\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 72, pp. 1--14.\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-2012/72\hfil Blow-up criterion] {Blow-up criterion for two-dimensional heat convection equations with zero heat conductivity} \author[Y.-Z. Wang, Z. Wei \hfil EJDE-2012/72\hfilneg] {Yu-Zhu Wang, Zhiqiang Wei} \address{Yu-Zhu Wang \newline School of Mathematics and Information Sciences \\ North China University of Water Resources and Electric Power, Zhengzhou 450011, China} \email{yuzhu108@163.com} \address{Zhiqiang Wei \newline School of Mathematics and Information Sciences \\ North China University of Water Resources and Electric Power, Zhengzhou 450011, China} \email{weizhiqiang@ncwu.edu.cn} \thanks{Submitted February 28, 2012. Published May 10, 2012.} \subjclass[2000]{76D03, 35Q35} \keywords{Heat convection equations; smooth solutions; blow-up criterion} \begin{abstract} In this article we obtain a blow-up criterion of smooth solutions to Cauchy problem for the incompressible heat convection equations with zero heat conductivity in $\mathbb{R}^2$. Our proof is based on careful H\"oder estimates of heat and transport equations and the standard Littlewood-Paley theory. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} The incompressible heat convection equations in two space dimensions take the form $$\begin{gathered} \partial_t u+u\cdot\nabla u+\nabla \pi =\mu \Delta u+\theta e_2, \\ \partial_t\theta +u\cdot\nabla \theta-\nu\Delta \theta =\frac{\mu}{2}\sum^2_{i, j=1}(\partial_iu^j+\partial_ju^i)^2, \\ \nabla \cdot u=0, \end{gathered} \label{e1.1}$$ where $u=(u^1, u^2)^t$ is the fluid velocity, $\pi$ is the pressure, $\theta$ stands for the absolute temperature, $\mu$ is the coefficient of viscosity, $\nu$ is the coefficient of heat conductivity and $e_2=(0, 1)$. Some problems related to \eqref{e1.1} have been studied in recent years (see \cite{lamb}, \cite{fo}, \cite{hi}-\cite{rk} and \cite{lk}). Fan and Ozawa \cite{fo} obtained some regularity criteria of strong solutions to the Cauchy problem for the \eqref{e1.1} in $\mathbb{R}^3$. Hiroshi \cite{hi} proved the existence of the strong solutions for the initial boundary value problems for \eqref{e1.1}. Kagei and Skowron \cite{ks} discussed the existence and uniqueness of solutions of the initial-boundary value problem for the heat convection equations \eqref{e1.1} of incompressible asymmetric fluids in $\mathbb{R}^3$. Moreover, Kagei \cite{kage} considered global attractors for the initial-boundary value problem for \eqref{e1.1} in $\mathbb{R}^2$. Lukaszewicz and Krzyzanowski \cite{lk} treated the initial-boundary value problem for \eqref{e1.1} with moving boundaries in $\mathbb{R}^3$. Kakizawa \cite{rk} proved that \eqref{e1.1} has uniquely a mild solution. Moreover, a mild solution of \eqref{e1.1} can be a strong or classical solution under appropriate assumptions for initial data. It is well known that the Boussinesq approximation \cite{bj1} is a simplified model