本文研究了下列Landau方程的初边值问题

(1.1) $ \begin{eqnarray} \partial_t F+v\cdot \nabla_x F = Q(F, F), \end{eqnarray} $

给定初值条件

(1.2) $ \begin{equation} F(0, x, v) = F_0(x, v) \end{equation} $

及inflow边界条件

(1.3) $ \begin{equation} F(t, -1, \bar{x}, v)|_{v_1>0} = G_-(t, \bar{x}, v), \ \ F(t, 1, \bar{x}, v)|_{v_1<0} = G_+(t, \bar{x}, v), \end{equation} $

其中$ F = F(t, x, v)\geq 0 $为粒子密度分布函数,并且有空间变量$ x = (x_1, x_2, x_3)\in \Omega\subset {\Bbb R}^3 $和速度变量$ v = (v_1, \bar{v}) = (v _1, v_2, v_3)\in {\Bbb R}^{3} $,以及时间$ t\geq0 $,有界域$ \Omega = I\times {\Bbb T}^{2} $, $ x_{1}\in I = (-1, 1) $和$ \bar{x} = (x_2, x_3)\in {\Bbb T}^{2} $, $ {\Bbb T}^{2} $为周期域. Landau碰撞算子$ Q(\cdot, \cdot) $为

$ \begin{eqnarray*} Q(G, F)(v) = \nabla_v\cdot\left\{\int_{ {\Bbb R}^{3}}\psi(v-u)[G(u) \nabla_vF(v)-F(v) \nabla_uG(u)]{\rm d}u\right\}, \end{eqnarray*} $

Landau碰撞核$ \psi $是非负对称矩阵值函数,当$ 0\neq z = (z_1, z_2, z_3)\in {\Bbb R}^{3} $时,

$ \begin{eqnarray*} \psi^{jm}(z) = \left\{ \delta_{jm}-\frac{z_jz_m}{|z|^2}\right\}|z|^{ \gamma+2}, \quad j, m = 1, 2, 3, \end{eqnarray*} $

其中$ \delta_{jm} $是Kronecker符号, $ \gamma $是由粒子间相互作用势决定的参数^[1].当$ 0< \gamma\leq 1 $时称为硬位势;当$ \gamma = 0 $时称为Maxwell分子;当$ -2\leq \gamma<0 $时称为中等软位势;当$ -3 \le \gamma < -2 $时称为软位势;当$ \gamma = -3 $时对应于经典库伦势.考虑以下全局Maxwell平衡态

$ \begin{eqnarray*} \mu = \mu(v) = (2\pi)^{-\frac{3}{2}}e^{-\frac{|v|^{2}}{2}}. \end{eqnarray*} $

考虑方程(1.1)下述形式的解

$ \begin{eqnarray*} F(t, x, v) = \mu+{\mu}^{\frac{1}{2}}f(t, x, v). \end{eqnarray*} $

因此,关于扰动$ f = f(t, x, v) $满足的方程为

(1.4) $ \begin{equation} \partial_tf+v\cdot \nabla_xf+Lf = \Gamma(f, f), \end{equation} $

初始值满足

(1.5) $ \begin{equation} f(0, x, v) = f_0(x, v) = \mu^{-\frac{1}{2}}[F_0(x, v)-\mu], \end{equation} $

相应的,关于扰动$ f = f(t, x, v) $满足的inflow边界条件为

(1.6) $ \begin{equation} f(t, -1, \bar{x}, v)|_{v_1>0} = g_-(t, \bar{x}, v), \ \ f(t, 1, \bar{x}, v)|_{v_1<0} = g_+(t, \bar{x}, v). \end{equation} $

方程(1.4)中有$ Lf = -\mu^{-\frac{1}{2}}\left\{Q(\mu, \mu^{\frac{1}{2}}f)+Q(\mu^{\frac{1}{2}}f, \mu)\right\} $和$ \Gamma(f, f) = \mu^{-\frac{1}{2}}Q(\mu^{\frac{1}{2}}f, \mu^{\frac{1}{2}}f) $.定义$ {\mathcal{N}}(L) = \left\{f|Lf = 0\right\} $,对于任意的$ f\in {\mathcal{N}}(L) $,当且仅当

(1.7) $ \begin{equation} f = {\bf P}_{0}f = \left\{a+b\cdot v+\frac{1}{2}(|v|^2-3)c\right\}{\mu}^{\frac{1}{2}}, \end{equation} $

其中$ [a, b, c] $是一个向量. $ {\mathcal{N}}^\perp(L) $表示$ {\mathcal{N}}(L) $的正交补空间.

在稀薄气体中粒子的运动可用Landau方程来描述, Guo^[2]对Landau方程在周期域上的Cauchy问题,给出了在Maxwell附近构造全局经典解$ f(t, x, v)\in L^{\infty}(0, \infty;H_{x, v}^{8}) $的第一个结果.最近Carrapatoso等^[3]将Guo^[2]的工作推广到更大的函数空间中.另外Caflisch^[4], Strain等^[5-7],以及Sohinger等^[8]对于Landau方程在软位势条件下解的大时间行为进行了研究.

目前Landau方程在$ L^{\infty}_{x, v} $上关于解的全局存在性的理论不多,主要的原因是因为Landau碰撞算子具有微观耗散结构,因此用特征方法来求解在变量$ x $上的$ L^{\infty} $的界非常困难.我们注意到Kim等^[9]发展了从$ L^2 $到$ L^{\infty} $的求解周期区域的Landau方程,其中初始值要求在$ L^{\infty}_{x, v} $上足够的小,并且属于$ H^{1}_{x, v} $空间.最近Guo等^[10]将此方法应用到镜面反射边界条件的Landau方程研究中. Duan等^[11]发展了一类新的函数空间来处理有限管道的Landau方程和非截断的Boltzman方程的inflow边界和镜面反射边界问题.受Duan等^[11]的启发,本文中我们引入了一个新的函数空间,它的主要特征是在空间变量法向方向具有较低的正则性,我们用能量估计的方法,证明Landau方程在这样一个新的具有一定正则性的函数空间中存在全局唯一解,并且该文还研究了解的大时间性态和正则性传播的问题.

在本文中$ C $ (一般大)和$ \lambda $ (一般小)表示与主要参数无关的正的常数,并且$ C $和$ \lambda $在不同的地方可以取不同的值. $ D\lesssim E $表示存在常数$ C>0 $使得$ D\leq CE $.

定义指标$ \alpha = ( \alpha_{1}, \bar{ \alpha}) $,其中$ \bar{ \alpha} = ( \alpha_{2}, \alpha_{3}) $,且$ \partial^{ \alpha} = \partial^{ \alpha_{1}}_{x_1} \partial^{\bar{ \alpha}}_{\bar{x}} = \partial^{ \alpha_{1}}_{x_1} \partial^{ \alpha_{2}}_{x_2} \partial^{ \alpha_{3}}_{x_3} $.

定义指标集$ {\Lambda} = \left\{ \alpha\big| \alpha_{1}\leq 1, |\bar{ \alpha}|\leq 2\right\} $.对于任意正整数$ m $,定义另一指标集$ {\Lambda}_{m} = \left\{ \alpha\big| \alpha_{1}\leq 1, |\bar{ \alpha}|\leq m\right\} $.

在本文中,若无特殊说明,通常符号$ (\cdot, \cdot) $表示在$ L^{2}_{x, v} $或$ L^{2}_{x} $上的内积, $ \langle\cdot, \cdot\rangle $表示在$ L^{2}_{\bar{x}, v} $上的内积, $ \|\cdot\| $表示在$ L^{2}_{x, v} $或$ L^{2}_{x} $上的范数.

对于任意$ 0<T\leq\infty $,定义函数空间

$ L^\infty_TH^{1}_{x_{1}}H^2_{\bar{x}}L^2_v = \left\{f\Big|\|f\|_{L^\infty_TH^{1}_{x_{1}}H^2_{\bar{x}}L^2_v} = \sum\limits_{ \alpha\in {\Lambda}}\sup\limits_{0\leq t\leq T}\| \partial^{ \alpha}f\|_{L^2_xL^{2}_{v}}<+\infty\right\}. $

定义速度权函数

$ w = w_{q, \theta}(v) = e^{\frac{q\langle v \rangle^{ \theta}}{4}}, \langle v \rangle = \sqrt{1+|v|^2}, $

并且参数$ (q, \theta) $满足

$ \begin{eqnarray*} \mbox{(H)}{\quad}\left\{\begin{array}{rl} &\mbox{如果$-2\leq \gamma\leq 1$有$q = 0$;}\\ &\mbox{如果$-3\leq \gamma<-2$有$q> 0$和$0< \theta\leq 2$,并且当$ \theta = 2$有$0< q<1$.} \end{array}\right. \end{eqnarray*} $

定义参数

(1.8) $ \begin{equation} \kappa = \left\{ \begin{array}{ll} 1, &\mbox{当$q = 0$, $-2\leq \gamma\leq 1$};\\ { } \frac{ \theta}{ \theta+| \gamma+2|}, &\mbox{当$q>0$, $-3\leq \gamma<-2$}. \end{array}\right. \end{equation} $

记

$ \sigma_{jm} = \sigma_{jm}(v) = \int_{ {\Bbb R}^3}\psi^{jm}(v-u)\mu(u){\rm d}u. $

