\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 06, 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 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/06\hfil A regularity criterio] {A regularity criterion for the Navier-Stokes equations in terms of the horizontal derivatives of the two velocity components} \author[W. Chen, S. Gala\hfil EJDE-2011/06\hfilneg] {Wenying Chen, Sadek Gala} % in alphabetical order \address{Wenying Chen \newline College of Mathematics and Computer Science, Chongqing Three Gorges University, Wanzhou 404000, Chongqing, China} \email{wenyingchenmath@gmail.com} \address{Sadek Gala \newline Department of Mathematics, University of Mostaganem, Box 227, Mostaganem 27000, Algeria} \email{sadek.gala@gmail.com} \thanks{Submitted October 18, 2010. Published January 12, 2011.} \subjclass[2000]{35Q30, 76F65} \keywords{Navier-Stokes equations; Leray-Hopf weak solutions; \hfill\break\indent regularity criterion} \begin{abstract} In this article, we consider the regularity for weak solutions to the Navier-Stokes equations in $\mathbb{R}^3$. It is proved that if the horizontal derivatives of the two velocity components $\nabla _h\widetilde{u}\in L^{2/(2-r)}(0,T;\dot{\mathcal{M}}_{2,3/r} (\mathbb{R}^3)),\quad \text{for }00. \end{gather*} The following imbedding holds \[ L^{3/r}\subset \dot{X}_{r},\quad 0\leq r<\frac{3}{2}\,.$ Now we recall the definition of Morrey-Campanato spaces (see e.g. \cite{Kat}). \begin{definition} \label{def2.2} \rm For $10} R^{3/q-3/p}\| f\|_{L^p(B(x,R))}<\infty \big\}. \label{eq1.13} \end{definition} It is easy to check that $\| f(\lambda .)\| _{\dot{\mathcal{M}}_{p,q}} = \frac{1}{\lambda ^{3/q}}\|f\| _{\dot{ \mathcal{M}}_{p,q}}, \quad \lambda >0.$ We have the following comparison between Lorentz and Morrey-Campanato spaces: For$p\geq 2$, $L^{\frac{3}{r}}(\mathbb{R}^3)\subset \text{ }L^{3/r,\infty } (\mathbb{R}^3)\subset \dot{\mathcal{M}}_{p,3/r}(\mathbb{R}^3).$ The relation $L^{\frac{3}{r},\infty }(\mathbb{R}^3)\subset \dot{ \mathcal{M}}_{p,\frac{3}{r}}(\mathbb{R}^3)$ is shown as follows. Let$f\in L^{3/r,\infty }(\mathbb{R} ^3). Then \begin{align*} \| f\| _{\dot{\mathcal{M}}_{p,\frac{3}{r}}} &\leq \sup_{E}| E|^{\frac{r}{3}-\frac{1}{2}} \Big(\int_{E}| f(y)| ^pdy\Big)^{1/p} \\ &= \Big(\sup_E | E| ^{\frac{pr}{3}-1}\int_{E}| f(y)| ^pdy\Big)^{1/p} \\ &\cong \Big(\sup_{R>0} R| \{ x\in \mathbb{R} ^3:| f(y)| ^p>R\} |^{\frac{pr}{3}}Big)^{1/p} \\ &= \underset{R>0}{\sup }R\big| \big\{ x\in \mathbb{R}^p:| f(y)| >R\big\} \big| ^{r/3} \\ &\cong \| f\| _{L^{3/r,\infty }}. \end{align*} For0k}2^{j(r-1)}2^{j}\| \Delta _{j}f\| _{L^2} \\ &\leq \Big(\sum_{j\leq k}2^{2jr}\Big)^{1/2} \Big( \sum_{j\leq k}\| \Delta _{j}f\| _{L^2}^2\Big)^{1/2} +\Big(\sum_{j>k}2^{2j(r-1)}\Big)^{1/2} \Big(\sum_{j>k}2^{2j}\| \Delta _{j}f\| _{L^2}^2\Big)^{1/2} \\ &\leq C\big(2^{rk}\| f\|_{L^2}+2^{k(r-1)}\|f\| _{\dot{H}^{1}}\big)\\ &= C\big(2^{rk}A^{-r}+2^{k(r-1)}A^{1-r}\big)\| f\| _{L^2}^{1-r}\| f\| _{\dot{H}^{1}}^r, \end{align*} where $A=\| f\| _{\dot{H}^{1}}/ \|f\| _{L^2}$. Choose $k$ such that $2^{rk}A^{-r}\leq 1$; that is, $k\leq [ \log A^r]$, we thus obtain $\| f\| _{\dot{B}_{2,1}^r} \leq C\big(1+2^{k(r-1)}A^{1-r}\big)\| f\| _{L^2}^{1-r}\|f\| _{\dot{H}^{1}}^r \\ \leq C\| f\| _{L^2}^{1-r}\| \nabla f\|_{L^2}^r.