1,\ldots ,k_n)}^{(n,m)}(\pm 1,t)&= - \left( 1- \sqrt{\frac{ T_* }{ T^* }} \frac{1}{2} \right) \int \limits _{ s_1+\ldots +s_n = \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} } \prod _{l=1}^n H \left( \textstyle \frac{s_l \sqrt{2RT_{(k_1,\ldots ,k_l)} }}{k_l} \right) \frac{\sqrt{2RT_{(k_1,\ldots ,k_l)}}}{k_l}\nonumber \\&\quad \, \times j \left( \pm 1_{(k_1,\ldots ,k_n)}, t- \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} \right) ds_1 \cdots ds_{n-1} \quad \text {when } d=1. \end{aligned}$$
(57)
The second term \(\mathcal V_{(k_1,\ldots ,k_n)}^{(n,m)}\) is the volume term:
$$\begin{aligned} \mathcal V_{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ,t)&= \frac{1}{n} \int \int ^t_{ t- \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} } \int \limits _{s_1+\ldots +s_n=t-s} \sum _{l=1}^n G \left( \phi _1, \frac{s_1 \sqrt{2RT_{(k_1)} }}{k_l} \right) \times \cdots \nonumber \\&\quad \, \times \frac{\sqrt{2RT_{(k_1,\ldots ,k_l)}}}{k_l} \frac{ \partial G }{ \partial \sigma } \left( \phi _l, \frac{s_l \sqrt{2RT_{(k_1,\ldots ,k_l)}}}{k_l} \right) \nonumber \\&\quad \times \cdots \times G \left( \phi _n, \frac{s_n \sqrt{2RT_{(k_1,\ldots ,k_n)}}}{k_n} \right) \nonumber \\&\quad \, \times d s_1 \cdots ds_{n-1} ~ j( \mathbf y _{(k_1,\ldots ,k_n)},s) d s \frac{\sqrt{2RT_{(k_1)}}}{k_1} \cdots \frac{\sqrt{2RT_{(k_1,\ldots ,k_n)}}}{k_n} d^n\phi \nonumber \\&\quad \text {when } d=2, \end{aligned}$$
(58)
$$\begin{aligned} \mathcal V_{(k_1,\ldots ,k_n)}^{(n,m)}(\pm 1,t)&=\frac{1}{n} \int \int ^t_{ t- \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} } \int \limits _{s_1+\ldots +s_n=t-s} \sum _{l=1}^n H \left( \frac{s_1 \sqrt{2RT_{(k_1)} }}{k_l} \right) \times \cdots \nonumber \\&\quad \, \times \frac{\sqrt{2RT_{(k_1,\ldots ,k_l)}}}{k_l} \frac{ \partial H }{ \partial \sigma } \left( \frac{s_l \sqrt{2RT_{(k_1,\ldots ,k_l)}}}{k_l} \right) \nonumber \\&\quad \times \cdots \times H \left( \frac{s_n \sqrt{2RT_{(k_1,\ldots ,k_n)}}}{k_n} \right) \nonumber \\&\quad \, \times d s_1 \cdots ds_{n-1} ~ j(\pm 1_{(k_1,\ldots ,k_n)},s) d s \frac{\sqrt{2RT_{(k_1)}}}{k_1} \cdots \frac{\sqrt{2RT_{(k_1,\ldots ,k_n)}}}{k_n}\nonumber \\&\quad \text {when } d=1. \end{aligned}$$
(59)
The lemma can be proved by a similar argument as in [10, 11] and we omit it. The boundary term \(\mathcal B_n\) can be easily bounded as following. First, we have
$$\begin{aligned} |\mathcal B_{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ,t)|&=O(1) \sup _{ \frac{t}{2}<s<t } \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s)\nonumber \\&\quad \!\times \! \int \limits _{ s_1\!+\!\ldots +s_n \!=\! \frac{t}{2} } \left( \prod _{l=1}^n G \left( \textstyle \phi _l, \frac{s_l \sqrt{2RT_{(k_1,\ldots ,k_l)}}}{k_l} \right) \frac{\sqrt{2RT_{(k_1,\ldots ,k_l)}}}{k_l} \right) ds_1 \cdots ds_{n-1}d^n\phi \nonumber \\&= O(1) \sup _{ \frac{t}{2}<s<t } \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s)\nonumber \\&\quad \,\times \int \limits _{\frac{k_1 \sigma _1}{2RT_{(k_1)}} + \ldots + \frac{k_n \sigma _n}{2RT_{(k_1,\ldots ,k_n)}} = \frac{t}{2}} \prod _{l=1}^n G \left( \textstyle \phi _l,\sigma _l \right) ds_1 \cdots ds_{n-1}d^n\phi \quad \text {when } d=2, \end{aligned}$$
(60a)
$$\begin{aligned} |\mathcal B_{(k_1,\ldots ,k_n)}^{(n,m)}(\pm 1,t)|&= O(1) \sup _{ \frac{t}{2}<s<t } \Big ( \left| j_+(s) \right| + \left| j_-(s) \right| \Big )\nonumber \\&\quad \, \times \int \limits _{ s_1+\ldots +s_n = \frac{t}{2} } \left( \prod _{l=1}^n H \left( \textstyle \frac{s_l \sqrt{2RT_{(k_1,\ldots ,k_l)}}}{k_l} \right) \frac{\sqrt{2RT_{(k_1,\ldots ,k_l)}}}{k_l} \right) ds_1 \cdots ds_{n-1} \nonumber \\&= O(1) \sup _{ \frac{t}{2}<s<t } \Big ( \left| j_+(s) \right| + \left| j_-(s) \right| \Big )\nonumber \\&\quad \, \times \int \limits _{\frac{k_1 \sigma _1}{2RT_{(k_1)}} + \ldots + \frac{k_n \sigma _n}{2RT_{(k_1,\ldots ,k_n)}} = \frac{t}{2}} \prod _{l=1}^n H \left( \textstyle \sigma _l \right) ds_1 \cdots ds_{n-1} \quad \text {when } d=1. \end{aligned}$$
(60b)
Hence given \(t'<t\) with \(t'/mn \gg 1\), by the law of large numbers, Theorem 7,
$$\begin{aligned} \int _{t'}^t |\mathcal B_{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ,s)| ds&=O(1) \sup _{ \frac{t}{2}<s<t } \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s)\nonumber \\&\quad \times \frac{m^{3}n^2\log (t'+1)}{t'^{3}}, \text {when } d=2, \end{aligned}$$
(61a)
$$\begin{aligned} \int _{t'}^t |\mathcal B_{(k_1,\ldots ,k_n)}^{(n,m)}(\pm 1,s)| ds&=O(1) \sup _{ \frac{t}{2}<s<t } \Big ( \left| j_+(s) \right| + \left| j_-(s) \right| \Big )\nonumber \\&\quad \times \frac{m^{2}n\log (t'+1)}{t'^{2}}, \quad \text {when } d=1. \end{aligned}$$
(61b)
We next turn to the major term \(\mathcal V_n\), First, as above, we have
$$\begin{aligned}&|\mathcal V_{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ,t)| \le \frac{1}{n} \sup _{ \frac{t}{2}<s<t } \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s)\nonumber \\&\quad \, \times \int \left| \sum _{l=1}^n \frac{\sqrt{2RT_{(k_1,\ldots ,k_l)}}}{k_l} G(\phi _1,\sigma _1) \cdots \frac{\partial G}{\partial \sigma }(\phi _l,\sigma _l) \cdots G(\phi _n,\sigma _n) \right| d^n\sigma d^n\phi&\quad \text {when } d=2, \end{aligned}$$
(62a)
$$\begin{aligned} |\mathcal V_{(k_1,\ldots ,k_n)}^{(n,m)}(\pm 1,t)|&\le \frac{1}{n} \sup _{ \frac{t}{2}<s<t } \Big ( \left| j_+(s) \right| + \left| j_-(s) \right| \Big )\nonumber \\&\quad \times \int \left| \sum _{l=1}^n \frac{\sqrt{2RT_{(k_1,\ldots ,k_l)}}}{k_l} H(\sigma _1) \!\cdots \! \frac{\partial H}{\partial \sigma }(\sigma _l) \!\cdots \! H(\sigma _n) \right| d^n\sigma&\quad \text {when } d\!=\!1. \end{aligned}$$
(62b)
To estimate (62), we use the following lemma.
Lemma 4
For any integer \(n>1\),
$$\begin{aligned}&\int \left| \sum _{l=1}^n \frac{\sqrt{2RT_{(k_1,\ldots ,k_l)}}}{k_l} G(\phi _1,\sigma _1) \cdots \frac{\partial G}{\partial \sigma }(\phi _l,\sigma _l) \cdots G(\phi _n,\sigma _n) \right| d^n\sigma d^n\phi \nonumber \\&\qquad \qquad = O \left( \left( n\log n \right) ^\frac{1}{2} \right) , \end{aligned}$$
(63a)
$$\begin{aligned}&\int \left| \sum _{l=1}^n \frac{\sqrt{2RT_{(k_1,\ldots ,k_l)}}}{k_l} H(\sigma _1) \cdots \frac{\partial H}{ \partial \sigma }(\sigma _l) \cdots H(\sigma _n) \right| d^n\sigma \nonumber \\&\qquad \qquad = O \left( \left( n \right) ^\frac{1}{2} \right) , \end{aligned}$$
(63b)
Consequently,
$$\begin{aligned} |\mathcal V_{(k_1,\ldots ,k_n)}^{(n,m)}( \mathbf y ,t)|&= O(1) \sup _{ \frac{t}{2}<s<t } \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s) \times \left( \frac{\log n}{n} \right) ^\frac{1}{2},\text { when } d=2,\\ |\mathcal V_{(k_1,\ldots ,k_n)}^{(n,m)}(\pm 1,t)|&= O(1) \sup _{ \frac{t}{2}<s<t } \Big ( \left| j_+(s) \right| + \left| j_-(s) \right| \Big ) \times \left( \frac{1}{n} \right) ^\frac{1}{2}, \text { when } d=1. \end{aligned}$$
Proof
Note that \(k_l\ge 1\) for each l, so
$$\begin{aligned}&\left| \sum _{l=1}^n \frac{\sqrt{2RT_{(k_1,\ldots ,k_l)}}}{k_l} G(\phi _1,\sigma _1) \cdots \frac{\partial G}{\partial \sigma }(\phi _l,\sigma _l) \cdots G(\phi _n,\sigma _n) \right| \\&\le \left| \sum _{l=1}^n \sqrt{2RT_{(k_1,\ldots ,k_l)}} G(\phi _1,\sigma _1) \cdots \frac{\partial G}{\partial \sigma }(\phi _l,\sigma _l) \cdots G(\phi _n,\sigma _n) \right| \end{aligned}$$
Then we follow the Lemma 3 in [11] to conclude the proof. \(\square \)
The following theorem follows from Lemma 4 together with (61).
