Regularity Criterion for the 3D Micropolar Fluid Equations in Terms of Pressure

Abstract

In this paper, we consider the regularity of weak solutions to the 3D incompressible micropolar fluid equations. It is proved that if the one directional derivative of the pressure, say $\partial _{3}P$, satisfies

$$\begin{aligned} \partial _{3}P \in L^{\beta }(0,T;L^{\alpha }(\mathbb {R}^{3})) \quad \text { with } \frac{2}{\beta }+\frac{3}{\alpha }\le 2, \frac{3}{2}\le \alpha <\infty , \end{aligned}$$

then the corresponding weak solution $(u,\omega )$ is regular on [0, T].

1 Introduction

This paper is concerned with the regularity of weak solutions to the following 3D incompressible micropolar fluid equations:

$$\begin{aligned}&{\partial _{t}}u-(\mu +\chi )\Delta u-\chi \nabla \times \omega +\nabla {P} +(u\cdot \nabla )u=0\quad \quad \text { in }\mathbb {R}^{3}\times (0,\infty ),\end{aligned}$$

(1.1)

$$\begin{aligned}&\partial _{t}\omega -\gamma \Delta \omega -\kappa \nabla {\text {div}}\omega +2\chi \omega +(u\cdot \nabla )\omega -\chi \nabla \times u=0 \quad \quad \text { in }\mathbb {R}^{3}\times (0,\infty ),\nonumber \\\end{aligned}$$

(1.2)

$$\begin{aligned}&{\text {div}} u=0\quad \quad \text { in }\mathbb {R}^{3}\times (0,\infty ) \end{aligned}$$

(1.3)

with the initial condition

$$\begin{aligned} (u,\omega )|_{t=0}=(u_{0},\omega _{0})\quad \text { in }\mathbb {R}^{3}. \end{aligned}$$

(1.4)

Here $u=u(x,t)=(u_{1}(x,t),u_{2}(x,t),u_{3}(x,t))$ is the unknown velocity vector field, $\omega =\omega (x,t)=(\omega _{1}(x,t),$$\omega _{2}(x,t),\omega _{3}(x,t))$ and $P=P(x,t)$ are, respectively, the unknown microrotational vector field and the unknown scalar pressure field. The initial data $(u_{0},\omega _{0})$ satisfy ${\text {div}}u_{0}=0$ in the sense of distributions. The constants $\mu ,\chi ,\gamma ,\kappa$ are positive numbers associated to the properties of the material: $\mu$ is the kinematic viscosity, $\chi$ is the vortex viscosity, and $\gamma$ and $\kappa$ are spin viscosities.

The micropolar fluid system was introduced by Eringen [14] in 1960s. It is a special model of microfluids which exhibits the microrotational effects and microrotational inertia, and can be viewed as a non-Newtonian fluid. In a physical sense, micropolar fluid may represent fluids that consists of rigid, randomly oriented (or spherical) particles suspended in a viscous medium, where the deformation of fluid particles is ignored. It describes many phenomena such as animal blood and certain anisotropic fluids, e.g., liquid crystals which cannot be characterized appropriately by the Navier–Stokes equations. For more detailed background, we refer the readers to see Lukaszewicz [23], Rojas-Medar [25] and the references therein. Besides their physical applications, the micropolar fluid system is also mathematically significant. Fundamental mathematical issues such as the existences of weak and strong solutions for micropolar fluid equations were treated by Galdi and Rionero [19] and Yamaguchi [30], respectively. We also refer the readers to see [10, 12] for well-posedness and regularity of solutions to the 2D micropolar fluid system. However, the problem of global regularity of weak solutions to the 3D micropolar fluids with any initial value still remains unsolve since the system (1.1)–(1.4) includes the 3D Navier–Stokes equations (the case $\omega =0$, see [13, 22]). Therefore, it is an interesting thing that regularity of a given weak solution of the 3D micropolar fluids or the 3D Navier–Stokes equations can be shown under some additional conditions, and over the years different criteria for regularity of the weak solutions have been proposed. For the Navier–Stokes equations, the well-known Prodi–Serrin condition (see [15, 24, 26, 27]) shows that any weak solution $u\in L^{p}(0,T; L^{q}(\mathbb {R}^{3}))$ with $\frac{2}{p}+\frac{3}{q}\le 1$, $3\le q\le \infty$ and $2\le p\le \infty$ is regular on $\mathbb {R}^{3}\times [0,T]$. Beirãa da Veiga [5] established a Serrin’s type regularity criterion on the gradient of the velocity field, i.e., $\nabla u\in L^{\beta }(0,T;L^{\alpha }(\mathbb {R}^{3}))$ with $\frac{3}{2}\le \alpha \le \infty$, $1\le \beta \le \infty$. The pressure criterion have been treated by Beirãa da Veiga [6], Berselli and Galdi [2] and Zhou [33,34,35,36], their results show that if the pressure satisfies

$$\begin{aligned} P\in L^{q}(0,T;L^{p}(\mathbb {R}^{3})) \text { with } \frac{3}{2}\le p\le \infty , \frac{2}{q}+\frac{3}{p}\le 2 \end{aligned}$$

