## Boundary Value Problems

Impact Factor 0.819

Open Access

# A logarithmically improved blow-up criterion for smooth solutions to the micropolar fluid equations in weak multiplier spaces

Boundary Value Problems20132013:122

DOI: 10.1186/1687-2770-2013-122

Accepted: 26 April 2013

Published: 10 May 2013

## Abstract

In this paper, we study the initial value problem for the three-dimensional micropolar fluid equations. A new logarithmically improved blow-up criterion for the three-dimensional micropolar fluid equations in a weak multiplier space is established.

MSC:35K15, 35K45.

### Keywords

micropolar fluid equations smooth solution blow-up criterion

## 1 Introduction

In the paper, we consider the initial value problem for the micropolar fluid equations in ${\mathbb{R}}^{3}$
$\left\{\begin{array}{c}{\partial }_{t}v-\left(\nu +\kappa \right)\mathrm{\Delta }v+v\cdot \mathrm{\nabla }v+\mathrm{\nabla }p-2\kappa \mathrm{\nabla }×w=0,\hfill \\ {\partial }_{t}w-\gamma \mathrm{\Delta }w-\left(\alpha +\beta \right)\mathrm{\nabla }\mathrm{\nabla }\cdot w+4\kappa w+v\cdot \mathrm{\nabla }w-2\kappa \mathrm{\nabla }×v=0,\hfill \\ \mathrm{\nabla }\cdot v=0\hfill \end{array}$
(1.1)
with the initial value
$t=0:\phantom{\rule{2em}{0ex}}v={v}_{0}\left(x\right),\phantom{\rule{1em}{0ex}}w={w}_{0}\left(x\right),$
(1.2)

where $v\left(t,x\right)$, $w\left(t,x\right)$ and $p\left(t,x\right)$ represent the divergence free velocity field, non-divergence free micro-rotation field and the scalar pressure, respectively. $\nu >0$ is the Newtonian kinetic viscosity and $\kappa >0$ is the dynamics micro-rotation viscosity, $\alpha ,\beta ,\gamma >0$ are the angular viscosity (see [1]).

The micropolar fluid equations were first proposed by Eringen [2]. The micropolar fluid equations are a generalization of the Navier-Stokes model. It takes into account the microstructure of the fluid, by which we mean the geometry and microrotation of particles. 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. Physically, it may represent adequately the fluids consisting of bar-like elements. Certain anisotropic fluids, e.g., liquid crystals that are made up of dumbbell molecules, are of the type. For more background, we refer to [1] and references therein.

Due to its importance in mathematics and physics, there is lots of literature devoted to the mathematical theory of the 3D micropolar fluid equations. Fundamental mathematical issues such as the global regularity of their solutions have generated extensive research and many interesting results have been established (see [312]). The regularity of weak solutions is examined by imposing some critical growth conditions only on the pressure field in the Lebesgue space, Morrey space, multiplier space, BMO space and Besov space, respectively (see [4]). A new logarithmically improved blow-up criterion for the 3D micropolar fluid equations in an appropriate homogeneous Besov space was obtained by Wang and Yuan [9]. A Serrin-type regularity criterion for the weak solutions to the micropolar fluid equations in ${\mathbb{R}}^{3}$ in the critical Morrey-Campanato space was built [10]. Wang and Zhao [12] established logarithmically improved blow-up criteria of a smooth solution to (1.1), (1.2) in the Morrey-Campanto space.

If $\kappa =0$ and $w=0$, then equations (1.1) reduce to be the Navier-Stokes equations. The Leray-Hopf weak solution was constructed by Leray [13] and Hopf [14], respectively. Later on, much effort has been devoted to establishing 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 [1524]).

Without loss of generality, we set $\nu =\kappa =\frac{1}{2}$, $\gamma =\alpha +\beta =1$ in the rest of the paper. The purpose of this paper is to establish a new logarithmically improved blow-up criterion to (1.1), (1.2) in a weak multiplier space. Now we state our results as follows.

Theorem 1.1 Assume that ${v}_{0},{w}_{0}\in {H}^{m}\left({\mathbb{R}}^{3}\right)$, $m\ge 3$ with $\mathrm{\nabla }\cdot {v}_{0}=0$. Let $\left(v,w\right)$ be a smooth solution to equations (1.1), (1.2) for $0\le t. If v satisfies
${\int }_{0}^{T}\frac{{\parallel \mathrm{\nabla }v\parallel }_{\mathcal{M}\left({\stackrel{˙}{H}}^{r}\to {\stackrel{˙}{H}}^{-r}\right)}^{1-r}}{1+ln\left(e+{\parallel \mathrm{\nabla }v\parallel }_{{L}^{\mathrm{\infty }}}\right)}\phantom{\rule{0.2em}{0ex}}dt<\mathrm{\infty },\phantom{\rule{1em}{0ex}}0\le r<1,$
(1.3)

then the solution $\left(v,w\right)$ can be extended beyond $t=T$.

We have the following corollary immediately.

Corollary 1.1 Assume that ${v}_{0},{w}_{0}\in {H}^{m}\left({\mathbb{R}}^{3}\right)$, $m\ge 3$ with $\mathrm{\nabla }\cdot {v}_{0}=0$. Let $\left(v,w\right)$ be a smooth solution to equations (1.1), (1.2) for $0\le t. Suppose that T is the maximal existence time, then
${\int }_{0}^{T}\frac{{\parallel \mathrm{\nabla }v\parallel }_{\mathcal{M}\left({\stackrel{˙}{H}}^{r}\to {\stackrel{˙}{H}}^{-r}\right)}^{1-r}}{1+ln\left(e+{\parallel \mathrm{\nabla }v\parallel }_{{L}^{\mathrm{\infty }}}\right)}\phantom{\rule{0.2em}{0ex}}dt=\mathrm{\infty },\phantom{\rule{1em}{0ex}}0\le r<1.$
(1.4)

The paper is organized as follows. We first state some preliminaries on functional settings and some important inequalities in Section 2, which play an important role in the proof of our main result. Then we prove the main result in Section 3.