Theorem 9
Let \(t'<t\), then, for \(1\ll t'/mn\),
$$\begin{aligned}&\Lambda ^{(n,m)}_{(k_1,\ldots ,k_n)}( \mathbf y ,t) - \Lambda ^{(n,m)}_{(k_1,\ldots ,k_n)}( \mathbf y ,t') = O(1) \sup _{ \frac{t'}{2}<s<t } \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s)\\&\quad \, \times \left( \frac{ m^{3}n^2\log (t'+1)}{ t'^{3} } + \left( \frac{\log n}{n} \right) ^\frac{1}{2} (t-t') \right) ,\quad \text { when } d=2, \end{aligned}$$
$$\begin{aligned}&\Lambda ^{(n,m)}_{(k_1,\ldots ,k_n)}(\pm 1,t) - \Lambda ^{(n,m)}_{(k_1,\ldots ,k_n)}(\pm 1,t') = O(1) \sup _{ \frac{t'}{2}<s<t } \Big ( \left| j_+(s) \right| + \left| j_-(s) \right| \Big ) \\&\quad \,\times \left( \frac{ m^{2}n\log (t'+1)}{ t'^{2} } + \left( \frac{1}{n} \right) ^\frac{1}{2} (t-t') \right) , \quad \text { when } d=1. \end{aligned}$$
From Theorem 9, (54) and (55), we obtain:
Corollary 2
(Temporal Fluctuation Estimate) Let \(t'<t\), then, for \(1\ll t'/mn\),
$$\begin{aligned}&j( \mathbf y ,t) - j( \mathbf y ,t') = O(1) \left( \frac{m(mn)^{3}\log (t'+1)}{t'^{3}}+(1-\alpha )^m \right) \left( \left\| g_{in} \right\| _{\infty ,\mu } + {\mathcal J} (t) \right) \\&\quad \,+\, O(1) \sup _{ \frac{t'}{2}<s<t } \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s) \times \left( \frac{\log n}{n} \right) ^\frac{1}{2} (t-t'), \text { for } d=2, \end{aligned}$$
$$\begin{aligned} j(\pm 1,t) - j(\pm 1,t')&= O(1) \left( \frac{m(mn)^{2}\log (t'+1)}{t'^{2}}+(1-\alpha )^m \right) \left( \left\| g_{in} \right\| _{\infty ,\mu } + {\mathcal J} (t) \right) \\&\quad \,+\, O(1) \sup _{ \frac{t'}{2}<s<t } \Big ( \left| j_+(s) \right| + \left| j_-(s) \right| \Big ) \times \left( \frac{1}{n} \right) ^\frac{1}{2} (t-t'), \text { for } d=1. \end{aligned}$$
2.5 Spacial Fluctuation Estimate
In this subsection, we investigate the spacial fluctuation.
Theorem 10
(Spacial Fluctuation Estimate) Suppose that \(t/mn,\ t/mN,\ N \gg 1, 0<q<1\), \( \mathbf y , \mathbf y '\in \partial D\). Then
$$\begin{aligned} j( \mathbf y ,t) - j( \mathbf y ',t)&= O(1) \left( \frac{ ((mn)^{3}+m^3N^2) \log (t+1) }{ t^{3} } +(1-\alpha )^m \right) \left( \left\| g_{in} \right\| _{\infty ,\mu } + {\mathcal J} (t) \right) \\&\quad \,+\,O(1) \sup _{\frac{t}{2}<s<t} \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s)\\&\quad \times \left( \left( \frac{\log n}{n} \right) ^\frac{1}{2} mN + \left( \frac{\log N}{N} \right) ^\frac{1}{2} \right) , \quad \text { when } d=2, \end{aligned}$$
$$\begin{aligned} |j(+1,t)-j(-1,t)|&=O(1) \left( \frac{m^3n^{2}\log (t+1)}{t^{2}}+(1-\alpha )^m \right) \left( {\mathcal J} (t)+ \left\| g_{in} \right\| _{\infty ,\mu } \right) \\&\quad \,+\,O(1) \left( \left( \frac{1}{n} \right) ^\frac{1}{2} t^q+ \left( \frac{m}{t^q} \right) ^2 \right) \\&\quad \sup _{ \frac{t}{2} < s < t } \Big ( \left| j_+(s) \right| + \left| j_-(s) \right| \Big ) \quad \text { when } d=1. \end{aligned}$$
We consider the one dimensional case first, which is much simpler than the multidimensional cases.
One dimensional case,
\(d=1\).
This case differs from the multidimensional cases in a fundamental sense: unlike the multidimensional cases, the boundary, comprising of two points, is discrete. This makes the one dimensional case much easier. Instead of processing \(\Lambda _{\pm ,n}\), we directly estimate: \(j(+1,t)-j(-1,t)\). Recall
$$\begin{aligned} j(+1,t)&=\sum \limits _{k=0}^{m-1}(1-\alpha )^k j_{in}^{(k)}(+1,t)\\&\quad \,+\,\alpha \sum \limits _{k=1}^{m}(1-\alpha )^{k-1} \int \limits _{0}^{t} j((-1)^{k},t-s)H\left( \frac{s\sqrt{2RT((-1)^{k})}}{k}\right) \frac{\sqrt{2RT((-1)^{k})}}{k} ds \\&\quad \,+\,(1-\alpha )^{m}E^{(m)}(+1,t). \end{aligned}$$
From
$$\begin{aligned} \int \limits _{0}^\infty H\left( \frac{s\sqrt{2RT((-1)^{k})}}{k}\right) \frac{\sqrt{2RT((-1)^{k})}}{k} ds=1, \end{aligned}$$
we have
$$\begin{aligned} j(-1,t)= & {} \alpha \sum \limits _{k=1}^{m}(1-\alpha )^{k-1} \int \limits _{0}^{\infty }j(-1,t)H\left( \frac{s\sqrt{2RT((-1)^{k})}}{k}\right) \frac{\sqrt{2RT((-1)^{k})}}{k} ds\nonumber \\&+\,(1-\alpha )^m j(-1,t). \end{aligned}$$
Now consider
$$\begin{aligned} j(+1,t)-j(-1,t)&=\sum \limits _{k=0}^{m-1}(1-\alpha )^k j_{in}^{(k)}(+1,t)\\&\quad \,+\,\alpha \sum \limits _{k=1}^{m}(1-\alpha )^{k-1} \int \limits _{0}^{t} \left( j((-1)^{k},t-s)-j(-1,t) \right) \\&\quad H\left( \frac{s\sqrt{2RT((-1)^{k})}}{k}\right) \frac{\sqrt{2RT((-1)^{k})}}{k} ds \,-\,\alpha \sum \limits _{k=1}^{m}(1-\alpha )^{k-1}\\&\quad \int \limits _{t}^{\infty } j(-1,t)H\left( \frac{s\sqrt{2RT((-1)^{k})}}{k}\right) \frac{\sqrt{2RT((-1)^{k})}}{k} ds \\&\quad \,+\,(1-\alpha )^{m} \left( E^{(m)}(+1,t)-j(-1,t) \right) . \end{aligned}$$
As before,
$$\begin{aligned}&\sum \limits _{k=0}^{m-1}(1-\alpha )^k j_{in}^{(k)}(+1,t) =O(1) \left\| g_{in} \right\| _{\infty ,\mu } \frac{m^2}{( t+1)^2}\\&(1-\alpha )^{m} \left( E^{(m)}(+1,t)-j(-1,t) \right) =O(1)(1-\alpha )^m \left( {\mathcal J} (t)+ \left\| g_{in} \right\| _{\infty ,\mu } \right) , \end{aligned}$$
and we decompose
$$\begin{aligned}&\int \limits _{0}^{t} \left( j((-1)^{k},t-s)-j(-1,t) \right) H\left( \frac{s\sqrt{2RT((-1)^{k})}}{k}\right) \frac{\sqrt{2RT((-1)^{k})}}{k} ds\\&=\int \limits _{0}^{t^q}\Big (\ldots \Big )+\int \limits _{t^q}^{t/2}\Big (\ldots \Big ) +\int \limits _{t/2}^{t}\Big (\ldots \Big ), \end{aligned}$$
where \(0<q<1\) is to be determined. And then
$$\begin{aligned}&\alpha \sum \limits _{k=1}^{m}(1-\alpha )^{k-1} \int \limits _{t}^{\infty } j(-1,t)H\left( \frac{s\sqrt{2RT((-1)^{k})}}{k}\right) \frac{\sqrt{2RT((-1)^{k})}}{k} ds\\&\quad =O(1) {\mathcal J} (t)\alpha \sum \limits _{k=1}^{m}(1-\alpha )^{k-1}\frac{k^2}{t^2} =O(1) {\mathcal J} (t)\frac{m^2}{ t^2},\\&\quad \alpha \sum \limits _{k=1}^{m}(1-\alpha )^{k-1}\int \limits _{t/2}^{t}\Big (\ldots \Big )= O(1) {\mathcal J} (t)\alpha \sum \limits _{k=1}^{m}(1-\alpha )^{k-1}\frac{k^2}{t^2} =O(1) {\mathcal J} (t)\frac{m^2}{ t^2},\\&\quad \alpha \sum \limits _{k=1}^{m}(1-\alpha )^{k-1}\int \limits _{t^q}^{t/2}\Big (\ldots \Big )= O(1) \left( \frac{1}{t^q } \right) ^2 \sup _{ \frac{t}{2} < s < t } \Big ( \left| j_+(s) \right| + \left| j_-(s) \right| \Big ) \alpha \sum \limits _{k=1}^{m}(1-\alpha )^{k-1}k^2\\&\quad =O(1) \left( \frac{m}{t^q} \right) ^2 \sup _{ \frac{t}{2} < s < t } \Big ( \left| j_+(s) \right| + \left| j_-(s) \right| \Big ). \end{aligned}$$
Finally, we consider
$$\begin{aligned}&\int \limits _{0}^{t^q} \left( j((-1)^{k},t-s)-j(-1,t) \right) H\left( \frac{s\sqrt{2RT((-1)^{k})}}{k}\right) \frac{\sqrt{2RT((-1)^{k})}}{k} ds\\&=\int \limits _{0}^{t^q} \left( j((-1)^{k},t-s)-j((-1)^{k},t)+j((-1)^{k},t)-j(-1,t) \right) \\&\quad H\left( \frac{s\sqrt{2RT((-1)^{k})}}{k}\right) \frac{\sqrt{2RT((-1)^{k})}}{k} ds, \end{aligned}$$
and it is easy to see that
Hence
$$\begin{aligned} \frac{1}{2-\alpha } \left( j(+1,t)-j(-1,t) \right)&=O(1) \left( \frac{m^2}{ t^2}+(1-\alpha )^m \right) \left( {\mathcal J} (t)+ \left\| g_{in} \right\| _{\infty ,\mu } \right) \\&\quad \,+\,\sup _{ t-t^q < t' < t } \left| j(\pm 1,t) - j(\pm 1,t') \right| \\&\quad \,+\,O(1) \left( \frac{m}{t^q} \right) ^2 \sup _{ \frac{t}{2} < s < t } \Big ( \left| j_+(s) \right| + \left| j_-(s) \right| \Big ), \end{aligned}$$
and it follows
$$\begin{aligned} |j(+1,t)-j(-1,t)|&=O(1) \left( \frac{m^3n^{2}\log (t+1)}{t^{2}}+\frac{m^2}{ t^2}+(1-\alpha )^m \right) \left( {\mathcal J} (t)+ \left\| g_{in} \right\| _{\infty ,\mu } \right) \\&\quad \,+\,O(1) \left( \frac{m}{t^q} \right) ^2 \sup _{ \frac{t}{2} < s < t } \Big ( \left| j_+(s) \right| + \left| j_-(s) \right| \Big )\\&\quad \,+\,O(1) \sup _{ \frac{t}{2}<s<t } \Big ( \left| j_+(s) \right| + \left| j_-(s) \right| \Big ) \times \left( \frac{1}{n} \right) ^\frac{1}{2} t^q. \end{aligned}$$
Subsequently, we can apply the temporal fluctuation estimate, Corollary 2, to the term \(\sup _{t-t^q< t'<t} |j(\pm 1,t)-j(\pm 1,t')|\). Hence Theorem 10 for \(d=1\) follows. Note that \(t-t^q>\frac{t}{2}\) so long as \(t\gg 1\).
