\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 107, pp. 1--22.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/107\hfil Serrin blow-up criterion] {Serrin blow-up criterion for strong solutions to the 3-D compressible nematic liquid crystal flows with vacuum} \author[Q. Liu \hfil EJDE-2013/107\hfilneg] {Qiao Liu} \address{Qiao Liu \newline College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China} \email{liuqao2005@163.com} \thanks{Submitted September 12, 2012. Published April 24, 2013.} \subjclass[2000]{76A15, 76N10, 35B65, 35Q35} \keywords{Compressible nematic liquid crystal flows; strong solution; \hfill\break\indent Serrin's criterion; blow-up criterion; compressible Navier-Stokes equations} \begin{abstract} In this article, we extend the well-known Serrin's blow-up criterion for solutions of the 3-D incompressible Navier-Stokes equations to the 3-D compressible nematic liquid crystal flows where the initial vacuum is allowed. It is proved that for the initial-boundary value problem of the 3-D compressible nematic liquid crystal flows in a bounded domain, the strong solution exists globally if the velocity satisfies the Serrin's condition and $L^1(0,T;L^{\infty})$-norm of the gradient of the velocity is bounded. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction}\label{Int} The time evolution of the density, the velocity and the orientation of a compressible nematic liquid crystal (NLC) flows occupying a bounded domain $\Omega$ of $\mathbb{R}^3$ can be described by the system \begin{gather} \label{eq1.1} \partial_t\rho+\operatorname{div} (\rho u)=0,\quad (x,t)\in \Omega\times(0,+\infty),\\ \label{eq1.2} \begin{aligned} &{\partial_t}(\rho u) +\operatorname{div}(\rho u\otimes u) +\nabla{P}\\ &=\mu\Delta u-\lambda\nabla\cdot(\nabla d \odot\nabla d -(\frac{1}{2}|\nabla d|^2+F(d))I), (x,t)\in \Omega\times(0,+\infty), \end{aligned}\\ \label{eq1.3} \partial_td+(u\cdot\nabla)d=\nu(\Delta d-f(d)), \quad (x,t)\in \Omega\times(0,+\infty), \end{gather} together with the initial value: $$\label{eq1.4} \rho(0,x)=\rho_0(x)\geq 0, \quad u(0,x)=u_0(x),\quad d(0,x)=d_0(x), \quad\forall x\in\Omega,$$ and the boundary conditions: $$\label{eq1.5} u(t,x)=0,\quad d(t,x)=d_0(x),\quad |d_0(x)|=1, \quad\forall (t,x)\in [0,+\infty)\times \partial\Omega.$$ Here we denote by $\rho$ the unknown density, $u=(u_1,u_2,u_3)$ the unknown velocity, $d=(d_1,d_2,d_3)$ the unknown orientation parameter of the nematic liquid crystal material, and $P=P(\rho)$ the pressure function. $\mu,\lambda$ and $\nu$ are positive viscosity coefficients. The unusual term $\nabla d\odot \nabla d$ denotes the $3\times 3$ matrix, whose $(i,j)$-th element is given by $\sum_{k=1}^3\partial_{i}d_{k}\partial_{j}d_{k}$. $I$ is the $3\times 3$ unit matrix. $f(d)$ is a polynomial function of $d$ which satisfies $f(d)=\frac{\partial}{\partial_{d}}F(d)$, where $F(d)$ is the bulk part of the elastic energy, usually we choose $F(d)$ to be the Ginzburg-Landau penalization; i.e., $F(d)=\frac{1}{4\sigma^2}(|d|^2-1)^2,\quad f(d)=\frac{1}{\sigma^2}(|d|^2-1)d,$ where $\sigma$ is a positive