## 不可压液晶方程组的Serrin解

1 上海健康医学院文理教学部 上海 201318

2 复旦大学数学科学学院 上海 200433

## Serrin's Type Solutions of the Incompressible Liquid Crystals System

Min Jianzhong1, Liu Xiangao,2, Liu Zixuan,2

1 Science and Arts Faculty, Shanghai University of Medicine and Health Sciences, Shanghai 201318

2 School of Mathematical Sciences, Fudan University, Shanghai 200433

 基金资助: 国家自然科学基金.  11631011国家自然科学基金.  11971113

 Fund supported: Supported by the NSFC.  11631011Supported by the NSFC.  11971113

Abstract

In this paper, we study the nematic liquid crystals system under the simplified Ginzburg-Landau model, which is probably the simplest mathematical model that one can derive, without destroying the basic nonlinear structure [1]. We get the local existence and uniquness of the Serrin's type of solutions provided the initial data $u_{0}\in L^{p}\cap H,$ $d_{0}\in W^{1, p}, p\geq n$. According to the Serrin's regularity criteria for the incompressible liquid crystals system [2], we actually prove the local existence of smooth solutions to liquid crystals system for big data and global existence of smooth solutions for small data.

Keywords： Existence ; Liquid crystal ; Serrin's criterion ; Uniqueness

Min Jianzhong, Liu Xiangao, Liu Zixuan. Serrin's Type Solutions of the Incompressible Liquid Crystals System. Acta Mathematica Scientia[J], 2021, 41(6): 1671-1683 doi:

## 1 引言

$$$\left \{\begin{array}{l} u_t-\Delta u+u\cdot\nabla u +\nabla P = -\rm{div}(\nabla d \odot \nabla d), \\ \rm{div}\ u = 0, \\ d_t+u\cdot \nabla d-\Delta d = f(d), \\ \end{array} \right.$$$

$$$\parallel u_{0}-u_{0\eta}\parallel_{n}+\parallel e_{0}-e_{0\eta}\parallel_{n}+CT^{\beta_{2}}(\parallel u_{0\eta}\parallel_{q}+\parallel e_{0\eta}\parallel_{q})< 1.$$$

$\begin{eqnarray} \parallel u_{k+1} \parallel_{2, \infty, T} & \leq &\parallel u_{0} \parallel_{2}+CT^{\beta_{1}}(\parallel u_{0} \parallel_{2}\parallel u_{0} \parallel_{\sigma}+\parallel e_{0} \parallel_{2}\parallel e_{0} \parallel_{\sigma} ){}\\ & \leq &(\parallel u_{0} \parallel_{2}+\parallel e_{0} \parallel_{2})(1+CT^{\beta_{1}}(\parallel u_{0} \parallel_{\sigma}+\parallel e_{0} \parallel_{\sigma})), \end{eqnarray}$

$\begin{eqnarray} \parallel e_{k+1} \parallel_{2, \infty, T} &\leq &\parallel e_{0} \parallel_{2}+CT^{\beta_{1}}\parallel u_{0} \parallel_{2}\parallel e_{0} \parallel_{\sigma} {}\\ &\leq &\parallel e_{0} \parallel_{2}(1+CT^{\beta_{1}}\parallel e_{0} \parallel_{\sigma}), \end{eqnarray}$

$\sigma = n$时, 在式(3.9), (3.10)中, 可以取适当的$\eta, T$使得$K_{2}, K_{4}$充分小, 使得

$$$\left \{\begin{array}{l} w_t-\Delta w+\nabla (P_{1}-P_{2}) = -w\cdot\nabla u_{1} -u_{2}\cdot\nabla w-{\rm{div}}(h^{T} e_{1} +e^{T}_{2}\cdot h ), \\ {\rm{div}}\ w = 0, \\ h_t-\Delta h = - e_{1}\nabla w-h\nabla u_{2}-w\cdot \nabla e_{1}-u_{2}\cdot \nabla h, \\ w(x, 0) = 0, e(x, 0) = 0, \\ \end{array} \right.$$$

$$$\parallel w(t)\parallel^{2}_{2}+2\int^{t}_{0}{\parallel\nabla w\parallel^{2}_{2} = \int^{t}_{0}{-(w\cdot\nabla u_{1}, w) -(u_{2}\cdot\nabla w, w)-({\rm{div}}(h^{T} e_{1} +e^{T}_{2}\cdot h), w)}},$$$

$$$\parallel h(t)\parallel^{2}_{2}+2\int^{t}_{0}{\parallel h\parallel^{2}_{2} = \int^{t}_{0}{-(e_{1}\nabla w, h)-(h\nabla u_{2}, h)-(w\cdot \nabla e_{1}, h)-(u_{2}\cdot \nabla h, h)}},$$$

$$$\int^{t}_{0}(v\nabla w, u)\leq C\left(\int^{t}_{0}\parallel \nabla w\parallel^{2}_{2}\right)^{\frac{1}{2}}\left(\int^{t}_{0}\parallel \nabla v\parallel^{2}_{2}\right)^{\frac{n}{2s}}\left(\int^{t}_{0}\parallel \nabla u\parallel^{r}_{s}\parallel v\parallel^{2}_{2}\right)^{\frac{1}{r}},$$$