Multidimensional cases,
\(d=2\).
As noted before, to estimate spacial fluctuation we invoke another variable N. From now on N will be the index of \(\Lambda ^{(N,m)}_{(k_1,\ldots ,k_N)}\).
The boundary \(\partial D\) is unit circle, so we parametrize it by the polar coordinates. Given two boundary points \( \mathbf y \) and \( \mathbf y '\), let \( \mathbf y '\) be point of degree zero, and denote the polar angle of \( \mathbf y \) by \(\theta \). Denote the relative polar angle of \( \mathbf y _{(k_1,\ldots ,k_{l-1},1)}\) with respect to \( \mathbf y _{(k_1,\ldots ,k_{l-1})}\) by \(\theta _l\). Then for \(1\le i \le k_l\), the relative polar angle of \( \mathbf y _{(k_1,\ldots ,k_{l-1},i)}\) with respect to \( \mathbf y _{(k_1,\ldots ,k_{l-1},i-1)}\) is also \(\theta _l\) because of the specular reflection, i.e. \(\theta +k_1\theta _1+\ldots +k_l\theta _l\) stands for the absolute polar angle of \( \mathbf y _{(k_1,\ldots ,k_l)}\). Since \(\partial D\) is the unit circle, \(\theta _l=\pi -2\phi _l\),
To simplify the notation, put \(T_{(k_1,\ldots ,k_l)}=T(\theta +k_1\theta _1+\ldots +k_l\theta _l)\), and \(T'_{(k_1,\ldots ,k_l)}=T(k_1\theta _1+\ldots +k_l\theta _l)\). Under this coordinate system, we have
$$\begin{aligned} \Lambda _{(k_1,\ldots ,k_N)}^{(N,m)}( \mathbf y ,t)&= \Lambda _{(k_1,\ldots ,k_N)}^{(N,m)}(\theta ,t) \nonumber \\&= \int \limits _{ \frac{k_1\sigma _1}{\sqrt{2RT_{(k_1)}}} + \ldots + \frac{k_N\sigma _N}{\sqrt{2RT_{(k_1,\ldots ,k_N)}}} < \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} } \prod _{l=1}^N G(\phi _l,\sigma _l)\nonumber \\&\quad \, \times j \left( \textstyle \theta + k_1\theta _1 + \ldots + k_N\theta _N , t - \frac{k_1\sigma _1}{\sqrt{2RT_{(k_1)}}} - \ldots - \frac{k_N\sigma _N}{\sqrt{2RT_{(k_1,\ldots ,k_N)}}} \right) \nonumber \\&\qquad d^N \sigma d^N \phi \nonumber \\&= \int \limits _{ \frac{k_1\sigma _1}{\sqrt{2R\tilde{T}_{(k_1)}}} + \ldots + \frac{k_N\sigma _N}{\sqrt{2R\tilde{T}_{(k_1,\ldots ,k_N)}}} < \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} } \prod _{l=1}^N G \left( \textstyle \phi _l + \frac{\theta }{2k_lN}, \sigma _l \right) \nonumber \\&\quad \, \times j \left( \textstyle k_1\theta _1 \!+\! \ldots \!+\! k_N\theta _N , t - \frac{k_1\sigma _1}{\sqrt{2R\tilde{T}_{(k_1)}}} \!-\! \ldots - \frac{k_N\sigma _N}{\sqrt{2R\tilde{T}_{(k_1,\ldots ,k_N)}}} \right) d^N \sigma d^N \phi , \end{aligned}$$
(64)
where \(\tilde{T}_{(k_1,\ldots ,k_l)} = T(\theta +k_1\theta _1+\cdots +k_N\theta _N-\frac{l\theta }{N})\).
Remark 4
In [10] and [11] we can deal with the case of spherical domain in \( \mathbb {R} ^d\) for \(d=1,2,3,\) but in this paper we only consider the the spherical domain in \( \mathbb {R} ^d\) for \(d=1,2\). As in our previous works [10] and [11] we need the symmetric property of domain to calculate exactly the spacial fluctuation. The symmetry of the boundary allows us to tract the exact location of the particle after multiple reflections. This is an essential ingredient of our analysis on treating spacial fluctuation. For two dimensional case, thanks to polar coordinate we are able to tract the exact location of the particle on a circle after mixed specular reflections and diffuse reflections. This allows us to conduct a change of variables (64), and to estimate the spacial fluctuation (65). However, in three dimensional case there is no universal coordinate to tract the exact location of the particle on a sphere after mixed specular reflections and diffuse reflections. The three dimensional case might require mathematical analysis different from ours. We hope to return to this problem in the future.
From (64), we have
$$\begin{aligned}&\Lambda _{(k_1,\ldots ,k_N)}^{(N,m)}( \mathbf y ,t) - \Lambda _{(k_1,\ldots ,k_N)}^{(N,m)}( \mathbf y ',t) = \Lambda _{(k_1,\ldots ,k_N)}^{(N,m)}(\theta ,t) - \Lambda _{(k_1,\ldots ,k_N)}^{(N,m)}(0,t)\nonumber \\&\quad = \int \limits _{ \frac{k_1\sigma _1}{\sqrt{2R\tilde{T}_{(k_1)}}} + \ldots + \frac{k_N\sigma _N}{\sqrt{2R\tilde{T}_{(k_1,\ldots ,k_N)}}} < \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} } \prod _{l=1}^N G \left( \textstyle \phi _l + \frac{\theta }{2k_lN}, \sigma _l \right) \nonumber \\&\qquad \, \times j \left( \textstyle k_1\theta _1 + \ldots + k_N\theta _N , t - \frac{k_1\sigma _1}{\sqrt{2R\tilde{T}_{(k_1)}}} - \ldots - \frac{k_N\sigma _N}{\sqrt{2R\tilde{T}_{(k_1,\ldots ,k_N)}}} \right) d^N \sigma d^N \phi \nonumber \\&\qquad \,-\, \int \limits _{ \frac{k_1\sigma _1}{\sqrt{2RT'_{(k_1)}}} + \ldots + \frac{k_N\sigma _N}{\sqrt{2RT'_{(k_1,\ldots ,k_N)}}} < \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} } \prod _{l=1}^N G(\phi _l,\sigma _l)\nonumber \\&\qquad \, \times j \left( \textstyle k_1\theta _1 + \ldots + k_N\theta _N , t - \frac{k_1\sigma _1}{\sqrt{2RT'_{(k_1)}}} - \ldots - \frac{k_N\sigma _N}{\sqrt{2RT'_{(k_1,\ldots ,k_N)}}} \right) d^N \sigma d^N \phi . \end{aligned}$$
(65)
The two terms in (65) differ from each other in three places: the domain of integration, the angular variable of the transition PDF G, and the time variable of j.
