1 引言
液晶介于液体与晶体之间, 其性质涉及物理、化学、材料等众多学科. 向列型液晶的特点是分子具有长程取向有序, 局部区域的分子取沿同一方向排列, 其应用最为广泛, 理论发展也较为完善, 其中最为著名的就是Ericksen-Lesile系统. Lin[1 ] 提出一般Ericksen-Lesile系统的简化, 该模型成功模拟向列型液晶的各种动力学行为, 宏观地描述了材料在流体影响下的演变速度场; 此外, 它还提供了棒状液晶的流体微观取向的宏观描述. 本文将从数学角度研究如下简化的Ericksen-Lesile系统
{ u ⋅ ∇ u + ∇ P = Δ u − ∇ ⋅ ( ∇ d ⊙ ∇ d ) , u ⋅ ∇ d = Δ d + | ∇ d | 2 d , ∇ ⋅ u = 0 ,
(1.1)
其中 u : R 3 → R 3 表示速度场, d : R 3 → S 2 表示液晶内部分子朝向, S 2 是 R 3 中球心在原点的单位球面. 向量 ∇ ⋅ ( ∇ d ⊙ ∇ d ) 的第 i 个分量为 3 ∑ j = 1 ∂ j ( ∂ i d ∂ j d ) .
该系统可以看作一个不可压缩Navier-Stokes方程与一个调和映照热流方程的耦合, Navier-Stokes方程决定了液晶的运动, 调和映照方程则控制内部分子的朝向. 当 ∇ d = 0 时, 这个系统简化为Navier-Stokes方程
目前关于稳态流体的Liouville定理引起了广泛的关注. 对于Navier-Stokes方程的一个公开问题是如果方程(1.2)的解满足条件
那么是否有u = 0 ?需要说明的是满足条件(1.3)的解称为D-解. 关于这个公开问题已经有了一些结果[2 ⇓ ⇓ -5 ] . 其中, Galdi[2 ] 证明了当 u \in {L^{\frac{9}{2}}}({\mathbb{R}^3}) 时, u \equiv 0 . Chae-Wolf[3 ] 将文献[2 ]条件推广到了如下情形
\int_{{\mathbb{R}^3}} |u|{^{\frac{9}{2}}}{\left\{ \log \left(2 + \frac{1}{{|u|}}\right) \right\} ^{ - 1}}{\rm d}x < \infty.
Seregin[4 ] 证明了速度场满足 u \in {L^6}({\mathbb{R}^3}) \cap BM{O^{ - 1}}({\mathbb{R}^3}) , 则Liouville定理也成立. 最近, Chae[5 ] 证明了若
u \in {L^6}({\mathbb{R}^3}) \cap {L^q}({\mathbb{R}^3}), {u_i} \in L_{{x_i}}^{\frac{q}{{q - 2}}}L_{{{\tilde x}_i}}^s(\mathbb{R} \times {\mathbb{R}^2}), \forall i = 1,2,3,
\frac{2}{q} + \frac{1}{s} \ge \frac{1}{2}, 2 < q < \infty, 1 \le s \le \infty.
对于液晶系统(1.1), 在二维空间下, 文献[6 ⇓ ⇓ -9 ]得到了关于初边值问题Leray-Hopf弱解的全局存在性. 对于三维空间, Lin-Wang[10 ] 证明了在条件满足{d_0} \in S_ + ^2 ( {d_0} 表示在上半球面取值)时, 初边值问题Leray-Hopf弱解的全局存在性. 目前, 对于系统(1.1)Liouville定理的研究相对较少[11 ,12 ] . Jarríin[11 ] 证明了此系统在局部Morrey空间中的Liouville定理. Hao[12 ] 证明了 u \in D_0^1, d \in {L^\infty } \cap \mathring{H} , 且满足条件 |u| + |\nabla d| \in {L^q}({\mathbb{R}^3}) 时, 可以得到 u = 0, \nabla d = 0 .
\begin{align*} D_0^1={\{u \in L^{6} ({\mathbb{R}^3})| \|\nabla u \| _{L^{2}({\mathbb{R}^3})} < \infty\}},\\ \mathring{H}={\{d \in L_{loc}^1({\mathbb{R}^3})} | \|\nabla d \| _{L^{2}({\mathbb{R}^3})} < \infty\}. \end{align*}
文献[11 ,12 ]对于液晶系统的研究结果是各向同性的. 受文献[5 ,11 ,12 ]启发, 本文主要研究关于速度各分量各向异性的Liouville定理.
定理 1.1 设 2 < q < \infty,1 \le s \le \infty 且 \frac{2}{q} + \frac{1}{s} \ge \frac{1}{2} . 若 u 和 d 是系统(1.1)的解, 满足条件
\begin{align*} & u \in{{L^6}({\mathbb{R}^3})}\cap {L^q}({\mathbb{R}^3}), \nabla d \in {{L^2}({\mathbb{R}^3})}\cap{L^q}({\mathbb{R}^3}),\\ &{u_i} \in L_{{x_i}}^{\frac{q}{{q - 2}}}L_{{{\tilde x}_i}}^s(\mathbb{R} \times {\mathbb{R}^2}), \forall i = 1,2,3, \end{align*}
u = 0, \nabla d = 0.
注 1.1 上述情况的一个特例, 当 s=q=6 时, 定理 1.1 中关于 u, \nabla d 的条件转化为
\begin{aligned} {u_i} \in (L_{{x_i}}^{\frac{{\rm{3}}}{{\rm{2}}}} \cap L_{{x_i}}^{\rm{6}})L_{{{\tilde x}_i}}^{\rm{6}}(\mathbb{R} \times {\mathbb{R}^2}), \nabla d\in {L^{\rm{6}}}({\mathbb{R}^3}) \cap {L^2}({\mathbb{R}^3}). \end{aligned}
(1.4)
\begin{aligned}{u \in {L^6}({\mathbb{R}^3}) \cap {L^{\frac{{\rm{9}}}{{\rm{2}}}}}({\mathbb{R}^3}), \nabla u \in {L^2}({\mathbb{R}^3}), \nabla d\in{L^{\frac{9}{2}}}({\mathbb{R}^3}) \cap {L^2}({\mathbb{R}^3})\cap L_{loc}^1} ({\mathbb{R}^3}),\end{aligned}
(1.5)
2 预备知识
{\delta _{jk}} = \left\{ \begin{array}{l} 1, {j} = {k},\\ 0, {j} \ne {k}. \end{array} \right.