定义速度加权$ D $ -范数

$ \left|w_{q, \theta}f\right|_{D}^{2} = \sum\limits_{j, m = 1}^3\int_{ {\Bbb R}^{3}}w^{2}_{q, \theta}\left\{ \sigma_{jm} \partial_{v_j}f \partial_{v_m}f +\frac{1}{4} \sigma_{jm}v_{j}v_{m}f^2\right\}{\rm d}v $

和

$ \|w_{q, \theta}f\|_{D}^{2} = \sum\limits_{j, m = 1}^3\int_{I\times {\Bbb T}^2}\int_{ {\Bbb R}^{3}}w^{2}_{q, \theta}\left\{ \sigma_{jm} \partial_{v_j}f \partial_{v_m}f +\frac{1}{4} \sigma_{jm}v_{j}v_{m}f^2\right\}{\rm d}v{\rm d}x. $

定义加权总能量泛函和能量耗散泛函分别为

$ {\cal E}_{T, w}(f) = \sum\limits_{ \alpha\in {\Lambda}}\|w_{q, \theta} \partial^{ \alpha}f\|_{L^{\infty}_{T}L^2_xL^{2}_{v}} = \sum\limits_{ \alpha\in {\Lambda}}\sup\limits_{0\leq t\leq T}\|w_{q, \theta} \partial^{ \alpha}f\|_{L^2_xL^{2}_{v}} $

和

$ \begin{eqnarray*} {\cal D}_{T, w}(f) & = &\sum\limits_{ \alpha\in {\Lambda}}\| \partial^{ \alpha}[a, b, c]\|_{L^2_{T}L^{2}_{x}}+ \sum\limits_{ \alpha\in {\Lambda}}\|w_{q, \theta}\left\{{\bf I}-{\bf P_{0}}\right\} \partial^{ \alpha}f\|_{L^2_TL^2_xL^{2}_{v, D}}\\& = &\sum\limits_{ \alpha\in {\Lambda}}\left(\int^{T}_{0}\| \partial^{ \alpha}[a, b, c]\|^{2}_{L^{2}_{x}}{\rm d}t\right)^{\frac{1}{2}}+ \sum\limits_{ \alpha\in {\Lambda}}\left(\int^{T}_{0}\|w_{q, \theta}\left\{{\bf I}-{\bf P_{0}}\right\} \partial^{ \alpha}f\|^{2}_{L^2_xL^{2}_{v, D}}{\rm d}t\right)^{\frac{1}{2}}, \end{eqnarray*} $

并且当$ q = 0 $时,简记为$ {\cal E}_{T} $和$ {\cal D}_{T} $.定义了以下泛函来控制给定函数$ g_{\pm} $的边界效应.

$ \begin{eqnarray*} E^{2}( \partial^{ \alpha}g_{\pm}) & = &\int_0^T\int_{ {\Bbb T}^{2}}\int_{\pm v_1<0}|v_1|| \partial^{ \alpha}g_\pm|^2{\rm d}v{\rm d}\bar{x}{\rm d}t \\&& +\int_0^T\int_{ {\Bbb T}^{2}}\int_{\pm v_1<0}|v_1|^{-1}|\bar{v}\cdot \nabla_{\bar{x}} \partial^{ \alpha}g_{\pm}|^2{\rm d}v{\rm d}\bar{x}{\rm d}t \\&& +\int_0^T\int_{ {\Bbb T}^{2}}\int_{\pm v_1<0}|v_1|^{-1}| \partial_t \partial^{ \alpha}g_\pm|^2{\rm d}v{\rm d}\bar{x}{\rm d}t \\&& +\int_0^T\int_{ {\Bbb T}^{2}}\int_{\pm v_1<0}|v_1|^{-1}|L{ \partial^{ \alpha}g_\pm}|^2{\rm d}v{\rm d}\bar{x}{\rm d}t \\& & +\int_0^T\int_{ {\Bbb T}^{2}}\int_{\pm v_1<0}|v_1|^{-1}| \partial^{ \alpha} \Gamma(g_\pm, g_\pm)|^2{\rm d}v{\rm d}\bar{x}{\rm d}t. \end{eqnarray*} $

本文的主要结论如下.

定理1.1 设$ \Omega = I\times {\Bbb T}^2 $, $ \alpha\in {\Lambda} $, $ (q, \theta) $满足(H),对于$ \epsilon_0>0 $和$ C>0 $,如果$ F_0(x_1, \bar{x}, v) = \mu+\mu^{\frac{1}{2}}f_0(x_1, \bar{x}, v)\geq0 $, $ F(t, \pm1, \bar{x}, v) = \mu+\mu^{\frac{1}{2}}g_\pm(t, \bar{x}, v)\geq0 $,并且有$ \sum\limits_{ \alpha\in {\Lambda}}\|w_{q, \theta} \partial^{ \alpha}f_{0}\|_{L^{2}_{x, v}}+\sum\limits_{ \alpha\in {\Lambda}}E(w_{q, \theta} \partial^{ \alpha}g_\pm)\leq \epsilon_0 $成立,对任意$ T>0 $,方程(1.1), (1.2)和(1.3)存在一个整体解$ F(t, x_1, \bar{x}, v) = \mu+\mu^{\frac{1}{2}}f(t, x_1, \bar{x}, v)\geq0 $和

(1.9) $ \begin{equation} {\cal E}_{T, w}(f)+{\cal D}_{T, w}(f)\leq C \left\{\sum\limits_{ \alpha\in {\Lambda}}\|w_{q, \theta} \partial^{ \alpha}f_0\|_{L^2_{x, v}} +\sum\limits_{ \alpha\in {\Lambda}}E(w_{q, \theta} \partial^{ \alpha}g_\pm)\right\} \end{equation} $

成立.假设$ \kappa $满足(1.8)式,则存在$ \lambda>0 $,如果有$ \sum\limits_{ \alpha\in {\Lambda}}E(w_{q, \theta} \partial^{ \alpha}g_\pm)+\sum\limits_{ \alpha\in {\Lambda}}\sup\limits_{s>0} E_(e^{ \lambda s^ \kappa} \partial^{ \alpha}g_{\pm})\leq \epsilon_0 $成立,其中$ \epsilon_0>0 $足够小,对任意的$ t\geq0 $,上面构建的解就会满足

(1.10) $ \begin{eqnarray} \sum\limits_{ \alpha\in {\Lambda}}\| \partial^{ \alpha}f(t)\|_{L^2_{x, v}} &\lesssim &e^{- \lambda t^{ \kappa}}\sum\limits_{ \alpha\in {\Lambda}}\|w_{q, \theta} \partial^{ \alpha}f_0\|_{L^2_{x, v}} {}\\ &&+e^{- \lambda t^{ \kappa}}\left\{\sum\limits_{ \alpha\in {\Lambda}}E(w_{q, \theta} \partial^{ \alpha}g_\pm) +\sum\limits_{ \alpha\in {\Lambda}}\sup\limits_{s> 0} E(e^{ \lambda s^ \kappa} \partial^{ \alpha}g_{\pm})\right\}. \end{eqnarray} $

定理1.2(沿空间变量$ \bar{x} $方向的正则性传播) 假设$ \alpha\in {\Lambda}_{m} $,在定理1.1的条件下,对于$ \epsilon_0>0 $和$ C>0 $,如果有$ \sum\limits_{ \alpha\in {\Lambda}_{m}}\|w_{q, \theta} \partial^{ \alpha}f_{0}\|_{L^{2}_{x, v}}+ \sum\limits_{ \alpha\in {\Lambda}_{m}}E(w_{q, \theta} \partial^{ \alpha}g_\pm)\leq \epsilon_0 $成立,我们就有

$ \begin{eqnarray*} {\nonumber} &&\sum\limits_{ \alpha\in {\Lambda}_{m}}\|w_{q, \theta} \partial^{ \alpha}f\|_{L^\infty_TL^2_{x, v}}+\sum\limits_{ \alpha\in {\Lambda}_{m}}\|w_{q, \theta} \partial^{ \alpha}f\|_{L^{2}_{T}L^{2}_{x}L^2_{v, D}} \\ &\leq& C\left\{\sum\limits_{ \alpha\in {\Lambda}_{m}}\|w_{q, \theta} \partial^{ \alpha}f_{0}\|_{L^{2}_{x, v}} +\sum\limits_{ \alpha\in {\Lambda}_{m}}E(w_{q, \theta} \partial^{ \alpha}g_\pm)\right\} \end{eqnarray*} $

成立.

本节主要讨论了非线性项$ \Gamma(f, f) $的估计,首先提出一个最基本的估计可参见文献[5, p327,引理10].

引理2.1 设$ (q, \theta) $满足(H),有

$ \left|\left\langle \Gamma(f, g), w^{2}_{q, \theta}h\right\rangle_{L^{2}_{v}}\right| \lesssim\left(\left|w_{q, \theta}f\right|_{2}\left|w_{q, \theta}g\right|_{D} +\left|w_{q, \theta}f\right|_{D}\left|w_{q, \theta}g\right|_{2}\right)\left|w_{q, \theta}h\right|_{D}. $

基于引理2.1,有下述引理成立.