We now break the spatial fluctuation into three parts:
$$\begin{aligned} \Lambda _{(k_1,\ldots ,k_N)}^{(N,m)}( \mathbf y ,t) - \Lambda _{(k_1,\ldots ,k_N)}^{(N,m)}( \mathbf y ',t) \!=\! \Lambda _{(k_1,\ldots ,k_N)}^{(N,m)}(\theta ,0) \!-\! \Lambda _{(k_1,\ldots ,k_N)}^{(N,m)}(0,t) =U_1+U_2+U_3, \end{aligned}$$
where
$$\begin{aligned}&U_1 \equiv \left( \int \limits _{ \frac{k_1\sigma _1}{\sqrt{2R\tilde{T}_{(k_1)}}} + \ldots + \frac{k_N\sigma _N}{\sqrt{2R\tilde{T}_{(k_1,\ldots ,k_N)}}} < \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} } -\int \limits _{ \frac{k_1\sigma _1}{\sqrt{2RT'_{(k_1)}}} + \ldots + \frac{k_N\sigma _N}{\sqrt{2RT'_{(k_1,\ldots ,k_N)}}} < \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} } \right) \times \\&\left( \prod _{l=1}^N G \left( \textstyle \phi _l + \frac{\theta }{2k_lN}, \sigma _l \right) \right) j \left( \textstyle k_1\theta _1 + \ldots + k_N\theta _N , t - \frac{k_1\sigma _1}{\sqrt{2R\tilde{T}_{(k_1)}}} - \ldots - \frac{k_N\sigma _N}{\sqrt{2R\tilde{T}_{(k_1,\ldots ,k_N)}}} \right) \\&\quad d^N\sigma d^N\phi , \end{aligned}$$
$$\begin{aligned} U_2&\equiv \int \limits _{ \frac{k_1\sigma _1}{\sqrt{2RT'_{(k_1)}}} + \ldots + \frac{k_N\sigma _N}{\sqrt{2RT'_{(k_1,\ldots ,k_N)}}} < \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} } \prod _{l=1}^N G \left( \textstyle \phi _l + \frac{\theta }{2k_lN}, \sigma _l \right) \\&\quad \,\times \left[ j \left( \textstyle k_1\theta _1 + \ldots + k_N\theta _N , t - \frac{k_1\sigma _1}{\sqrt{2R\tilde{T}_{(k_1)}}} - \ldots - \frac{k_N\sigma _N}{\sqrt{2R\tilde{T}_{(k_1,\ldots ,k_N)}}} \right) \right. \\&\left. \quad \,- \, j \left( \textstyle k_1\theta _1 + \ldots + k_N\theta _N , t - \frac{k_1\sigma _1}{\sqrt{2RT'_{(k_1)}}} - \ldots - \frac{k_N\sigma _N}{\sqrt{2RT'_{(k_1,\ldots ,k_N)}}} \right) \right] d^N\sigma d^N\phi , \end{aligned}$$
$$\begin{aligned} U_3&\equiv \int \limits _{ \frac{k_1\sigma _1}{\sqrt{2RT'_{(k_1)}}} + \ldots + \frac{k_N\sigma _N}{\sqrt{2RT'_{(k_1,\ldots ,k_N)}}} < \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} } \left( \prod _{l=1}^N G \left( \textstyle \phi _l + \frac{\theta }{2k_lN}, \sigma _l \right) - \prod _{l=1}^N G(\phi _l,\sigma _l) \right) \\&\quad \,\times j \left( \textstyle k_1\theta _1 + \ldots + k_N\theta _N , t - \frac{k_1\sigma _1}{\sqrt{2RT'_{(k_1)}}} - \ldots - \frac{k_N\sigma _N}{\sqrt{2RT'_{(k_1,\ldots ,k_N)}}} \right) d^N\sigma d^N\phi . \end{aligned}$$
\(U_1,U_2,U_3\) register the difference in domain of integration, the time variable of j, and angular variable of the transition PDF G, respectively.
We now proceed to estimate \(U_1\), \(U_2\), and \(U_3\). Consider first \(U_1\). As noted before, \(U_1\) registers the difference in domain of integration. Denote by \(A\ominus B\) the symmetric difference \((A\setminus B)\cup (B\setminus A)\). Since \(k_i\le m\) for each i and \(T\ge T_*\) on the boundary, one can observe that both of the events
$$\begin{aligned} \mathscr {E}_1 \equiv&\left\{ \frac{k_1\sigma _1}{\sqrt{2R\tilde{T}_{(k_1)}}} + \ldots + \frac{k_N\sigma _N}{\sqrt{2R\tilde{T}_{(k_1,\ldots ,k_N)}}} < \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} \right\} \\ \mathscr {E}_2 \equiv&\left\{ \frac{k_1\sigma _1}{\sqrt{2RT'_{(k_1)}}} + \ldots + \frac{k_N\sigma _N}{\sqrt{2RT'_{(k_1,\ldots ,k_N)}}} < \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} \right\} \end{aligned}$$
contain
$$\begin{aligned} \left\{ \sigma _1+\ldots +\sigma _N < \sqrt{2R T_* } \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2m} \right\} . \end{aligned}$$
So we have
$$\begin{aligned} \mathscr {E}_1 \ominus \mathscr {E}_2 \subset \left\{ \sigma _1+\ldots +\sigma _N \ge \sqrt{2R T_* } \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2m} \right\} . \end{aligned}$$
(66)
This implies that \(\mathscr {E}_1\ominus \mathscr {E}_2\), and thereby \(U_1\), is a rare event, and can be estimated by the law of large numbers, Theorem 7.
Lemma 5
Assume \(t/(Nm) \gg 1\),
$$\begin{aligned} U_1 =\,&O(1) \frac{m^3N^2\log (t+1)}{t^{3}} \times {\mathcal J} (t). \end{aligned}$$
(67)
Proof
From (66),
$$\begin{aligned} |U_1| \le {\mathcal J} (t) \times 2 \int \limits _{\sigma _1+\ldots +\sigma _N \ge \sqrt{2R T_* } \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2m} } G(\phi _1,\sigma _1) \cdots G(\phi _N,\sigma _N) d^N\sigma d^n\phi . \end{aligned}$$
Applying law of large numbers Theorem 7 with \(\gamma = \sqrt{2R T_* } \sqrt{\frac{ T_* }{ T^* }} \frac{t}{3}\), we conclude (67). \(\square \)
\(U_2\) records the difference in time variable of j, which is exactly the temporal fluctuation of j. Therefore, we can apply our previous estimate in temporal fluctuation to this part.
Lemma 6
Suppose that \(t/mn,t/mN,N \gg 1\). Then
$$\begin{aligned} U_2 =\,&O(1) \left( \frac{(mn)^{3}\log (t+1)}{t^{3}} +(1-\alpha )^m \right) \left( \left\| g_{in} \right\| _{\infty ,\mu } + {\mathcal J} (t) \right) \nonumber \\&+ O(1) \sup _{\frac{t}{2}<s<t} \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s) \times \left( \frac{\log N}{N} + \left( \frac{\log n}{n} \right) ^\frac{1}{2} mN \right) . \end{aligned}$$
(68)
Proof
$$\begin{aligned} U_2&= \int \limits _{ \frac{k_1\sigma _1}{\sqrt{2RT'_{(k_1)}}} + \ldots + \frac{k_N\sigma _N}{\sqrt{2RT'_{(k_1,\ldots ,k_N)}}} < \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} } \prod _{l=1}^N G \left( \textstyle \phi _l+\frac{\theta }{2k_lN}, \sigma _l \right) \\&\quad \,\times \Bigg [ j \left( \textstyle k_1\theta _1 + \ldots + k_N\theta _N, t - \frac{k_1\sigma _1}{\sqrt{2R\tilde{T}_{(k_1)}}} - \ldots - \frac{k_N\sigma _N}{\sqrt{2R\tilde{T}_{(k_1,\ldots ,k_N)}}} \right) \\&\quad \,-\, j \left( \textstyle k_1\theta _1 + \ldots + k_N\theta _N, t - \frac{k_1\sigma _1}{\sqrt{2RT'_{(k_1)}}} - \ldots - \frac{k_N\sigma _N}{\sqrt{2RT'_{(k_1,\ldots ,k_N)}}} \right) \Bigg ] d^N\sigma d^N\phi \end{aligned}$$
$$\begin{aligned} \quad \quad = \left( \mathop { \int _{ |\sigma _1+\ldots +\sigma _N - N{{\mathrm{E}}}[X_1]| > N } + \int _{ |\sigma _1+\ldots +\sigma _N - N{{\mathrm{E}}}[X_1]| < N } } _{ \frac{k_1\sigma _1}{\sqrt{2RT'_{(k_1)}}} + \ldots + \frac{k_N\sigma _N}{\sqrt{2RT'_{(k_1,\ldots ,k_N)}}} < \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} } \right) (\cdots ) \equiv U_{21} + U_{22}, \end{aligned}$$
For \(U_{21}\), since
$$\begin{aligned}&\left\{ \frac{k_1\sigma _1}{\sqrt{2RT'_{(k_1)}}} + \ldots + \frac{k_N\sigma _N}{\sqrt{2RT'_{(k_1,\ldots ,k_N)}}} < \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} \right\} \subset \\&\left\{ k_1\sigma _1+\ldots + k_N\sigma _N < \sqrt{2R T^* } \sqrt{\frac{ T_* }{ T^* }} \frac{t}{2} \right\} \subset \left\{ \frac{k_1\sigma _1}{\sqrt{2R\tilde{T}_{(k_1)}}} + \ldots + \frac{k_N\sigma _N}{\sqrt{2R\tilde{T}_{(k_1,\ldots ,k_N)}}} < \frac{t}{2} \right\} , \end{aligned}$$
we have
$$\begin{aligned} |U_{21}| \le&\sup _{ \frac{t}{2}<s<t } \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s)\\&\times 2 \int \limits _{ |\sigma _1+\ldots +\sigma _N - N{{\mathrm{E}}}[X_1]| > N} G(\phi _1,\sigma _1) \cdots G(\phi _N,\sigma _N) d^N\sigma d^N\phi . \end{aligned}$$
Applying Theorem 7 with \(\gamma = N\), we obtain
$$\begin{aligned} U_{21} = O(1)\sup \limits _{ \frac{t}{2}<s<t } \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s) \times \frac{\log N}{N}. \end{aligned}$$
(69)
Note that the prerequisite \(N/N^{\frac{1}{3}} \gg 1\) of Theorem 7 is satisfied since \(N \gg 1\). Next, for \(U_{22}\), by Corollary 2,
$$\begin{aligned} |U_{22}|&\le \sup \left\{ \left| j( \mathbf y ,s) - j( \mathbf y ,s') \right| : s,s' \in \left( t - mN\frac{1+{{\mathrm{E}}}(X_1)}{\sqrt{2R T_* }} ,t \right) , \ \mathbf y \in \partial D \right\} \nonumber \\&= O(1) \left( \frac{ (nm)^{3}\log (t+1) }{ t^{3} } +(1-\alpha )^m \right) \left( \left\| g_{in} \right\| _{\infty ,\mu } + {\mathcal J} (t) \right) \nonumber \\&\quad \,+\, O(1) \sup _{ \frac{t}{2}<s<t } \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s) \times \left( \frac{\log n}{n} \right) ^\frac{1}{2} mN. \end{aligned}$$
(70)
From (69) and (70) we conclude (68). \(\square \)
Finally, we investigate \(U_3\). \(U_3\) involves only the angular difference of those PDF G. No difference in boundary temperature are included. Therefore, the estimate of \(U_3\) is reduced to the constant boundary temperature case, as in [10]. Since all the \(\theta \) dependences appear only in the N copy of G, \(U_3\) is a \(C^1\) function of \(\theta \). Moreover, \(U_3|_{\theta =0}=0\). By direct computations,
$$\begin{aligned} \left| \frac{dU_3}{d\theta } \right|&\le \sup _{\frac{t}{2}<s<t} \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s)\nonumber \\&\quad \,\times \frac{1}{N} \int \frac{1}{2} \left| \sum _{l=1}^N G(\phi _1,\sigma _1) \cdots \frac{1}{k_l}\frac{\partial G}{\partial \phi }(\phi _l,\sigma _l) \cdots G(\phi _N,\sigma _N) \right| d^N\phi d^N\sigma . \end{aligned}$$
(71)
The RHS of (71) can be derived by the similar argument as in [10]. Thanks to \(k_l\ge 1\) for each l, the following lemma allows to obtain a decay of \(U_3\) in N. For a proof, see [10].