of heat convection of incompressible viscous fluids. There is no doubt that many investigations on the Boussinesq approximation have been carried out for a hundred years. For regularity criteria of weak solutions and blow up criteria of smooth solutions, we refer to \cite{fz} and so on. Equation \eqref{e1.1} is the Navier-Stokes equations coupled with the heat equation. Due to its importance in mathematics and physics, there is lots of literature devoted to the mathematical theory of the Navier-Stokes equations. Leray-Hopf weak solution were constructed by Leray \cite{leray} and Hopf \cite{hopf}, respectively. Later on, much effort has been devoted to establish the global existence and uniqueness of smooth solutions to the Navier-Stokes equations. Different criteria for regularity of the weak solutions have been proposed and many interesting results were established (see \cite{cg}, \cite{fo}-\cite{fjnz}, \cite{gala1}, \cite{he}, \cite{serrin} and \cite{zhou}-\cite{zg}). Serrin-type regularity criteria of Leray weak solutions in terms of pressure in Besov space were obtained in \cite{gg} and \cite{hg}. In this paper, we consider \eqref{e1.1} with the zero heat conductivity; i.e., $\nu=0$. Without loss of generality, we take $\mu=1$. The corresponding heat convection equations thus reads $$\begin{gathered} \partial_t u+u\cdot\nabla u+\nabla \pi = \Delta u+\theta e_2, \\ \partial_t\theta +u\cdot\nabla \theta =\frac{1}{2}\sum^2_{i, j=1}(\partial_iu^j+\partial_ju^i)^2, \\ \nabla \cdot u=0. \end{gathered} \label{e1.3}$$ Due to the term $\frac{1}{2}\sum^2_{i,j=1}(\partial_iu^j+\partial_ju^i)^2$, it is very difficult to deal with \eqref{e1.3}. The local well-posedness of the Cauchy problem for \eqref{e1.3} is rather standard, which can be obtained by standard Galerkin method and energy estimates (for example see \cite{fo}). In the absence of global well-posedness, the development of blow-up/ non blow-up theory (see \cite{bkm}) is of major importance for both theoretical and pratical purposes. In this paper, we obtain a blow-up criterion of smooth solutions to the Cauchy problem for \eqref{e1.3}. Our main theorem is as follows. \begin{theorem} \label{thm1.1} Assume that $(u, \theta)$ is a local smooth solution to the heat convection equations with zero heat conductivity \eqref{e1.3} on $[0, T)$ and $\|u(0)\|_{H^1\cap\dot{C}^{1+\alpha}}+\|\theta(0)\|_{L^2\cap \dot{C}^\alpha}<\infty$ for some $\alpha \in (0, 1)$. Then $$\|u(t)\|_{\dot{C}^{1+\alpha}}+\|\theta(t)\|_{\dot{C}^\alpha}<\infty$$ for all $0\leq t\leq T$ provided that $$\|u\|_{L^2_T(\dot{B}^0_{\infty, \infty})}<\infty, \quad \|\theta\|_{L^1_T(\dot{B}^0_{\infty, \infty})}<\infty. \label{e1.4}$$ \end{theorem} This article is organized as follows. We first state some preliminary on functional settings and some important inequalities in Section 2 and then prove the blow-up criterion of smooth solutions of \eqref{e1.3} in Section 3. \section{Preliminaries} Let $\mathcal{S}(\mathbb{R}^2)$ be the Schwartz class of rapidly decreasing functions. Given $f \in \mathcal{S}(\mathbb{R}^2)$, its Fourier transform $\mathcal{F}f=\hat{f}$ is defined by $$\hat{f}(\xi)=\int_{\mathbb{R}^2} e^{-ix\cdot\xi}f(x)dx$$ and for any given $g \in \mathcal{S}(\mathbb{R}^2)$, its inverse Fourier transform $\mathcal{F}^{-1}g=\check{g}$ is defined by $$\check{g}(x)=\int_{\mathbb{R}^2} e^{ix\cdot\xi}g(\xi)d\xi.