\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 188, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/188\hfil Blow-up criterion for strong solutions] {Blow-up criterion for strong solutions to the 3D magneto-micropolar fluid equations in the multiplier space} \author[H. Zhang, Y. Zhao\hfil EJDE-2012/188\hfilneg] {Hui Zhang, Yongye Zhao} % in alphabetical order \address{Hui Zhang \newline School of Mathematical Science, Xiangtan University, Xiangtan 411105, China} \email{zhangxtu@126.com} \address{Yongye Zhao \newline Department of Mathematics, South China University of Technology, Guangzhou 510640, China} \email{zhao.yongye@mail.scut.edu.cn} \thanks{Submitted March 30, 2012. Published October 28, 2012.} \subjclass[2000]{35Q30, 76D05} \keywords{Magneto-micropolar equations; blow-up criterion; multiplier spaces} \begin{abstract} In this article, we study the blow-up of strong solutions to the magneto-micropolar (MMP) fluid equations in $\mathbb{R}^3$. It is proved that if the gradient field of velocity satisfies $$\nabla u\in L^{2/(2-r)}(0,T;\dot{X}_{r}(\mathbb{R}^3))\quad \text{with }r\in[0,1],$$ then the strong solution $(u,w,b)$ can be extended beyond $t=T$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In this article, we consider the 3D magneto-micropolar (MMP) fluid equations $$\label{e1.1} \begin{gathered} \partial _{t}u+(u\cdot \nabla )u-(\mu+\chi)\Delta u-(b\cdot\nabla)b +\nabla(p+|b|^2)-\chi\nabla\times w = 0, \\ \partial_{t}w-\gamma\Delta w-\kappa\nabla\nabla\cdot w+2\chi w +u\cdot\nabla w-\chi\nabla\times u =0, \\ \partial_{t}b-\nu\Delta b+(u\cdot\nabla)b-(b\cdot\nabla)u =0, \\ \nabla\cdot u=\nabla\cdot b =0, \\ u(x,0)=u_0(x),b(x,0)=b_0(x),w(x,0)=w_0(x). \end{gathered}$$ where $u(x,t)\in \mathbb{R}^3$, $w(x,t)\in \mathbb{R}^3$, $b(x,t)\in\mathbb{R}^3$, $p=p(x,t)\in \mathbb{R}$ denote the velocity of the fluid, the micro-rotational velocity, magnetic field and pressure, respectively. $\mu$ is the kinematic viscosity, $\chi$ is the vortex viscosity, $\kappa$ and $\gamma$ are spin viscosities and $\nu$ is the magnetic diffusivity. $(u_0,w_0,b_0)$ are the given initial data with $\nabla\cdot u_0=\nabla\cdot b_0=0$. The MMP fluid system \eqref{e1.1} was first studied by Galdi and Rionero in \cite{g1}. Rojas-Medar and Boldrin \cite{M1} proved the existence of weak solutions by the Galerkin method, and in 2D case, also proved the uniqueness of the weak solutions. Ortega-Torres and Rojas-Medar \cite{M2,E1} established the local in time existence and uniqueness of strong solutions and proved global in time existence of strong solution for small initial data. However, whether the local strong solutions can exist globally or the global weak solution is regular and