引理2.2 设$ \alpha\in {\Lambda} $,对任意的$ \eta>0 $有

$ \begin{eqnarray*} &&\sum\limits_{ \alpha\in {\Lambda}}\left(\int_{0}^{T}|\left( \partial^{ \alpha} \Gamma(f, g), w^{2}_{q, \theta}h\right)_{L^{2}_{x, v}}|{\rm d}t\right)^{\frac{1}{2}} \\& \lesssim& \eta\|w_{q, \theta}h\|_{L^2_TL^2_{x}L^2_{v, D}} +C_\eta\left\|w_{q, \theta}f\right\|_{L^\infty_TH^1_{x_1}H^2_{\bar{x}}L^2_v} \left\|w_{q, \theta}g\right\|_{L^2_TH^1_{x_1}H^2_{\bar{x}}L^2_{v, D}} \\&& +C_\eta\left\|w_{q, \theta}f\right\|_{L^2_TH^1_{x_1}H^2_{\bar{x}}L^2_{v, D}} \left\|w_{q, \theta}g\right\|_{L^\infty_TH^1_{x_1}H^2_{\bar{x}}L^2_{v}} \end{eqnarray*} $

成立.

证 下面证明中只考虑$ |\bar{ \alpha}| = 2 $的情形,此时

(2.1) $ \begin{eqnarray} \sum\limits_{ \alpha\in {\Lambda}}\left(\int_{0}^{T}|\left( \partial^{ \alpha} \Gamma(f, g), w^{2}_{q, \theta}h\right)|{\rm d}t\right)^{\frac{1}{2}} & = &\sum\limits_{|\bar{ \alpha}| = 2}\left(\int_{0}^{T}|\left( \partial_{\bar{x}}^{\bar{ \alpha}} \Gamma(f, g), w^{2}_{q, \theta}h\right)|{\rm d}t\right)^{\frac{1}{2}} {}\\&& +\sum\limits_{|\bar{ \alpha}| = 2}\left(\int_{0}^{T}|\left( \partial_{x_1} \partial_{\bar{x}}^{\bar{ \alpha}} \Gamma(f, g), w^{2}_{q, \theta}h\right)|{\rm d}t\right)^{\frac{1}{2}}, \end{eqnarray} $

其中有

(2.2) $ \begin{eqnarray} &&\sum\limits_{|\bar{ \alpha}| = 2}\left(\int_{0}^{T}|\left( \partial_{\bar{x}}^{\bar{ \alpha}} \Gamma(f, g), w^{2}_{q, \theta}h\right)|{\rm d}t\right)^{\frac{1}{2}} {}\\& \leq&\sum\limits_{2\leq i, j\leq3}\left(\int_{0}^{T}|\left( \Gamma( \partial_{x_i} \partial_{x_j}f, g), w^{2}_{q, \theta}h\right)|{\rm d}t\right)^{\frac{1}{2}} +\sum\limits_{2\leq i, j\leq3}\left(\int_{0}^{T}|\left( \Gamma(f, \partial_{x_i} \partial_{x_j}g), w^{2}_{q, \theta}h\right)|{\rm d}t\right)^{\frac{1}{2}} {}\\&& +\sum\limits_{2\leq i, j\leq3}\left(\int_{0}^{T}|\left( \Gamma( \partial_{x_i}f, \partial_{x_j}g), w^{2}_{q, \theta}h\right)|{\rm d}t\right)^{\frac{1}{2}} {}\\& = &\sum\limits_{2\leq i, j\leq3}A^{\frac{1}{2}}_{i, j}+\sum\limits_{2\leq i, j\leq3}B^{\frac{1}{2}}_{i, j}+\sum\limits_{2\leq i, j\leq3}C^{\frac{1}{2}}_{i, j}. \end{eqnarray} $

估计$ A_{2, 2} $,由引理2.1得到

(2.3) $ \begin{eqnarray} A_{2, 2} &\lesssim&\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}\left|w_{q, \theta} \partial^{2}_{x_2}f\right|_{2}|w_{q, \theta}g|_{D}|w_{q, \theta}h|_{D}{\rm d}\bar{x}{\rm d}x_1{\rm d}t {}\\ &&+\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}\left|w_{q, \theta} \partial^{2}_{x_2}f\right|_D|w_{q, \theta}g|_2|w_{q, \theta}h|_D{\rm d}\bar{x}{\rm d}x_1{\rm d}t, \end{eqnarray} $

(2.3)式右端第一项,由Young不等式和Sobolev嵌入$ H^1_{x_1}\hookrightarrow L^{\infty}_{x_1} , H^2_{\bar{x}}\hookrightarrow L^{\infty}_{\bar{x}} $得到

$ \begin{eqnarray*} &&\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}\left|w_{q, \theta} \partial^{2}_{x_2}f\right|_{2}|w_{q, \theta}g|_{D}|w_{q, \theta}h|_{D}{\rm d}\bar{x}{\rm d}x_1{\rm d}t {\nonumber}\\&\lesssim &C_\eta\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}\left|w_{q, \theta} \partial^{2}_{x_2}f\right|^2_{2}|w_{q, \theta}g|^2_{D}{\rm d}\bar{x}{\rm d}x_1{\rm d}t +\eta\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}|w_{q, \theta}h|^2_{D}{\rm d}\bar{x}{\rm d}x_1{\rm d}t {\nonumber}\\&\lesssim& C_\eta\int_{0}^{T}\|w_{q, \theta}g\|^2_{L^\infty_{x}L^2_{v, D}}\left\|w_{q, \theta} \partial^{2}_{x_2}f\right\|^2_{L^2_{x}L^2_{v}}{\rm d}t +\eta\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}|w_{q, \theta}h|^2_{D}{\rm d}\bar{x}{\rm d}x_1{\rm d}t {\nonumber}\\&\lesssim& C_\eta\int_{0}^{T}\|w_{q, \theta}g\|^2_{H^1_{x_1}H^2_{\bar{x}}L^2_{v, D}}\|w_{q, \theta}f\|^2_{H^1_{x_1}H^2_{\bar{x}}L^2_{v}}{\rm d}t +\eta\|w_{q, \theta}h\|^2_{L^2_TL^2_xL^2_{v, D}} {\nonumber}\\&\lesssim& C_\eta\|w_{q, \theta}g\|^2_{L^2_TH^1_{x_1}H^2_{\bar{x}}L^2_{v, D}}\|w_{q, \theta}f\|^2_{L^\infty_TH^1_{x_1}H^2_{\bar{x}}L^2_{v}} +\eta\|w_{q, \theta}h\|^2_{L^2_TL^2_xL^2_{v, D}}, \end{eqnarray*} $

(2.3)式右端第二项,类似可得

$ \begin{eqnarray*} &&\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}\left|w_{q, \theta} \partial^{2}_{x_2}f\right|_D|w_{q, \theta}g|_2|w_{q, \theta}h|_D{\rm d}\bar{x}{\rm d}x_1{\rm d}t \\ & \lesssim& C_\eta\|w_{q, \theta}g\|^2_{L^\infty_TH^1_{x_1}H^2_{\bar{x}}L^2_{v}}\|w_{q, \theta}f\|^2_{L^2_TH^1_{x_1}H^2_{\bar{x}}L^2_{v, D}} +\eta\|w_{q, \theta}h\|^2_{L^2_TL^2_xL^2_{v, D}}, \end{eqnarray*} $

因此有

$ \begin{eqnarray*} A^{\frac{1}{2}}_{2, 2}&\lesssim& \eta\|w_{q, \theta}h\|_{L^2_TL^2_{x}L^2_{v, D}} +C_\eta\left\|w_{q, \theta}f\right\|_{L^\infty_TH^1_{x_1}H^2_{\bar{x}}L^2_v} \left\|w_{q, \theta}g\right\|_{L^2_TH^1_{x_1}H^2_{\bar{x}}L^2_{v, D}} \\&& +C_\eta\left\|w_{q, \theta}f\right\|_{L^2_TH^1_{x_1}H^2_{\bar{x}}L^2_{v, D}} \left\|w_{q, \theta}g\right\|_{L^\infty_TH^1_{x_1}H^2_{\bar{x}}L^2_{v}}. \end{eqnarray*} $

估计$ C_{2, 2} $,由引理2.1得到

(2.4) $ \begin{eqnarray} C_{2, 2} &\lesssim&\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}\left|w_{q, \theta} \partial_{x_2}f\right|_{2}|w_{q, \theta} \partial_{x_2}g|_{D}|w_{q, \theta}h|_{D}{\rm d}\bar{x}{\rm d}x_1{\rm d}t {}\\ &&+\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}\left|w_{q, \theta} \partial_{x_2}f\right|_D|w_{q, \theta} \partial_{x_2}g|_2|w_{q, \theta}h|_D{\rm d}\bar{x}{\rm d}x_1{\rm d}t, \end{eqnarray} $

(2.4)式右端第一项,由Young不等式, Holder不等式和Sobolev嵌入$ H^1_{\bar{x}}\hookrightarrow L^4_{\bar{x}} $可得