## 2 Preliminaries

Definition 2.1 [25]

For $0, $\mathcal{M}\left({\stackrel{˙}{H}}^{r}\to {\stackrel{˙}{H}}^{-r}\right)$ is a Banach space of all distributions f on ${\mathbb{R}}^{3}$ such that there exists a constant C such that for all $u\in \mathcal{D}$, we have $fu\in {\stackrel{˙}{H}}^{r}$ and
${\parallel fu\parallel }_{{\stackrel{˙}{H}}^{r}}\le C{\parallel u\parallel }_{{\stackrel{˙}{H}}^{-r}},$

where we denote by ${\stackrel{˙}{H}}^{r}$ the completion of the space ${C}_{0}^{\mathrm{\infty }}\left({\mathbb{R}}^{3}\right)$ with respect to the norm ${\parallel u\parallel }_{{\stackrel{˙}{H}}^{r}}={\parallel {\left(-\mathrm{\Delta }\right)}^{\frac{r}{2}}u\parallel }_{{L}^{2}}$.

The norm of $\mathcal{M}\left({\stackrel{˙}{H}}^{r}\to {\stackrel{˙}{H}}^{-r}\right)$ is given by the operator norm of pointwise multiplication
${\parallel f\parallel }_{\mathcal{M}\left({\stackrel{˙}{H}}^{r}\to {\stackrel{˙}{H}}^{-r}\right)}=sup\left\{{\parallel fu\parallel }_{{\stackrel{˙}{H}}^{r}}:{\parallel u\parallel }_{{\stackrel{˙}{H}}^{r}}\le 1,u\in \mathcal{D}\right\}.$

The following lemma comes from [26].

Lemma 2.2 Assume that $1. For $f,g\in {W}^{m,p}$, and $1, $1, we have
${\parallel {\mathrm{\nabla }}^{\alpha }\left(fg\right)-f{\mathrm{\nabla }}^{\alpha }g\parallel }_{{L}^{p}}\le C\left({\parallel \mathrm{\nabla }f\parallel }_{{L}^{{q}_{1}}}{\parallel {\mathrm{\nabla }}^{\alpha -1}g\parallel }_{{L}^{{r}_{1}}}+{\parallel g\parallel }_{{L}^{{q}_{2}}}{\parallel {\mathrm{\nabla }}^{\alpha }f\parallel }_{{L}^{{r}_{2}}}\right),$
(2.1)

where $1\le \alpha \le m$ and $\frac{1}{p}=\frac{1}{{q}_{1}}+\frac{1}{{r}_{1}}=\frac{1}{{q}_{2}}+\frac{1}{{r}_{2}}$.

We also need the following interpolation inequalities in three space dimensions.

Lemma 2.3 In three space dimensions, the following inequalities hold:
$\left\{\begin{array}{c}{\parallel \mathrm{\nabla }f\parallel }_{{L}^{4}}\le C{\parallel f\parallel }_{{L}^{2}}^{\frac{1}{8}}{\parallel {\mathrm{\nabla }}^{2}f\parallel }_{{L}^{2}}^{\frac{7}{8}},\hfill \\ {\parallel f\parallel }_{{L}^{4}}\le C{\parallel f\parallel }_{{L}^{2}}^{\frac{5}{8}}{\parallel {\mathrm{\nabla }}^{2}f\parallel }_{{L}^{2}}^{\frac{3}{8}},\hfill \\ {\parallel {\mathrm{\nabla }}^{2}f\parallel }_{{L}^{4}}\le C{\parallel f\parallel }_{{L}^{2}}^{\frac{1}{12}}{\parallel {\mathrm{\nabla }}^{3}f\parallel }_{{L}^{2}}^{\frac{11}{12}},\hfill \\ {\parallel {\mathrm{\nabla }}^{2}f\parallel }_{{L}^{2}}\le C{\parallel f\parallel }_{{L}^{2}}^{\frac{1}{3}}{\parallel {\mathrm{\nabla }}^{3}f\parallel }_{{L}^{2}}^{\frac{2}{3}}.\hfill \end{array}$
(2.2)