Lemma 7
$$\begin{aligned} \int \left| \sum _{l=1}^N G(\phi _1,\sigma _1) \cdots \frac{\partial G}{\partial \phi }(\phi _l,\sigma _l) \cdots G(\phi _N,\sigma _N) \right| d^N\phi =O \left( (N\log N)^\frac{1}{2} \right) , \end{aligned}$$
Therefore,
$$\begin{aligned} |U_3| \le \int _0^\pi \left| \frac{dU_3}{d\theta } \right| d\theta = O(1) \sup _{\frac{t}{2}<s<t} \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s) \times \left( \frac{\log N}{N} \right) ^\frac{1}{2}. \end{aligned}$$
(72)
Under the assumption \(t/mn,t/mN,N \gg 1\), patching (67), (68), and (72) together we have
$$\begin{aligned}&\Lambda _{(k_1,\ldots ,k_N)}^{(N,m)}( \mathbf y ,t) - \Lambda _{(k_1,\ldots ,k_N)}^{(N,m)}( \mathbf y ',t)\\&\quad \qquad = O(1) \left( \frac{ (mn)^{3}\log (t+1) }{ t^{3} } +(1-\alpha )^m \right) \left( \left\| g_{in} \right\| _{\infty ,\mu } + {\mathcal J} (t) \right) \\&\qquad \qquad \,+\, O(1) \sup _{\frac{t}{2}<s<t} \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s) \times \left( \left( \frac{\log N}{N} \right) ^\frac{1}{2} + \left( \frac{\log n}{n} \right) ^\frac{1}{2} mN \right) . \end{aligned}$$
Plugging this to (54), we conclude Theorem 10.
2.6 Convergence of Boundary Flux
In this subsection, we prove our main theorem, Theorem 6, for free molecular flow.
To apply a priori estimate, we need to establish the boundedness of j first: \( \left\| j \right\| _{L^\infty _ \mathbf y }=O(1) \left\| g_{in} \right\| _{\infty ,\mu }\), cf. Lemma 8. We apply a priori estimate twice: in the first time we obtain a rougher estimate, the boundedness of j, and in the second time we use the boundedness of j to obtain the convergence rate \((\alpha t+1)^{-d}+(1-\alpha )^{t^{\frac{1}{400}}}\) of j.
Now we recall \(j( \mathbf y ,t)=j_{in}( \mathbf y ,t)+j_{mid}( \mathbf y ,t)+j_{fl}( \mathbf y ,t)\), (49), (51), (52) and (53).
Proposition 1
For \(t>1\),
$$\begin{aligned} j_{in}( \mathbf y ,t)&= \frac{O(1)}{t^d} \left( \left\| g_{in} \right\| _{\infty ,\mu } + {\mathcal J} (t) \right) ,\end{aligned}$$
(73a)
$$\begin{aligned} j_{mid}( \mathbf y ,t) \!&=\! O(1) \left( \left( \alpha \sum \limits _{i=1}^{\lfloor K/2\rfloor }(1\!-\!\alpha )^{i-1}i^{d-1}\!+\! \sum \limits _{k=1}^K(1\!-\!\alpha )^kk^{d-1}\right) \frac{1}{t^d}\!+\!(1-\alpha )^{K/2}\frac{K^d}{t^d} \right) {\mathcal J} (t) \end{aligned}$$
(73b)
$$\begin{aligned}&\quad \,+\,O(1)\sum \limits _{k=1}^K(1\!-\!\alpha )^k\frac{k^{d-1}}{t^d} \left\| g_{in} \right\| _{\infty ,\mu } + O(1)\sup _{\frac{t}{2}<s<t} \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s)\frac{K^d}{t^d},\nonumber \\ j_{fl}( \mathbf y ,t)&= O(1)\sup _{ \begin{array}{c} t'\in \left( t-K^p\log (t+1), t \right) \\ \mathbf y , \mathbf y '\in \partial D \end{array} } \left| j( \mathbf y ,t) - j( \mathbf y ',t') \right| +O(1)(1-\alpha )^{K^p}\nonumber \\&\quad \left( {\mathcal J} (t)+ \left\| g_{in} \right\| _{\infty ,\mu } \right) . \end{aligned}$$
(73c)
Proof
For \(j_{in}\), since \(| {\varvec{\xi }}| < \frac{| \mathbf x - \mathbf x _{(1)}|}{t} \le \frac{{{\mathrm{diam}}}(D)}{t} = \frac{2}{t}\),
$$\begin{aligned}&|j_{in}( \mathbf y ,t)|\\&= \left| \int _{| {\varvec{\xi }}|<\frac{| \mathbf x \!-\! \mathbf x _{(1)}|}{t}} \left[ \alpha \sum \limits _{i=1}^{\infty }(1\!-\!\alpha )^{i-1}j( \mathbf y ,t) \left( \frac{2\pi }{RT( \mathbf x _{(i)})} \right) ^{\frac{1}{2}} M_{T( \mathbf x _{(i)})}( {\varvec{\xi }}) \!-\! \bar{g}_{in}( \mathbf x - {\varvec{\xi }}t, {\varvec{\xi }}) \right] d \mathbf x d {\varvec{\xi }} \right| \\&= O(1)\int _{ | {\varvec{\xi }}| < \frac{2}{t} } \left[ {\mathcal J} (t) M( {\varvec{\xi }}) + \left\| g_{in} \right\| _{\infty ,\mu } \left( \int \frac{1}{(1+| {\varvec{\zeta }}|)^\mu } d {\varvec{\eta }} \right) \right] d {\varvec{\xi }}\\&= \frac{O(1)}{t^d} \left( {\mathcal J} (t) + \left\| g_{in} \right\| _{\infty ,\mu } \right) . \end{aligned}$$
By (52),
$$\begin{aligned}&j_{mid}( \mathbf y ,t)\le \frac{1}{C_S|D|} \sum \limits _{k=1}^{K} \int _{A_k} \left\{ \alpha \sum \limits _{i=1}^{k}(1-\alpha )^{i-1} \left[ j( \mathbf y ,t)-j( \mathbf x _{(i)},t-t_1-...-t_i) \right] \right. \\&\left. \left( \frac{2\pi }{RT( \mathbf x _{(i)})} \right) ^{\frac{1}{2}}M_{T( \mathbf x _{(i)})}( {\varvec{\xi }}) +(1-\alpha )^k\Bigg (\alpha \sum \limits _{i=1}^{\infty }(1-\alpha )^{i-1}j( \mathbf y ,t) \left( \frac{2\pi }{RT( \mathbf x _{(k+i)})} \right) ^{\frac{1}{2}}\right. \\&\left. M_{T( \mathbf x _{(k+i)})}( {\varvec{\xi }}) \,-\, \bar{g}_{in}\left( \mathbf x _{(k)}- {\varvec{\xi }}^k(t-t_1-...-t_k), {\varvec{\xi }}^k\right) \Bigg )\right\} d \mathbf x d {\varvec{\xi }}\equiv I+II. \end{aligned}$$
Direct computations yield
$$\begin{aligned} II&=\sum \limits _{k=1}^K\int _{A_k}(1-\alpha )^k\left\{ \alpha \sum \limits _{i=1}^{\infty }(1-\alpha )^{i-1}j( \mathbf y ,t) \left( \frac{2\pi }{RT( \mathbf x _{(k+i)})} \right) ^{\frac{1}{2}} M_{T( \mathbf x _{(k+i)})}( {\varvec{\xi }})\right. \\&\left. \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - \bar{g}_{in}\left( \mathbf x _{(k)}- {\varvec{\xi }}^k(t-t_1-...-t_k), {\varvec{\xi }}^k\right) \right\} d \mathbf x d {\varvec{\xi }}\\&=O(1)\sum \limits _{k=1}^K(1-\alpha )^k\int _{A_k} \left( {\mathcal J} (t) + \left\| g_{in} \right\| _{\infty ,\mu } \right) d \mathbf x d {\varvec{\xi }}\\&= O(1) \left( {\mathcal J} (t)+ \left\| g_{in} \right\| _{\infty ,\mu } \right) {\left\{ \begin{array}{ll} \sum \limits _{k=1}^K(1-\alpha )^k\frac{1}{t} \quad \text { if } d=1,\\ \sum \limits _{k=1}^K(1-\alpha )^k\frac{k}{t^2}\quad \text { if } d=2.\\ \end{array}\right. } \end{aligned}$$
It is easy to show that
$$\begin{aligned} t-t_1-...-t_i\ge \frac{1}{2}t \iff k\ge 2i \quad \text { on } A_k, \end{aligned}$$
and therefore, we rewrite I as