$ \begin{eqnarray*} &&\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}\left|w_{q, \theta} \partial_{x_2}f\right|_{2}|w_{q, \theta} \partial_{x_2}g|_{D}|w_{q, \theta}h|_{D}{\rm d}\bar{x}{\rm d}x_1{\rm d}t \\&\lesssim& C_\eta\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}\left|w_{q, \theta} \partial_{x_2}f\right|^2_{2}|w_{q, \theta} \partial_{x_2}g|^2_{D}{\rm d}\bar{x}{\rm d}x_1{\rm d}t +\eta\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}|w_{q, \theta}h|^2_{D}{\rm d}\bar{x}{\rm d}x_1{\rm d}t \\&\lesssim& C_\eta\int_{0}^{T}\int_{I}\|w_{q, \theta} \partial_{x_2}g\|^2_{L^4_{\bar{x}}L^2_{v, D}}\left\|w_{q, \theta} \partial_{x_2}f\right\|^2_{L^4_{\bar{x}}L^2_{v}}{\rm d}x_1{\rm d}t +\eta\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}|w_{q, \theta}h|^2_{D}{\rm d}\bar{x}{\rm d}x_1{\rm d}t \\&\lesssim& C_\eta\int_{0}^{T}\int_{I}\|w_{q, \theta}g\|^2_{H^2_{\bar{x}}L^2_{v, D}}\|w_{q, \theta}f\|^2_{H^2_{\bar{x}}L^2_{v}}{\rm d}x_1{\rm d}t +\eta\|w_{q, \theta}h\|^2_{L^2_TL^2_xL^2_{v, D}} \\&\lesssim& C_\eta\int_{0}^{T}\|w_{q, \theta}g\|^2_{L^\infty_{x_1}H^2_{\bar{x}}L^2_{v, D}}\|w_{q, \theta}f\|^2_{L^2_{x_1}H^2_{\bar{x}}L^2_{v}}{\rm d}t +\eta\|w_{q, \theta}h\|^2_{L^2_TL^2_xL^2_{v, D}} \\&\lesssim& C_\eta\int_{0}^{T}\|w_{q, \theta}g\|^2_{H^1_{x_1}H^2_{\bar{x}}L^2_{v, D}}\|w_{q, \theta}f\|^2_{H^1_{x_1}H^2_{\bar{x}}L^2_{v}}{\rm d}t +\eta\|w_{q, \theta}h\|^2_{L^2_TL^2_xL^2_{v, D}} \\&\lesssim& C_\eta\|w_{q, \theta}g\|^2_{L^2_TH^1_{x_1}H^2_{\bar{x}}L^2_{v, D}}\|w_{q, \theta}f\|^2_{L^\infty_TH^1_{x_1}H^2_{\bar{x}}L^2_{v}} +\eta\|w_{q, \theta}h\|^2_{L^2_TL^2_xL^2_{v, D}}, \end{eqnarray*} $

(2.4)式右端第二项,类似可得

$ \begin{eqnarray*} &&\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}\left|w_{q, \theta} \partial_{x_2}f\right|_D|w_{q, \theta} \partial_{x_2}g|_2|w_{q, \theta}h|_D{\rm d}\bar{x}{\rm d}x_1{\rm d}t \\ &\lesssim& C_\eta\|w_{q, \theta}g\|^2_{L^\infty_TH^1_{x_1}H^2_{\bar{x}}L^2_{v}}\|w_{q, \theta}f\|^2_{L^2_TH^1_{x_1}H^2_{\bar{x}}L^2_{v, D}} +\eta\|w_{q, \theta}h\|^2_{L^2_TL^2_xL^2_{v, D}}, \end{eqnarray*} $

因此有

$ \begin{eqnarray*} C^{\frac{1}{2}}_{2, 2}&\lesssim& \eta\|w_{q, \theta}h\|_{L^2_TL^2_{x}L^2_{v, D}} +C_\eta\left\|w_{q, \theta}f\right\|_{L^\infty_TH^1_{x_1}H^2_{\bar{x}}L^2_v} \left\|w_{q, \theta}g\right\|_{L^2_TH^1_{x_1}H^2_{\bar{x}}L^2_{v, D}} \\&& +C_\eta\left\|w_{q, \theta}f\right\|_{L^2_TH^1_{x_1}H^2_{\bar{x}}L^2_{v, D}} \left\|w_{q, \theta}g\right\|_{L^\infty_TH^1_{x_1}H^2_{\bar{x}}L^2_{v}}. \end{eqnarray*} $

(2.2)式中的其余项用类似方法处理.

(2.1)式右端第二项有

(2.5) $ \begin{equation} \sum\limits_{|\bar{ \alpha}| = 2}\left(\int_{0}^{T}|\left( \partial_{x_1} \partial_{\bar{x}}^{\bar{ \alpha}} \Gamma(f, g), w^{2}_{q, \theta}h\right)|{\rm d}t\right)^{\frac{1}{2}} \leq\sum\limits_{2\leq i, j\leq3}\left\{H^{\frac{1}{2}}_{i, j}+I^{\frac{1}{2}}_{i, j}+J^{\frac{1}{2}}_{i, j} +K^{\frac{1}{2}}_{i, j}+N^{\frac{1}{2}}_{i, j}+M^{\frac{1}{2}}_{i, j}\right\}, \end{equation} $

其中

$ \begin{eqnarray*} \left\{\begin{array}{l} { } H_{i, j} = \int_{0}^{T}|\left( \Gamma( \partial_{x_1} \partial_{x_i} \partial_{x_j}f, g), w^{2}_{q, \theta}h\right)|{\rm d}t, \quad { } I_{i, j} = \int_{0}^{T}|\left( \Gamma( \partial_{x_i} \partial_{x_j}f, \partial_{x_1}g), w^{2}_{q, \theta}h\right)|{\rm d}t, \\ { } J_{i, j} = \int_{0}^{T}|\left( \Gamma( \partial_{x_1}f, \partial_{x_i} \partial_{x_j}g), w^{2}_{q, \theta}h\right)|{\rm d}t, \quad K_{i, j} = \int_{0}^{T}|\left( \Gamma(f, \partial_{x_1} \partial_{x_i} \partial_{x_j}g), w^{2}_{q, \theta}h\right)|{\rm d}t, \\ { } M_{i, j} = \int_{0}^{T}|\left( \Gamma( \partial_{x_1} \partial_{x_i}f, \partial_{x_j}g), w^{2}_{q, \theta}h\right)|{\rm d}t, \quad N_{i, j} = \int_{0}^{T}|\left( \Gamma( \partial_{x_i}f, \partial_{x_1} \partial_{x_j}g), w^{2}_{q, \theta}h\right)|{\rm d}t. \end{array}\right. \end{eqnarray*} $

估计$ H_{2, 2} $,由引理2.1得到

(2.6) $ \begin{eqnarray} H_{2, 2} &\lesssim&\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}\left|w_{q, \theta} \partial_{x_1} \partial^{2}_{x_2}f\right|_{2}|w_{q, \theta}g|_{D}|w_{q, \theta}h|_{D}{\rm d}\bar{x}{\rm d}x_1{\rm d}t {}\\ &&+\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}\left|w_{q, \theta} \partial_{x_1} \partial^{2}_{x_2}f\right|_D|w_{q, \theta}g|_2|w_{q, \theta}h|_D{\rm d}\bar{x}{\rm d}x_1{\rm d}t, \end{eqnarray} $

(2.6)式右端第一项,由Young不等式和Sobolev嵌入得到

$ \begin{eqnarray*} &&\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}\left|w_{q, \theta} \partial_{x_1} \partial^{2}_{x_2}f\right|_{2}|w_{q, \theta}g|_{D}|w_{q, \theta}h|_{D}{\rm d}\bar{x}{\rm d}x_1{\rm d}t \\&\lesssim &C_\eta\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}\left|w_{q, \theta} \partial_{x_1} \partial^{2}_{x_2}f\right|^2_{2}|w_{q, \theta}g|^2_{D}{\rm d}\bar{x}{\rm d}x_1{\rm d}t +\eta\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}|w_{q, \theta}h|^2_{D}{\rm d}\bar{x}{\rm d}x_1{\rm d}t \\&\lesssim& C_\eta\int_{0}^{T}\|w_{q, \theta}g\|^2_{L^\infty_{x}L^2_{v, D}}\left\|w_{q, \theta} \partial_{x_1} \partial^{2}_{x_2}f\right\|^2_{L^2_{x}L^2_{v}}{\rm d}t +\eta\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}|w_{q, \theta}h|^2_{D}{\rm d}\bar{x}{\rm d}x_1{\rm d}t \\&\lesssim& C_\eta\int_{0}^{T}\|w_{q, \theta}g\|^2_{H^1_{x_1}H^2_{\bar{x}}L^2_{v, D}}\|w_{q, \theta}f\|^2_{H^1_{x_1}H^2_{\bar{x}}L^2_{v}}{\rm d}t +\eta\|w_{q, \theta}h\|^2_{L^2_TL^2_xL^2_{v, D}} \\&\lesssim& C_\eta\|w_{q, \theta}g\|^2_{L^2_TH^1_{x_1}H^2_{\bar{x}}L^2_{v, D}}\|w_{q, \theta}f\|^2_{L^\infty_TH^1_{x_1}H^2_{\bar{x}}L^2_{v}} +\eta\|w_{q, \theta}h\|^2_{L^2_TL^2_xL^2_{v, D}}, \end{eqnarray*} $

(2.6)式右端第二项,类似可得

$ \begin{eqnarray*} &&\int_{0}^{T}\int_{I}\int_{ {\Bbb T}^{2}}\left|w_{q, \theta} \partial_{x_1} \partial^{2}_{x_2}f\right|_D|w_{q, \theta}g|_2|w_{q, \theta}h|_D{\rm d}\bar{x}{\rm d}x_1{\rm d}t \\ &\lesssim& C_\eta\|w_{q, \theta}g\|^2_{L^\infty_TH^1_{x_1}H^2_{\bar{x}}L^2_{v}}\|w_{q, \theta}f\|^2_{L^2_TH^1_{x_1}H^2_{\bar{x}}L^2_{v, D}} +\eta\|w_{q, \theta}h\|^2_{L^2_TL^2_xL^2_{v, D}}, \end{eqnarray*} $

因此有

$ \begin{eqnarray*} H^{\frac{1}{2}}_{2, 2}&\lesssim& \eta\|w_{q, \theta}h\|_{L^2_TL^2_{x}L^2_{v, D}} +C_\eta\left\|w_{q, \theta}f\right\|_{L^\infty_TH^1_{x_1}H^2_{\bar{x}}L^2_v} \left\|w_{q, \theta}g\right\|_{L^2_TH^1_{x_1}H^2_{\bar{x}}L^2_{v, D}} \\&& +C_\eta\left\|w_{q, \theta}f\right\|_{L^2_TH^1_{x_1}H^2_{\bar{x}}L^2_{v, D}} \left\|w_{q, \theta}g\right\|_{L^\infty_TH^1_{x_1}H^2_{\bar{x}}L^2_{v}}. \end{eqnarray*} $

(2.5)式中的其余项用类似方法处理.证毕.