## 3 Proof of Theorem 1.1

Multiplying the first equation of (1.1) by v and integrating with x respect to on ${\mathbb{R}}^{3}$, using integration by parts, we obtain
$\frac{1}{2}\frac{d}{dt}{\parallel v\left(t\right)\parallel }_{{L}^{2}}^{2}+{\parallel \mathrm{\nabla }v\left(t\right)\parallel }_{{L}^{2}}^{2}={\int }_{{\mathbb{R}}^{3}}\left(\mathrm{\nabla }×w\right)\cdot v\phantom{\rule{0.2em}{0ex}}dx.$
(3.1)
Similarly, we get
$\frac{1}{2}\frac{d}{dt}{\parallel w\left(t\right)\parallel }_{{L}^{2}}^{2}+{\parallel \mathrm{\nabla }w\left(t\right)\parallel }_{{L}^{2}}^{2}+{\parallel \mathrm{\nabla }\cdot w\parallel }_{{L}^{2}}^{2}+2{\parallel w\parallel }_{{L}^{2}}^{2}={\int }_{{\mathbb{R}}^{3}}\left(\mathrm{\nabla }×v\right)\cdot w\phantom{\rule{0.2em}{0ex}}dx.$
(3.2)
Summing up (3.1)-(3.2), we deduce that
(3.3)
We apply integration by parts and the Cauchy inequality. This yields
${\int }_{{\mathbb{R}}^{3}}\left(\mathrm{\nabla }×w\right)\cdot v\phantom{\rule{0.2em}{0ex}}dx+{\int }_{{\mathbb{R}}^{3}}\left(\mathrm{\nabla }×v\right)\cdot w\phantom{\rule{0.2em}{0ex}}dx\le \frac{1}{2}{\parallel \mathrm{\nabla }v\parallel }_{{L}^{2}}^{2}+2{\parallel w\parallel }_{{L}^{2}}^{2}.$
(3.4)
Substituting (3.3) into (3.4) yields
$\frac{1}{2}\frac{d}{dt}\left({\parallel v\left(t\right)\parallel }_{{L}^{2}}^{2}+{\parallel w\left(t\right)\parallel }_{{L}^{2}}^{2}\right)+\frac{1}{2}{\parallel \mathrm{\nabla }v\left(t\right)\parallel }_{{L}^{2}}^{2}+{\parallel \mathrm{\nabla }w\left(t\right)\parallel }_{{L}^{2}}^{2}+{\parallel \mathrm{\nabla }\cdot w\parallel }_{{L}^{2}}^{2}\le 0.$
Integrating with respect to t, we have
(3.5)
Multiplying the first equation of (1.1) by ${|v|}^{2}v$, then integrating the resulting equation with respect to x over ${\mathbb{R}}^{3}$ and using integrating by parts, we obtain
(3.6)
We multiply the second equation of (1.1) by ${|w|}^{2}w$, then integrate the resulting equation with respect to x over ${\mathbb{R}}^{3}$ and use integrating by parts. This yields
$\frac{1}{4}\frac{d}{dt}{\parallel w\parallel }_{{L}^{4}}^{4}+{\parallel |w|\mathrm{\nabla }w\parallel }_{{L}^{2}}^{2}+\frac{1}{2}{\parallel |w|\mathrm{\nabla }\cdot w\parallel }_{{L}^{2}}^{2}+2{\parallel w\parallel }_{{L}^{4}}^{4}\le {\int }_{{\mathbb{R}}^{3}}v\mathrm{\nabla }×\left({|w|}^{2}w\right)\phantom{\rule{0.2em}{0ex}}dx,$
(3.7)
where we have used
$\begin{array}{rl}{\int }_{{\mathbb{R}}^{3}}\left(\mathrm{\nabla }\cdot w\right)\mathrm{\nabla }\cdot \left(w{|w|}^{2}\right)\phantom{\rule{0.2em}{0ex}}dx& ={\int }_{{\mathbb{R}}^{3}}{|\mathrm{\nabla }\cdot w|}^{2}{|w|}^{2}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{{\mathbb{R}}^{3}}\left(\mathrm{\nabla }\cdot w\right)w\cdot \mathrm{\nabla }{|w|}^{2}\phantom{\rule{0.2em}{0ex}}dx\\ \ge \frac{1}{2}{\int }_{{\mathbb{R}}^{3}}{|\mathrm{\nabla }\cdot w|}^{2}{|w|}^{2}\phantom{\rule{0.2em}{0ex}}dx-\frac{1}{2}{\int }_{{\mathbb{R}}^{3}}{|\mathrm{\nabla }{|w|}^{2}|}^{2}\phantom{\rule{0.2em}{0ex}}dx.\end{array}$
Equations (3.6) and (3.7) give
(3.8)
Making use of the Young inequality, we have
(3.9)
Applying the divergence operator to the first equation of (1.1) produces the expression of the pressure
$p={\left(-\mathrm{\Delta }\right)}^{-1}\mathrm{\nabla }\cdot \left(v\cdot \mathrm{\nabla }v\right).$
(3.10)
It follows from (3.10) that
${\parallel p\parallel }_{{L}^{2}}\le C{\parallel v\parallel }_{{L}^{4}}^{2},\phantom{\rule{2em}{0ex}}{\parallel \mathrm{\nabla }p\parallel }_{{L}^{2}}\le C{\parallel |v|\mathrm{\nabla }v\parallel }_{{L}^{2}}.$
(3.11)
By integration by parts and (3.11), we obtain
