| 刘桥.带粗糙初始值向列型液晶流的适定性[J].数学年刊A辑,2014,35(5):591~612 |
| 带粗糙初始值向列型液晶流的适定性 |
| Well-Posedness for the Nematic Liquid Crystal Flow with Rough Initial Data |
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| DOI： |
| 中文关键词: 向列型液晶流, 适定性, 唯一性, Navier--Stokes 方程组, $Q$-空间 |
| 英文关键词:Nematic liquid crystal flow, Well-posedness, Uniqueness, Navier-
Stokes equations, Q-space |
| 基金项目:国家自然科学基金 (No.11401202, No.11171357),数学天元基金 (No.11326155) 和湖南省自然科学基金(No.13JJ4043) |
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| 中文摘要: |
| 考虑了$\mathbb{R}^{n}$上$n\ (n\geq2)$维向列型液晶流 $(u,d)$ 当初值属于$Q_{\alpha}^{-1}(\mathbb{R}^{n},\mathbb{R}^{n})\times
Q_{\alpha}(\mathbb{R}^{n},\mathbb{S}^{2})$ (其中 $\alpha\in (0,1)$)时 Cauchy 问题的适定性, 这里的
$Q_{\alpha}(\mathbb{R}^{n})$ 最早由 Essen, Janson,
Peng 和 Xiao (见 [Essen M, Janson S, Peng L, Xiao J.
$Q$ space of several real variables, {\it Indiana Univ Math J},
2000, 49:575--615])引入, 是指由 $\mathbb{R}^{n}$ 中满足
\begin{align*}
\sup_{I}\Big((l(I))^{2\alpha-n}\int_{I}\int_{I}
\frac{|f(x)-f(y)|^{2}}{|x-y|^{n+2\alpha}}\text{d}x\text{d}y\Big)^{\frac{1}{2}}<\infty
\end{align*}
的所有可测函数 $f$ 全体所组成的空间. 上式左端在取遍$\mathbb{R}^{n}$中所有以 $l(I)$ 为边长
且边平行于坐标轴的立方体 $I$的全体中取上确界, 而$Q_{\alpha}^{-1}(\mathbb{R}^{n}):=\nabla\cdot
Q_{\alpha}(\mathbb{R}^{n})$. 最后证明了解$(u, d)$在类$C([0,T);Q_{\alpha,T}^{-1}
(\mathbb{R}^{n}$, $\mathbb{R}^{n}))\cap
L^{\infty}_{\rm loc}((0,T);L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n}))\times
C([0,T);Q_{\alpha,T}(\mathbb{R}^{n}, \mathbb{S}^{2})) \cap
L^{\infty}_{\rm loc}((0,T); \dot{W}^{1,\infty}(\mathbb{R}^{n},\mathbb{S}^{2}))$
(其中 $0 |
| 英文摘要: |
| The author investigates the well-posedness of the Cauchy problem of the $n$-dimensional ($n\geq 2$)
hydrodynamic flow $(u, d)$ of nematic liquid crystal materials on
$\mathbb{R}^{n}$ with the initial data in
$Q_{\alpha}^{-1}(\mathbb{R}^{n},\mathbb{R}^{n})\times
Q_{\alpha}(\mathbb{R}^{n},\mathbb{S}^{2})$ with $\alpha\in (0,1)$.
Here, $Q_{\alpha}(\mathbb{R}^{n})$, introduced by Essen, Janson,
Peng and Xiao (see [Essen M, Janson S, Peng L, Xiao J.
$Q$ space of several real variables, {\it Indiana Univ Math J},
2000, 49:575--615]), is the space of all measurable functions $f$ on
$\mathbb{R}^{n}$, satisfying
\begin{align*}
\sup_{I}\Big((l(I))^{2\alpha-n}\int_{I}\int_{I}
\frac{|f(x)-f(y)|^{2}}{|x-y|^{n+2\alpha}}\text{d}x\text{d}y\Big)^{\frac{1}{2}}<\infty,
\end{align*}
where the supremum is taken over all cubes $I$ with the edge length
$l(I)$ and the edges parallel to the coordinate axes in
$\mathbb{R}^{n}$, and $Q_{\alpha}^{-1}(\mathbb{R}^{n}):=\nabla\cdot
Q_{\alpha}(\mathbb{R}^{n})$. %More precisely, we prove the existence
%of a global mild solution in
%$Q^{-1}_{\alpha}(\mathbb{R}^{n},\mathbb{R}^{n})\times
%Q_{\alpha}(\mathbb{R}^{n},\mathbb{S}^{2})$ for small initial data.
Moreover, for the nematic liquid crystal flow $(u,d)$, it is shown that
the solution is unique in the class
$C([0,T);Q_{\alpha,T}^{-1}(\mathbb{R}^{n},\mathbb{R}^{n}))\cap
L^{\infty}_{\rm loc}((0,T);L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n}))\times
C([0,T);Q_{\alpha,T}(\mathbb{R}^{n},\mathbb{S}^{2}))\cap
L^{\infty}_{\rm loc}((0,T); \linebreak \dot{W}^{1,\infty}(\mathbb{R}^{n},\mathbb{S}^{2}))$
for $0 |
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