对$ f $作宏-微观分解,有$ f = {\bf P}_{0}f+\left\{{\bf I}-{\bf P}_{0}\right\}f $,其中$ {\bf I}f = f $.在本节中,我们推导了方程(1.4)–(1.6)解的宏观部分$ [a, b, c] $的重要估计.在此之前,首先定义边界积分泛函

$ \begin{eqnarray*} {\nonumber} \vert\Upsilon_{T, w}^+(h)\vert^2& = &\int_0^T\int_{ {\Bbb T}^{2}}\int_{v_1>0}|v_1|w^2_{q, \theta}|h(t, 1, \bar{x}, v)|^2{\rm d}v{\rm d}{\bar{x}}{\rm d}t \\ &&+\int_0^T\int_{ {\Bbb T}^{2}}\int_{v_1<0}|v_1|w^2_{q, \theta}|h(t, -1, \bar{x}, v)|^2{\rm d}v{\rm d}{\bar{x}}{\rm d}t, \\ \vert \Upsilon_{T, w}^-(h) \vert^2& = &\int_0^T\int_{ {\Bbb T}^{2}}\int_{v_1< 0}|v_1|w^2_{q, \theta}|h(t, 1, \bar{x}, v)|^2{\rm d}v{\rm d}{\bar{x}}{\rm d}t \\ &&+\int_0^T\int_{ {\Bbb T}^{2}}\int_{v_1>0}|v_1|w^2_{q, \theta}|h(t, -1, \bar{x}, v)|^2{\rm d}v{\rm d}{\bar{x}}{\rm d}t, \end{eqnarray*} $

其中$ h = h(t, x_1, \bar{x}, v) $是在边界上定义好的分布函数.当$ q = 0 $时,简记$ \Upsilon_{T, w}^\pm(\cdot) = \Upsilon_T^\pm(\cdot) $.

定理3.1 假设在定理1.1的条件下,当$ \alpha\in {\Lambda} $时,有

(3.1) $ \begin{eqnarray} \sum\limits_{ \alpha\in {\Lambda}}\| \partial^{ \alpha}[a, b, c]\|_{L^2_TL^2_{x}}{} &\lesssim&\sum\limits_{ \alpha\in {\Lambda}}\|\left\{\bf I-\bf P_{0}\right\} \partial^{ \alpha} f\|_{L^2_TL^{2}_{x}L^2_{v, D}} +\sum\limits_{ \alpha\in\Lambda}\| \partial^{ \alpha}f\|_{L^\infty_TL^2_{x, v}} {\nonumber} \\&&+\sum\limits_{ \alpha\in {\Lambda}}\| \partial^{ \alpha}f_0\|_{L^2_{x, v}}+\sum\limits_{ \alpha\in {\Lambda}}\left(\int_0^T\left\|\left({ \partial^ \alpha \Gamma(f, f)}, \mu^{\frac{1}{4}}\right)_{L^{2}_{v}}\right\|^2{\rm d}t\right)^{1/2} {} \\&&+\sum\limits_{ \alpha\in {\Lambda}}E({ \partial^{ \alpha}g_\pm}) +\sum\limits_{ \alpha\in {\Lambda}}|\Upsilon_{T}^+({ \partial^ \alpha f})| +\sum\limits_{ \alpha\in {\Lambda}}\left|\Upsilon_{T}^-\left(\frac{ \partial^{ \alpha}\Gamma(f, f)}{|v_1|}\right)\right| \end{eqnarray} $

成立.

证 给定速度矩: $ \mu^{\frac{1}{2}} $, $ v_{j}\mu^{\frac{1}{2}} $, $ {\frac{1}{6}}(|v|^{2}-3)\mu^{\frac{1}{2}} $, $ (v_{j}v_{m}-1)\mu^{\frac{1}{2}} $, $ {\frac{1}{10}}(|v|^{2}-5)v_{j}\mu^{\frac{1}{2}} $,其中$ 1\leq j, $$ m\leq3 $,将方程(1.4)分别与上述速度矩关于$ v $作内积,得到系数函数$ [a, b, c] $所满足的流体系统

(3.2) $ \begin{equation} \left\{\begin{array}{l} \partial_t a + \partial_{x_1}b_1+ \partial_{\bar{x}}\cdot \bar{b} = 0, \ \bar{b} = (b_2, b_3), \\ \partial_t b + \nabla_x (a+2c)+ \nabla_x\cdot \Theta (\left\{{\bf I}-{\bf P_{0}}\right\} f) = 0, \\ { } \partial_t c +\frac{1}{3} \nabla_x\cdot b +\frac{1}{6} \nabla_x\cdot {\Lambda} (\left\{{\bf I}-{\bf P_{0}}\right\} f) = 0, \\ \partial_t[ \Theta_{{ jm}}(\left\{{\bf I}-{\bf P_{0}}\right\} f)+2c \delta_{{ jm}}]+ \partial_jb_m+ \partial_m b_j = \Theta_{jm}({\bf r}+{\bf h}), \\ \partial_t {\Lambda}_j(\left\{{\bf I}-{\bf P_{0}}\right\} f)+ \partial_j c = {\Lambda}_j(\bf{r}+{\bf h}), \end{array}\right. \end{equation} $

其中$ \Theta = ( \Theta_{jm}(\cdot))_{3\times 3} $和$ \Lambda = ( \Lambda_j(\cdot))_{1\leq j\leq 3} $分别定义为

$ \begin{eqnarray} \Theta_{jm}(f) = \left ((v_jv_m-1)\mu^{\frac{1}{2}}, f\right)_{L^2_v}, \ \Lambda_j(f) = \frac{1}{10}\left ((|v|^2-5)v_j\mu^{\frac{1}{2}}, f\right)_{L^2_v}, {} \end{eqnarray} $

并且有$ {\bf r} = -{v}\cdot \nabla_{{x}}\left\{{\bf I}-{\bf P_{0}}\right\}f $和$ {\bf h} = -L \left\{{\bf I}-{\bf P_{0}}\right\}f+\Gamma(f, f) $.方程(1.4)两边作用$ \partial^{ \alpha} $,再取$ \Psi(t, x_1, \bar{x}, v)\in C^1((0, +\infty)\times\Omega\times {\Bbb R}^3) $作为测试函数,在$ [0, T] $上积分,有

$ -\int_{0}^{T}\left( \partial^{ \alpha}{\bf P_{0}}f, v\cdot \nabla\Psi\right){\rm d}t = \sum\limits^{5}_{j = 1}S_{j}, $

其中$ S_{j}(1\leq j\leq 5) $为

$ \begin{eqnarray*} \left\{\begin{array}{ll} { } S_1 = -( \partial^{ \alpha}f, \Psi)(T)+( \partial^{ \alpha}f, \Psi)(0), \ S_2 = \int_{0}^{T}( \partial^{ \alpha}f, \partial_{t}\Psi){\rm d}t, \\ { } S_3 = \int_{0}^{T}\left(\left\{{\bf I}-{\bf P_{0}}\right\} \partial^{ \alpha}f, v\cdot \nabla\Psi\right){\rm d}t, \\ { } S_4 = -\int_{0}^{T}\left\langle v_{1} \partial^{ \alpha}f(1), \Psi(1)\right\rangle {\rm d}t+\int_{0}^{T}\big\langle v_{1} \partial^{ \alpha}f(-1), \Psi(-1)\big\rangle {\rm d}t , \\ { } S_5 = -\int_{0}^{T}(L \partial^{ \alpha}f, \Psi){\rm d}t+\int_{0}^{T}( \partial^{ \alpha}\Gamma(f, f), \Psi){\rm d}t. \end{array}\right. \end{eqnarray*} $

估计$ c(t, x_1, \bar{x}) $:取测试函数$ \Psi = \Psi_c = (\vert v \vert^2-5)\left\{v\cdot \nabla\Phi_c(t, x_1, \bar{x})\right\}\mu^{\frac{1}{2}} $,其中$ \Phi_{c} $满足椭圆方程

(3.3) $ \begin{equation} -\triangle\Phi_{c} = \partial^{ \alpha}c. \end{equation} $

由椭圆的$ H^{2} $估计得

(3.4) $ \begin{equation} \|\Phi_{c}\|_{H_{x}^{2}}\lesssim\| \partial^{ \alpha}c\|_{L_{x}^{2}}. \end{equation} $