$\begin{array}{rcl}-{\int }_{{\mathbb{R}}^{3}}\left(v\cdot \mathrm{\nabla }p\right){|v|}^{2}\phantom{\rule{0.2em}{0ex}}dx& \le & C{\parallel {|v|}^{2}\mathrm{\nabla }v\parallel }_{{\stackrel{˙}{H}}^{-r}}{\parallel p\parallel }_{{\stackrel{˙}{H}}^{r}}\\ \le & C{\parallel \mathrm{\nabla }v\parallel }_{\mathcal{M}\left({\stackrel{˙}{H}}^{r}\to {\stackrel{˙}{H}}^{-r}\right)}{\parallel {|v|}^{2}\parallel }_{{\stackrel{˙}{H}}^{r}}{\parallel p\parallel }_{{L}^{2}}^{1-r}{\parallel \mathrm{\nabla }p\parallel }_{{L}^{2}}^{r}\\ \le & C{\parallel \mathrm{\nabla }v\parallel }_{\mathcal{M}\left({\stackrel{˙}{H}}^{r}\to {\stackrel{˙}{H}}^{-r}\right)}{\parallel {|v|}^{2}\parallel }_{{L}^{2}}^{1-r}{\parallel \mathrm{\nabla }{|v|}^{2}\parallel }_{{L}^{2}}^{r}{\parallel p\parallel }_{{L}^{2}}^{1-r}{\parallel \mathrm{\nabla }p\parallel }_{{L}^{2}}^{r}\\ \le & C{\parallel \mathrm{\nabla }v\parallel }_{\mathcal{M}\left({\stackrel{˙}{H}}^{r}\to {\stackrel{˙}{H}}^{-r}\right)}{\parallel v\parallel }_{{L}^{4}}^{4\left(1-r\right)}{\parallel |v|\mathrm{\nabla }v\parallel }_{{L}^{2}}^{2r}\\ \le & ϵ{\parallel |v|\mathrm{\nabla }v\parallel }_{{L}^{2}}^{2}+C{\parallel \mathrm{\nabla }v\parallel }_{\mathcal{M}\left({\stackrel{˙}{H}}^{r}\to {\stackrel{˙}{H}}^{-r}\right)}^{1-r}{\parallel v\parallel }_{{L}^{4}}^{4}.\end{array}$
(3.12)
Combining (3.8), (3.9), (3.12) and (3.5) yields
(3.13)
where we have used
${H}^{2}↪{L}^{\mathrm{\infty }}.$
By (1.3), we know that for any small constant $\epsilon >0$, there exists ${T}_{\ast } such that
${\int }_{{T}_{\ast }}^{T}\frac{{\parallel \mathrm{\nabla }v\parallel }_{\mathcal{M}\left({\stackrel{˙}{H}}^{r}\to {\stackrel{˙}{H}}^{-r}\right)}^{1-r}}{1+ln\left(e+{\parallel \mathrm{\nabla }v\parallel }_{{L}^{\mathrm{\infty }}}\right)}\phantom{\rule{0.2em}{0ex}}dt\le \epsilon .$
(3.14)
Let
$A\left(t\right)=\underset{{T}_{\ast }\le \tau \le t}{sup}\left({\parallel {\mathrm{\nabla }}^{3}v\parallel }_{{L}^{2}}^{2}+{\parallel {\mathrm{\nabla }}^{3}w\parallel }_{{L}^{2}}^{2}\right),\phantom{\rule{1em}{0ex}}{T}_{\ast }\le t
(3.15)
The Gronwall inequality and (3.13)-(3.15) give
$\begin{array}{rl}{\parallel v\parallel }_{{L}^{4}}^{4}+{\parallel w\parallel }_{{L}^{4}}^{4}& \le Cexp\left\{Cϵ\left(1+ln\left(e+A\left(t\right)\right)\right)\right\}\\ \le Cexp\left\{2Cϵln\left(e+A\left(t\right)\right)\right\}\\ \le C{\left(e+A\left(t\right)\right)}^{2C\epsilon },\phantom{\rule{1em}{0ex}}{T}_{\ast }\le t
(3.16)
Applying to the first equation, then multiplying the resulting equation by v and using integration by parts, the Hölder inequality, (2.2) and the Young inequality, we obtain
$\begin{array}{rl}\frac{1}{2}\frac{d}{dt}{\parallel \mathrm{\nabla }v\parallel }_{{L}^{2}}^{2}+{\parallel {\mathrm{\nabla }}^{2}v\parallel }_{{L}^{2}}^{2}& ={\int }_{{\mathbb{R}}^{3}}\mathrm{\nabla }v\cdot \mathrm{\nabla }v\mathrm{\nabla }v\phantom{\rule{0.2em}{0ex}}dx+{\int }_{{\mathbb{R}}^{3}}\left(\mathrm{\nabla }×w\right){\mathrm{\nabla }}^{2}v\phantom{\rule{0.2em}{0ex}}dx\\ \le C{\parallel v\parallel }_{{L}^{4}}{\parallel \mathrm{\nabla }v\parallel }_{{L}^{4}}{\parallel {\mathrm{\nabla }}^{2}v\parallel }_{{L}^{2}}+C{\parallel \mathrm{\nabla }w\parallel }_{{L}^{2}}{\parallel {\mathrm{\nabla }}^{2}v\parallel }_{{L}^{2}}\\ \le C{\parallel v\parallel }_{{L}^{4}}{\parallel v\parallel }_{{L}^{2}}^{\frac{1}{8}}{\parallel {\mathrm{\nabla }}^{2}v\parallel }_{{L}^{2}}^{\frac{15}{8}}+\frac{1}{8}{\parallel {\mathrm{\nabla }}^{2}v\parallel }_{{L}^{2}}^{2}+C{\parallel w\parallel }_{{L}^{2}}{\parallel {\mathrm{\nabla }}^{2}w\parallel }_{{L}^{2}}\\ \le \frac{1}{4}{\parallel {\mathrm{\nabla }}^{2}v\parallel }_{{L}^{2}}^{2}+\frac{1}{4}{\parallel {\mathrm{\nabla }}^{2}w\parallel }_{{L}^{2}}^{2}+C{\parallel v\parallel }_{{L}^{4}}^{16}{\parallel v\parallel }_{{L}^{2}}^{2}+C{\parallel w\parallel }_{{L}^{2}}^{2}.\end{array}$
(3.17)
Similarly, we have
$\begin{array}{r}\frac{1}{2}\frac{d}{dt}{\parallel \mathrm{\nabla }w\parallel }_{{L}^{2}}^{2}+{\parallel {\mathrm{\nabla }}^{2}w\parallel }_{{L}^{2}}^{2}+{\parallel \mathrm{\nabla }\mathrm{\nabla }\cdot w\parallel }_{{L}^{2}}^{2}+2{\parallel \mathrm{\nabla }w\parallel }_{{L}^{2}}^{2}\\ \phantom{\rule{1em}{0ex}}={\int }_{{\mathbb{R}}^{3}}\mathrm{\nabla }v\cdot \mathrm{\nabla }w\mathrm{\nabla }w\phantom{\rule{0.2em}{0ex}}dx+{\int }_{{\mathbb{R}}^{3}}\left(\mathrm{\nabla }×v\right){\mathrm{\nabla }}^{2}w\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C{\parallel v\parallel }_{{L}^{4}}{\parallel \mathrm{\nabla }w\parallel }_{{L}^{4}}{\parallel {\mathrm{\nabla }}^{2}w\parallel }_{{L}^{2}}+C{\parallel \mathrm{\nabla }v\parallel }_{{L}^{2}}{\parallel {\mathrm{\nabla }}^{2}w\parallel }_{{L}^{2}}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel v\parallel }_{{L}^{4}}{\parallel w\parallel }_{{L}^{2}}^{\frac{1}{8}}{\parallel {\mathrm{\nabla }}^{2}w\parallel }_{{L}^{2}}^{\frac{15}{8}}+\frac{1}{8}{\parallel {\mathrm{\nabla }}^{2}w\parallel }_{{L}^{2}}^{2}+C{\parallel v\parallel }_{{L}^{2}}{\parallel {\mathrm{\nabla }}^{2}v\parallel }_{{L}^{2}}\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{4}{\parallel {\mathrm{\nabla }}^{2}w\parallel }_{{L}^{2}}^{2}+\frac{1}{4}{\parallel {\mathrm{\nabla }}^{2}v\parallel }_{{L}^{2}}^{2}+C{\parallel v\parallel }_{{L}^{4}}^{16}{\parallel w\parallel }_{{L}^{2}}^{2}+C{\parallel v\parallel }_{{L}^{2}}^{2}.\end{array}$