$ \end{proof} Additionally, for $20$, $u_{0}\in H^{1}(\mathbb{R}^3)$ and $\nabla \cdot u_{0}=0$ in the sense of distributions. Assume that $u$ is a smooth solution of \eqref{eq1.1} on $\mathbb{R}^3\times (0,T)$ and satisfies any one of of the three degree $-1$ growth conditions \eqref{eq1.11}. Then \begin{aligned} &\sup_{0\leq t0$. \end{theorem} To prove this theorem, we need the following lemma. \begin{lemma}[\cite{BZ}] \label{lem1} Let$u$be a smooth solution to the Navier-Stokes system \eqref{eq1.1} in$\mathbb{R}^3$. Furthermore, let$\widetilde{u}=(u_1,u_2,0)$and$\nabla _h\widetilde{u}=(\partial _1 \widetilde{u},\partial _2\widetilde{u},0)$. Then $\big| \sum_{i,j,k=1}^3\int_{\mathbb{R}^3}u_{i}\partial _{i}u_{j}\partial _{kk}u_{j}dx\big| \leq C\int_{\mathbb{R} ^3}| \nabla _h\widetilde{u}| | \nabla u| ^2$ for some constant$C>0$. \end{lemma} The proof of this lemma is simple; see \cite[Lemma 2.2]{BZ}). \begin{proof}[Proof of Theorem \ref{th2}] Multiply the first equation of \eqref{eq1.1} by$\Delta u$, and integrating on$\mathbb{R}^3$, after suitable integration by parts, we obtain for$t\in (0,T), $$\frac{1}{2}\frac{d}{dt}\| \nabla u\| _{L^2}^2+\| \Delta u(t)\| _{L^2}^2\leq 2\big| \sum_{i,j,k=1}^3\int_{\mathbb{R}^3}u_{i}\partial _{i}u_{j}\partial _{kk}u_{j}dx\big| . \label{eq2.11}$$ Due to H\"{o}lder' s inequality and Lemma \ref{lem3}, the right-hand side \eqref{eq2.11} can be estimated as \begin{align*} \big| \sum_{i,j,k=1}^3\int_{\mathbb{R}^3}u_{i}\partial _{i}u_{j}\partial _{kk}u_{j}dx\big| &\leq \| \nabla_h \widetilde{u} \cdot \nabla u\| _{L^2}\| \nabla u\| _{L^2} \\ &\leq C\| \nabla _h\widetilde{u}\| _{\dot{ \mathcal{M}}_{2,3/r}}\| \nabla u\| _{_{\dot{B }_{2,1}^r}}\| \nabla u\| _{L^2} \\ &\leq C\| \nabla _h\widetilde{u}\| _{\dot{ \mathcal{M}}_{2,3/r}}\| \nabla u\| _{L^2}^{2-r}\| \Delta u\| _{L^2}^r \\ &\leq C\Big(\| \nabla _h\widetilde{u}\| _{\dot{ \mathcal{M}}_{2,3/r}}^{2/(2-r)}\| \nabla u\| _{L^2}^2\Big)^{(2-r)/2}(\| \Delta u\| _{L^2}^2)^{r/2}. \end{align*} By Young' s inequality, we obtain $$\big| \sum_{i,j,k=1}^3\int_{\mathbb{R}^3}u_{i}\partial _{i}u_{j}\partial _{kk}u_{j}dx\big| \leq \frac{1}{2}\| \Delta u\| _{L^2}^2+C\| \nabla _h\widetilde{u}\| _{ \dot{\mathcal{M}}_{2,3/r}}^{2/(2-r)}\| \nabla u\| _{L^2}^2 \label{eq6}$$ Substituting \eqref{eq6} into \eqref{eq2.11}, it follows that $\frac{d}{dt}\| \nabla u(.,t)\| _{L^2}^2+\| \Delta u(.,t)\| _{L^2}^2\leq C\| \nabla _h\widetilde{u }\| _{\dot{\mathcal{M}}_{2,3/r}}^{2/(2-r) }\| \nabla u\| _{L^2}^2.$ Then Gronwall' s inequality yields \begin{align*} &\| \nabla u(t)\| _{L^2}^2+\int_{0}^{T}\int_{\mathbb{R} ^3}\| \Delta u(x,s)\| _{L^2}^2\,dx\,ds\\ &\leq C\| \nabla u_{0}\| _{L^2}^2\exp \Big(\int_{0}^{t}\| \nabla _h\widetilde{u}(.,s)\| _{\dot{\mathcal{M}}_{2,3/r}} ^{2/(2-r)}ds\Big). \end{align*} This completes the proof . \end{proof} \section{Proof of Theorem \ref{th1}} After we established the key estimate in section 2, the proof of Theorem \ref{th1} is straightforward. It is well known \cite{FK} that there is a unique strong solution\overline{u}\in C([0,T^{\ast }),H^{1}(\mathbb{R}^3))\cap C^{1}([0,T^{\ast }),H^{1}(\mathbb{R}^3))\cap C([0,T^{\ast }),H^3(\mathbb{R}^3))$to \eqref{eq1.1} for some$T^{\ast }>0$, for any given$u_{0}\in H^{1}(\mathbb{R}^3)$with$\nabla.u_{0}=0$. Since$u$is a Leray-Hopf weak solution which satisfies the energy inequality \eqref{eq1.9}, we have according to the Serrin's uniqueness criterion \cite{Se}, $\bar{u}\equiv u\quad \text{on } [0,T^{\ast }).$ By the assumption \eqref{eq1.11} and standard continuation argument, the local strong solution can be extended to time$T$. So we have proved$u$actually is a strong solution on$[0,T)$. This completes the proof of Theorem \ref{th1}. \begin{thebibliography}{00} \bibitem{Bdv} H. Beir\~{a}o da Veiga; \emph{A new regularity class for the Navier-Stokes equations in$R^{n}$}. Chin. Ann. Math. 16 (1995) 407-412. \bibitem{BZ} B. Dong and Z. 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