unique is an outstanding open problem. \par If $b=0$, \eqref{e1.1} reduce to micropolar fluid equations.The micropolar fluid equations were first proposed by Eringen \cite{E3}. It is a type of fluids which exhibit the micro-rotational effects and micro-rotational inertia, and can be viewed as a non-Newtonian fluid. It can describe some physical phenomena that can't be treated by the classical Navier-Stokes equations for the viscous incompressible fluids, such as the motion of animal blood, liquid crystals and dilute aqueous polymer solutions, etc. The existences of weak and strong solutions were treated by Galdi and Rionero \cite{G2}, as well as Yamaguchi \cite{Y2}, respectively. Recently, Ferreira and Villamizar-Roa \cite{V3} considered the existence and stability of solutions to the micropolar fluids in exterior domains. Villamizar-Roa and Rodr\'iguez-Bellido \cite{V1} studied the micropolar system in a bounded domain by using the semigroup approach in $L^{p}$-space, showing the global existence of strong solutions for small data and the asymptotic behavior and stability of the solutions. The purpose of this article is to study the breakdown criteria of smooth solutions to the MMP fluid system \eqref{e1.1}. Some fundamental Serrin's-type regularity criteria was done in \cite{E2} and \cite{Y1} independently. Torres \cite{E2} showed the uniqueness of weak solution if $(u,w,b)\in L^{p}(0,T;L^{q}(\Omega))$; $\frac{2}{p}+\frac{3}{q}\leq1$, $q>3$ in a bounded domain with the no-slip boundary conditions, Yuan \cite{Y1} proved that if $\nabla u\in L^{p}(0,T;L^{q}(\mathbb{R}^3))$; $\frac{2}{p}+\frac{3}{q}=2$, $\frac{3}{2}0$, $(u,w,b)$ be a strong solution of 3D MMP equation \eqref{e1.1} on $(0,T)$ with the initial data $(u_0,w_0,b_0)\in H^{1}(\mathbb{R}^3)$ with $\nabla\cdot u_0=\nabla\cdot b_0=0$. If the gradient field of velocity satisfies $$\label{e1.2} \nabla u\in L^{2/(2-r)}(0,T;\dot{X}_{r}(\mathbb{R}^3)),\quad 0\leq r\leq 1$$ then the solution $(u,w,b)$ can be extended smoothly beyond $t=T$. \end{theorem} \begin{definition} \label{def1.2} \rm Let $T>0$, $(u_0,w_0,b_0)\in H^{1}(\mathbb{R}^3)$ with $\nabla\cdot u_0=\nabla\cdot b_0=0$. A measurable $R^3$-valued triple $(u,w,b)$ is said to be a weak solution of the MMP equation on $(0,T]$ if the following conditions hold: \begin{itemize} \item[(1)] $(u,w,b)\in L^{\infty}(0,T;L^2(\mathbb{R}^3)) \cap L^2(0,T;H^{1}(\mathbb{R}^3))$, \item[(2)] $(u,w,b)$ verifies \eqref{e1.1} in the sense of distribution, \item[(3)] the following energy inequality is satisfied, \label{e1.3} \begin{aligned} &\|(u,w,b)\|_2^2+2(\mu+\chi)\int_0^{t}\|\nabla u\|_2^2ds\\ &+2\gamma\int_0^{t}\|\nabla w\|_2^2ds+2\nu\int_0^{t}\|\nabla b\|_2^2ds +2\chi\int_0^{t}\|w\|_2^2ds \leq \|(u_0,b_0,w_0)\|_2^2. \end{aligned} \end{itemize} \end {definition} For the convenience, we set $\mu=\chi=1/2$, $\kappa=\gamma=\nu=1$, throughout this article, the $L^{p}$-norm of a function denoted by $\|\cdot\|_{p}$, and the $H^{s}$-norm by $\|\cdot\|_{H^{s}}$. \section{The multiplier space} In this section, we describe the multiplier space $\dot{X}_{r}$ introduced by Lemarie-Rieusset \cite{L1} . \begin{definition} \label{def2.1} \rm For $0\leq r<3/2$, the space $\dot{X}_{r}$ is defined as the space of $f(x)\in L_{loc}^2(\mathbb{R}^3)$ such that $$\|f\|_{\dot{X}_{r}}=\sup_t{\|g\|_{\dot{H}^r\leq 1}} \|f g\|_2<\infty,$$ where we denote by $\dot{H}^r(R^d)$ the completion of the space $D(\mathbb{R}^d)$ with respect to the norm $\|u\|_{\dot{H}^r}=\|(-\Delta)^{\frac{r}{2}}u\|_2=\||\xi|^r\hat{u}(\xi)\|_2$, where $\hat{u}(\xi)$ denotes the Fourier transform of $u$. \end{definition} For any function $f(x,t)$ defined for both spatial and time variables, we have $$\|f_{\lambda}\|_{L^{\frac{2}{1-r}}(0,\frac{T}{\lambda^2},\dot{X}_{r})} =\|f_{\lambda}\|_{L^{\frac{2}{1-r}}(0,T,\dot{X}_{r})}$$ for any $\lambda>0$, with $f_{\lambda}(x,t)=\lambda f(\lambda x,\lambda^2t)$. So, if $(u,w,b)$ solves the MMP equation, then so does $(u_{\lambda},w_{\lambda},b_{\lambda})$ for all $\lambda>0$. This is so called scaling dimension zero property. For more details, we refer the reader to \cite{L1,L2,Z2}. In particular, we have the imbedding $$L^{3/r}(\mathbb{R}^3)\subset\dot{X}_{r}(\mathbb{R}^3),\quad 0\leq r<\frac{3}{2}$$ holds, $r = 0$, it is clear that, \cite{G1}, $$\dot{X}_0\cong BMO.$$ \section{Proof of main theorem} We take gradient of the both sides of \eqref{e1.1} and take the $L^2$ inner product of the resulting equation with $(\nabla u, \nabla w,\nabla b)$. With help of integrating by parts, we have \label{e3.1} \begin{aligned} &\frac{1}{2}\frac{d}{dt}(\|\nabla u\|_2^2+\|\nabla w\|_2^2+\|\nabla b\|_2^2) +(\|\nabla^2 u\|_2^2+\|\nabla^2 w\|_2^2\\ &+\|\nabla^2 b\|_2^2) +\|\nabla\nabla\cdot w\|_2^2+\|\nabla w\|_2^2dx \\ &=-\int_{\mathbb{R}^3}\nabla[(u\cdot\nabla)u]\nabla udx +\int_{\mathbb{R}^3}\nabla[(b\cdot\nabla)b]\nabla udx -\int_{\mathbb{R}^3}\nabla[(u\cdot\nabla)w]\nabla wdx\\ &\quad +\frac{1}{2}\int_{\mathbb{R}^3}\nabla(\nabla\times w)\nabla udx +\frac{1}{2}\int_{\mathbb{R}^3}\nabla(\nabla\times u)\nabla wdx\\ &\quad -\int_{\mathbb{R}^3}\nabla[(u\cdot\nabla)b]\nabla bdx +\int_{\mathbb{R}^3}\nabla[(b\cdot\nabla)u]\nabla bdx\hspace{4cm}\\ &=\sum_{i=1}^{7}I_{i}. \end{aligned} To estimate $I_{1}$, we integrate by parts and apply Holder's inequality: $$\label{e3.2} \begin{split} I_{1}&= -\int_{\mathbb{R}^3}\nabla[(u\cdot\nabla)u]\nabla u dx \\ &= -\sum_{i,j,k=1}^3\int_{\mathbb{R}^3}\partial_{k}(u_{i} \partial_{i}u_j)\partial_{k}u_jdx \\ &= -\sum_{i,j,k=1}^3\int_{\mathbb{R}^3}\partial_{k}u_{i} \partial_{i}u_j\partial_{k}u_jdx -\sum_{i,j,k=1}^3\int_{\mathbb{R}^3}u_{i}\partial_{i} \partial_{k}u_j\partial_{k}u_jdx\\ &= I_{11}+I_{12}. \end{split}$$ $$\label{e3.3} \begin{split} |I_{11}| &\leq \int_{\mathbb{R}^3}|\nabla u|^3dx\hspace{7cm} \\ &\leq \|\nabla u \cdot\nabla u\|_2 \|\nabla u\|_2\\ &\leq \|\nabla u\|_{\dot{X}_{r}}\|\nabla u\|_2^{2-r}\|\nabla^2u\|_2^r \\ &\leq C \|\nabla u\|_{\dot{X}_{r}}^{2/(2-r)}\|\nabla u\|_2^2 +\frac{1}{4}\|\nabla^2u\|_2^2. \end{split}$$ Here we have used the inequality $\|f\|_{\dot{H}^r}\leq \|f\|_2^{1-r}\|\nabla f\|_2^r$. Using the incompressible condition $\nabla\cdot u=0$, we obtain $$\label{e3.4} |I_{12}|=0.$$ Similarly, for $I_2$ one can deduce $$\label{e3.5} \begin{split} I_2&= \int_{\mathbb{R}^3}\nabla[(b\cdot\nabla)b]\nabla u dx \\ &= \sum_{i,j,k=1}^3\int_{\mathbb{R}^3}\partial_{k}(b_{i}\partial_{i}b_j)\partial_{k}u_jdx \\ &= \sum_{i,j,k=1}^3\int_{\mathbb{R}^3}\partial_{k}b_{i}\partial_{i}b_j\partial_{k}u_jdx+\sum_{i,j,k=1}^3\int_{\mathbb{R}^3}b_{i}\partial_{i}\partial_{k}b_j\partial_{k}u_jdx\\ &= I_{21}+I_{22}. \end{split}$$ $$\label{e3.6} \begin{split} |I_{21}| &\leq \int_{\mathbb{R}^3}|\nabla b|^2|\nabla u|dx\hspace{6cm} \\ &\leq \|\nabla u \cdot\nabla b\|_2 \|\nabla b\|_2\\ &\leq \|\nabla u\|_{\dot{X}_{r}}\|\nabla b\|_2^{2-r}\|\nabla^2b\|_2^{\gamma} \\ &\leq C \|\nabla u\|_{\dot{X}_{r}}^{2/(2-r)}\|\nabla b\|_2^2 +\frac{1}{6}\|\nabla^2b\|_2^2. \end{split}$$ For $I_{22}$, we will give a result of $I_{22}+I_{72}=0$ later. In the same way, for $I_3$, we have $$\label{e3.7} \begin{split} I_3&= -\int_{\mathbb{R}^3}\nabla[(u\cdot\nabla)w]\nabla w dx \\ &= -\sum_{i,j,k=1}^3\int_{\mathbb{R}^3}\partial_{k}(u_{i}\partial_{i}w_j)\partial_{k}w_j dx\\ &= -\sum_{i,j,k=1}^3\int_{\mathbb{R}^3}\partial_{k}u_{i}\partial_{i}w_j \partial_{k}w_jdx -\sum_{i,j,k=1}^3\int_{\mathbb{R}^3}u_{i}\partial_{i}\partial_{k}w_j\partial_{k}w_jdx\\ &= I_{31}+I_{32}. \end{split}$$ $$\label{e3.8} |I_{31}|\leq C \|\nabla u\|_{\dot{X}_{r}}^{2/(2-r)}\|\nabla w\|_2^2 +\frac{1}{2}\|\nabla^2w\|_2^2.$$ Using the incompressible condition $\nabla\cdot u=0$, we obtain $$\label{e3.9} |I_{32}|=0.$$ Integrating by parts and Holder's inequality, we find $$\label{e3.10} \begin{split} |I_{4}+I_{5}| &= |\frac{1}{2}\int_{\mathbb{R}^3}\nabla(\nabla\times w)\nabla u +\nabla(\nabla\times u)\nabla w dx |\\ &\leq \int_{\mathbb{R}^3}|\nabla^2u||\nabla w|dx\\ &\leq \frac{1}{4}\|\nabla^2 u\|_2^2+\|\nabla w\|_2^2. \end{split}$$ With the similar derivation as of $I_{1}$, one has $$\label{e3.11} \begin{split} I_{6}&= -\int_{\mathbb{R}^3}\nabla[(u\cdot\nabla)b]\nabla b dx \\ &= -\sum_{i,j,k=1}^3\int_{\mathbb{R}^3}\partial_{k}(u_{i}\partial_{i}b_j) \partial_{k}b_j dx\\ &= -\sum_{i,j,k=1}^3\int_{\mathbb{R}^3}\partial_{k}u_{i}\partial_{i} b_j\partial_{k}b_jdx-\sum_{i,j,k=1}^3 \int_{\mathbb{R}^3}u_{i}\partial_{i}\partial_{k}b_j\partial_{k}b_jdx\\ &= I_{61}+I_{62}. \end{split}$$ $$\label{e3.12} \begin{split} |I_{61}| &\leq \int_{\mathbb{R}^3}|\nabla b|^2|\nabla u|dx\hspace{7cm} \\ &\leq \|\nabla u \cdot\nabla b\|_2 \|\nabla b\|_2\\ &\leq \|\nabla u\|_{\dot{X}_{r}}\|\nabla b\|_2^{2-r}\|\nabla^2b\|_2^r \\ &\leq C \|\nabla u\|_{\dot{X}_{r}}^{2/(2-r)}\|\nabla b\|_2^2 +\frac{1}{6}\|\nabla^2b\|_2^2. \end{split}$$ Using the incompressible condition $\nabla\cdot u=0$, we have $$\label{e3.13} |I_{62}|=0.$$ Similarly, $$\label{e3.14} \begin{split} I_{7}&= \int_{\mathbb{R}^3}\nabla[(b\cdot\nabla)u]\nabla b dx \\ &= \sum_{i,j,k=1}^3\int_{\mathbb{R}^3}\partial_{k}(b_{i}\partial_{i}u_j)\partial_{k}b_jdx \\ &= \sum_{i,j,k=1}^3\int_{\mathbb{R}^3}\partial_{k}b_{i}\partial_{i}u_j\partial_{k}b_jdx+\sum_{i,j,k=1}^3\int_{\mathbb{R}^3}b_{i}\partial_{i}\partial_{k}u_j\partial_{k}b_jdx\\ &= I_{71}+I_{72}. \end{split}$$ $$\label{e3.15} |I_{71}|\leq C \|\nabla u\|_{\dot{X}_{r}}^{2/(2-r)} \|\nabla b\|_2^2+\frac{1}{6}\|\nabla^2b\|_2^2.$$ Now, we give a simple result $$\label{e3.16} \begin{split} I_{22}+I_{72} &= \sum_{i,j,k=1}^3\int_{\mathbb{R}^3}b_{i}\partial_{i}(\partial_{k} b_j\partial_{k}u_j)dx \\ &= \sum_{i,j,k=1}^3\int_{\mathbb{R}^3}\partial_{i}[b_{i} (\partial_{k}b_j\partial_{k}u_j)]dx - \sum_{i,j,k=1}^3\int_{\mathbb{R}^3}\partial_{i}b_{i} (\partial_{k}b_j\partial_{k}u_j) = 0. \end{split}$$ Here we have used the incompressible condition and the $|(u,w,b)|\to 0$ as $|x|\to \infty$. Now combing the estimates of \eqref{e3.1}--\eqref{e3.16}, we obtain $$\label{e3.17} \begin{split} &\frac{d}{dt}(\|\nabla u\|_2^2+\|\nabla w\|_2^2+\|\nabla b\|_2^2) +(\|\nabla^2 u\|_2^2+\|\nabla^2 w\|_2^2+\|\nabla^2 b\|_2^2) +\|\nabla\nabla\cdot w\|_2^2\\ &\leq C(\|\nabla u\|_2^2+\|\nabla w\|_2^2+\|\nabla b\|_2^2)\|\nabla u\|_{\dot{X}_{r}}^{2/(2-r)}. \end{split}$$ Applying Gronwall's inequality, we have $$\label{e3.18} (\|\nabla u\|_2^2+\|\nabla w\|_2^2+\|\nabla b\|_2^2) \leq C \exp\{{\int_0^{T}\|\nabla u\|_{\dot{X}_{r}}^{2/(2-r)}dt}\}.$$ Combining the a priori estimate \eqref{e3.18} with the energy inequality \eqref{e1.3} and by standard arguments of continuation of local solutions, we conclude that the solutions $(u,w,b)$ can be extended beyond $t = T$ provided that $\nabla u\in L^{2/(2-r)}(0,T;\dot{X}_{r}(R^3)),\,r\in[0,1]$. This completes the proof of Theorem 1.1. \noindent\textbf{Remark.} We point out that the above methods do not seem to work for a bounded domains. 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