由(1.7)式有

$ \begin{eqnarray*} &&-\int_0^T( \partial^{ \alpha}{{\bf P_{0}}f}, v\cdot \nabla\Psi_{c}){\rm d}t \\ & = &-\sum\limits_{j, n}\int_0^T\left (\left\{ \partial^{ \alpha}{a}+ \partial^{ \alpha}{b}\cdot v +\frac{1}{2}(|v|^2-3) \partial^{ \alpha}{c}\right\}\mu^{\frac{1}{2}}, v_jv_n(|v|^2-5)\mu^{\frac{1}{2}} \partial_j \partial_n\Phi_c \right){\rm d}t\\ & = &15\sum\limits_{j}\int_0^T( \partial^{ \alpha}c, - \partial^{2}_{j}\Phi_c){\rm d}t = 15\int_0^T\| \partial^{ \alpha}c\|^2_{L^{2}_{x}}{\rm d}t. \end{eqnarray*} $

接下来逐步估计$ S_{j}(1\leq j\leq 5) $.对于$ S_{1} $,由Cauchy-Schwarz不等式和椭圆估计(3.4)式得到

$ |S_1|\leq\left|\left( \partial^{ \alpha}f, \Psi_{c}\right)(0)\right|+\left|\left( \partial^{ \alpha}f, \Psi_{c}\right)(T)\right| \lesssim\left\| \partial^{ \alpha}f(T)\right\|^{2}+\left\| \partial^{ \alpha}f(0)\right\|^{2}. $

对于$ S_{2} $,首先由椭圆方程(3.3)得到

(3.5) $ \begin{equation} \| \partial_{t}\Phi_{c}\|_{H^{1}_{x}}\lesssim\| \partial_{t} \partial^{ \alpha}c\|_{H^{-1}_{x}}. \end{equation} $

再由(3.2)式中的第三个方程得到

(3.6) $ \begin{equation} \| \partial_{t} \partial^{ \alpha}c\|_{H^{-1}_{x}}\lesssim\| \partial^{ \alpha}b\|+\|\left\{{\bf I}-{\bf P_{0}}\right\} \partial^{ \alpha}f\|_{D}. \end{equation} $

由Cauchy-Schwarz不等式, (3.5)式和(3.6)式得到

$ \begin{eqnarray*} |S_2|&\leq& \int_0^T|(\left\{{\bf I}-{\bf P_{0}}\right\} \partial^{ \alpha}f, \partial_{t}\Psi_{c})|{\rm d}t \leq\int_0^T\|\left\{{\bf I}-{\bf P_{0}}\right\} \partial^{ \alpha}f\|\| \partial_{t}\Psi_{c}\|{\rm d}t \\& \lesssim &\eta\int_0^T\| \partial_{t}\Psi_{c}\|^2{\rm d}t+C_\eta\int_0^T\|\left\{{\bf I}-{\bf P_{0}}\right\} \partial^{ \alpha}f\|^2_{D}{\rm d}t \\& \lesssim &\eta\int_0^T\| \partial_{t} \nabla\Phi_{c}\|^2{\rm d}t+C_\eta\int_0^T\|\left\{{\bf I}-{\bf P_{0}}\right\} \partial^{ \alpha}f\|^2_{D}{\rm d}t \\& \lesssim &\eta\int_0^T\| \partial^{ \alpha}b\|^2{\rm d}t+C_\eta\int_0^T\|\left\{{\bf I}-{\bf P_{0}}\right\} \partial^{ \alpha}f\|^2_{D}{\rm d}t. \end{eqnarray*} $

对于$ S_{3} $,由Cauchy-Schwarz不等式和椭圆估计(3.4)式得到

$ \begin{eqnarray*} |S_3|\lesssim \eta\int_0^T\| \partial^{ \alpha}c\|^2{\rm d}t+C_\eta\int_0^T\|\left\{{\bf I}-{\bf P_{0}}\right\} \partial^{ \alpha}f\|_{D}^2{\rm d}t. \end{eqnarray*} $

对于$ S_{4} $,由迹定理和椭圆估计(3.4)式得

(3.7) $ \begin{equation} \left\| \nabla\Phi_{c}(t, \pm1, \bar{x})\right\|\lesssim\left\|\Phi_{c}(t, \pm1, \bar{x})\right\|_{H^{2}_{x}}\lesssim\| \partial^{ \alpha}c\|, \end{equation} $

由Cauchy-Schwarz不等式, Young不等式和(3.7)式得到

$ \begin{eqnarray*} |S_{4}| \lesssim C_{\eta}|\Upsilon_{T}^+({ \partial^ \alpha f})|^{2} +C_{\eta}|\Upsilon_{T}^-({ \partial^ \alpha f})|^{2} +\eta\int_{0}^{T}\| \partial^{ \alpha}c\|^{2}{\rm d}t. \end{eqnarray*} $

由$ \partial^{ \alpha} = \partial^{ \alpha_{1}}_{x_{1}} \partial^{\bar{ \alpha}}_{\bar{x}} $,其中$ \alpha = ( \alpha_{1}, \bar{ \alpha})\in {\Lambda} = \left\{ \alpha\big| \alpha_{1}\leq1, |\bar{ \alpha}|\leq2\right\} $.当$ \alpha_1 = 0 $时,有

$ \begin{eqnarray*} \partial^ \alpha f(t, -1, \bar{x}, v)|_{v_1>0} = \partial^{\bar{ \alpha}}g_{-}(t, \bar{x}, v), \ \ \partial^ \alpha f(t, 1, \bar{x}, v)|_{v_1<0} = \partial^{\bar{ \alpha}}g_+(t, \bar{x}, v), \end{eqnarray*} $

因此有

$ |\Upsilon_{T}^-({ \partial^ \alpha f})|^{2}\lesssim E^{2}( \partial^{\bar{ \alpha}}g_{\pm}). $

当$ \alpha_1 = 1 $时,由(1.4)式得到

$ \partial^ \alpha f(t, -1, \bar{x}, v)|_{v_1>0} = -\frac{1}{v_1}\left\{ \partial_{t} \partial^{\bar{ \alpha}}g_{-} +\bar{v}\cdot \nabla_{\bar{x}} \partial^{\bar{ \alpha}}g_-+L{ \partial^{\bar{ \alpha}}g_-}- \partial^{\bar{ \alpha}}\Gamma(g_-, g_-)\right\}, $

$ \partial^ \alpha f(t, 1, \bar{x}, v)|_{v_1<0} = -\frac{1}{v_1}\left\{ \partial_{t} \partial^{\bar{ \alpha}}g_{+} +\bar{v}\cdot \nabla_{\bar{x}} \partial^{\bar{ \alpha}}g_++L{ \partial^{\bar{ \alpha}}g_+}- \partial^{\bar{ \alpha}}\Gamma(g_+, g_+)\right\}. $

因此有

$ |\Upsilon_{T}^-({ \partial^ \alpha f})|^{2}\lesssim E^{2}( \partial^{\bar{ \alpha}}g_{\pm})+\left|\Upsilon_{T}^-\left(\frac{ \partial^{\bar{ \alpha}}\Gamma(f, f)}{|v_1|}\right)\right|^{2}. $

综上得

$ \begin{eqnarray*} |S_{4}|\lesssim C_{\eta}\left|\Upsilon_{T}^-\left(\frac{ \partial^{\bar{ \alpha}}\Gamma(f, f)}{|v_1|}\right)\right|^2 +C_{\eta}E^2( \partial^{\bar{ \alpha}}g_{\pm})+C_{\eta}|\Upsilon_{T}^+({ \partial^ \alpha f})|^2 +\eta\int_{0}^{T}\| \partial^{ \alpha}c\|^{2}{\rm d}t. \end{eqnarray*} $

对于$ S_{5} $,由Cauchy-Schwarz不等式和椭圆估计(3.4)式得到

$ |S_{5}|\lesssim\eta\int_0^T\| \partial^{ \alpha}c\|^2{\rm d}t+C_\eta\int_0^T\|\left\{{\bf I}-{\bf P_{0}}\right\} \partial^{ \alpha}f\|_{D}^2{\rm d}t+ C_{\eta}\int_{0}^{T}\left\|\left( \partial^{ \alpha}\Gamma(f, f), \mu^{\frac{1}{4}}\right)_{L^{2}_{v}}\right\|^{2}{\rm d}t. $

综上$ S_{j}(1\leq j\leq 5) $的估计得

$ \begin{eqnarray*} \int_0^T\| \partial^{ \alpha}c\|^2{\rm d}t &\lesssim&\| \partial^{ \alpha}f(T)\|^{2}+\| \partial^{ \alpha}f(0)\|^{2}+ C_{\eta}\left|\Upsilon_{T}^-\left(\frac{ \partial^{\bar{ \alpha}}\Gamma(f, f)}{|v_1|}\right)\right|^2 +C_{\eta}E^2( \partial^{\bar{ \alpha}}g_{\pm}) \\&& +C_{\eta}|\Upsilon_{T}^+({ \partial^ \alpha f})|^2 +C_{\eta}\int_{0}^{T}\left\|\left( \partial^{ \alpha} \Gamma(f, f), \mu^{\frac{1}{4}}\right)_{L^{2}_{v}}\right\|^{2}{\rm d}t \\&& +\eta\int_0^T\| \partial^{ \alpha}b\|^2{\rm d}t +C_\eta\int_0^T\|\left\{{\bf I}-{\bf P_{0}}\right\} \partial^{ \alpha}f\|_{D}^2{\rm d}t. \end{eqnarray*} $

使用类似的方法可得到$ a, b $的估计,这里就不再赘述.证毕.