对于 \Omega \subset {\mathbb{R}^2} , 记
\int_\Omega f {\rm d}{{\tilde x}_{\rm{1}}} = \int_\Omega f {\rm d}{x_2}{\rm d}{x_3}, \int_\Omega f {\rm d}{{\tilde x}_{\rm{2}}} = \int_\Omega f {\rm d}{x_3}{\rm d}{x_1}, \int_\Omega f {\rm d}{{\tilde x}_{\rm{3}}} = \int_\Omega f {\rm d}{x_1}{\rm d}{x_2},
其中{{\tilde x}_{\rm{1}}}: = ({x_2},{x_3}), {{\tilde x}_{\rm{2}}}: = ({x_3},{x_1}), {{\tilde x}_{\rm{3}}}: = ({x_1},{x_2}).
定义 2.1 设 1 \leq r,s \le + \infty,i \in \{ 1,2,3\} , 若
\| f \| _{L_{{x_i}}^rL_{{{\tilde x}_i}}^s}: = {\left\{ {\int_\mathbb{R} {{{\left( {\int_{{\mathbb{R}^2}} | f{|^s}{\rm d}{{\tilde x}_i}} \right)}^{\frac{r}{s}}}} {\rm d}x_i} \right\}^{\frac{1}{r}}} < + \infty.
则称f \in L_{{x_i}}^rL_{{{\tilde x}_i}}^s(\mathbb{R} \times {\mathbb{R}^2}). 此处, r 或 s=\infty 时, 可以理解为本性上界.
定义 2.2 取一个光滑的实函数 \psi :[\infty ) \to [0,1] , 满足
\psi (s) = \left\{ \begin{array}{l} 1, 0 \le s \le 1,\\ 0, s \ge 4. \end{array} \right.
对于集合{{\cal J}_1} = \{ 2,3\}, \quad {{\cal J}_2} = \{ {\rm{1}},3\}, \quad {{\cal J}_3} = \{ {\rm{1,2}}\}, 当 R > 0 , 定义截断函数
{\varphi _R}(x) = \prod\limits_{j = 1}^3 \psi \left( {\frac{{x_j^2}}{{{R^2}}}} \right), {\tilde \varphi _{i,R}}(x) = \prod\limits_{j \in {{\cal J}_i}} \psi \left( {\frac{{x_j^2}}{{{R^2}}}} \right),\quad i \in \{ 1,2,3\}.
{D_i}: = \left\{ {{{\tilde x}_i} \in {\mathbb{R}^2}\big||{x_j}| < 2R,\quad \forall j \in {{\cal J}_i}} \right\},\quad i \in \left\{ {1,2,3} \right\}.
3 定理1.1的证明
证 将 (1.1)1 式两边同乘以 u{\varphi _R} 并在 \mathbb{R}^3 上积分, 得
\begin{equation} \underbrace{- \int_{{\mathbb{R}^3}} \Delta u \cdot {\varphi _R}u{\rm d}x}_{ J_1}+ \underbrace{\int_{{\mathbb{R}^3}} {{\mathop{\rm div}\nolimits} } (\nabla d \odot \nabla d) \cdot {\varphi _R}u{\rm d}x}_{J_2}+ \underbrace{\int_{{\mathbb{R}^3}} {u \cdot \nabla u \cdot {\varphi _R}u{\rm d}x}}_{J_3}+ \underbrace{\int_{{\mathbb{R}^3}} {\nabla P} \cdot {\varphi _R}u{\rm d}x}_{J_4}=0, \label{1} \end{equation}
(3.1)
下面分别处理 J_{1},J_{2},J_{3} 和 J_{4}. 首先, 由分部积分公式, 得
\begin{matrix} J_1 &= - \sum\limits_{i,j = 1}^3 {\int_{{\mathbb{R}^3}} {(\partial _j^2{u_i})({\varphi _R}{u_i})} } {\rm d}x \\ &= \frac{1}{2}\sum\limits_{i,j = 1}^3 {\int_{{\mathbb{R}^3}} {({\partial _j}{\varphi _R})} } {\partial _j}(u_i^2){\rm d}x + \int_{\mathbb{R}^3} | \nabla u{|^2}{\varphi _R}{\rm d}x \\ &= - \frac{1}{2}\int_{{\mathbb{R}^3}} | u{|^2}\Delta {\varphi _R}{\rm d}x + \int_{{\mathbb{R}^3}} | \nabla u{|^2}{\varphi _R}{\rm d}x. \label{x1} \end{matrix}
(3.2)
对于 J_{2} , 由 \nabla \cdot u = 0 , 得到
\begin{matrix} J_2 &= \sum\limits_{i,j = 1}^3 \int_{{\mathbb{R}^3}} {{u_i}{\partial _j}({\partial _i}d \cdot {\partial _j}d){\varphi _R}{\rm d}x} \\ &=\sum\limits_{i,j = 1}^3 \int_{{\mathbb{R}^3}} {{u_i}({\partial _i}{\partial _j}d \cdot {\partial _j}d + {\partial _i}d \cdot \partial _j^2d){\varphi _R}{\rm d}x} \\ &=\sum\limits_{i,j = 1}^3 \int_{{\mathbb{R}^3}} {\frac{1}{2}{u_i}{\partial _i}|\nabla d{|^2}{\varphi _R} + (u \cdot \nabla d) \cdot \Delta d{\varphi _R}{\rm d}x}.\label{x2} \end{matrix}
(3.3)
对于 J_{3} , 同样由分部积分公式及 \nabla \cdot u = 0 , 可得
\begin{equation} J_3 = \frac{1}{2}\int_{{\mathbb{R}^3}} u \cdot \nabla |u{|^2}{\varphi _R}{\rm d}x =- \frac{1}{2}\int_{{\mathbb{R}^3}}| u{|^2}u \cdot \nabla {\varphi _R}{\rm d}x.\label{x3} \end{equation}
(3.4)