$$\begin{aligned} I&=\sum \limits _{k=1}^K\int _{A_k} \alpha \sum \limits _{i=1}^{k}(1-\alpha )^{i-1} \Big (j( \mathbf y ,t)-j( \mathbf x _{(i)},t-t_1-..-t_i)\Big )\\&\quad \, \times \left( \frac{2\pi }{RT( \mathbf x _{(i)})} \right) ^{\frac{1}{2}} M_{T( \mathbf x _{(i)})}( {\varvec{\xi }}) d \mathbf x d {\varvec{\xi }}\\&= \alpha \sum \limits _{i=1}^{K}(1-\alpha )^{i-1}\sum \limits _{k=i}^{K}\int _{A_k}\Big (\ldots \Big )\\&= \alpha \left\{ \sum \limits _{i=1}^{\lfloor K/2\rfloor }(1-\alpha )^{i-1} \left( \sum \limits _{k=i}^{2i-1}+\sum \limits _{k=2i}^{K} \right) +\sum \limits _{i=\lfloor K/2\rfloor +1}^{K}(1-\alpha )^{i-1}\sum \limits _{k=i}^{K}\right\} \Big (\ldots \Big ). \end{aligned}$$
Direct computations yield
$$\begin{aligned}&\alpha \sum \limits _{i=\lfloor K/2\rfloor +1}^{K}(1-\alpha )^{i-1}\sum \limits _{k=i}^{K}\Big (\ldots \Big )\\&=O(1) {\mathcal J} (t)\sum \limits _{i=\lfloor K/2 \rfloor +1}^{K}(1-\alpha )^{i-1}\alpha \sum \limits _{k=i}^{K}\int _{A_k} \left( \frac{2\pi }{R T_* } \right) ^{\frac{1}{2}} M( {\varvec{\xi }}) d {\varvec{\xi }}d \mathbf x \\&=O(1) {\mathcal J} (t)(1-\alpha )^{K/2}\frac{K^d}{t^d}, \end{aligned}$$
$$\begin{aligned}&\sum \limits _{i=1}^{\lfloor K/2\rfloor }(1-\alpha )^{i-1}\alpha \sum \limits _{k=2i}^{K}\int _{A_k} \Big (j( \mathbf y ,t)-j( \mathbf x _{(i)},t-t_1-..-t_i)\Big ) \left( \frac{2\pi }{RT( \mathbf x _{(i)})} \right) ^{\frac{1}{2}} \\&\qquad M_{T( \mathbf x _{(i)})}( {\varvec{\xi }}) d {\varvec{\xi }}d \mathbf x \\&\quad = O(1)\sup _{\frac{t}{2}<s<t} \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s) \sum \limits _{i=1}^{\lfloor K/2\rfloor }(1-\alpha )^{i-1}\alpha \sum \limits _{k=2i}^{K}\int _{A_k} \left( \frac{2\pi }{R T_* } \right) ^{\frac{1}{2}} M( {\varvec{\xi }}) d {\varvec{\xi }}d \mathbf x \\&\quad = O(1)\sup _{\frac{t}{2}<s<t} \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s)\frac{K^d}{t^d}, \end{aligned}$$
$$\begin{aligned}&\sum \limits _{i=1}^{\lfloor K/2\rfloor }(1-\alpha )^{i-1}\alpha \sum \limits _{k=i}^{2i-1}\int _{A_k} \Big (j( \mathbf y ,t)-j( \mathbf x _{(i)},t-t_1-..-t_i)\Big ) \left( \frac{2\pi }{RT( \mathbf x _{(i)})} \right) ^{\frac{1}{2}}\\&\qquad M_{T( \mathbf x _{(i)})}( {\varvec{\xi }}) d {\varvec{\xi }}d \mathbf x \\&\quad =O(1) {\mathcal J} (t)\sum \limits _{i=1}^{\lfloor K/2\rfloor }(1-\alpha )^{i-1}\alpha \sum \limits _{k=i}^{2i-1}\int _{A_k} \left( \frac{2\pi }{R T_* } \right) ^{\frac{1}{2}} M( {\varvec{\xi }}) d {\varvec{\xi }}d \mathbf x \\&\quad =O(1) {\mathcal J} (t)\sum \limits _{i=1}^{\lfloor K/2\rfloor }(1-\alpha )^{i-1}\alpha {\left\{ \begin{array}{ll} \frac{i}{t} \quad \text { if } d=1,\\ \frac{i^2}{t^2} \quad \text { if } d=2. \end{array}\right. } \end{aligned}$$
For \(j_{fl}\),
$$\begin{aligned}&\int _{| {\varvec{\xi }}|>\frac{| \mathbf x _{(1)}- \mathbf x _{(2)}|}{\log (t+1)}} \left\{ \alpha \sum \limits _{i=K+1}^{\infty }(1-\alpha )^{i-1}j( \mathbf y ,t) \left( \frac{2\pi }{RT( \mathbf x _{(i)})} \right) ^{\frac{1}{2}}M_{T( \mathbf x _{(i)})}( {\varvec{\xi }})\right. \\&\left. \,-\,(1-\alpha )^K \bar{g}( \mathbf x _{(K)}- {\varvec{\xi }}^K(t-t_1-...-t_K), {\varvec{\xi }}^K)\right\} d \mathbf x d {\varvec{\xi }}\\&= O(1)(1-\alpha )^{K} \left( {\mathcal J} (t)+ \left\| g_{in} \right\| _{\infty ,\mu } \right) , \end{aligned}$$
$$\begin{aligned}&\int \limits _{| {\varvec{\xi }}|>\frac{| \mathbf x _{(1)}- \mathbf x _{(2)}|}{\log (t+1)}} \alpha \sum \limits _{i=1}^{K}(1-\alpha )^{i-1}\Big (j( \mathbf y ,t)-j( \mathbf x _{(i)},t-t_1-...-t_i)\Big ) \left( \frac{2\pi }{RT( \mathbf x _{(i)})} \right) ^{\frac{1}{2}}\\&\qquad M_{T( \mathbf x _{(i)})}( {\varvec{\xi }})d \mathbf x d {\varvec{\xi }}\\&\quad =\int \limits _{| {\varvec{\xi }}|>\frac{| \mathbf x _{(1)}- \mathbf x _{(2)}|}{\log (t+1)}} \alpha \left( \sum \limits _{i=1}^{\lfloor K^p\rfloor }+\sum \limits _{i=\lfloor K^p\rfloor }^{K}\right) \Big (\ldots \Big )\\&\quad =O(1)\sup _{ \begin{array}{c} t'\in \left( t-K^p\log (t+1), t \right) \\ \mathbf y , \mathbf y '\in \partial D \end{array} } \left| j( \mathbf y ,t) - j( \mathbf y ',t') \right| +O(1)(1-\alpha )^{K^p} \left( {\mathcal J} (t)+ \left\| g_{in} \right\| _{\infty ,\mu } \right) , \end{aligned}$$
where \(0<p<1\) is any fixed number. \(\square \)
With the estimates (73a), (73b), and (73c), the main task is to study the RHS of (73c), the fluctuation of j. We have treated this in Sects. 2.2, 2.4, and 2.5. The fluctuation estimate, the following theorem, follows directly from Corollary 2 on the temporal fluctuation estimate, and Theorem 10 on the spacial fluctuation estimate.
Theorem 11
(Fluctuation Estimate) Let \(t'<t\). \( \mathbf y , \mathbf y '\in \partial D\), then for sufficiently large \(t'/mn,\ t'/mN\) and N,
$$\begin{aligned}&j( \mathbf y ,t) - j( \mathbf y ',t')\nonumber \\&= O(1) \left( \sup _{\frac{t'}{2}<s<t} \left( \textstyle \left\| j \right\| _{L^{\infty }_{ \mathbf y }} \right) (s) \right) \times \left( \left( \frac{\log N}{N} \right) ^\frac{1}{2} + \left( \frac{\log n}{n} \right) ^\frac{1}{2} mN \right) (t-t')\nonumber \\&\quad \,+\, O(1) \left( \frac{ (m^4n^{3}+m^3N^2)\log (t'+1) }{ t'^{3} } +(1-\alpha )^m \right) \left( \left\| g_{in} \right\| _{\infty ,\mu } + {\mathcal J} (t) \right) ,\nonumber \\ \text { when } d=2, \end{aligned}$$
(74a)
$$\begin{aligned}&|j( \mathbf y ,t)-j( \mathbf y ',t')|=O(1) \left( \frac{m^3n^{2}\log (t'+1)}{t'^{2}}+(1-\alpha )^m \right) \Big ( {\mathcal J} (t)+ \left\| g_{in} \right\| _{\infty ,\mu }\Big )\nonumber \\&\qquad \,+\,O(1) \left( \left( \frac{(t-t')+t^q}{n^\frac{1}{2}} \right) + \left( \frac{m}{t^q} \right) ^2 \right) \sup _{ \frac{t'}{2} < s < t } \Big ( \left| j_+(s) \right| + \left| j_-(s) \right| \Big ) \nonumber \\&\text { when } d=1. \end{aligned}$$
(74b)
Here we will complete the proof Theorem 6 using Theorem 11.