(3.18)
Adding (3.17) and (3.18), we arrive at
$\begin{array}{r}\frac{d}{dt}\left({\parallel \mathrm{\nabla }v\parallel }_{{L}^{2}}^{2}+{\parallel \mathrm{\nabla }w\parallel }_{{L}^{2}}^{2}\right)+{\parallel {\mathrm{\nabla }}^{2}v\parallel }_{{L}^{2}}^{2}+{\parallel {\mathrm{\nabla }}^{2}w\parallel }_{{L}^{2}}^{2}+{\parallel \mathrm{\nabla }\mathrm{\nabla }\cdot w\parallel }_{{L}^{2}}^{2}+{\parallel \mathrm{\nabla }w\parallel }_{{L}^{2}}^{2}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel v\parallel }_{{L}^{4}}^{16}\left({\parallel v\parallel }_{{L}^{2}}^{2}+{\parallel w\parallel }_{{L}^{2}}^{2}\right)+C\left({\parallel v\parallel }_{{L}^{2}}^{2}+{\parallel w\parallel }_{{L}^{2}}^{2}\right).\end{array}$
(3.19)
Equations (3.5), (3.16), (3.19) and the Gronwall inequality give
${\parallel \mathrm{\nabla }v\parallel }_{{L}^{2}}^{2}+{\parallel \mathrm{\nabla }w\parallel }_{{L}^{2}}^{2}\le C{\left(e+A\left(t\right)\right)}^{8C\epsilon },\phantom{\rule{1em}{0ex}}{T}_{\ast }\le t
(3.20)
Applying ${\mathrm{\nabla }}^{m}$ to the first equation in (1.1), then taking ${L}^{2}$ inner product of the resulting equation with ${\mathrm{\nabla }}^{m}v$ and using integration by parts, we have
$\begin{array}{r}\frac{1}{2}\frac{d}{dt}{\parallel {\mathrm{\nabla }}^{m}v\parallel }_{{L}^{2}}^{2}+{\parallel {\mathrm{\nabla }}^{m+1}v\parallel }_{{L}^{2}}^{2}\\ \phantom{\rule{1em}{0ex}}=-{\int }_{{\mathbb{R}}^{3}}{\mathrm{\nabla }}^{m}\left(v\cdot \mathrm{\nabla }v\right){\mathrm{\nabla }}^{m}v\phantom{\rule{0.2em}{0ex}}dx+{\int }_{{\mathbb{R}}^{3}}{\mathrm{\nabla }}^{m}\left(\mathrm{\nabla }×w\right){\mathrm{\nabla }}^{m}v\phantom{\rule{0.2em}{0ex}}dx.\end{array}$
(3.21)
Similarly, we obtain
$\begin{array}{r}\frac{1}{2}\frac{d}{dt}{\parallel {\mathrm{\nabla }}^{m}w\parallel }_{{L}^{2}}^{2}+{\parallel {\mathrm{\nabla }}^{m+1}w\parallel }_{{L}^{2}}^{2}+{\parallel {\mathrm{\nabla }}^{m}\mathrm{\nabla }\cdot w\parallel }_{{L}^{2}}^{2}+2{\parallel {\mathrm{\nabla }}^{m}w\parallel }_{{L}^{2}}^{2}\\ \phantom{\rule{1em}{0ex}}=-{\int }_{{\mathbb{R}}^{3}}{\mathrm{\nabla }}^{m}\left(v\cdot \mathrm{\nabla }w\right){\mathrm{\nabla }}^{m}w\phantom{\rule{0.2em}{0ex}}dx+{\int }_{{\mathbb{R}}^{3}}{\mathrm{\nabla }}^{m}\left(\mathrm{\nabla }×v\right){\mathrm{\nabla }}^{m}w\phantom{\rule{0.2em}{0ex}}dx.\end{array}$
(3.22)
Summing (3.21), (3.22) and using $\mathrm{\nabla }\cdot v=0$, we get
$\begin{array}{r}\frac{1}{2}\frac{d}{dt}\left({\parallel {\mathrm{\nabla }}^{m}v\parallel }_{{L}^{2}}^{2}+{\parallel {\mathrm{\nabla }}^{m}w\parallel }_{{L}^{2}}^{2}\right)+{\parallel {\mathrm{\nabla }}^{m+1}v\parallel }_{{L}^{2}}^{2}+{\parallel {\mathrm{\nabla }}^{m+1}w\parallel }_{{L}^{2}}^{2}+{\parallel {\mathrm{\nabla }}^{m}\mathrm{\nabla }\cdot w\parallel }_{{L}^{2}}^{2}+2{\parallel {\mathrm{\nabla }}^{m}w\parallel }_{{L}^{2}}^{2}\\ \phantom{\rule{1em}{0ex}}=-{\int }_{{\mathbb{R}}^{3}}\left[{\mathrm{\nabla }}^{m}\left(v\cdot \mathrm{\nabla }v\right)-v\cdot {\mathrm{\nabla }}^{m}\mathrm{\nabla }v\right]{\mathrm{\nabla }}^{m}v\phantom{\rule{0.2em}{0ex}}dx-{\int }_{{\mathbb{R}}^{3}}\left[{\mathrm{\nabla }}^{m}\left(v\cdot \mathrm{\nabla }w\right)-v\cdot {\mathrm{\nabla }}^{m}\mathrm{\nabla }w\right]{\mathrm{\nabla }}^{m}w\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+{\int }_{{\mathbb{R}}^{3}}{\mathrm{\nabla }}^{m}\left(\mathrm{\nabla }×w\right){\mathrm{\nabla }}^{m}v\phantom{\rule{0.2em}{0ex}}dx+{\int }_{{\mathbb{R}}^{3}}{\mathrm{\nabla }}^{m}\left(\mathrm{\nabla }×v\right){\mathrm{\nabla }}^{m}w\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\triangleq {I}_{1}+{I}_{2}+{I}_{3}+{I}_{4}.\end{array}$
(3.23)