\begin{equation} J_4={ \sum\limits_{i = 1}^3 {\int_{{\mathbb{R}^3}} {({\partial _i}P)} } {\varphi _R}{u_i}{\rm d}x = - \int_{{\mathbb{R}^3}} P (u \cdot \nabla {\varphi _R}){\rm d}x}. \label{x4} \end{equation}
(3.5)
将(3.2), (3.3), (3.4)和(3.5)式代入(3.1)式得
\begin{matrix} \int_{{\mathbb{R}^3}} {(|\nabla u{|^2} + |\Delta d{|^2})} {\varphi _R}{\rm d}x = &\int_{{\mathbb{R}^3}} {\left(\frac{1}{2}|u{|^2} + \frac{1}{2}|\nabla d{|^2} + P\right)(u \cdot \nabla {\varphi _R})} {\rm d}x \\ &+ \int_{{\mathbb{R}^3}} {\frac{1}{2}|u{|^2}\Delta {\varphi _R}{\rm d}x} - \int_{{\mathbb{R}^3}} {(u \cdot \nabla d) \cdot \Delta {\rm d}{\varphi _R}} {\rm d}x.\label{4} \end{matrix}
(3.6)
将(1.1)2 式两边同乘以 - \Delta d{\varphi _R} 并在 \mathbb{R}^3 上积分, 得
\begin{equation} \int_{{\mathbb{R}^3}} {|\Delta d{|^2}{\varphi _R}{\rm d}x} = - \int_{{\mathbb{R}^3}} {|\nabla d{|^2}d \cdot \Delta {\rm d}{\varphi _R}{\rm d}x} + \int_{{\mathbb{R}^3}} {u \cdot \nabla d \cdot \Delta {\rm d}{\varphi _R}{\rm d}x}. \label{2} \end{equation}
(3.7)
\begin{matrix} \int_{{\mathbb{R}^3}} {(|\nabla u{|^2} + |\Delta d{|^2})} {\varphi _R}{\rm d}x = & \int_{{\mathbb{R}^3}} {\left(\frac{1}{2}|u{|^2} +\frac{1}{2}|\nabla d{|^2} + P\right)(u \cdot \nabla {\varphi _R})} {\rm d}x \\ & + \int_{{\mathbb{R}^3}} {\frac{1}{2}|u{|^2}\Delta {\varphi _R}{\rm d}x} - \int_{{\mathbb{R}^3}} | \nabla |d{|^2}(d \cdot \Delta d){\varphi _R}{\rm d}x. \label{3} \end{matrix}
(3.8)
由于 |d|{\rm{ = 1}} , 可得 \Delta |d{|^{\rm{2}}}{\rm{ = 0}} , 从而
\begin{matrix} - |\nabla d{|^2} &= - \sum\limits_{i,j = 1}^3 {({\partial _i}{d_j})^2} + \frac{1}{2}\Delta |d{|^2} = - \sum\limits_{i,j = 1}^3 {({\partial _i}{d_j})^2} + \sum\limits_{i,j = 1}^3 \frac{1}{2}{\partial _j}{\partial _j}({d_i}^2) \\ &= - \sum\limits_{i,j = 1}^3 {({\partial _i}{d_j})^2} +\sum\limits_{i,j = 1}^3 \frac{1}{2}{\partial _j}(2({\partial _j}{d_i}){d_i}) \\ & = - \sum\limits_{i,j = 1}^3 {({\partial _i}{d_j})^2} +\sum\limits_{i,j = 1}^3 (({\partial _j}{\partial _j}{d_i}){d_i} + {({\partial _j}d)^2}) \\ &= \sum\limits_{i,j = 1}^3 ({\partial _j}{\partial _j}{d_i}){d_i}= d \cdot\Delta d.\label{d} \end{matrix}
(3.9)
\begin{matrix} - \int_{{\mathbb{R}^3}} {|\nabla d{|^2}(d \cdot \Delta d)} {\varphi _R}{\rm d}x &= \int_{{\mathbb{R}^3}} {|d \cdot \Delta d{|^2}} {\varphi _R}{\rm d}x \\ & \le \int_{{\mathbb{R}^3}} {|d{|^2}|\Delta d{|^2}{\varphi _R}{\rm d}x \le } \int_{{\mathbb{R}^3}} {|\Delta d{|^2}{\varphi _R}{\rm d}x}.\label{e} \end{matrix}
(3.10)
\begin{matrix} \int_{{\mathbb{R}^3}} {|\nabla u{|^2}} {\varphi _R}{\rm d}x \le &\int_{{\mathbb{R}^3}} {\left(\frac{{|u{|^2}}}{2} + P\right)(u \cdot \nabla {\varphi _R}){\rm d}x} \\ &+ \frac{1}{2}\int_{{\mathbb{R}^3}} {|u{|^2}\Delta {\varphi _R}} {\rm d}x + \frac{1}{2}\int_{{\mathbb{R}^3}} {|\nabla d{|^2}(u \cdot \nabla {\varphi _R}){\rm d}x}.\label{Z} \end{matrix}
(3.11)
下面对压力项 P 作估计. 将(1.1)1 式同时作用散度算子, 由于 \nabla \cdot u = 0 , 可得
{\rm{div}}\left( {\left( {u \cdot \nabla } \right)u} \right) + {\rm{div}}\left( {{\rm{div}}\left( {\nabla d \odot \nabla d} \right)} \right) + \Delta P = 0.
将 (u \cdot \nabla )u 和 {\mathop{\rm div}\nolimits} (\nabla d \odot \nabla d) 分别用分量表示为
{[(u \cdot \nabla )u]_i} = \sum\limits_{j = 1}^3 {{\partial _j}} ({u_j}{u_i}), {[{\mathop{\rm div}\nolimits} (\nabla d \odot \nabla d)]_i} = \sum\limits_{k = 1}^3 {\sum\limits_{j = 1}^3 {{\partial _j}({\partial _i}{d_k}{\partial _j}{d_k}), i = 1,2,3} }.