Recall \(K\approx \frac{t}{\log (t+1)}\) as \(t\gg 1\). And hence for each \(0<p<1\), \(K^p\log (t+1)=o(t)\) as \(t\gg 1\). Plugging Theorem 11 into (73c), we obtain the following estimate of \(j_{fl}\):
$$\begin{aligned} j_{fl}( \mathbf y ,t)&= O(1) \left( \frac{ (m^4n^{3}+m^3N^2)\log (t+1) }{ t^{3} } +(1-\alpha )^m +(1-\alpha )^{ \left( \frac{t}{\log (t+1)} \right) ^p} \right) \nonumber \\&\quad \quad \qquad \qquad \,\times \left( \left\| g_{in} \right\| _{\infty ,\mu } + {\mathcal J} (t) \right) \nonumber \\&\quad \,+\, O(1) \left( \sup _{ t/2 < s < t } \left( \textstyle \left\| j \right\| _{L^{\infty }_{ \mathbf y }} \right) (s) \right) \left( \left( \frac{\log N}{N} \right) ^\frac{1}{2} + \left( \frac{\log n}{n} \right) ^\frac{1}{2} mN \right) \nonumber \\&\quad \left( \frac{t}{\log (t+1)} \right) ^p\log (t+1),\nonumber \\ \text { when } d=2, \end{aligned}$$
(75a)
$$\begin{aligned}&j_{fl}( \mathbf y ,t)=O(1) \left( \frac{m^3n^{2}\log (t+1)}{t^{2}}+(1-\alpha )^m+(1-\alpha )^{ \left( \frac{t}{\log (t+1)} \right) ^p} \right) \left( {\mathcal J} (t)+ \left\| g_{in} \right\| _{\infty ,\mu } \right) \nonumber \\&\quad \,+\,O(1) \left( \frac{ \left( \left( \frac{t}{\log (t+1)} \right) ^p\log (t+1)+t^q \right) }{n^\frac{1}{2}} \right) + \left( \frac{m}{t^q} \right) ^2 \sup _{ t/2 < s < t } \Big ( \left| j_+(s) \right| + \left| j_-(s) \right| \Big )\nonumber \\&\text { when } d=1. \end{aligned}$$
(75b)
We first establish the uniform boundedness of j:
Lemma 8
The boundary flux j is uniformly bounded:
$$\begin{aligned} {\mathcal J} (t) = O(1) \left\| g_{in} \right\| _{\infty ,\mu }, \ t \ge 0. \end{aligned}$$
(76)
Proof
From (73a), (73b), and (75), we have for \(t>1\),
$$\begin{aligned} j_{in}( \mathbf y ,t)&= \frac{O(1)}{t^d} \left( \left\| g_{in} \right\| _{\infty ,\mu } + {\mathcal J} (t) \right) ,\\ j_{mid}( \mathbf y ,t)&= O(1)\frac{1}{(\log (1+t))^d} \left( \left\| g_{in} \right\| _{\infty ,\mu }+ {\mathcal J} (t) \right) ,\\ j_{fl}( \mathbf y ,t)&= O(1) \Bigg (\frac{ m^{d+2}n^{d+1}\log (t+1) }{ t^{d+1} } \!+\!(1-\alpha )^m \!+\!(1-\alpha )^{ \left( \frac{t}{\log (t+1)} \right) ^p}\\&\quad \,+\, \left\{ \begin{array}{c@{,~}l} \frac{ m^3N^2\log (t+1) }{ t^{3} } &{} d=2,\\ 0 &{} d=1 \end{array} \right\} \Bigg ) \left( \left\| g_{in} \right\| _{\infty ,\mu }+ {\mathcal J} (t) \right) \\&\quad \,+\, \left\{ \begin{array}{c@{,~}l} \frac{ m^3N^2\log (t+1) }{ t^{3} }\!+\! \left( \left( \frac{\log N}{N} \right) ^\frac{1}{2} + \left( \frac{\log n}{n} \right) ^\frac{1}{2} mN \right) \left( \frac{t}{\log (t+1)} \right) ^p\log (t+1) &{} d\!=\!2,\\ \frac{ \left( \frac{t}{\log (t+1)} \right) ^p\log (t+1)+t^q}{n^{1/2}}+ \left( \frac{m}{t^q} \right) ^2 &{} d=1 \end{array} \right\} {\mathcal J} (t). \end{aligned}$$
Therefore,
$$\begin{aligned} j( \mathbf y ,t)&= j_{in}( \mathbf y ,t) + j_{mid}( \mathbf y ,t) + j_{fl}( \mathbf y ,t)\\&= O(1) \left[ \frac{1}{(\log (t+1))^d}+(1-\alpha )^{ \left( \frac{t}{\log (t+1)} \right) ^p} +(1-\alpha )^m + \frac{ m^{d+2}n^{d+1} \log (t+1) }{ t^{d+1} } \right. \\&\left. \quad \,+\, \left\{ \begin{array}{c@{,~}l} \frac{ m^{3}N^2 \log (t+1) }{ t^{3} } &{}\quad d=2,\\ 0 &{} \quad d=1 \end{array} \right\} \right] \left\| g_{in} \right\| _{\infty ,\mu }\\&\quad \,+\, O(1) \left[ \frac{1}{(\log (1+t))^d}+(1-\alpha )^{ \left( \frac{t}{\log (t+1)} \right) ^p} +(1-\alpha )^m + \frac{ m^{d+2}n^{d+1} \log (t+1) }{ t^{d+1} }\right. \\&\left. \quad \,+\, \left\{ \begin{array}{c@{,~}l} \frac{ m^{3}N^2 \log (t+1) }{ t^{d+1} }+ \left( \left( \frac{\log n}{n} \right) ^\frac{1}{2} mN + \left( \frac{\log N}{N} \right) ^\frac{1}{2} \right) \left( \frac{t}{\log (t+1)} \right) ^{p}\log (t+1) &{} d=2,\\ \left( \frac{ \left( \frac{t}{\log (t+1)} \right) ^p\log (t+1)+t^q}{n^\frac{1}{2}} \right) + \left( \frac{m}{t^q} \right) ^2 &{} d=1 \end{array} \right\} \right] {\mathcal J} (t). \end{aligned}$$
So far we only have to assume \(t/mn,\ t/Nm,\ N\gg 1\). Now we set \(n=n(t)=\lfloor t^{r_1}\rfloor \), \(m=m(t)=\lfloor t^{r_2}\rfloor \) and \(N=N(t)=\lfloor t^{r_3}\rfloor \) , where \(0<r_1,r_2,r_3<1\) are to be determined. In order to get
$$\begin{aligned}&\lim _{t\rightarrow \infty } \left( \frac{ 1 }{ (\log (1+ t))^d } +(1-\alpha )^{ \left( \frac{t}{\log (t+1)} \right) ^p} +(1-\alpha )^m + \frac{ m^{d+2}n^{d+1} \log (t+1) }{ t^{d+1} } \right) = 0,\\&\lim _{t\rightarrow \infty } \left\{ \begin{array}{c@{,~}l} \frac{ m^{3}N^2 \log (t+1) }{ t^{d+1} }+ \left( \left( \frac{\log n}{n} \right) ^\frac{1}{2} mN + \left( \frac{\log N}{N} \right) ^\frac{1}{2} \right) \left( \frac{t}{\log (1\!+\!t)} \right) ^p\log (1+t) &{} d\!=\!2,\\ \left( \frac{ \left( \frac{t}{\log (t+1)} \right) ^p\log (t+1)+t^q}{n^\frac{1}{2}} \right) + \left( \frac{m}{t^q} \right) ^2 &{} d\!=\!1 \end{array} \right\} =0, \end{aligned}$$
we need
$$\begin{aligned}&r_2+r_3+p<\frac{1}{2}r_1,\\&p<\frac{1}{2}r_3,\\&r_1+r_2<1,\\&r_2+r_3<1,\\&r_2<q,\\&p,q<\frac{1}{2}r_1. \end{aligned}$$
This can be done by choosing any \(0<r_1<6/7\) and setting \(q=r_1/3, r_2=r_1/6, r_3=r_1/12, p=r_1/36\). Therefore, there exists \(t_*>0\) such that, for all \(t>t_*,\)
$$\begin{aligned} \left\{ \begin{array}{c@{,~}l} \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (t) &{} d=2 \\ \Big ( \left| j_+(t) \right| + \left| j_-(t) \right| \Big ) &{} d=1 \end{array} \right\} \le O(1) \left\| g_{in} \right\| _{\infty ,\mu } + \frac{1}{2} {\mathcal J} (t). \end{aligned}$$
(77)
Moreover, from (31), now that \(t_*\) is fixed, j is bounded by a constant multiple of \( \left\| g_{in} \right\| _{\infty ,\mu }\) for all \(0\le t\le t_*\). Hence, (77) actually holds for all t, and this implies that \(\frac{1}{2} {\mathcal J} (t) = O(1) \left\| g_{in} \right\| _{\infty ,\mu }\) and the lemma is proved. \(\square \)
With the boundedness of j, (76), we can perform the second a priori estimate to obtain the \((\alpha t+1)^{-d}+(1-\alpha )^{t^{\frac{1}{400}}}\) decay of j, Theorem 6. First, since \( {\mathcal J} = O(1)\Vert g_{in}\Vert _{\infty ,\mu }\) and the following Lemma 9, we can rewrite our previous estimates (73a), (73b), and (75) as the following in Proposition 2. We will need the following identities, whose simple proof is omitted.
Lemma 9
For \(0<x<1\),
$$\begin{aligned}&\sum \limits _{k=1}^\infty kx^k = \frac{x}{(1-x)^2},\\&\sum \limits _{k=1}^\infty k^2x^k = \frac{x(x+1)}{(1-x)^3}.\\ \end{aligned}$$
Proposition 2
For \(t>1\),
$$\begin{aligned} j_{in}( \mathbf y ,t)&= \frac{O(1)}{ t^d} \left\| g_{in} \right\| _{\infty ,\mu },\end{aligned}$$
(78a)
$$\begin{aligned} j_{mid}( \mathbf y ,t)&= O(1) \left( \frac{1}{(\alpha t)^d} +(1-\alpha )^{ \left( \frac{t}{\log (t+1)} \right) }\frac{1}{(\log (1+ t))^d} \right) \left\| g_{in} \right\| _{\infty ,\mu }\nonumber \\&\quad \,+\, O(1)\frac{1}{(\log (1+t))^d} \sup _{\frac{t}{2}<s<t} \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s), \end{aligned}$$
(78b)
and
$$\begin{aligned} j_{fl}( \mathbf y ,t)&=O(1) \left( \frac{ (m^4n^{3}+m^3N^2)\log (t+1) }{ t^{3} } +(1-\alpha )^m +(1-\alpha )^{ \left( \frac{t}{\log (t+1)} \right) ^p} \right) \left\| g_{in} \right\| _{\infty ,\mu }\nonumber \\&\quad \,+\, O(1) \left( \sup _{ t/2 < s < t } \left( \textstyle \left\| j \right\| _{L^{\infty }_{ \mathbf y }} \right) (s) \right) \left( \left( \frac{\log N}{N} \right) ^\frac{1}{2} + \left( \frac{\log n}{n} \right) ^\frac{1}{2} mN \right) \nonumber \\&\quad \left( \frac{t}{\log (1+t)} \right) ^p\log (1+t),\nonumber \\&\text { when } d=2, \end{aligned}$$
(78c)
$$\begin{aligned} j_{fl}( \mathbf y ,t)&=O(1) \left( \frac{m^3n^{2}\log (t+1)}{t^{2}}+(1-\alpha )^m+(1-\alpha )^{ \left( \frac{t}{\log (t+1)} \right) ^p} \right) \left\| g_{in} \right\| _{\infty ,\mu } \nonumber \\&\quad \,+\,O(1) \left( \frac{ \left( \left( \frac{t}{\log (t+1)} \right) ^p\log (1+t)+t^q \right) }{n^\frac{1}{2}} + \left( \frac{m}{t^q} \right) ^2 \right) \sup _{ t/2 < s < t }\nonumber \\&\quad \Big ( \left| j_+(s) \right| + \left| j_-(s) \right| \Big )\nonumber \\&\text { when } d=1.&\quad \,\,\, (\mathrm{78c}) \end{aligned}$$
Now we are ready to prove Theorem 6.