In what follows, for simplicity, we set $m=3$.

By the Hölder inequality, (2.1), (2.2) and the Young inequality, we obtain
$\begin{array}{r}-{\int }_{{\mathbb{R}}^{3}}\left[{\mathrm{\nabla }}^{3}\left(v\cdot \mathrm{\nabla }v\right)-v\cdot \mathrm{\nabla }{\mathrm{\nabla }}^{3}v\right]{\mathrm{\nabla }}^{3}v\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le {\parallel {\mathrm{\nabla }}^{3}\left(v\cdot \mathrm{\nabla }v\right)-v\cdot \mathrm{\nabla }{\mathrm{\nabla }}^{3}v\parallel }_{{L}^{2}}{\parallel {\mathrm{\nabla }}^{3}v\parallel }_{{L}^{2}}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel \mathrm{\nabla }v\parallel }_{{L}^{4}}{\parallel {\mathrm{\nabla }}^{3}v\parallel }_{{L}^{4}}{\parallel {\mathrm{\nabla }}^{3}v\parallel }_{{L}^{2}}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel \mathrm{\nabla }v\parallel }_{{L}^{2}}^{\frac{5}{8}}{\parallel {\mathrm{\nabla }}^{3}v\parallel }_{{L}^{2}}^{\frac{3}{8}}{\parallel \mathrm{\nabla }v\parallel }_{{L}^{2}}^{\frac{1}{12}}{\parallel {\mathrm{\nabla }}^{4}v\parallel }_{{L}^{2}}^{\frac{11}{12}}{\parallel \mathrm{\nabla }v\parallel }_{{L}^{2}}^{\frac{1}{3}}{\parallel {\mathrm{\nabla }}^{4}v\parallel }_{{L}^{2}}^{\frac{2}{3}}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel \mathrm{\nabla }v\parallel }_{{L}^{2}}^{\frac{25}{24}}{\parallel {\mathrm{\nabla }}^{3}v\parallel }_{{L}^{2}}^{\frac{3}{8}}{\parallel {\mathrm{\nabla }}^{4}v\parallel }_{{L}^{2}}^{\frac{19}{12}}\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{8}{\parallel {\mathrm{\nabla }}^{4}v\parallel }_{{L}^{2}}^{2}+C{\parallel \mathrm{\nabla }v\parallel }_{{L}^{2}}^{5}{\parallel {\mathrm{\nabla }}^{3}v\parallel }_{{L}^{2}}^{\frac{9}{5}}\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{8}{\parallel {\mathrm{\nabla }}^{4}v\parallel }_{{L}^{2}}^{2}+C{\left(e+A\left(t\right)\right)}^{20C\epsilon +\frac{9}{10}}\end{array}$
(3.24)
and
$\begin{array}{r}-{\int }_{{\mathbb{R}}^{3}}\left[{\mathrm{\nabla }}^{3}\left(v\cdot \mathrm{\nabla }w\right)-v\cdot \mathrm{\nabla }{\mathrm{\nabla }}^{3}w\right]{\mathrm{\nabla }}^{3}w\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le {\parallel {\mathrm{\nabla }}^{3}\left(v\cdot \mathrm{\nabla }w\right)-v\cdot \mathrm{\nabla }{\mathrm{\nabla }}^{3}w\parallel }_{{L}^{2}}{\parallel {\mathrm{\nabla }}^{3}w\parallel }_{{L}^{2}}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel \mathrm{\nabla }v\parallel }_{{L}^{4}}{\parallel {\mathrm{\nabla }}^{3}w\parallel }_{{L}^{4}}{\parallel {\mathrm{\nabla }}^{3}w\parallel }_{{L}^{2}}+{\parallel \mathrm{\nabla }w\parallel }_{{L}^{4}}{\parallel {\mathrm{\nabla }}^{3}v\parallel }_{{L}^{4}}{\parallel {\mathrm{\nabla }}^{3}w\parallel }_{{L}^{2}}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel \mathrm{\nabla }v\parallel }_{{L}^{2}}^{\frac{5}{8}}{\parallel {\mathrm{\nabla }}^{3}v\parallel }_{{L}^{2}}^{\frac{3}{8}}{\parallel \mathrm{\nabla }w\parallel }_{{L}^{2}}^{\frac{1}{12}}{\parallel {\mathrm{\nabla }}^{4}w\parallel }_{{L}^{2}}^{\frac{11}{12}}{\parallel \mathrm{\nabla }w\parallel }_{{L}^{2}}^{\frac{1}{3}}{\parallel {\mathrm{\nabla }}^{4}w\parallel }_{{L}^{2}}^{\frac{2}{3}}\\ \phantom{\rule{2em}{0ex}}+C{\parallel \mathrm{\nabla }w\parallel }_{{L}^{2}}^{\frac{5}{8}}{\parallel {\mathrm{\nabla }}^{3}w\parallel }_{{L}^{2}}^{\frac{3}{8}}{\parallel \mathrm{\nabla }v\parallel }_{{L}^{2}}^{\frac{1}{12}}{\parallel {\mathrm{\nabla }}^{4}v\parallel }_{{L}^{2}}^{\frac{11}{12}}{\parallel \mathrm{\nabla }w\parallel }_{{L}^{2}}^{\frac{1}{3}}{\parallel {\mathrm{\nabla }}^{4}w\parallel }_{{L}^{2}}^{\frac{2}{3}}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel \mathrm{\nabla }v\parallel }_{{L}^{2}}^{\frac{5}{8}}{\parallel {\mathrm{\nabla }}^{3}v\parallel }_{{L}^{2}}^{\frac{3}{8}}{\parallel \mathrm{\nabla }w\parallel }_{{L}^{2}}^{\frac{5}{12}}{\parallel {\mathrm{\nabla }}^{4}w\parallel }_{{L}^{2}}^{\frac{19}{12}}\\ \phantom{\rule{2em}{0ex}}+C{\parallel \mathrm{\nabla }w\parallel }_{{L}^{2}}^{\frac{23}{24}}{\parallel {\mathrm{\nabla }}^{3}w\parallel }_{{L}^{2}}^{\frac{3}{8}}{\parallel \mathrm{\nabla }v\parallel }_{{L}^{2}}^{\frac{1}{12}}{\parallel {\mathrm{\nabla }}^{4}v\parallel }_{{L}^{2}}^{\frac{11}{12}}{\parallel {\mathrm{\nabla }}^{4}w\parallel }_{{L}^{2}}^{\frac{2}{3}}\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{8}{\parallel {\mathrm{\nabla }}^{4}v\parallel }_{{L}^{2}}^{2}+\frac{1}{8}{\parallel {\mathrm{\nabla }}^{4}w\parallel }_{{L}^{2}}^{2}+C{\parallel \mathrm{\nabla }v\parallel }_{{L}^{2}}^{3}{\parallel \mathrm{\nabla }w\parallel }_{{L}^{2}}^{2}{\parallel {\mathrm{\nabla }}^{3}v\parallel }_{{L}^{2}}^{\frac{9}{5}}\\ \phantom{\rule{2em}{0ex}}+C{\parallel \mathrm{\nabla }v\parallel }_{{L}^{2}}^{\frac{2}{5}}{\parallel \mathrm{\nabla }w\parallel }_{{L}^{2}}^{\frac{23}{5}}{\parallel {\mathrm{\nabla }}^{3}w\parallel }_{{L}^{2}}^{\frac{9}{5}}\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{8}{\parallel {\mathrm{\nabla }}^{4}v\parallel }_{{L}^{2}}^{2}+\frac{1}{8}{\parallel {\mathrm{\nabla }}^{4}w\parallel }_{{L}^{2}}^{2}+C{\left(e+A\left(t\right)\right)}^{20C\epsilon +\frac{9}{10}}.\end{array}$
(3.25)
From the Young inequality and (2.2), we deduce that
$\begin{array}{rcl}{I}_{3}& \le & C{\parallel {\mathrm{\nabla }}^{4}v\parallel }_{{L}^{2}}{\parallel {\mathrm{\nabla }}^{3}w\parallel }_{{L}^{2}}\\ \le & \frac{1}{8}{\parallel {\mathrm{\nabla }}^{4}v\parallel }_{{L}^{2}}^{2}+C{\parallel {\mathrm{\nabla }}^{3}w\parallel }_{{L}^{2}}^{2}\\ \le & \frac{1}{8}{\parallel {\mathrm{\nabla }}^{4}v\parallel }_{{L}^{2}}^{2}+C{\parallel \mathrm{\nabla }w\parallel }_{{L}^{2}}^{\frac{2}{3}}{\parallel {\mathrm{\nabla }}^{4}w\parallel }_{{L}^{2}}^{\frac{4}{3}}\\ \le & \frac{1}{8}{\parallel {\mathrm{\nabla }}^{4}v\parallel }_{{L}^{2}}^{2}+\frac{1}{8}{\parallel {\mathrm{\nabla }}^{4}w\parallel }_{{L}^{2}}^{2}+C{\parallel \mathrm{\nabla }w\parallel }_{{L}^{2}}^{2}\\ \le & \frac{1}{8}{\parallel {\mathrm{\nabla }}^{4}v\parallel }_{{L}^{2}}^{2}+\frac{1}{8}{\parallel {\mathrm{\nabla }}^{4}w\parallel }_{{L}^{2}}^{2}+C{\left(e+A\left(t\right)\right)}^{8C\epsilon },\phantom{\rule{1em}{0ex}}{T}_{\ast }\le t
(3.26)
Similarly, we have
${I}_{4}\le \frac{1}{8}{\parallel {\mathrm{\nabla }}^{4}v\parallel }_{{L}^{2}}^{2}+\frac{1}{8}{\parallel {\mathrm{\nabla }}^{4}w\parallel }_{{L}^{2}}^{2}+C{\left(e+A\left(t\right)\right)}^{8C\epsilon },\phantom{\rule{1em}{0ex}}{T}_{\ast }\le t
(3.27)
Inserting (3.24)-(3.27) into (3.23) and taking ε small enough such that $20C\epsilon <\frac{1}{10}$, we obtain
$\frac{d}{dt}\left({\parallel {\mathrm{\nabla }}^{3}v\parallel }_{{L}^{2}}^{2}+{\parallel {\mathrm{\nabla }}^{3}w\parallel }_{{L}^{2}}^{2}\right)\le C\left(e+A\left(t\right)\right),\phantom{\rule{1em}{0ex}}{T}_{\ast }\le t
(3.28)