定理1.1的证明 我们将证明分为以下三个部分,首先证明解的全局存在性,然后证明解的时间衰减,最后证明解的唯一性和非负性.

全局存在.证明解的全局存在,在这里我们只证明解的先验估计,因为解的全局存在性可以通过局部解的构造和连续性技巧得到.而解的局部存在性的证明与文献[11]中的证明完全类似,故省略.

假设$ \alpha\in {\Lambda} $,方程(1.4)两边同时作用$ \partial^{\alpha} $得

(4.1) $ \begin{equation} \partial_t \partial^ \alpha f+v\cdot \nabla_{x} \partial^ \alpha f+L \partial^ \alpha f = \partial^ \alpha \Gamma(f, f). \end{equation} $

方程(4.1)与$ \partial^{ \alpha}f $关于$ (x, v) $作内积,由引理5.1,存在$ \delta_{0}>0 $使得

(4.2) $ \begin{eqnarray} &&{\frac{1}{2}}{\frac{\rm d}{{\rm d}t}\| \partial^{ \alpha}f}\|^{2} +{\frac{1}{2}}\int_{ {\Bbb T}^2}\int_{ {\Bbb R}^3}v_1\left[| \partial^{ \alpha}f(1)|^{2}-| \partial^{ \alpha}f(-1)|^{2}\right]{\rm d}v{\rm d}\bar{x} +\delta_{0}\|\left\{{\bf I}-{\bf P_{0}}\right\} \partial^{ \alpha}f\|_{D}^{2} {}\\&\leq&( \partial^{ \alpha}\Gamma(f, f), \left\{{\bf I}-{\bf P_{0}}\right\} \partial^{ \alpha}f), \end{eqnarray} $

对任意的$ 0\leq t \leq T $, (4.2)式在$ [0, t] $上积分并再对$ t $取上确界,注意$ |\Upsilon_{T}^{\pm}(\cdot)|^{2} $的定义,得到

$ \begin{eqnarray*} &&\sup\limits_{0\leq t\leq T}\| \partial^{ \alpha}f(t)\|^{2}+|\Upsilon_{T}^{+}( \partial^{ \alpha}f)|^{2} +2\delta_0\int_{0}^{T}\|\left\{{\bf I}-{\bf P_{0}}\right\}\partial^{\alpha}f\|_{D}^{2}{\rm d}t \\& \lesssim&\| \partial^{ \alpha}f_{0}\|^{2}+|\Upsilon_{T}^{-}( \partial^{ \alpha}f)|^{2} +\int_{0}^{T}( \partial^{ \alpha}\Gamma(f, f), \left\{{\bf I}-{\bf P_{0}}\right\} \partial^{ \alpha}f){\rm d}t, \end{eqnarray*} $

由初等不等式和引理2.2得到

(4.3) $ \begin{eqnarray} &&\sum\limits_{ \alpha\in {\Lambda}}\sup\limits_{0\leq t\leq T}\| \partial^{ \alpha}f(t)\|+\sum\limits_{ \alpha\in {\Lambda}}|\Upsilon_{T}^{+}( \partial^{ \alpha}f)| +\sum\limits_{ \alpha\in {\Lambda}}\left(\int_{0}^{T}\|\left\{{\bf I}-{\bf P_{0}}\right\} \partial^{ \alpha}f\|_{D}^{2}{\rm d}t\right)^{\frac{1}{2}} {}\\& \lesssim&\sum\limits_{ \alpha\in {\Lambda}}\| \partial^{ \alpha}f_{0}\|+\sum\limits_{ \alpha\in {\Lambda}}E( \partial^{ \alpha}g_\pm) +C_{\eta}\|f\|_{L^{\infty}_{T}H^{1}_{x_{1}}H^{2}_{\bar{x}}L^{2}_{v}}\|f\|_{L^{2}_{T}H^{1}_{x_{1}}H^{2}_{\bar{x}}L^{2}_{v, D}} {}\\&& +\eta\sum\limits_{ \alpha\in {\Lambda}}\|\left\{{\bf I}-{\bf P_{0}}\right\} \partial^{ \alpha}f\|_{L^{2}_{T}L^{2}_{x}L^{2}_{v, D}}. \end{eqnarray} $

由定理3.1, (4.3)式和(3.1)式适当的线性组合得到

$ {\cal E}_T(f)+{\cal D}_T(f)\lesssim\sum\limits_{ \alpha\in {\Lambda}}\| \partial^{ \alpha}f_{0}\|_{L^{2}_{x, v}}+{\cal E}_T(f){\cal D}_T(f)+\sum\limits_{ \alpha\in {\Lambda}}E( \partial^{ \alpha}g_\pm). $

方程(4.1)与$ w_{q, \theta}^{2} \partial^{ \alpha}f $关于$ (x, v) $作内积,使用引理5.2经过类似的计算可以得到加权的估计

$ {\cal E}_{T, w}(f)+{\cal D}_{T, w}(f)\lesssim\sum\limits_{ \alpha\in {\Lambda}}\|w_{q, \theta} \partial^{ \alpha}f_{0}\|_{L^{2}_{x, v}}+{\cal E}_{T, w}(f){\cal D}_{T, w}(f)+\sum\limits_{ \alpha\in {\Lambda}}E(w_{q, \theta} \partial^{ \alpha}g_\pm). $

时间衰减率.这里只证明软势$ -3\leq \gamma<-2 $的情况.取$ h = e^{ \lambda t^{p}}f $,其中$ \lambda>1.1>p>0 $,关于$ \partial^{ \alpha}h $满足的方程为

(4.4) $ \begin{equation} \partial_{t} \partial^{ \alpha}h+v_{1} \partial_{x_1} \partial^{ \alpha}h+\bar{v}\cdot \nabla_{\bar{x}} \partial^{ \alpha}h +L \partial^{ \alpha}h = \lambda pt^{p-1} \partial^{ \alpha}h+e^{- \lambda t^{p}} \partial^{ \alpha}\Gamma(h, h). \end{equation} $

对方程(4.4)作类似于全局存在性的估计可得

(4.5) $ \begin{eqnarray} &&{\frac{\rm d}{{\rm d}t}}\| \partial^{ \alpha}h\|^{2}+\int_{ {\Bbb T}^2}\int_{ {\Bbb R}^3}v_1| \partial^{ \alpha}h(1)|^2{\rm d}v{\rm d}\bar{x}-\int_{ {\Bbb T}^2}\int_{ {\Bbb R}^3}v_1| \partial^{ \alpha}h(-1)|^2{\rm d}v{\rm d}\bar{x} +\| \partial^{ \alpha}h\|^{2}_{D} {}\\&\lesssim& \lambda pt^{p-1}\| \partial^{ \alpha}h\|^{2}+\sum\limits_{ \alpha\in {\Lambda}}E(e^{ \lambda t^{p}} \partial^{ \alpha}g_\pm). \end{eqnarray} $

对任意的$ 0\leq t\leq T $,由(4.5)式进一步可得到

(4.6) $ \begin{eqnarray} &&\sup\limits_{0\leq t\leq T}\| \partial^{ \alpha}h\|+\left(\int_{0}^{T}\| \partial^{ \alpha}h\|^{2}_{D}{\rm d}t\right)^{\frac{1}{2}}+|\Upsilon_{T}^{+}( \partial^{ \alpha}h)| {}\\ &\lesssim&\| \partial^{ \alpha}f_0\|+\sup\limits_{0\leq t\leq T}E(e^{ \lambda t^{p}} \partial^{ \alpha}g_{\pm})+\left(\int_{0}^{T} \lambda pt^{p-1}\| \partial^{ \alpha}h\|^{2}{\rm d}t\right)^{\frac{1}{2}}+\sum\limits_{ \alpha\in {\Lambda}}E(w_{q, \theta} \partial^{ \alpha}g_\pm). \end{eqnarray} $

定义集合$ E = \left\{\langle v\rangle\leqslant\rho t^{p'}\right\} $,其中$ \rho>0 $足够小, $ p' $是一个与$ p $有关的正数, (4.6)式右端的第三项有

$ \begin{eqnarray*} \sqrt{ \lambda p}\left(\int_{0}^{T}t^{p-1}\| \partial^{ \alpha}h\|^{2}{\rm d}t\right)^{\frac{1}{2}} &\lesssim&\sqrt{ \lambda p}\left(\int_{0}^{T}\int_{I\times {\Bbb T}^2}\int_{E}t^{p-1}e^{2 \lambda t^{p}}| \partial^{ \alpha}f|^{2}{\rm d}v{\rm d}x{\rm d}t\right)^{\frac{1}{2}} \\&& +\sqrt{ \lambda p}\left(\int_{0}^{T}\int_{I\times {\Bbb T}^2}\int_{E^{c}}t^{p-1}e^{2 \lambda t^{p}}| \partial^{ \alpha}f|^{2}{\rm d}v{\rm d}x{\rm d}t\right)^{\frac{1}{2}} \\& = & I_{1}+I_{2}, \end{eqnarray*} $