因此, 由Riesz算子 {{\cal R}_i}=\frac{{{\partial _i}}}{{\sqrt { - \Delta } }} , 有
\begin{matrix} P &= \sum\limits_{i = 1}^3 {\sum\limits_{i = j}^3 {\frac{1}{{ - \Delta }}} } {{\partial _i}{\partial _j}({u_j}{u_i})} + \sum\limits_{i = 1}^3 {\sum\limits_{j = 1}^3 {\sum\limits_{k = 1}^3 {\frac{1}{{ - \Delta }} {{\partial _i}{\partial _j}({\partial _i}{d_k}{\partial _j}{d_k})} } } } \\ &= \sum\limits_{i = 1}^3 {\sum\limits_{i = j}^3 { {\frac{{{\partial _i}}}{{\sqrt { - \Delta } }}\frac{{{\partial _j}}}{{\sqrt { - \Delta } }}({u_j}{u_i})} } } + \sum\limits_{i = 1}^3 {\sum\limits_{j = 1}^3 {\sum\limits_{k = 1}^3 { {\frac{{{\partial _i}}}{{\sqrt { - \Delta } }}\frac{{{\partial _j}}}{{\sqrt { - \Delta } }}({\partial _i}{d_k}{\partial _j}{d_k})} } } } \\ &= \sum\limits_{i = 1}^3 {\sum\limits_{i = j}^3 { {{{\cal R}_i}{{\cal R}_j}({u_j}{u_i})} } } + \sum\limits_{i = 1}^3 {\sum\limits_{j = 1}^3 {\sum\limits_{k = 1}^3 { {{{\cal R}_i}{{\cal R}_j}({\partial _i}{d_k}{\partial _j}{d_k})} } } } \\ & =: P_{1} + P_{2}\label{P}. \end{matrix}
(3.12)
由标准的Calderon-Zygmund不等式[13 ] , 存在常数 C , 使得
\begin{array}{l} \| {P_1}\| _{{L^q}({\mathbb{R}^3})} \le C\| u\| _{{L^{2q}}({\mathbb{R}^3})}^2\quad \forall q \in (1, + \infty ),\\ \| {P_2}\| _{{L^q}({\mathbb{R}^3})} \le C\| \nabla d\| _{{L^{2q}}({\mathbb{R}^3})}^2\quad \forall q \in (1, + \infty ). \end{array}
\| P\| _{{L^q}({\mathbb{R}^3})} \le \| P_1\| _{{L^q}({\mathbb{R}^3})}{\rm{ + }}\| {P_2}\| _{{L^q}({\mathbb{R}^3})} \le C\| u\| _{{L^{2q}}({\mathbb{R}^3})}^2{\rm{ + }}C\| \nabla d\| _{{L^{2q}}({\mathbb{R}^3})}^2, \forall q \in (1, + \infty ).
记 Q=\frac{{|u{|^2}}}{2} + {P_1} , 则
\begin{array}{l} \| {Q}{\| _{{L^q}({\mathbb{R}^3})}} \le C\| u\| _{{L^{2q}}({\mathbb{R}^3})}^2, \quad \forall q \in (1, + \infty ).\\ \end{array}
\begin{matrix} \int_{{\mathbb{R}^3}} {|\nabla u{|^2}} {\varphi _R}{\rm d}x & \le \frac{1}{2}\int_{{\mathbb{R}^3}} {|u{|^2}\Delta {\varphi _R}} {\rm d}x + \int_{{\mathbb{R}^3}} {\left(\frac{{|u{|^2}}}{2} + {P_1}\right)(u \cdot \nabla {\varphi _R}){\rm d}x} \\ & \ + \int_{{\mathbb{R}^3}} {{P_2}(u \cdot \nabla {\varphi _R}){\rm d}x} + \frac{1}{2}\int_{{\mathbb{R}^3}} {|\nabla d{|^2}(u \cdot \nabla {\varphi _R}){\rm d}x} \\ & \le \frac{1}{2}\int_{{\mathbb{R}^3}} {|u{|^2}\Delta {\varphi _R}} {\rm d}x + \int_{{\mathbb{R}^3}} {Q(u \cdot \nabla {\varphi _R}){\rm d}x} + \int_{{\mathbb{R}^3}} {{P_2}(u \cdot \nabla {\varphi _R}){\rm d}x} \\ & \ + \frac{1}{2}\int_{{\mathbb{R}^3}} {|\nabla d{|^2}(u \cdot \nabla {\varphi _R}){\rm d}x} =: \sum\limits_{i = 1}^4 {{I_i}}. \end{matrix}
(3.13)
下面分别对 {I_i}\ (i = 1,2,3,4) 各项进行估计. 对于 {I_1} , 利用Hölder不等式, 有
\begin{align*} {I_1} & = \frac{1}{2}\sum\limits_{i = 1}^3 {\int_{_{ R \le |{x_i}| \le 2R }} {\int_{{D_i}} {|u{|^2}{{\tilde \varphi }_{i,R}}\left\{ {\frac{2}{{{R^2}}}\psi '\left( {\frac{{x_i^2}}{{{R^2}}}} \right) + \frac{{4\tilde x_i^2}}{{{R^4}}}\psi ''\left( {\frac{{x_i^2}}{{{R^2}}}} \right)} \right\}{\rm d}{{\tilde x}_i}{\rm d}x_i} } } \\ & \le \frac{C}{{{R^2}}}\sum\limits_{i = 1}^3 {{{\left( {\int_{ R \le |{x_i}| \le 2R } {\int_{{D_i}} {|u{|^6}} } {\rm d}x} \right)}^{\frac{1}{3}}}} {\left( {\int_{ R \le |{x_i}| \le 2R } {\int_{{D_i}} 1{\rm d}{{\tilde x}_i}{\rm d}x_i } } \right)^{\frac{2}{3}}}\\ & \le \frac{C}{{{R^2}}}\sum\limits_{i = 1}^3 {{{\left( {\int_{ R \le |{x_i}| \le 2R } {\int_{{\mathbb{R}^2}} {|u{|^6}} } {\rm d}x} \right)}^{\frac{1}{3}}}}. \end{align*}
当 R\rightarrow+\infty 时, 则 I_{1}\rightarrow0.