for all ${T}_{\ast }\le t.

Integrating (3.28) with respect to time from ${T}_{\ast }$ to τ, we have
$\begin{array}{r}e+{\parallel {\mathrm{\nabla }}^{3}v\left(\tau \right)\parallel }_{{L}^{2}}^{2}+{\parallel {\mathrm{\nabla }}^{3}w\left(\tau \right)\parallel }_{{L}^{2}}^{2}\\ \phantom{\rule{1em}{0ex}}\le e+{\parallel {\mathrm{\nabla }}^{3}v\left({T}_{\ast }\right)\parallel }_{{L}^{2}}^{2}+{\parallel {\mathrm{\nabla }}^{3}w\left({T}_{\ast }\right)\parallel }_{{L}^{2}}^{2}+{C}_{2}{\int }_{{T}_{\ast }}^{\tau }\left(e+A\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds.\end{array}$
(3.29)
We get from (3.29)
$e+A\left(t\right)\le e+{\parallel {\mathrm{\nabla }}^{3}v\left({T}_{\ast }\right)\parallel }_{{L}^{2}}^{2}+{\parallel {\mathrm{\nabla }}^{3}w\left({T}_{\ast }\right)\parallel }_{{L}^{2}}^{2}+{C}_{2}{\int }_{{T}_{\ast }}^{t}\left(e+A\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau .$
(3.30)
For all ${T}_{\ast }\le t, with the help of Gronwall inequality and (3.30), we have
$e+{\parallel {\mathrm{\nabla }}^{3}v\left(t\right)\parallel }_{{L}^{2}}^{2}+{\parallel {\mathrm{\nabla }}^{3}w\left(t\right)\parallel }_{{L}^{2}}^{2}\le C,$
(3.31)