constant. In what follows, we will assume $\sigma=1$ since it does not play a special role in our discussion. Throughout this paper, we adopt the following simplified notations for standard Sobolev spaces $L^q:=L^q(\Omega),\quad W^{k,p}:=W^{k,p}(\Omega),\quad H^{k}:=H^{k}(\Omega)=W^{k,2},\quad H_0^1:=H_0^1(\Omega),$ where $1\leq p$, $q\leq \infty$ and $k\in \mathbb{N}$. System \eqref{eq1.1}--\eqref{eq1.5} is a simplified version of Ericksen-Lesile system modeling the flow of compressible nematic liquid crystals materials, and the hydrodynamic theory of liquid crystals was established by Ericksen \cite{ER1,ER2} and Leslie \cite{LE} in the 1960's. In \cite{WY}, Wang and Yu established the global existence and large-time behavior of weak solutions for the initial-boundary value problem \eqref{eq1.1}--\eqref{eq1.5}. When the direction $d$ does not appear, the system \eqref{eq1.1}--\eqref{eq1.5} becomes the compressible Navier-Stokes (CNS) equations. Matsumura and Nishida \cite{MN} obtained global existence of smooth solutions for the initial data is a small perturbation of a non--vacuum equilibrium. For the existence of solutions for arbitrary initial value, Lions \cite{PL} and Feireisl \cite{EF1,EF} established the global existence of weak solution to the CNS equations. Cho et al \cite{CK1,CCK,CK2} proved that the existence and uniqueness of local strong solutions of the CNS equations in the case where initial density need not to be positive and may vanish in an open set. Xin in \cite{X} showed that there is no global smooth solution to the Cauchy problem of the CNS equations with a nontrivial compactly supported initial density. Hence, there are many works \cite{CCK,FJ,FJO,HX,HLX1,HLX2,SWZ1,SWZ2} trying to establish blow-up criterion for the strong solution to the CNS equations. In particular, it is proved in \cite{HLX2} by Huang, Li and Xin that the Serrin's blow--up criterion (see \cite{JS}) for the incompressible Navier--Stokes equations still holds for the CNS equations; i.e., if $T^{*}$ is the maximal time of existence strong solution, then $$\label{eq1.6} \lim_{T\to T^{*}}(\|\operatorname{div} u\|_{L^1(0,T;L^{\infty})}+\|\rho^{1/2} u\|_{L^s(0,T;L^r)})=\infty$$ or $$\label{eq1.7} \lim_{T\to T^{*}}(\|\rho\|_{L^1(0,T;L^{\infty})}+\|\rho^{1/2} u\|_{L^s(0,T;L^r)})=\infty,$$ where $r$ and $s$ satisfy $\frac{2}{s}+\frac{3}{r}\leq 1$, $3< r\leq \infty$. Huang et al \cite{HX,HLX1} established that the Beale-Kato-Majda criterion (see \cite{BKM}) for the ideal incompressible flows still holds for the CNS equations; that is, $\lim_{T\to T^{*}}\int_0^T\|\nabla u\|_{L^{\infty}}\,\mathrm{d} t=\infty.$ Sun, Wang and Zhang in \cite{SWZ1} (see also \cite{HLX2}) obtained another Beale--Kato--Majda criterion in terms of the density, i.e., $\lim\sup_{T\to T^{*}}\|\rho \|_{L^{\infty}(0,T;L^{\infty})}=\infty.$ Recently, Wen and Zhu in \cite{WZ} established a blow-up criterion of the strong solution for the CNS equations in terms of the density, $\limsup_{T\to T^{*}}\|\rho \|_{L^{\infty}(0,T;L^q)}=\infty,$ for some $11$, it follows that $\partial_t(\rho^p) +\operatorname{div}(\rho^pu)+(p-1)\rho^p\operatorname{div}u=0.