$$ Next let us recall the Littlewood-Paley decomposition. Choose two non-negative radial functions $\chi, \phi \in \mathcal{S}(\mathbb{R}^2)$, supported respectively in $\mathbb{B}=\{\xi \in \mathbb{R}^2: |\xi|\leq \frac{4}{3}\}$ and $\mathcal{C}=\{ \xi \in \mathbb{R}^2: \frac{3}{4}\leq|\xi|\leq \frac{8}{3}\}$ such that $$\chi(\xi)+\sum_{k\geq 0}\phi(2^{-k}\xi)=1, \quad \forall \xi \in \mathbb{R}^2$$ and $$\sum ^{\infty}_{k=-\infty}\phi (2^{-k}\xi)=1, \quad \forall \xi \in \mathbb{R}^2\backslash\{0\}.$$ The frequency localization operator is defined by $$\Delta_kf =\int_{\mathbb{R}^2}\check{\phi}(y)f(x-2^{-k}y)dy, \quad S_kf=\sum_{k'\leq k-1}\Delta_{k'}f.$$ Let us now recall homogeneous Besov spaces (for example, see \cite{bl} and \cite{ht}). For $(p, q)\in [1, \infty]^2$ and $s \in \mathbb{R}$, the homogeneous Besov space $\dot{B}^s_{p, q}$ is defined as the set of $f$ up to polynomials such that $$\|f\|_{\dot{B}^s_{p,q}} = \|2^{ks} \|\Delta_kf\|_{L^p}\|_{l^q(\mathbb{Z})}< \infty.$$ Finally, we recall the following space, which is defined in \cite{cjy2}. Let $p$ be in $[1, \infty]$ and $r \in \mathbb{R}$; the space $\tilde{L}^p_T(C^r)$ is the space of the distributions $f$ such that $$\|f\|_{\tilde{L}^p_T(C^r)}=\sup_k 2^{kr} \|\Delta_kf\|_{L^p_T(L^\infty)}<\infty.$$ The open ball with radius $R$ centered at $x_0 \in \mathbb{R}^2$ is denoted by $\textbf{B}(x_0,R)$. The ring $\{\xi\in \mathbb{R}^2 | R_1 \leq |\xi|\leq R_2\}$ is denoted by $\textbf{C}(0,R_1,R_2)$. In what follows, we shall use Bernstein inequalities, which can be found in \cite{cjy}. \begin{lemma} \label{lem2.1} Let $k$ a positive integer and $\sigma$ any smooth homogeneous function of degree $m\in \mathbb{R}$. A constant $C$ exists such that, for any positive real number $\lambda$ and any function $f$ in $L^p(\mathbb{R}^2)$, we have \begin{gather} \operatorname{supp} \hat{f}\subset \lambda\textbf{B}\Rightarrow \sup_{|\beta|=k}\|\partial^\beta f\|_{L^q}\leq C\lambda^{k+2(\frac{1}{p}-\frac{1}{q})}\|f\|_{L^p},\label{e2.1} \\ \operatorname{supp} \hat{f}\subset \lambda \textbf{C} \Rightarrow C^{-1}\lambda^{k}\|f\|_{L^p}\leq \sup_{|\beta|=k} \|\partial^\beta f\|_{L^p}\leq C \lambda^{k}\|f\|_{L^p}. \label{e2.2} \end{gather} Moreover, if $\sigma$ is a smooth function on $\mathbb{R}^2$ which is homogeneous of degree $m$ outside a fixed ball, then we have $$\operatorname{supp} \hat{f}\subset \lambda\textbf{C} \Rightarrow \|\sigma(D)f\|_{L^q}\leq C \lambda^{(m+2(\frac{1}{p}-\frac{1}{q}))}\|f\|_{L^p}. \label{e2.3}$$ \end{lemma} \begin{lemma} \label{lem2.2} For any $f \in L^p(\mathbb{R}^2)(p>1)$ and any