带粗糙初始值向列型液晶流的适定性

考虑了R~n上n(n≥2)维向列型液晶流(u,d)当初值属于Q_α~(-1)(R~n,R~n)×Q_α(R~n,S~2)(其中α∈(0,1))时Cauchy问题的适定性,这里的Q_α(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,Indiana Univ Math J,2000,49:575-615])引入,是指由R~n中满足的所有可测函数f全体所组成的空间.上式左端在取遍Rn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_α~(-1)(R~n):=▽·Q_α(R~n).最后证明了解(u,d)在类C(0,T);Q_(α,T)~(-1)(R~n,R~n))∩L_(loc)~∞((0,T);L~∞(R~n,R~n))×C(0,T);Q_α,T(R~n,S~2))∩L_(loc)~∞((0,T);W~(1,∞)(R~n,S~2))(其中0T≤∞)中是唯一的.

Well-Posedness for the Nematic Liquid Crystal Flow with Rough Initial Data

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