对于$ I_{1} $,由引理5.1,取$ {\frac{p-1}{p'}} = \gamma+2 $,有

$ \begin{eqnarray*} I_{1}& \leq&\sqrt{ \lambda p}\rho^{-{\frac{p-1}{p'}}}\left(\int_{0}^{T}\int_{I\times {\Bbb T}^2}\int_{E}\langle v\rangle^{{\frac{p-1}{p'}}} e^{2 \lambda t^{p}}| \partial^{ \alpha}f|^{2}{\rm d}v{\rm d}x{\rm d}t\right)^{\frac{1}{2}} \\& \leq&\sqrt{ \lambda p}\rho^{-{\frac{p-1}{p'}}}\left(\int_{0}^{T}\| \partial^{ \alpha}h\|_{D}^{2}{\rm d}t\right)^{\frac{1}{2}}. \end{eqnarray*} $

取$ p = p' \theta $,即有$ p = {\frac{ \theta}{ \theta-(\gamma+2)}} $,令$ 2 \lambda <{\frac{1}{2}}q\rho^{ \theta} $,有$ w_{q, \theta}^{-2}\leq e^{-{\frac{1}{2}}q\rho^{ \theta}t^{p' \theta}} $,

$ \begin{eqnarray*} I_{2}&\leq&\sqrt{ \lambda p}\left(\int_{0}^{T}\int_{I\times {\Bbb T}^2}\int_{E^{c}}t^{p-1}e^{2 \lambda t^{p}}e^{-{\frac{1}{2}}q\rho^{ \theta}t^{p' \theta}}w_{q, \theta}^{2}| \partial^{ \alpha}f|^{2}{\rm d}v{\rm d}x{\rm d}t\right)^{\frac{1}{2}} \\& \leq&\sqrt{ \lambda p}\sup\limits_{0\leq t\leq T}\|w_{q, \theta} \partial^{ \alpha}f\|\left(\int_{0}^{T}t^{p-1}e^{2 \lambda t^{p}}e^{-{\frac{1}{2}}q\rho^{ \theta}t^{p' \theta}}{\rm d}t\right)^{\frac{1}{2}} \\& \leq&\sqrt{ \lambda p}\sup\limits_{0\leq t\leq T}\|w_{q, \theta} \partial^{ \alpha}f\| \lesssim\|w_{q, \theta} \partial^{ \alpha}f_0\|+E(w_{q, \theta} \partial^{ \alpha}g_{\pm}). \end{eqnarray*} $

将$ I_1 $和$ I_2 $的估计带入(4.6)式,即可得到时间衰减估计(1.10)式.

唯一性和非负性.假设$ f $和$ g $是方程(1.4)–(1.6)满足(1.9)式的两个解,关于$ f-g $满足的方程为

$ \partial_{t}(f-g)+v\cdot \nabla_{x}(f-g)+L[f-g] = \Gamma(f-g, f)+ \Gamma(g, f-g). $

且$ [f-g](0, x, v) = 0 $,

$ \begin{eqnarray*} \label{ifb} [f-g](t, -1, \bar{x}, v)|_{v_1>0} = 0, \ \ [f-g](t, 1, \bar{x}, v)|_{v_1<0} = 0. \end{eqnarray*} $

上述方程两边乘以$ f-g $并在$ (0, t)\times \Omega\times{ {\Bbb R}^3} $上积分得到

$ {\frac{1}{2}}\|f-g\|^{2}+\int_{0}^{t}(L[f-g], f-g){\rm d}\tau = \int_{0}^{t}( \Gamma(f-g, f)+ \Gamma(g, f-g), f-g){\rm d}\tau, $

由引理2.2,有

$ \|f-g\|^{2}+\int_{0}^{t}\|f-g\|^{2}_{D}{\rm d}\tau \leq C\epsilon_{0}\int_{0}^{t}\|f-g\|^{2}_{D}{\rm d}\tau+C\int_{0}^{t}\|f-g\|^{2}{\rm d}\tau, $

其中$ \epsilon_{0} $由定理1.1给出,由$ \epsilon_{0} $任意小,由Gronwall不等式有$ f = g $.因为有$ F(t, \pm1, \bar{x}, v) = \mu+\sqrt{\mu}g_{\pm}(t, \bar{x}, v)\geq0 $成立,由极大值原理有$ F(t, \pm1, \bar{x}, v)\geqslant0 $,具体证明可参见文献[2].证毕.

定理1.2的证明 我们证明初始数据或边界数据的正则性可以沿切向从边界传播到管道内部.设$ \alpha\in {\Lambda}_{m} $,对于非线性项有

$ \begin{eqnarray*} &&\sum\limits_{ \alpha\in {\Lambda}_{m}}\left(\int_{0}^{T}|\left( \partial^{ \alpha} \Gamma(f, g), w^{2}_{q, \theta}h\right)_{L^{2}_{x, v}}|{\rm d}t\right)^{\frac{1}{2}} \\& \lesssim& \eta\|w_{q, \theta}h\|_{L^2_TL^2_{x}L^2_{v, D}} +C_\eta\left\|w_{q, \theta}f\right\|_{L^\infty_TH^1_{x_1}H^m_{\bar{x}}L^2_v} \left\|w_{q, \theta}g\right\|_{L^2_TH^1_{x_1}H^m_{\bar{x}}L^2_{v, D}} \\&& +C_\eta\left\|w_{q, \theta}f\right\|_{L^2_TH^1_{x_1}H^m_{\bar{x}}L^2_{v, D}} \left\|w_{q, \theta}g\right\|_{L^\infty_TH^1_{x_1}H^m_{\bar{x}}L^2_{v}}, \end{eqnarray*} $

宏观估计有

$ \begin{eqnarray*} \sum\limits_{ \alpha\in {\Lambda}_{m}}\| \partial^{ \alpha}[a, b, c]\|_{L^2_TL^2_{x}}{\nonumber} &\lesssim&\sum\limits_{ \alpha\in {\Lambda}_{m}}\|\left\{\bf I-\bf P_{0}\right\} \partial^{ \alpha} f\|_{L^2_TL^{2}_{x}L^2_{v, D}} +\sum\limits_{ \alpha\in {\Lambda}_{m}}\| \partial^{ \alpha}f\|_{L^\infty_TL^2_{x, v}} {\nonumber} \\&\quad&+\sum\limits_{ \alpha\in {\Lambda}_{m}}\| \partial^{ \alpha}f_0\|_{L^2_{x, v}}+\sum\limits_{ \alpha\in {\Lambda}_{m}}\left(\int_0^T\left\|\left({ \partial^ \alpha \Gamma(f, f)}, \mu^{\frac{1}{4}}\right)_{L^{2}_{v}}\right\|^2{\rm d}t\right)^{1/2} {\nonumber} \\&\quad&+\sum\limits_{ \alpha\in {\Lambda}_{m}}E({ \partial^{ \alpha}g_\pm}) +\sum\limits_{ \alpha\in {\Lambda}_{m}}|\Upsilon_{T}^+({ \partial^ \alpha f})| +\sum\limits_{ \alpha\in {\Lambda}_{m}}\left|\Upsilon_{T}^-\left(\frac{ \partial^{ \alpha}\Gamma(f, f)}{|v_1|}\right)\right|. \end{eqnarray*} $

方程(1.4)两边同时作用$ \partial^{ \alpha} $得

(4.7) $ \begin{equation} \partial_t \partial^{ \alpha}f+v\cdot \nabla_{x} \partial^{ \alpha} f+L \partial^{ \alpha} f = \partial^{ \alpha} \Gamma(f, f). \end{equation} $

(4.7)式与$ w_{q, \theta}^{2} \partial^{ \alpha}f $在$ (x, v) $上作内积,类似于定理1.1中全局存在性的证明,由引理5.2得到定理1.2成立.

附录中列出了线性算子$ L $的基本估计,它们的证明可以参见文献[2, p400,推论1,引理5]和[5, p323,引理9].

引理5.1 设$ \gamma\geq-3 $,则存在常数$ \delta_0, \ C>0 $,使得

$ \delta_0|\left\{{\bf I}-{\bf P_{0}}\right\}g|_{D}^2\leq ( Lg, g)_{L^2_v} $

和

$ |g|_{D}^2\geq C\left\{|\langle v\rangle^{\frac{ \gamma}{2}}\left\{{\bf P}_v \partial_jg\right\}|_2^2 +|\langle v\rangle^{\frac{ \gamma+2}{2}}\left\{({\bf I}-{\bf P}_v) \partial_jg\right\}|_2^2 +|\langle v\rangle^{\frac{ \gamma+2}{2}}g|_2^2\right\} $

成立,其中映射$ {\bf P}_v $被定义为:对任意的向量值函数$ h(v) = [h_1(v), h_2(v), h_3(v)] $,有

$ {\bf P}_vh_j = \sum\limits_{m = 1}^{3} {h_mv_m}\frac{v_j}{|v|^2}, \ \ j\in {1, 2, 3}. $

引理5.2(加权估计) 设$ Lf = -\mu^{-\frac{1}{2}}\left\{Q(\mu, \mu^{\frac{1}{2}}f)+Q(\mu^{\frac{1}{2}}f, \mu)\right\} $,有

$ \begin{eqnarray*} ( Lg, w^{2}_{q, \theta}g)_{L^2_v}\geq \delta_q|w_{q, \theta}g|^2_{D} -C|g|_{L^2({B_R})}^2, \end{eqnarray*} $

其中$ \delta_q, C>0 $, $ B_R $是$ {\Bbb R}^3_v $中以原点为圆心半径$ R>0 $的闭球,并且$ (q, \theta) $满足(H).