\begin{align*} {I_2} &\le \sum\limits_{i = 1}^3 {\frac{2}{{{R^2}}}} \left| {\int_{{\mathbb{R}^3}} {{x_i}} {{\tilde \varphi }_{i,R}}Q{u_i} \cdot \psi '\left( {\frac{{x_i^2}}{{{R^2}}}} \right){\rm d}x} \right|\\ & \le \frac{C}{R}\sum\limits_{i = 1}^3 {\int_{ R < |{x_i}| < 2R} {\int_{{D_i}} | } } Q||{u_i}|{\rm d}{{\tilde x}_i}{\rm d}x_i\\ & \le \frac{C}{R}\sum\limits_{i = 1}^3 {\int_{ R < |{x_i}| < 2R} {{{\left( {\int_{{D_i}} | Q{|^{\frac{q}{2}}}{\rm d}{{\tilde x}_i}} \right)}^{\frac{2}{q}}}} } {\left( {\int_{{D_i}} {|{u_i}{|^s}{\rm d}{{\tilde x}_i}} } \right)^{\frac{1}{s}}}{\left( {\int_{D_{i}} {1{\rm d}{{\tilde x}_i}} } \right)^{\frac{{qs - q - 2s}}{{qs}}}}{\rm d}x_i\\ &\le C{R^{\frac{{2(qs - q - 2s)}}{{qs}} - 1}}\sum\limits_{i = 1}^3 {\int_{ R < |{x_i}| < 2R} {{{\left( {\int_{{D_i}} | Q{|^{\frac{q}{2}}}{\rm d}{{\tilde x}_i}} \right)}^{\frac{2}{q}}}} } {\left( {\int_{{D_i}} | {u_i}{|^s}{\rm d}{{\tilde x}_i}} \right)^{\frac{1}{s}}}{\rm d}x_i\\ & \le C{R^{\frac{{2(qs - q - 2s)}}{{qs}} - 1}}\\ & \ \times \sum\limits_{i = 1}^3 {{{\left( {\int_{ R < |{x_i}| < 2R} {\int_{{D_i}} | } Q{|^{\frac{q}{2}}}{\rm d}{{\tilde x}_i}{\rm d}x_i} \right)}^{\frac{2}{q}}}} {\left\{ {\int_{ R < |{x_i}| < 2R} {{{\left( {\int_{{D_i}} | {u_i}{|^s}{\rm d}{{\tilde x}_i}} \right)}^{\frac{q}{{s(q - 2)}}}}} {\rm d}x_i} \right\}^{\frac{{q - 2}}{q}}}\\ & \le C{R^{\frac{{2(qs - q - 2s)}}{{qs}} - 1}} {{{\left( {\int_{{\mathbb{R}^3}} {|Q{|^{\frac{q}{2}}}{\rm d}x} } \right)}^{\frac{2}{q}}}} {\sum\limits_{i = 1}^3 {\left\{ {\int_{ R < |{x_i}| < 2R} {{{\left( {\int_{{\mathbb{R}^2}} | {u_i}{|^s}{\rm d}{{\tilde x}_i}} \right)}^{\frac{q}{{s(q - 2)}}}}} {\rm d}x_i} \right\}} ^{\frac{{q - 2}}{q}}}\\ & \le C{R^{\frac{{2(qs - q - 2s)}}{{qs}} - 1}} {{{\left( {\int_{{\mathbb{R}^3}} {|u{|^q}{\rm d}x} } \right)}^{\frac{2}{q}}}} {\sum\limits_{i = 1}^3 {\left\{ {\int_{ R < |{x_i}| < 2R} {{{\left( {\int_{{\mathbb{R}^2}} | {u_i}{|^s}{\rm d}{{\tilde x}_i}} \right)}^{\frac{q}{{s(q - 2)}}}}} {\rm d}x_i} \right\}} ^{\frac{{q - 2}}{q}}}. \end{align*}
对于 {I_3} 和 {I_4} , 类似于 {I_2} 的估计, 有
\begin{align*} {I_3} &\le \sum\limits_{i = 1}^3 {\frac{2}{{{R^2}}}} \left| {\int_{{\mathbb{R}^3}} {{x_i}} {{\tilde \varphi }_{i,R}}{P_2}{u_i} \cdot \psi '\left( {\frac{{x_i^2}}{{{R^2}}}} \right){\rm d}x} \right|\\ & \le \frac{C}{R}\sum\limits_{i = 1}^3 {\int_{ R < |{x_i}| < 2R} {\int_{{D_i}} | } } {P_2}||{u_i}|{\rm d}{{\tilde x}_i}{\rm d}x_i\\ & \le \frac{C}{R}\sum\limits_{i = 1}^3 {\int_{ R < |{x_i}| < 2R} {{{\left( {\int_{{D_i}} | {P_2}{|^{\frac{q}{2}}}{\rm d}{{\tilde x}_i}} \right)}^{\frac{2}{q}}}} } {\left( {\int_{{D_i}} {|{u_i}{|^s}{\rm d}{{\tilde x}_i}} } \right)^{\frac{1}{s}}}{\left( {\int_{D_{i}} {1{\rm d}{{\tilde x}_i}} } \right)^{\frac{{qs - q - 2s}}{{qs}}}}{\rm d}x_i\\ & \le C{R^{\frac{{2(qs - q - 2s)}}{{qs}} - 1}}\sum\limits_{i = 1}^3 {\int_{ R < |{x_i}| < 2R} {{{\left( {\int_{{D_i}} | {P_2}{|^{\frac{q}{2}}}{\rm d}{{\tilde x}_i}} \right)}^{\frac{2}{q}}}} } {\left( {\int_{{D_i}} | {u_i}{|^s}{\rm d}{{\tilde x}_i}} \right)^{\frac{1}{s}}}{\rm d}x_i\\ & \le C{R^{\frac{{2(qs - q - 2s)}}{{qs}} - 1}}\\ & \ \times \sum\limits_{i = 1}^3 {{{\left( {\int_{ R < |{x_i}| < 2R} {\int_{{D_i}} | } {P_2}{|^{\frac{q}{2}}}|{\rm d}{{\tilde x}_i}{\rm d}x_i} \right)}^{\frac{2}{q}}}} {\left\{ {\int_{ R < |{x_i}| < 2R} {{{\left( {\int_{{D_i}} | {u_i}{|^s}{\rm d}{{\tilde x}_i}} \right)}^{\frac{q}{{s(q - 2)}}}}} {\rm d}x_i} \right\}^{\frac{{q - 2}}{q}}}\\ & \le C{R^{\frac{{2(qs - q - 2s)}}{{qs}} - 1}} {{{\left( {\int_{{\mathbb{R}^3}} {|{P_2}{|^{\frac{q}{2}}}{\rm d}x} } \right)}^{\frac{2}{q}}}} {\sum\limits_{i = 1}^3 {\left\{ {\int_{ R < |{x_i}| < 2R} {{{\left( {\int_{{D_i}} | {u_i}{|^s}{\rm d}{{\tilde x}_i}} \right)}^{\frac{q}{{s(q - 2)}}}}} {\rm d}x_i} \right\}} ^{\frac{{q - 2}}{q}}}\\ & \le C{R^{\frac{{2(qs - q - 2s)}}{{qs}} - 1}} {{{\left( {\int_{{\mathbb{R}^3}} {|\nabla d{|^q}{\rm d}x} } \right)}^{\frac{2}{q}}}} {\sum\limits_{i = 1}^3 {\left\{ {\int_{ R < |{x_i}| < 2R} {{{\left( {\int_{{D_i}} | {u_i}{|^s}{\rm d}{{\tilde x}_i}} \right)}^{\frac{q}{{s(q - 2)}}}}} {\rm d}x_i} \right\}} ^{\frac{{q - 2}}{q}}}. \\ {I_4}&\le \sum\limits_{i = 1}^3 {\frac{2}{{{R^2}}}} \left| {\int_{{\mathbb{R}^3}} {{x_i}} {{\tilde \varphi }_{i,R}}|\nabla