where C depends on ${\parallel \mathrm{\nabla }v\left({T}_{\ast }\right)\parallel }_{{L}^{2}}^{2}+{\parallel \mathrm{\nabla }w\left({T}_{\ast }\right)\parallel }_{{L}^{2}}^{2}$. (3.31) and (3.5) imply $\left(v,w\right)\in {L}^{\mathrm{\infty }}\left(0,T;{H}^{3}\left({\mathbb{R}}^{3}\right)\right)$. Thus, $\left(v,w\right)$ can be extended smoothly beyond $t=T$. We have completed the proof of Theorem 1.1.

## Authors’ Affiliations

(1)
School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power

## References

1. Lukaszewicz G: Micropolar Fluids. Theory and Applications. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston, Boston; 1999.Google Scholar
2. Eringen A: Theory of micropolar fluids. J. Math. Mech. 1966, 16: 1–18.
3. Dong B, Chen Z: Regularity criteria of weak solutions to the three-dimensional micropolar flows. J. Math. Phys. 2009., 50: Article ID 103525Google Scholar
4. Dong B, Jia Y, Chen Z: Pressure regularity criteria of the three-dimensional micropolar fluid flows. Math. Methods Appl. Sci. 2011, 34: 595–606. 10.1002/mma.1383
5. He X, Fan J: A regularity criterion for 3D micropolar fluid flows. Appl. Math. Lett. 2012, 25: 47–51. 10.1016/j.aml.2011.07.007
6. Jia Y, Zhang W, Dong B: Remarks on the regularity criterion of the 3D micropolar fluid flows in terms of the pressure. Appl. Math. Lett. 2011, 24: 199–203. 10.1016/j.aml.2010.09.003
7. Ortega-Torres E, Rojas-Medar M: On the regularity for solutions of the micropolar fluid equations. Rend. Semin. Mat. Univ. Padova 2009, 122: 27–37.
8. Wang Y, Chen Z: Regularity criterion for weak solution to the 3D micropolar fluid equations. J. Appl. Math. 2011., 2011: Article ID 456547Google Scholar
9. Wang Y, Yuan H: A logarithmically improved blow up criterion of smooth solutions to the 3D micropolar fluid equations. Nonlinear Anal., Real World Appl. 2012, 13: 1904–1912. 10.1016/j.nonrwa.2011.12.018
10. Gala S: On regularity criteria for the three-dimensional micropolar fluid equations in the critical Morrey-Campanato space. Nonlinear Anal., Real World Appl. 2011, 12: 2142–2150. 10.1016/j.nonrwa.2010.12.028
11. Gala S, Yan J: Two regularity criteria via the logarithmic of the weak solutions to the micropolar fluid equations. J. Partial Differ. Equ. 2012, 25: 32–40.
12. Wang Y, Zhao H: Logarithmically improved blow up criterion of smooth solutions to the 3D micropolar fluid equations. J. Appl. Math. 2012., 2012: Article ID 541203Google Scholar
13. Leray J: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 1934, 63: 183–248.
14. Hopf E: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 1951, 4: 213–231.
15. Fan J, Ozawa T: Regularity criterion for weak solutions to the Navier-Stokes equations in terms of the gradient of the pressure. J. Inequal. Appl. 2008., 2008: Article ID 412678Google Scholar
16. Fan J, Jiang S, Nakamura G, Zhou Y: Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations. J. Math. Fluid Mech. (Printed Ed.) 2011, 13: 557–571. 10.1007/s00021-010-0039-5
17. He C: New sufficient conditions for regularity of solutions to the Navier-Stokes equations. Adv. Math. Sci. Appl. 2002, 12: 535–548.
18. Serrin J: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 1962, 9: 187–195.
19. Zhang Z, Chen Q:Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in ${\mathbb{R}}^{3}$. J. Differ. Equ. 2005, 216: 470–481. 10.1016/j.jde.2005.06.001
20. Zhou Y, Gala S: Logarithmically improved regularity criteria for the Navier-Stokes equations in multiplier spaces. J. Math. Anal. Appl. 2009, 356: 498–501. 10.1016/j.jmaa.2009.03.038
21. Zhou Y, Pokorný M: On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component. J. Math. Phys. 2009., 50: Article ID 123514Google Scholar
22. Zhou Y, Pokorný M: On the regularity of the solutions of the Navier-Stokes equations via one velocity component. Nonlinearity 2010, 23: 1097–1107. 10.1088/0951-7715/23/5/004
23. Guo Z, Gala S: Remarks on logarithmical regularity criteria for the Navier-Stokes equations. J. Math. Phys. 2011., 52: Article ID 063503Google Scholar
24. He X, Gala S:Regularity criterion for weak solutions to the Navier-Stokes equations in terms of the pressure in the class ${L}^{2}\left(0,T;{\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{-1}\left({\mathbb{R}}^{3}\right)\right)$. Nonlinear Anal., Real World Appl. 2011, 12: 3602–3607. 10.1016/j.nonrwa.2011.06.018
25. Jiang Z, Gala S, Ni L: On the regularity criterion for the solutions of 3D Navier-Stokes equations in weak multiplier spaces. Math. Methods Appl. Sci. 2011, 34: 2060–2064. 10.1002/mma.1506
26. Kato T, Ponce G: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 1988, 41: 891–907. 10.1002/cpa.3160410704