positive real number $\lambda$, $$\operatorname{supp} \hat{f}\subset \lambda \textbf{C} \Rightarrow \|e^{t\Delta}f\|_{L^p}\leq Ce^{-c \lambda^{2}t}\|f\|_{L^p}, \label{e2.4}$$ where $C$ and $c$ are positive constants. See \cite{cjy1} for the proof of \eqref{e2.4}. \end{lemma} The following lemma plays an important role in the proof of Theorem \ref{thm1.1} (see also \cite {mzz} and \cite{m} where similar estimate were established). \begin{lemma} \label{lem2.3} Assume that $\gamma>0$, then there exists a positive constant $C>0$ such that $$\|f\|_{L^\infty}\leq C \left(1+\|f\|_{L^2}+\|f\|_{\dot{B}^0_{\infty, \infty}}\ln(e+\|f\|_{\dot{C}^\gamma})\right) \label{e2.5}$$ and $$\begin{split} \int^T_0\|\nabla f(\tau)\|_{L^\infty}d\tau &\leq C\Big(1+\int^T_0\| f(\tau) \|_{L^2}d\tau + \sup_k\int^T_0\|\Delta_k\nabla f(\tau)\|_{L^\infty}d\tau\\ &\quad\times \ln\big(e+\int^T_0\|\nabla f(\tau)\|_{\dot{C}^\gamma}d\tau\big)\Big). \end{split}\label{e2.6}$$ \end{lemma} \begin{proof} If $f\in W^{m,p}$, $m>\frac{2}{p}$, $C^\gamma$ in \eqref{e2.5} is replaced by $W^{m, p}$, then \eqref{e2.5} still holds. For example, see \cite{bkm,kt}. It is not difficult to prove \eqref{e2.5} (see \cite{ww}). For the reader convenience, we give a detail proof. It follows from Littlewood-Paley composition that $$f=\sum^0_{k=-\infty }\Delta_k f+\sum^A_{k=1}\Delta_kf+\sum^\infty_{k= A+1}\Delta_k f. \label{e2.7}$$ Using \eqref{e2.7} and \eqref{e2.3}, we obtain \begin{align*} { \| f \|_{L^\infty}} &\leq { \sum^0_{k=-\infty} \|\Delta_k f\|_{L^\infty}+ A\max_{1\leq k\leq A}\|\Delta_k f\|_{L^\infty} +\sum^\infty_{k= A+1}\|\Delta_kf\|_{L^\infty}} \\ &\leq { C\sum^0_{k=-\infty } 2^k\|\Delta_k f\|_{L^2}+A\|f\|_{\dot{B}^0_{\infty, \infty}}+ \sum^\infty_{k= A+1}2^{-\gamma k}2^{\gamma k}\|\Delta_kf\|_{L^\infty}} \\ &\leq { C\|f\|_{L^2}+A\|f\|_{\dot{B}^0_{\infty, \infty}} + \sum^\infty_{k= A+1}2^{-\gamma k}\| f\|_{\dot{C}^\gamma}} \\ &\leq { C\| f\|_{L^2}+A\|f\|_{\dot{B}^0_{\infty, \infty}}+2^{-\gamma A} \| f\|_{\dot{C}^\gamma}.} \end{align*} Equation \eqref{e2.5} follows immediately by choosing $$A=\frac{1}{\gamma}\log_2(e+\| f\|_{\dot{C}^\gamma}) \leq C\ln (e+\|f\|_{\dot{C}^\gamma}).$$ Similar to the proof of \eqref{e2.5}, we can obtain \eqref{e2.6} (see also \cite{lmz}). Thus the proof is complete. \end{proof} To prove Theorem \ref{thm1.1}, we need the following interpolation inequalities in two space dimensions. \begin{lemma} \label{lem2.4} The following inequalities hold $$\|f\|_{L^p}\leq C\|f\|^{1-\frac{2}{q} +\frac{2}{p}}_{L^q}\|\nabla f\|^{\frac{2}{q}-\frac{2}{p}}_{L^q},\quad -\frac{2}{p}\leq 1-\frac{2}{q}, \quad p\geq q. \label{e2.8}$$ \end{lemma} \begin{proof} Noting $-\frac{2}{p}\leq 1-\frac{2}{q}$, $p\geq q$ and using the Sobolev embedding theorem, we obtain $$\|f\|_{L^p}\leq C(\|f\|_{L^q}+\|\nabla f\|_{L^q}). \label{e2.9}$$ Let $f_\lambda(x)=f(\lambda x)$. From \eqref{e2.9}, we obtain $$\|f_\lambda\|_{L^p}\leq C(\|f_\lambda\|_{L^q}+\|\nabla f_\lambda\|_{L^q}),$$ which implies $$\|f\|_{L^p}\leq C(\lambda^{\frac{2}{p}-\frac{2}{q}}\|f\|_{L^q} +\lambda^{1+\frac{2}{p}-\frac{2}{q}}\|\nabla f\|_{L^q}). \label{e2.10}$$ Taking $\lambda=\|f\|_{L^q}\|\nabla f\|^{-1}_{L^q}$, from \eqref{e2.10}, we immediately obtain \eqref{e2.8}. Thus, the proof is complete. \end{proof} \section{Proof of main results} This section is devoted to the proof of Theorem \ref{thm1.1}, for which we need the following Lemma that is basically established in \cite{fo}. For completeness, the proof is also sketched here. \begin{lemma} \label{lem3.1} Assume $\|u(0)\|_{ H^1}+ \|\theta(0)\|_{L^2}<\infty$ and assume furthermore that $(u, \theta)$ is a smooth solution to the Cauchy problem for \eqref{e1.3} on $\times [0, T)$. If $$u\in L^2\left(0, T; \dot{B}^0_{\infty, \infty}(\mathbb{R}^2)\right), \label{e3.1}$$ then \begin{aligned} &\|u(t)\|^2_{L^2}+\|\nabla u(t)\|^2_{L^2}+\|\theta(t)\|^2_{L^2} +\int^T_0(\|\nabla u(t)\|^2_{L^2}+\|\Delta u(t)\|^2_{L^2})dt\\ &\leq C(\|u(0)\|^2_{H^1}+\|\theta(0)\|^2_{L^2}). \end{aligned} \label{e3.2} \end{lemma} \begin{proof} Multiplying the first equation in \eqref{e1.3} by $u$ and using Cauchy inequality, we obtain $$\frac{1}{2}\frac{d}{dt}\|u(t)\|^2_{L^2}+\|\nabla u(t)\|^2_{L^2} \leq \frac{1}{2}\int_{\mathbb{R}^2}(|\theta|^2+|u|^2)(x, t)dx. \label{e3.3}$$ Multiplying the first equation in \eqref{e1.3} by $-\Delta u,$ using integration by parts, we obtain $$\frac{1}{2}\frac{d}{dt}\|\nabla u(t)\|^2_{L^2}+\|\Delta u(t)\|^2_{L^2} =-\int_{\mathbb{R}^2}\theta e_2\cdot \Delta u dx +\int_{\mathbb{R}^2}u\cdot\nabla u\cdot \Delta udx. \label{e3.4}$$ Note that (see \cite{zf}) $$-\Delta u=\nabla \times (\nabla \times u), \quad \nabla \times (u\cdot\nabla u)=u\cdot\nabla (\nabla\times u)$$ provided that $\nabla\cdot u=0$. Using integration by parts, we obtain \begin{aligned} { \int_{\mathbb{R}^2}u\cdot\nabla u\cdot \Delta udx}&= -\int_{\mathbb{R}^2} (u\cdot \nabla u)\cdot \nabla \times (\nabla \times u)dx \\ &= -\int_{\mathbb{R}^2} \nabla \times (u\cdot\nabla u)\cdot\nabla \times udx \\ &= -\int_{\mathbb{R}^2}u\cdot\nabla (\nabla \times u)\cdot(\nabla \times u)dx = 0. \end{aligned} \label{e3.5} It follows from \eqref{e3.4}, \eqref{e3.5} and Young inequality that $$\frac{1}{2}\frac{d}{dt}\|\nabla u(t)\|^2_{L^2}+\frac{1}{2}\|\Delta u(t)\|^2_{L^2} \leq C\|\theta(t)\|^2_{L^2}. \label{e3.6}$$ Multiplying the second equation in \eqref{e1.3} by $\theta$, using H\"{o}lder inequality and Young inequality, it holds that \begin{aligned} \frac{1}{2}\frac{d}{dt}\|\theta(t)\|^2_{L^2} &= -\frac{1}{2}\sum^2_{i, j=1}\int_{\mathbb{R}^2} \theta(\partial_iu_j+\partial_ju_i)^2dx \\ &\leq C\|\theta(t)\|_{L^2}\|\nabla u(t)\|^2_{L^4} \\ &\leq C\|\theta(t)\|_{L^2} \|u(t)\| _{\dot{B}^0_{\infty, \infty}} \|\Delta u(t)\|_{L^2} \\ &\leq \frac{1}{6}\|\Delta u(t)\|^2_{L^2}+C\|u(t)\|^2 _{\dot{B}^0_{\infty, \infty}} \|\theta(t)\|^2_{L^2}, \end{aligned} \label{e3.7} where we have used the interpolation inequality (see for example \cite{mo}) $$\|\nabla u(t)\|_{L^4} \leq C \|u(t)\|^{1/2} _{\dot{B}^0_{\infty, \infty}}\|\Delta u(t)\|^{1/2}_{L^2}. \label{e3.8}$$ Collecting \eqref{e3.3}, \eqref{e3.6} and \eqref{e3.7} gives \begin{aligned} &\frac{d}{dt}(\|u(t)\|^2_{L^2}+\|\nabla u(t)\|^2_{L^2}+\|\theta(t)\|^2_{L^2})+\|\nabla u(t)\|^2_{L^2}+\|\Delta u(t)\|^2_{L^2}\\ & \leq C\left(\|u(t)\|^2_{L^2}+\|\theta(t)\|^2_{L^2}+\|u(t)\|^2_{\dot{B}^{0}_{\infty, \infty}}(\|\nabla u(t)\|^2_{L^2}+\|\theta(t)\|^2_{L^2})\right). \end{aligned} \label{e3.9} Inequality \eqref{e3.2} follows immediately from \eqref{e3.1}, \eqref{e3.9} and Gronwall's inequality. Thus, the proof complete. \end{proof} We also need the following lemma (see also \cite{cjy2,lmz} where similar estimates were established). \begin{lemma} \label{lem3.2} Assume that $F\in \tilde{L}^1_T(C^{-1})\cap L^2_T({L}^{2})$ and $u_0\in L^2$. Let $u$ be a solution of the Navier-Stokes equations $$\begin{gathered} \partial_t u+u\cdot\nabla u+\nabla \pi = \Delta u+F, \\ \nabla \cdot u=0, \\ t=0:\quad u=u_0(x). \end{gathered} \label{e3.10}$$ Then it holds that \begin{aligned} \|u\|_{\tilde{L}^1_T(C^{1})} &\leq C(\sup_k\|\Delta_ku_0\|_{L^2}(1-\exp\{-c2^{2k}T\}) +(\|u_0\|_{L^2} \\ &\quad +\|F\|_{L^2_T({L}^{2})})\|\nabla u\|^2_{L^2_T(L^2)} + \sup_k\int^T_02^{-k}\|\Delta_kF(\tau)\|_{L^\infty}d\tau ). \end{aligned} \label{e3.11} \end{lemma} \begin{proof} Applying $\Delta_k$ to \eqref{e3.10}, we obtain $$\Delta_ku=e^{\Delta t}\Delta_ku_0 +\int^t_0e^{\Delta(t-\tau)}\Delta_k \mathbb{P} \big(\nabla \cdot(u\otimes u)+F\big)(\tau)d\tau, \label{e3.12}$$ where operator $\mathbb{P}$ satisfies $(\hat{\mathbb{P}u})^i=\sum^2_{j=1} (\delta_{ij}-\frac{\xi^i\xi^j}{|\xi|^2})\hat{u}^j(\xi).$ It follows from \eqref{e2.3} and \eqref{e2.4} that \begin{aligned} &\|\Delta_ku(t)\|_{L^\infty}\\ &\leq C \Big(e^{-c2^{2k}t}\|\Delta_ku_0\|_{L^\infty} +\int^t_0e^{-c2^{2k}(t-\tau)}\|\Delta_k\nabla \cdot(u\otimes u)(\tau) \|_{L^\infty}d\tau\Big)\\ &\quad + C\int^t_0e^{-c2^{2k}(t-\tau)}\|\Delta_k F(\tau)\|_{L^\infty}d\tau. \end{aligned} \label{e3.13} This implies that \begin{aligned} &\|u\|_{\tilde{L}^1_T(C^{1})}\\ &\leq C\sup_k\int^T_0 2^ke^{-c2^{2k}t}\|\Delta_ku_0\|_{L^\infty}dt \\ &\quad + C\sup_k\int^T_0\int^t_02^{2k}e^{-c2^{2k}(t-\tau)} \|\Delta_k u\otimes u(\tau)\|_{L^\infty}d\tau dt \\ &\quad + C\sup_k\int^T_0\int^t_02^{k}e^{-c2^{2k}(t-\tau)} \|\Delta_k F(\tau)\|_{L^\infty}d\tau dt \\ &\leq C\sup_k\|\Delta_ku_0\|_{L^2}(1-e^{-c2^{2k}T})\\ &\quad + C\sup_k\int^T_0\|\Delta_k(u\otimes u)(\tau)\|_{L^\infty}d\tau +\sup_k\int^T_02^{-k}\|\Delta_kF(\tau)\|_{L^\infty}d\tau . \end{aligned} \label{e3.14} It follows from Bony decomposition that \begin{align*} &\|\Delta_k(u\otimes u)(\tau)\|_{L^\infty}\\ &= \sum_{|m-n|\leq 1}\|\Delta_k(\Delta_mu\otimes\Delta_nu)(\tau)\|_{L^\infty} +\sum_{m-n \geq 2}\|\Delta_k(\Delta_mu\otimes\Delta_nu)(\tau)\|_{L^\infty}\\ &\quad + \sum_{n-m\geq 2}\|\Delta_k(\Delta_mu\otimes\Delta_nu)(\tau)\|_{L^\infty} \end{align*} By \eqref{e2.1} and \eqref{e2.2}, a straight computation gives \begin{align*} & \int^T_0\sum_{|m-n|\leq 1}\|\Delta_k(\Delta_mu\otimes\Delta_nu) (\tau)\|_{L^\infty}d\tau \\ &\leq C\int^T_0\sum_{|m-n|\leq 1}2^k\|\Delta_k(\Delta_mu\otimes\Delta_nu)(\tau) \|_{L^2}d\tau \\ &\leq C\int^t_0\sum_{|m-n|\leq 1, m\geq k-3}2^{k-\frac{m+n}{2}} \|2^m\Delta_m u(\tau)\|^{1/2}_{L^\infty} \|\Delta_n u(\tau)\|^{1/2}_{L^2}\|\Delta_m u(\tau)\|^{1/2}_{L^\infty}\\ &\quad \times \|2^n\Delta_nu(\tau)\|^{1/2}_{L^2}d\tau \\ &\leq C\int^t_0\sum_{|m-n|\leq 1, m\geq k-3}2^{k-\frac{m+n}{2}} \|2^m\Delta_m u(\tau)\|^{1/2}_{L^\infty} \|\Delta_n u(\tau)\|^{1/2}_{L^2}\|2^m\Delta_m u(\tau)\|^{1/2}_{L^2}\\ &\quad\times \|2^n\Delta_nu(\tau)\|^{1/2}_{L^2}d\tau \\ & \leq C\|u\|^{1/2}_{L^\infty_T(L^2)}\|\nabla u\|_{L^2_T(L^2)} \|u\|^{1/2}_{\tilde{L}^1_T(C^1)}. \end{align*} Similarly, we obtain \begin{align*} &\int^T_0\Big(\sum_{m-n\geq 2}\|\Delta_k(\Delta_mu\otimes \Delta_nu)(\tau)\|_{L^\infty} +\sum_{n-m\geq 2}\|\Delta_k(\Delta_mu\otimes \Delta_nu)(\tau)\|_{L^\infty}\Big)d\tau \\ & \leq C\int^T_0\sum_{m-n\geq 2, |m-k|\leq 2}\|\Delta_m u(\tau) \|_{L^\infty}\|\Delta_n u(\tau)\|_{L^\infty}d\tau \\ &\leq C\sum_{m-n\geq 2, |m-k|\leq 2}\|2^m\Delta_m u(\tau)\| ^{1/2}_{L^\infty}\|2^m\Delta_m u(\tau)\|^{1/2}_{L^2}2^{n-\frac{m}{2}} \|\Delta_n u(\tau)\|_{L^2}d\tau \\ &\leq C\|u\|^{1/2}_{L^\infty_T(L^2)}\|\nabla u\|_{L^2_T(L^2)} \|u\|^{1/2}_{\tilde{L}^1_T(C^1)}. \end{align*} Using the above two estimates, from \eqref{e3.14} and Young inequality, we obtain \begin{aligned} \|u\|_{\tilde{L}^1_T(C^1)} &\leq C(\sup_k\|\Delta_ku_0\|_{L^2}(1-\exp\{-c2^{2k}T\}) +\|u\|_{L^\infty_T(L^2)}\|\nabla u\|^2_{L^2_T(L^2)} \\ &\quad + \sup_k\int^T_02^{-k}\|\Delta_kF(\tau)\|_{L^\infty}d\tau ). \end{aligned} \label{e3.15} Combining \eqref{e3.15} and the basic energy estimate $$\|u\|^2_{L^\infty_T(L^2)}+\|\nabla u\|^2_{L^2_T(L^2)}\leq C(\|u_0\|^2_{L^2}+\|F\|^2_{L^2_T({L}^{2})}) \label{e3.16}$$ gives \eqref{e3.11}. Thus, the proof is complete. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.1}] Set $F=u\cdot\nabla u+\theta e_2$. It follows from \eqref{e1.4} and \eqref{e3.2} that $F\in \tilde{L}^1_T(C^{-1})\cap L^2_T({L}^{2})$. Applying $\Delta_k$ to both sides of \eqref{e3.10} and using standard energy estimate, \eqref{e2.2} and Young inequality, we have \begin{align*} &\frac{1}{2}\frac{d}{dt}\|\Delta_ku\|^2_{L^2}+c2^{2k}\|\Delta_ku\|^2_{L^2}\\ &\leq \frac{c}{2}2^{2k}\|\Delta_ku\|^2_{L^2} +C(\|\Delta_ku\|_{L^2}+\|\Delta_kF\|^2_{L^2}+\|\Delta_k(u\otimes u)\|^2_{L^2}). \end{align*} Integrating the above inequality with respect to $t$ and summing over $k$, we obtain \begin{aligned} &\sum_{k}\|\Delta_ku\|^2_{L^\infty_T(L^2)} +\sum_{k}\int^t_02^{2k}\|\Delta_ku(\tau)\|^2_{L^2}d\tau \\ &\leq C(\|u_0\|^2_{L^2}+\|F\|^2_{L^2_T({L}^{2})}+ \|u\|^2_{L^\infty_T(L^2)}\|\nabla u\|^2_{L^2_T(L^2)}), \end{aligned}\label{e3.17} where we used the interpolation inequality (see Lemma \ref{lem2.4}) $$\|u\|_{L^4}\leq C\|u\|^{1/2}_{L^2}\|\nabla u\|^{1/2}_{L^2}.