d{|^2}{u_i} \cdot \psi '\left( {\frac{{x_i^2}}{{{R^2}}}} \right){\rm d}x} \right|\\ & \le \frac{C}{R}\sum\limits_{i = 1}^3 {\int_{ R < |{x_i}| < 2R} {\int_{{D_i}} | } } \nabla d{|^2}|{u_i}|{\rm d}{{\tilde x}_i}{\rm d}x_i\\ &\le \frac{C}{R}\sum\limits_{i = 1}^3 {\int_{ R < |{x_i}| < 2R} {{{\left( {\int_{{D_i}} | \nabla d{|^q}{\rm d}{{\tilde x}_i}} \right)}^{\frac{2}{q}}}} } {\left( {\int_{{D_i}} {|{u_i}{|^s}{\rm d}{{\tilde x}_i}} } \right)^{\frac{1}{s}}}{\left( {\int_{D_{i}} {1{\rm d}{{\tilde x}_i}} } \right)^{\frac{{qs - q - 2s}}{{qs}}}}{\rm d}x_i\\ & \le C{R^{\frac{{2(qs - q - 2s)}}{{qs}} - 1}}\sum\limits_{i = 1}^3 {\int_{ R < |{x_i}| < 2R} {{{\left( {\int_{{D_i}} | \nabla d{|^q}{\rm d}{{\tilde x}_i}} \right)}^{\frac{2}{q}}}} } {\left( {\int_{{D_i}} | {u_i}{|^s}{\rm d}{{\tilde x}_i}} \right)^{\frac{1}{s}}}{\rm d}x_i\\ &\le C{R^{\frac{{2(qs - q - 2s)}}{{qs}} - 1}}\\ & \ \times\sum\limits_{i = 1}^3 {{{\left( {\int_{ R < |{x_i}| < 2R} {\int_{{D_i}} | } \nabla d{|^q}{\rm d}{{\tilde x}_i}{\rm d}x_i} \right)}^{\frac{2}{q}}}} {\left\{ {\int_{ R < |{x_i}| < 2R} {{{\left( {\int_{{D_i}} | {u_i}{|^s}{\rm d}{{\tilde x}_i}} \right)}^{\frac{q}{{s(q - 2)}}}}} {\rm d}x_i} \right\}^{\frac{{q - 2}}{q}}}\\ & \le C{R^{\frac{{2(qs - q - 2s)}}{{qs}} - 1}} {{{\left( {\int_{{\mathbb{R}^3}} {|\nabla d{|^q}{\rm d}x} } \right)}^{\frac{2}{q}}}} {\sum\limits_{i = 1}^3 {\left\{ {\int_{ R < |{x_i}| < 2R} {{{\left( {\int_{{\mathbb{R}^2}} | {u_i}{|^s}{\rm d}{{\tilde x}_i}} \right)}^{\frac{q}{{s(q - 2)}}}}} {\rm d}x_i} \right\}} ^{\frac{{q - 2}}{q}}}\\ &\le C{R^{\frac{{2(qs - q - 2s)}}{{qs}} - 1}} {{{\left( {\int_{{\mathbb{R}^3}} {|\nabla d{|^q}{\rm d}x} } \right)}^{\frac{2}{q}}}} {\sum\limits_{i = 1}^3 {\left\{ {\int_{ R < |{x_i}| < 2R} {{{\left( {\int_{{\mathbb{R}^2}} | {u_i}{|^s}{\rm d}{{\tilde x}_i}} \right)}^{\frac{q}{{s(q - 2)}}}}} {\rm d}x_i} \right\}} ^{\frac{{q - 2}}{q}}}. \end{align*}
由定理 1.1 中的条件 \frac{2}{q} + \frac{1}{s} \ge \frac{1}{2} 可知
\frac{{2(qs - q - 2s)}}{{qs}} - 1 \le 0.
当 R\rightarrow+\infty 时, 则 I_{2},I_{3},I_{4}\rightarrow0 . 从而,
\int_{{\mathbb{R}^3}} {|\nabla u{|^2}}{\rm d}x=0.
又 u\in{L^6}({\mathbb{R}^3}) , 则
u=0.
记 d_{i,j}=\frac{{\partial {d_i}}}{{\partial {x_j}}} , 将 u = 0 代入(1.1)2 式可得
\Delta d = -|\nabla d{|^2}d.
(3.14)
将(3.14)式两边同时乘以 x \cdot \nabla d{\varphi _R} 并在 {\mathbb{R}^3} 积分, 由于 {\partial _{{x_j}}}|d{|^2} =\sum\limits_{i = 1}^3 2 d_{i}d_{i,j} = 0 (j = 1,2,3), 可知
\sum\limits_{i,j = 1}^3\int_{{\mathbb{R}^3}} {\Delta {d_i}{x_j}{d_{i,j}}{\varphi _R}{\rm d}x} = -\sum\limits_{i,j = 1}^3\int_{{\mathbb{R}^3}} {|\nabla d{|^2}{x_j}{d_i}{d_{i,j}}{\varphi _R}{\rm d}x = 0,}
\begin{align*} 0&=\sum\limits_{i,j = 1}^3\int_{{\mathbb{R}^3}} \Delta {d_i}{x_j}{d_{i,j}}{\varphi _R}{\rm d}x\\ &= - \sum\limits_{i,j,k = 1}^3\int_{{\mathbb{R}^3}} {{d_{i,k}}{\delta _{j,k}}{d_{i,j}}{\varphi _R} +{d_{i,k}}{x_j}{d_{i,j}}{\partial _{{x_k}}}{\varphi _R}} + {d_{i,k}}{x_j}{d_{i,jk}}{\varphi _R}{\rm d}x\\ &{\rm{ = }} - \sum\limits_{i,j,k = 1}^3\int_{{\mathbb{R}^3}} {{d_{i,k}}{\delta _{j,k}}{d_{i,j}}{\varphi _R} +{d_{i,k}}{x_j}{d_{i,j}}{\partial _{{x_k}}}{\varphi _R}} + {\frac{{\rm{1}}}{{\rm{2}}}\partial_{x_j}{{({d_{i,k}})}^2}{x_j}{\varphi _R}}{\rm d}x\\ &{\rm{ = }} -\sum\limits_{i,j,k = 1}^3 \int_{{\mathbb{R}^3}} {\left( {|\nabla d{|^2}{\varphi _R} + {d_{i,k}}{x_j}{d_{i,j}}{\partial _{{x_k}}}{\varphi _R}} \right)} {\rm d}x\\ & \ + \sum\limits_{i,j,k = 1}^3\int_{{\mathbb{R}^3}} {\left( {\frac{3}{2}|\nabla d{|^2}{\varphi _R} + \frac{1}{2}|\nabla d{|^2}{x_j}{\partial _{{x_j}}}{\varphi _R}} \right)} {\rm d}x\\ &{\rm{ = }} \frac{1}{2}\int_{{\mathbb{R}^3}} | \nabla d{|^2}{\varphi _R}{\rm d}x + \int_{{\mathbb{R}^3}} \sum\limits_{i,j,k = 1}^3{\left(\frac{1}{2}|\nabla d{|^2}{x_j}{\partial _{{x_j}}}{\varphi _R} - {d_{i,k}}{x_j}{d_{i,j}}{\partial _{{x_k}}}{\varphi _R}\right){\rm d}x}, \end{align*}
\int_{{\mathbb{R}^3}} | \nabla d{|^2}{\varphi _R}{\rm d}x =- \sum\limits_{i,j,k = 1}^3\int_{{\mathbb{R}^3}} {\left(|\nabla d{|^2}{x_j}{\partial _{{x_j}}}{\varphi _R} - 2 {d_{i,k}}{x_j}{d_{i,j}}{\partial _{{x_k}}}{\varphi _R}\right){\rm d}x},
|{x_j}{\partial _{{x_i}}} {\varphi _k}| \le \frac{C}{R}|{x_j}| \le C, |{x_j}{\partial _{{x_k}}} {\varphi _k}| \le \frac{C}{R}|{x_j}| \le C.