$\begin{eqnarray} \int^{t}_{0}-(w\nabla u_{1}, w)& = &\int^{t}_{0}(w\nabla w, u_{1}){}\\ &\leq& C\left(\int^{t}_{0}\parallel \nabla w\parallel^{2}_{2}\right)^{\frac{1}{2}}\left(\int^{t}_{0}\parallel \nabla w\parallel^{2}_{2}\right)^{\frac{n}{2s}}\left(\int^{t}_{0}\parallel u_{1}\parallel^{r}_{s}\parallel w\parallel^{2}_{2}\right)^{\frac{1}{r}}\\ &\leq& \varepsilon\left(\int^{t}_{0}\parallel \nabla w\parallel^{2}_{2}\right)+C(\varepsilon)\left(\int^{t}_{0}\parallel u_{1}\parallel^{r}_{s}\parallel w\parallel^{2}_{2}\right)^{\frac{2}{r}}, \end{eqnarray}$

$\begin{eqnarray} &&\int^{t}_{0}{-({\rm{div}}(h^{T} e_{1} +e^{T}_{2}\cdot h), w)}{}\\ & = &\int^{t}_{0}{((h^{T} e_{1} +e^{T}_{2}\cdot h), \nabla w)}\\ &\leq & C\left(\int^{t}_{0}\parallel \nabla w\parallel^{2}_{2}\right)^{\frac{1}{2}}\left(\int^{t}_{0}\parallel \nabla h\parallel^{2}_{2}\right)^{\frac{n}{2s}}\left(\int^{t}_{0}(\parallel e_{1} \parallel^{r}_{s}+\parallel e_{2} \parallel^{r}_{s})\parallel h\parallel^{2}_{2}\right)^{\frac{1}{r}} \\ &\leq& \varepsilon\left(\int^{t}_{0}\parallel \nabla w\parallel^{2}_{2}+\int^{t}_{0}\parallel \nabla h\parallel^{2}_{2}\right)+C(\varepsilon)\left(\int^{t}_{0}(\parallel e_{1}\parallel^{r}_{s}+\parallel e_{2}\parallel^{r}_{s})\parallel h\parallel^{2}_{2}\right), \end{eqnarray}$

$\begin{eqnarray} &&\int^{t}_{0}{-( e_{1}\nabla w +h\nabla u_{2}+w\nabla e_{1}, h)}{}\\ & = &\int^{t}_{0}{-( e_{1}\nabla w, h)+(u_{2}\nabla h, h)+(w\nabla h, e_{1})}\\ &\leq & C\left(\int^{t}_{0}\parallel \nabla w\parallel^{2}_{2}\right)^{\frac{1}{2}}\left(\int^{t}_{0}\parallel \nabla h\parallel^{2}_{2}\right)^{\frac{n}{2s}}\left(\int^{t}_{0}(\parallel e_{1} \parallel^{r}_{s})\parallel h\parallel^{2}_{2}\right)^{\frac{1}{r}} \nonumber\\ &&+C\left(\int^{t}_{0}\parallel \nabla h\parallel^{2}_{2}\right)^{\frac{1}{2}}\left(\int^{t}_{0}\parallel \nabla h\parallel^{2}_{2}\right)^{\frac{n}{2s}}\left(\int^{t}_{0}(\parallel u_{2} \parallel^{r}_{s})\parallel h\parallel^{2}_{2}\right)^{\frac{1}{r}}\\ &&+C\left(\int^{t}_{0}\parallel \nabla h\parallel^{2}_{2}\right)^{\frac{1}{2}}\left(\int^{t}_{0}\parallel \nabla w\parallel^{2}_{2}\right)^{\frac{n}{2s}}\left(\int^{t}_{0}(\parallel e_{1} \parallel^{r}_{s})\parallel w\parallel^{2}_{2}\right)^{\frac{1}{r}}\\ &\leq& \varepsilon\left(\int^{t}_{0}\parallel \nabla w\parallel^{2}_{2}+\int^{t}_{0}\parallel \nabla h\parallel^{2}_{2}\right)\\ &&+C(\varepsilon)\left(\int^{t}_{0}(\parallel e_{1}\parallel^{r}_{s}+\parallel u_{2}\parallel^{r}_{s})\parallel h\parallel^{2}_{2}+\parallel e_{1} \parallel^{r}_{s}\parallel w\parallel^{2}_{2}\right). \end{eqnarray}$

$$$Y(t)\leq C(\varepsilon)\left(\int^{t}_{0}a(t)Y(t)\right), \ Y(0) = 0,$$$

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