$ Integrating the above equality over $\Omega$ yields $\partial_t\|\rho\|_{L^p}^p\leq (p-1)\|\operatorname{div}u\|_{L^{\infty}}\|\rho\|_{L^p}^p;$ i.e., $$\label{eq2.6} \partial_t\|\rho\|_{L^p}\leq \frac{(p-1)}{p}\|\operatorname{div}u\|_{L^{\infty}}\|\rho\|_{L^p}.$$ Condition \eqref{eq2.3} and estimate \eqref{eq2.6} imply $\sup_{0\leq t\leq T}\|\rho\|_{L^p}\leq C\quad \forall p>1,$ where $C$ is a positive constant independent of $p$, letting $p\to\infty$, we obtain \eqref{eq2.4}, and this completes the first part of the proof. To prove the estimate \eqref{eq2.5}, we multiply the liquid crystal equation \eqref{eq1.3} by $q|d|^{q-2}d$ with $q\geq 2$, integrate it over $\Omega$, and make use of the boundary condition \eqref{eq1.5}, then there holds \label{eq2.7} \begin{aligned} &\frac{d}{dt}\|d\|_{L^q}^q+\int_{\Omega}(q\nu |\nabla d|^2|d|^{q-2}+q(q-2)\nu|d|^{q-2}|\nabla |d||^2)\,\mathrm{d}x+q\nu\int_{\Omega } |d|^{q+2}\,\mathrm{d}x \\ &= -\sum_{i=1}^3\int_{\Omega} u_{i}\partial_{i} (|d|^q)\,\mathrm{d}x+q\nu\int_{\Omega}|d|^q\,\mathrm{d}x \\ &\leq \int_{\Omega} |\operatorname{div}u| |d|^q\,\mathrm{d}x +\Big(\frac{q\nu}{2}\int_{\Omega}|d|^{q+2}\,\mathrm{d}x\Big)^{\frac{q}{q+2}} (2|\Omega|)^{\frac{2}{q+2}} \\ &\leq C\|\operatorname{div} u\|_{L^{\infty}}\|d\|_{q}^q+\frac{q\nu}{2}\int_{\Omega}|d|^{q+2}\,\mathrm{d}x+C, \end{aligned} where we have used the fact that $d\nabla d=|d|\nabla|d|$ implies \begin{align*} -q\int_{\Omega}\Delta d |d|^{q-2}d\,\mathrm{d}x &= q\int_{\Omega}|\nabla d|^2|d|^{q-2}\,\mathrm{d}x+q\int_{\Omega}\nabla d \cdot d\nabla |d|^{q-2}\,\mathrm{d}x\\ &= q\int_{\Omega}|\nabla d|^2|d|^{q-2}\,\mathrm{d}x+q\int_{\Omega}|\nabla d| |d|(q-2)|d|^{q-2}\nabla |d|\,\mathrm{d}x\\ &=q\int_{\Omega}|\nabla d|^2|d|^{q-2}\,\mathrm{d}x+q(q-2)\int_{\Omega}|d|^{q-1}|\nabla |d||^2\,\mathrm{d}x \end{align*} and the fact that $f(d)=(|d|^2-1)d$. It follows from inequality \eqref{eq2.7} that $\frac{d}{dt}\|d\|_{L^q}^q\leq C\|\operatorname{div} u\|_{L^{\infty}}\|d\|_{L^q}^q+C,$ which together with the Gronwall's inequality imply $$\label{eq2.8} \sup_{0\leq t\leq T}\|d\|_{L^q}\leq C\quad\text{ for all }q\geq 2,$$ where $C$ is a positive constant independent of $q$. By letting $q\to\infty$, we notice that the estimate \eqref{eq2.5} still holds. \end{proof} By assumption \eqref{eq1.8} on the pressure $P$ and the estimate \eqref{eq2.4}, it is easy to obtain $$\label{eq2.9} \sup_{0\leq t\leq T}\{\|P(\rho)\|_{L^{\infty}},\|P'(\rho)\|_{L^{\infty}}\}\leq C<\infty.$$ Now, let us derive the stand energy inequality. \begin{lemma}\label{lem2.2} There holds $$\label{eq2.10} \sup_{0\leq t\leq T}\int_{\Omega}(\rho|u|^2+|\nabla d|^2+2F(d))\,\mathrm{d}x +\int_0^T\int_{\Omega}|\nabla u|^2+ |\Delta d-f(d)|^2\,\mathrm{d}x\, \mathrm{d}t\leq C.