\int_{{\mathbb{R}^3}} {|\nabla d{|^2}{\rm d}x} \le C\int_{R \le |x| \le 2R} {|\nabla d{|^2}{\rm d}x} \to 0.
由 \nabla d \in L^{2}(\mathbb{R}^3) , 则 \nabla d = 0.
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Nonlinear theory of defects in nematic liquid crystals; Phase transition and flow phenomena
1
1989
... 液晶介于液体与晶体之间, 其性质涉及物理、化学、材料等众多学科. 向列型液晶的特点是分子具有长程取向有序, 局部区域的分子取沿同一方向排列, 其应用最为广泛, 理论发展也较为完善, 其中最为著名的就是Ericksen-Lesile系统. Lin[1 ] 提出一般Ericksen-Lesile系统的简化, 该模型成功模拟向列型液晶的各种动力学行为, 宏观地描述了材料在流体影响下的演变速度场; 此外, 它还提供了棒状液晶的流体微观取向的宏观描述. 本文将从数学角度研究如下简化的Ericksen-Lesile系统 ...
3
2011
... 那么是否有u = 0 ?需要说明的是满足条件(1.3)的解称为D-解. 关于这个公开问题已经有了一些结果[2 ⇓ ⇓ -5 ] . 其中, Galdi[2 ] 证明了当 u \in {L^{\frac{9}{2}}}({\mathbb{R}^3}) 时, u \equiv 0 . Chae-Wolf[3 ] 将文献[2 ]条件推广到了如下情形 ...
... [2 ]证明了当 u \in {L^{\frac{9}{2}}}({\mathbb{R}^3}) 时, u \equiv 0 . Chae-Wolf[3 ] 将文献[2 ]条件推广到了如下情形 ...
... 将文献[2 ]条件推广到了如下情形 ...
On Liouville type theorems for the steady Navier-Stokes equations in \mathbb{R}^3
2
2016
... 那么是否有u = 0 ?需要说明的是满足条件(1.3)的解称为D-解. 关于这个公开问题已经有了一些结果[2 ⇓ ⇓ -5 ] . 其中, Galdi[2 ] 证明了当 u \in {L^{\frac{9}{2}}}({\mathbb{R}^3}) 时, u \equiv 0 . Chae-Wolf[3 ] 将文献[2 ]条件推广到了如下情形 ...
... [3 ]将文献[2 ]条件推广到了如下情形 ...
A Liouville type theorem for steady-state Navier-Stokes equations
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2016
... 那么是否有u = 0 ?需要说明的是满足条件(1.3)的解称为D-解. 关于这个公开问题已经有了一些结果[2 ⇓ ⇓ -5 ] . 其中, Galdi[2 ] 证明了当 u \in {L^{\frac{9}{2}}}({\mathbb{R}^3}) 时, u \equiv 0 . Chae-Wolf[3 ] 将文献[2 ]条件推广到了如下情形 ...
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Anisotropic Liouville type theorem for the stationary Navier-Stokes equations in \mathbb{R}^3
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2023
... 那么是否有u = 0 ?需要说明的是满足条件(1.3)的解称为D-解. 关于这个公开问题已经有了一些结果[2 ⇓ ⇓ -5 ] . 其中, Galdi[2 ] 证明了当 u \in {L^{\frac{9}{2}}}({\mathbb{R}^3}) 时, u \equiv 0 . Chae-Wolf[3 ] 将文献[2 ]条件推广到了如下情形 ...
... Seregin[4 ] 证明了速度场满足 u \in {L^6}({\mathbb{R}^3}) \cap BM{O^{ - 1}}({\mathbb{R}^3}) , 则Liouville定理也成立. 最近, Chae[5 ] 证明了若 ...
... 文献[11 ,12 ]对于液晶系统的研究结果是各向同性的. 受文献[5 ,11 ,12 ]启发, 本文主要研究关于速度各分量各向异性的Liouville定理. ...
Liquid crystal flows in two dimensions
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2010
... 对于液晶系统(1.1), 在二维空间下, 文献[6 ⇓ ⇓ -9 ]得到了关于初边值问题Leray-Hopf弱解的全局存在性. 对于三维空间, Lin-Wang[10 ] 证明了在条件满足{d_0} \in S_ + ^2 ( {d_0} 表示在上半球面取值)时, 初边值问题Leray-Hopf弱解的全局存在性. 目前, 对于系统(1.1)Liouville定理的研究相对较少[11 ,12 ] . Jarríin[11 ] 证明了此系统在局部Morrey空间中的Liouville定理. Hao[12 ] 证明了 u \in D_0^1, d \in {L^\infty } \cap \mathring{H} , 且满足条件 |u| + |\nabla d| \in {L^q}({\mathbb{R}^3}) 时, 可以得到 u = 0, \nabla d = 0 . ...