$$ It follows from \eqref{e3.16} and \eqref{e3.17} that \begin{aligned} &\sum_{k}\|\Delta_ku\|^2_{L^\infty_T(L^2)} +\sum_{k}\int^t_02^{2k}\|\Delta_ku(\tau)\|^2_{L^2}d\tau \\ &\leq C(\|u_0\|^2_{L^2} +\|F\|^2_{L^2_T({L}^{2})})(1+\|u_0\|^2_{L^2}+\|F\|^2_{L^2_T({L}^{2})}). \end{aligned}\label{e3.18} Using \eqref{e3.18}, for any $t_0\in [0, T)$, we can choose $k_0>0$ such that $$\sup_{k\geq k_0}\|\Delta_ku\|_{L^\infty_{[t_0, T]}(L^2)}\leq \frac{\varepsilon}{4C}.$$ By \eqref{e3.16}, we can choose $t_1\in[t_0, T]$ such that \begin{align*} &\sup_{t_1\leq t\leq T}\sup_{k\leq k_0}\|\Delta_ku(t)\|_{L^2}(1-\exp\{-c2^{2k}(T-t)\})\\ &\leq \sup_{t_1\leq t\leq T}2c2^{2k_0}(T-t_1)\|u(t)\|_{L^2} \\ &\leq C2^{2k_0}(\|u_0\|_{L^2}+\|F\|_{L^2_T({L}^{2})})(T-t_1) \leq \frac{\varepsilon}{4C}. \end{align*} Consequently, $$\sup_{t_1\leq t\leq T}\sup_k\|\Delta_ku(t)\|_{L^2}(1-\exp\{-c2^{2k}(T-t)\}) \leq \frac{\varepsilon}{2C}. \label{e3.19}$$ On the other hand, we can choose $t_2\in[t_1, T)$ such that \begin{aligned} &\Big(\sup_{t_2\leq t\leq T}\|u(t)\|_{L^2}+\|F\|_{L^2_{[t_2, T]}({L}^{2})}\Big) \|\nabla u\|^2_{L^2_{[t_2, T]}(L^2)}\\ &+ \sup_k\int^T_{t_2}2^{-k}\|\Delta_kF(\tau)\|_{L^\infty}d\tau )\\ &\leq \frac{\varepsilon}{2C}. \end{aligned}\label{e3.20} It follows from \eqref{e3.11} that \begin{aligned} \|u\|_{\tilde{L}^1_{[t_2, T]}(C^{1})} &\leq C\Big(\sup_k\|\Delta_ku(t_2)\|_{L^2}(1-\exp\{-c2^{2k}(T-t_2)\})\\ &\quad + \big(\|u(t_2)\|_{L^2}+\|F\|_{L^2_{[t_2, T]}({L}^{2})}\big)\|\nabla u\|^2_{L^2_{[t_2, T]}(L^2)}\\ &\quad + \sup_k\int^T_{t_2}2^{-k}\|\Delta_kF(\tau)\|_{L^\infty}d\tau \Big). \end{aligned}\label{e3.21} Combining \eqref{e3.19}-\eqref{e3.21} gives $$\|u\|_{\tilde{L}^1_{[t_2, T]}(C^{1})}\leq \varepsilon. \label{e3.22}$$ Using \eqref{e3.22} and \eqref{e1.4}, we can choose $t^\ast\in [t_2, T)$ such that $$\|u\|_{\tilde{L}^1_{[t^\ast, T]}(C^{1})}\leq \varepsilon, \quad \|\theta\|_{L^1_{[t^\ast, T]}(\dot{B}^0_{\infty, \infty})}\leq \varepsilon. \label{e3.23}$$ For $0\leq t< T$, define M(t)=\sup_{0\leq \tau 0 is suitably small. By Gronwall's inequality and \eqref{e3.38}, we obtain \begin{align*} &e+\|u(0)\|_{\dot{C}^{1+\alpha}}+N(t) \\ &\leq C_\star(e+\|u(0)\|_{\dot{C}^{1+\alpha}} +\|\theta(0)\|_{\dot{C}^\alpha}) \exp\{C\int^t_0(\|\nabla u(\tau)\|_{L^\infty} +\|\theta(\tau)\|_{L^\infty})d\tau\} \\ &\leq C_\star(e+\|u(0)\|_{\dot{C}^{1+\alpha}}+\|\theta(0)\|_{\dot{C}^\alpha}) \exp\{C_\star+C\varepsilon \ln(e+\|u(0)\|_{\dot{C}^{1+\alpha}}+N(t))\} \\ &\leq C_\star(e+\|u(0)\|_{\dot{C}^{1+\alpha}}+\|\theta(0)\|_{\dot{C}^\alpha}) (e+\|u(0)\|_{\dot{C}^{1+\alpha}}+N(t))^{C\varepsilon}. \end{align*} Choosing \varepsilon>0 suitably small, the above inequality and \eqref{e3.29} yields M(t)+N(t)\leq C_\star(1+\|u(0)\|_{\dot{C}^{1+\alpha}} +\|\theta(0)\|_{\dot{C}^\alpha})^2.  The proof is complete. \end{proof} \subsection*{Acknowledgements} The research is supported by grant 11101144 from the NNSF of China. \begin{thebibliography}{00} \bibitem{bkm} J. Beale, T. Kato, A. 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