$$ \end{lemma} \begin{proof} Multiplying the momentum equation \eqref{eq1.2} by $u$, integrating it over $\Omega$, making use of the mass conversation equation \eqref{eq1.1} and the boundary condition \eqref{eq1.5}, it follows that $$\label{eq2.11} \frac{1}{2}\frac{d}{dt}\int_{\Omega}\rho|u|^2\,\mathrm{d}x +\mu\int_{\Omega}|\nabla u|^2\,\mathrm{d}x =-\int_{\Omega} u\nabla P\,\mathrm{d}x -\lambda\int_{\Omega}(u\cdot\nabla)d\cdot(\Delta d -f(d))\,\mathrm{d}x,$$ where we have used the equality $\operatorname{div}(\nabla d\odot \nabla d)=(\nabla d)^T\Delta d +\nabla\left(\frac{|\nabla d|^2}{2}\right)$. Multiplying the liquid crystal equation \eqref{eq1.3} by $\Delta d-f(d)$ and integrating it over $\Omega$, we obtain $$\label{eq2.12} \frac{d}{dt}\int_{\Omega}(\frac{1}{2}|\nabla d|^2+F(d))\,\mathrm{d}x+\nu \int_{\Omega}|\Delta d-f(d)|^2\,\mathrm{d}x=\int_{\Omega}(u\cdot \nabla)d\cdot(\Delta d-f(d))\,\mathrm{d}x.$$ %----------------(eq2.12)-------------------- Combining \eqref{eq2.11} and \eqref{eq2.12} \label{eq2.13} \begin{aligned} &\frac{d}{dt}\int_{\Omega}[\frac{1}{2}(\rho |u|^2+\lambda |\nabla d|^2)+\lambda F(d)]\,\mathrm{d}x+\mu\int_{\Omega }|\nabla u|^2\,\mathrm{d}x+\lambda\nu \int_{\Omega }|\Delta d-f(d)|^2\,\mathrm{d}x \\ &= -\int_{\Omega } u\nabla P\,\mathrm{d}x\\ &=\int_{\Omega}P\operatorname{div}u\,\mathrm{d}x\leq \varepsilon\int_{\Omega }|\nabla u|^2\,\mathrm{d}x+C \varepsilon^{-1}, \end{aligned} where we have used the estimates \eqref{eq2.4}, \eqref{eq2.9} and the Young inequality. Taking $\varepsilon$ small enough in \eqref{eq2.13} and applying the Gronwall's inequality, we can establish the estimate \eqref{eq2.10} immediately. \end{proof} In the next lemma, we will derive some estimates of the direction field $d$. \begin{lemma}\label{lem2.3} Under the assumption \eqref{eq2.1}, it holds that for $0\leq T0$, gives \label{eq2.42} \begin{aligned} \xi\|\nabla \rho\|_{L^2}^2 &\leq C\xi(1+\int_0^T(\|\rho^{1/2}u_t\|_{L^2}^2+\|u\|_{L^{\infty}}^2\|\nabla u\|_{L^2}^2+\|\nabla \rho\|_{L^2}^2+\|\nabla d_t\|_{L^2}^2)\,\mathrm{d}t) \\ &\leq C\xi(1+\int_0^T(\|\rho^{1/2}u_t\|_{L^2}^2+\|u\|_{L^{\infty}}^2\|\nabla u\|_{L^2}^2+\|\nabla \rho\|_{L^2}^2)\,\mathrm{d}t), \end{aligned} where we have used the estimate \eqref{eq2.16} again. Combining \eqref{eq2.39} and \eqref{eq2.42}, and taking $\varepsilon<\xi$ small enough, it follows that \label{eq2.43} \begin{aligned} &\|\nabla u\|_{L^2}^2+\frac{\xi}{2}\|\nabla\rho\|_{L^2}^2 +\frac{1}{2}\int_0^T\int_{\Omega}\rho|u_t|^2\,\mathrm{d}x\,\mathrm{d}t \\ &\leq C(1+\xi)\int_0^T(\|\nabla u\|_{L^2}^2+\|\nabla \rho\|_{L^2}^2) (\|\nabla u\|_{L^{\infty}}+\|u\|_{L^r}^{\frac{2r}{r-3}} +\|u\|_{L^{\infty}}^2+1)\,\mathrm{d}t+C. \end{aligned} By using the Gronwall's inequality and noticing that the assumption \eqref{eq2.1}, we can deduce that $\sup_{0\leq t\leq T}(\|\nabla u\|_{L^2}^2+\|\nabla \rho\|_{L^2}^2)+\int_0^T\int_{\Omega}\rho|u_t|^2\,\mathrm{d}x\,\mathrm{d}t\leq C.$ For \eqref{eq2.25}, it follows from \eqref{eq2.32} that $\int_0^T\|u\|_{H^2}^2\,\mathrm{d}t\leq C\int_0^T(\|\rho^{1/2}u_t\|_{L^2}^2+\|\nabla u\|_{L^2}^2+\|\nabla d_t\|_{L^2}^2+1)\,\mathrm{d}t\leq C,$ where we have used the estimates \eqref{eq2.10}, \eqref{eq2.16} and \eqref{eq2.24} in the last inequality above. This completes the proof of Lemma \ref{lem2.4}. \end{proof} \begin{lemma}\label{lem2.5} Under the assumption \eqref{eq2.1}, it holds that for \$0\leq T