Global existence of solutions of the simplified Ericksen-Leslie system in dimension two
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2011
... 对于液晶系统(1.1), 在二维空间下, 文献[6 ⇓ ⇓ -9 ]得到了关于初边值问题Leray-Hopf弱解的全局存在性. 对于三维空间, Lin-Wang[10 ] 证明了在条件满足{d_0} \in S_ + ^2 ( {d_0} 表示在上半球面取值)时, 初边值问题Leray-Hopf弱解的全局存在性. 目前, 对于系统(1.1)Liouville定理的研究相对较少[11 ,12 ] . Jarríin[11 ] 证明了此系统在局部Morrey空间中的Liouville定理. Hao[12 ] 证明了 u \in D_0^1, d \in {L^\infty } \cap \mathring{H} , 且满足条件 |u| + |\nabla d| \in {L^q}({\mathbb{R}^3}) 时, 可以得到 u = 0, \nabla d = 0 . ...
Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows
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2012
... 对于液晶系统(1.1), 在二维空间下, 文献[6 ⇓ ⇓ -9 ]得到了关于初边值问题Leray-Hopf弱解的全局存在性. 对于三维空间, Lin-Wang[10 ] 证明了在条件满足{d_0} \in S_ + ^2 ( {d_0} 表示在上半球面取值)时, 初边值问题Leray-Hopf弱解的全局存在性. 目前, 对于系统(1.1)Liouville定理的研究相对较少[11 ,12 ] . Jarríin[11 ] 证明了此系统在局部Morrey空间中的Liouville定理. Hao[12 ] 证明了 u \in D_0^1, d \in {L^\infty } \cap \mathring{H} , 且满足条件 |u| + |\nabla d| \in {L^q}({\mathbb{R}^3}) 时, 可以得到 u = 0, \nabla d = 0 . ...
On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals
1
2010
... 对于液晶系统(1.1), 在二维空间下, 文献[6 ⇓ ⇓ -9 ]得到了关于初边值问题Leray-Hopf弱解的全局存在性. 对于三维空间, Lin-Wang[10 ] 证明了在条件满足{d_0} \in S_ + ^2 ( {d_0} 表示在上半球面取值)时, 初边值问题Leray-Hopf弱解的全局存在性. 目前, 对于系统(1.1)Liouville定理的研究相对较少[11 ,12 ] . Jarríin[11 ] 证明了此系统在局部Morrey空间中的Liouville定理. Hao[12 ] 证明了 u \in D_0^1, d \in {L^\infty } \cap \mathring{H} , 且满足条件 |u| + |\nabla d| \in {L^q}({\mathbb{R}^3}) 时, 可以得到 u = 0, \nabla d = 0 . ...
Global existence of weak solutions of the nematic liquid crystal flow in dimension three
1
2016
... 对于液晶系统(1.1), 在二维空间下, 文献[6 ⇓ ⇓ -9 ]得到了关于初边值问题Leray-Hopf弱解的全局存在性. 对于三维空间, Lin-Wang[10 ] 证明了在条件满足{d_0} \in S_ + ^2 ( {d_0} 表示在上半球面取值)时, 初边值问题Leray-Hopf弱解的全局存在性. 目前, 对于系统(1.1)Liouville定理的研究相对较少[11 ,12 ] . Jarríin[11 ] 证明了此系统在局部Morrey空间中的Liouville定理. Hao[12 ] 证明了 u \in D_0^1, d \in {L^\infty } \cap \mathring{H} , 且满足条件 |u| + |\nabla d| \in {L^q}({\mathbb{R}^3}) 时, 可以得到 u = 0, \nabla d = 0 . ...
Liouville theorems for a stationary and non-stationary coupled system of liquid crystal flows in local Morrey spaces
4
2022
... 对于液晶系统(1.1), 在二维空间下, 文献[6 ⇓ ⇓ -9 ]得到了关于初边值问题Leray-Hopf弱解的全局存在性. 对于三维空间, Lin-Wang[10 ] 证明了在条件满足{d_0} \in S_ + ^2 ( {d_0} 表示在上半球面取值)时, 初边值问题Leray-Hopf弱解的全局存在性. 目前, 对于系统(1.1)Liouville定理的研究相对较少[11 ,12 ] . Jarríin[11 ] 证明了此系统在局部Morrey空间中的Liouville定理. Hao[12 ] 证明了 u \in D_0^1, d \in {L^\infty } \cap \mathring{H} , 且满足条件 |u| + |\nabla d| \in {L^q}({\mathbb{R}^3}) 时, 可以得到 u = 0, \nabla d = 0 . ...
... [11 ]证明了此系统在局部Morrey空间中的Liouville定理. Hao[12 ] 证明了 u \in D_0^1, d \in {L^\infty } \cap \mathring{H} , 且满足条件 |u| + |\nabla d| \in {L^q}({\mathbb{R}^3}) 时, 可以得到 u = 0, \nabla d = 0 . ...
... 文献[11 ,12 ]对于液晶系统的研究结果是各向同性的. 受文献[5 ,11 ,12 ]启发, 本文主要研究关于速度各分量各向异性的Liouville定理. ...
... ,11 ,12 ]启发, 本文主要研究关于速度各分量各向异性的Liouville定理. ...
Liouville theorem for steady-state solutions of simplified Ericksen-Leslie system
5
2019
... 对于液晶系统(1.1), 在二维空间下, 文献[6 ⇓ ⇓ -9 ]得到了关于初边值问题Leray-Hopf弱解的全局存在性. 对于三维空间, Lin-Wang[10 ] 证明了在条件满足{d_0} \in S_ + ^2 ( {d_0} 表示在上半球面取值)时, 初边值问题Leray-Hopf弱解的全局存在性. 目前, 对于系统(1.1)Liouville定理的研究相对较少[11 ,12 ] . Jarríin[11 ] 证明了此系统在局部Morrey空间中的Liouville定理. Hao[12 ] 证明了 u \in D_0^1, d \in {L^\infty } \cap \mathring{H} , 且满足条件 |u| + |\nabla d| \in {L^q}({\mathbb{R}^3}) 时, 可以得到 u = 0, \nabla d = 0 . ...
... [12 ]证明了 u \in D_0^1, d \in {L^\infty } \cap \mathring{H} , 且满足条件 |u| + |\nabla d| \in {L^q}({\mathbb{R}^3}) 时, 可以得到 u = 0, \nabla d = 0 . ...
... 文献[11 ,12 ]对于液晶系统的研究结果是各向同性的. 受文献[5 ,11 ,12 ]启发, 本文主要研究关于速度各分量各向异性的Liouville定理. ...
... ,12 ]启发, 本文主要研究关于速度各分量各向异性的Liouville定理. ...
... 相比于Hao[12 ] 的结果 ...
1
1970
... 由标准的Calderon-Zygmund不等式[13 ] , 存在常数 C , 使得 ...