Definition 2
The a priori norm of j is a function of t defined as
$$\begin{aligned} {\mathcal N} (t) \equiv \sup _{ 0 \le s \le t } {\left\{ \begin{array}{ll} \left( (\alpha s)^{-2}+(1-\alpha )^{s^{\frac{1}{400}}} \right) ^{-1} \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (s) &{} \text { for } d=2, \\ \left( (\alpha s)^{-1}+(1-\alpha )^{s^{\frac{1}{400}}} \right) ^{-1} \Big ( \left| j_+(s) \right| + \left| j_-(s) \right| \Big ) &{} \text { for } d=1, \end{array}\right. } \end{aligned}$$
(79)
where \(j_\pm (s)=j(\pm 1,s)\).
Proof of of Theorem 6
From (78) and for \(t>1\),
$$\begin{aligned} j_{mid}( \mathbf y ,t)&= O(1) \left( \frac{1}{(\alpha t)^d} +(1-\alpha )^{ \left( \frac{t}{\log (t+1)} \right) }\frac{1}{(\log (1+ t))^d} \right) \left\| g_{in} \right\| _{\infty ,\mu }\\&\quad \,+\, O(1)\frac{1}{(\log (1+t))^d} \left( (\alpha t)^{-d}+(1-\alpha )^{t^{\frac{1}{400}}} \right) {\mathcal N} (t), \end{aligned}$$
and
$$\begin{aligned} j_{fl}( \mathbf y ,t)&= O(1) \left[ \frac{ m^{d+2}n^{d+1}\log (t+1) }{ t^{d+1} } +(1-\alpha )^m +(1-\alpha )^{K^p}\right. \\&\left. \quad + \left\{ \begin{array}{c@{,~}l} \frac{ m^{3}N^2\log (t+1) }{ t^{3} } &{} d=2\\ 0 &{} d=1 \end{array} \right\} \right] \left\| g_{in} \right\| _{\infty ,\mu }\\&\quad \,+\, O(1) \left( (\alpha t)^{-d}+(1-\alpha )^{t^{\frac{1}{400}}} \right) {\mathcal N} (t)\\&\quad \left\{ \begin{array}{c@{,~}l} \left( \left( \frac{\log N}{N} \right) ^\frac{1}{2} + \left( \frac{\log n}{n} \right) ^\frac{1}{2} mN \right) \left( \frac{t}{\log (1+t)} \right) ^p\log (1+t), &{} d=2\\ \left( \frac{ \left( \frac{t}{\log (1+t)} \right) ^p\log (1+t)+t^q}{n^\frac{1}{2}} \right) + \left( \frac{m}{t^q} \right) ^2 &{} d=1 \end{array} \right\} . \end{aligned}$$
Therefore,
$$\begin{aligned} j( \mathbf y ,t)&= O(1) \left[ \frac{1}{(\alpha t)^d} +(1-\alpha )^{ \left( \frac{t}{\log (t+1)} \right) }\frac{1}{(\log (1+t))^{d}}+\frac{ m^{d+2}n^{d+1}\log (t+1) }{ t^{d+1} } +(1-\alpha )^m \right. \\&\left. \quad \,+\,(1-\alpha )^{ \left( \frac{t}{\log (t+1)} \right) ^p} + \left\{ \begin{array}{c@{,~}l} \frac{ m^{3}N^2\log (t+1) }{ t^{3} } &{} d=2\\ 0 &{} d=1 \end{array} \right\} \right] \left\| g_{in} \right\| _{\infty ,\mu }\\&\quad \,+\, O(1) \left( (\alpha t)^{-d}+(1-\alpha )^{t^{\frac{1}{400}}} \right) {\mathcal N} (t)\\&\quad \, \times \left[ \frac{1}{(\log (1+t))^d} + \left\{ \begin{array}{c@{,~}l} \left( \left( \frac{\log N}{N} \right) ^\frac{1}{2} + \left( \frac{\log n}{n} \right) ^\frac{1}{2} mN \right) \left( \frac{t}{\log (1+t)} \right) ^p\log (1+t) &{} d=2\\ \left( \frac{ \left( \frac{t}{\log (1+t)} \right) ^p\log (1+t)+t^q}{n^\frac{1}{2}} \right) + \left( \frac{m}{t^q } \right) ^2 &{} d=1 \end{array} \right\} \right] . \end{aligned}$$
Now we choose \(0<r_1,r_2,r_3\ll 1\) and set \(n=n(t)=\lfloor t^{r_1}\rfloor \), \(m=m(t)=\lfloor t^{r_2}\rfloor \), \(N=N(t)=\lfloor t^{r_3}\rfloor \) so that
$$\begin{aligned}&\left[ (1-\alpha )^{ \left( \frac{t}{\log (t+1)} \right) }\frac{1}{(\log (1+t))^{d}}+\frac{ m^{d+2}n^{d+1}\log (t+1) }{ t^{d+1} } +(1-\alpha )^m +(1-\alpha )^{ \left( \frac{t}{\log (t+1)} \right) ^p}\right. \\&\left. \,+\, \left\{ \begin{array}{c@{,~}l} \frac{ m^{3}N^2\log (t+1) }{ t^{3} } &{} d=2\\ \frac{m^2}{ t^2} &{} d=1 \end{array} \right\} \right] = O\Big ((\alpha t)^{-d}+(1-\alpha )^{t^{\frac{p}{2}}}\Big ),\\&\lim _{t\rightarrow \infty } \frac{1}{(\log (1+t))^d} + \left\{ \begin{array}{c@{,~}l} \left( \left( \frac{\log N}{N} \right) ^\frac{1}{2} + \left( \frac{\log n}{n} \right) ^\frac{1}{2} mN \right) \left( \frac{t}{\log (1+t)} \right) ^p\log (1+t) &{} d\!=\!2\\ \left( \frac{ \left( \frac{t}{\log (1+t)} \right) ^p\log (1+t)+t^q}{n^\frac{1}{2}} \right) + \left( \frac{m}{t^q } \right) ^2 &{} d\!=\!1 \end{array} \right\} \!=\!0. \end{aligned}$$
This can be done if \(0<r_1,r_2,r_3,p\), and q satisfy
$$\begin{aligned}&r_2+r_3+p<\frac{1}{2}r_1,\\&p<\frac{1}{2}r_3,\\&\frac{p}{2}<r_2,\\&r_1+r_2<\frac{1}{d+2},\\&r_2+r_3<\frac{1}{d+1},\\&r_2<q,\\&p,q<\frac{1}{2}r_1. \end{aligned}$$
In fact, we may choose \(p=\frac{1}{200},r_3=\frac{1}{90},r_2=\frac{1}{30},q=\frac{1}{25},r_1=\frac{1}{10}\) so that the above inequalities hold. Consequently, we can find some sufficiently large \(t_*>0\) such that for all \(t>t_*\)
$$\begin{aligned} \left\{ \begin{array}{c@{,~}l} \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (t) &{} d=2 \\ \Big ( \left| j_+(t) \right| + \left| j_-(t) \right| \Big ) &{} d=1 \end{array} \right\} \le \Big ((\alpha t)^{-d}+(1-\alpha )^{t^{\frac{1}{400}}}\Big ) \left( O(1) \left\| g_{in} \right\| _{\infty ,\mu } +\frac{1}{2} {\mathcal N} (t) \right) . \end{aligned}$$
(80)
Therefore,
$$\begin{aligned} \left\{ \begin{array}{c@{,~}l} \Big ((\alpha t)^{-2}+(1-\alpha )^{t^{\frac{1}{400}}}\Big )^{-1} \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (t) &{} d=2 \\ \Big ((\alpha t)^{-1}+(1-\alpha )^{t^{\frac{1}{400}}}\Big )^{-1}\Big ( \left| j_+(t) \right| + \left| j_-(t) \right| \Big ) &{} d=1 \end{array} \right\} \le O(1) \left\| g_{in} \right\| _{\infty ,\mu } + \frac{1}{2} {\mathcal N} (t). \end{aligned}$$
(81)
Moreover, by Lemma 8, for all \(t<t_*\)
$$\begin{aligned} \left\{ \begin{array}{c@{,~}l} \Big ((\alpha t)^{-2}+(1-\alpha )^{t^{\frac{1}{400}}}\Big )^{-1} \left( \left\| j \right\| _{L^\infty _ \mathbf y } \right) (t) &{} d=2 \\ \Big ((\alpha t)^{-1}+(1-\alpha )^{t^{\frac{1}{400}}}\Big )^{-1}\Big ( \left| j_+(t) \right| + \left| j_-(t) \right| \Big ) &{} d=1 \end{array} \right\} =O(1) \left\| g_{in} \right\| _{\infty ,\mu }. \end{aligned}$$
(82)
Hence, (81) actually holds for all t, and this implies \( {\mathcal N} (t)= O(1) \left\| g_{in} \right\| _{\infty ,\mu }\). From this estimate, the definition of \( {\mathcal N} (t)\), (79), and the boundedness of \(j( \mathbf y ,t)\), (76), it is easy to see that
$$\begin{aligned} j( \mathbf y ,t)=O(1) \left\| g_{in} \right\| _{\infty ,\mu } \left( \frac{1}{(1+\alpha t)^d}+(1-\alpha )^{t^{\frac{1}{400}}} \right) , \end{aligned}$$
and the theorem is proved. \(\square \)
