$L^3_{\hbox{uloc}}(\R^3)$中向列相液晶流动的适定性
Well-Posedness of Nematic Liquid Crystal Flow in $L^3_{\hbox{uloc}}(\R^3)$
摘要：
本文对任意初始数据$(u_0,d_0)$具有$(u_0,\nabla d_0)$的小$L^3_{\hbox{uloc}}$ -范数，在$\mathbb R^3$中建立了液晶流体动力学流简化形式（\ref{LLF}）柯西问题的局部适定性.这里$L^3_{\hbox{uloc}}(\mathbb R^3)$是一致局部$L^3$ -可积函数空间。对于任何初始数据$(u_0, d_0)$与小$\displaystyle |(u_0,\nabla d_0)|_{L^3(\mathbb R^3)}$，我们表明，存在一个唯一的，全球性的解决方案（\ref{LLF}），这是顺利的$t>0$和单调递减$L^3$ -能量$t\ge 0$。
英文摘要：
In this paper, we establish the local well-posedness for the Cauchy problem of the simplified version of hydrodynamic flow of nematic liquid crystals (\ref{LLF}) in $\mathbb R^3$ for any initial data $(u_0,d_0)$ having small $L^3_{\hbox{uloc}}$-norm of $(u_0,\nabla d_0)$. Here $L^3_{\hbox{uloc}}(\mathbb R^3)$ is the space of uniformly locally $L^3$-integrable functions. For any initial data $(u_0, d_0)$ with small $\displaystyle |(u_0,\nabla d_0)|_{L^3(\mathbb R^3)}$, we show that there exists a unique, global solution to (\ref{LLF}) which is smooth for $t>0$ and has monotone deceasing $L^3$-energy for $t\ge 0$.
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