(1.5)

or

$$\begin{aligned} \nabla P\in L^{\beta }(0,T;L^{\alpha }(\mathbb {R}^{3})) \text { with } 1\le \alpha \le \infty , \frac{2}{\beta }+\frac{3}{\alpha }\le 3, \end{aligned}$$

(1.6)

then the weak solution u of the 3D Navier–Stokes equations is regular on $\mathbb {R}^{3}\times [0,T]$. Cao and Titi [3] established a regularity criterion for the 3D Navier–Stokes equations only in terms of one directional derivative of the pressure, more precisely, they proved that if

$$\begin{aligned} \partial _{3}P\in L^{\mu }(0,T; L^{\sigma }) \text { with } \frac{3}{\sigma }+\frac{2}{\mu }<\frac{20}{7}, \sigma>\frac{21}{16}, \mu >1, \end{aligned}$$

(1.7)

then the corresponding weak solution is regular up to time T. Similar regularity criteria for the MHD equations involving either the velocity or the pressure have been established by a number of researchers (see., e.g., [4, 16, 37] and the references therein). Cao and Wu in [4] established two regularity criteria in terms of one direction derivative of the velocity and one directional derivative of the pressure, more precisely, they proved that if

$$\begin{aligned}&\partial _{3} u\in L^{\beta }(0,T; L^{\alpha }) \text { with } \frac{2}{\beta }+\frac{3}{\alpha }\le 1, \alpha \ge 3, \end{aligned}$$

(1.8)

or

$$\begin{aligned}&\partial _{3} P\in L^{\beta }(0,T; L^{\alpha }) \text { with } \frac{2}{\beta }+\frac{3}{\alpha }\le \frac{7}{4}, \alpha \ge \frac{12}{7}, \end{aligned}$$

(1.9)

then the corresponding weak solution (u, b) to the 3D MHD equations is regular on $\mathbb {R}^{3}\times [0,T]$. Recently, Jia and Zhou [20] improved (1.9) as

$$\begin{aligned} \partial _{3} P\in L^{\beta }(0,T; L^{\alpha }) \text { with } \frac{2}{\beta }+\frac{3}{\alpha }\le 2 \text { and } 3\le \alpha <\infty . \end{aligned}$$

Zhang Li and Yu [31] still obtained a regularity criterion on $\partial _{3}P$, that is

$$\begin{aligned}&\partial _{3}P\in L^{\beta }(0,T; L^{\alpha }) \text { with } \frac{2}{\beta }+\frac{3}{\alpha }= \gamma \text { and } \max \left\{ \frac{9}{7(\gamma -1),\frac{3}{\gamma }}\right\} \\&\quad \le \frac{3}{\gamma -1} \text { for some } \gamma \in (1,2). \end{aligned}$$

As the case of the 3D micropolar fluids, there are still also many interesting results have been obtained, Dong and Chen [8] proved the regularity of weak solutions under the velocity condition

$$\begin{aligned} \nabla u\in L^{q}(0,T; \dot{B}^{0}_{p,r}(\mathbb {R}^{3})) \text { with } \frac{2}{q}+\frac{3}{p}\le 2, \frac{3}{2}< p\le \infty , 1\le r\le \frac{2p}{3}. \end{aligned}$$

In [9], Dong, Jia and Chen established that the Serrin’s condition on the pressure (i.e., (1.5)) or on the gradient of the pressure (i.e., (1.6)) still holds for the 3D micropolar fluid equations (1.1)–(1.4). Jia, Zhang and Dong [21] obtain the following pressure regularity criterion

$$\begin{aligned} P\in L^{q}(0,T;\dot{B}^{r}_{p,\infty }(\mathbb {R}^{3})) \text { with } \frac{2}{q}+\frac{3}{p}\le 2+r, \frac{3}{2+r}<p<\infty , -1<r<1. \end{aligned}$$

For other regularity criteria results for the 3D micropolar fluids, we refer the readers to see [11, 17, 18, 32] and their references therein. We also refer the readers to see [29] for regularity criteria results for the magneto-micropolar fluids.

Motivated by the above cited papers on the regularity criteria of fluid dynamics equations, the purpose of the present paper is focused on the regularity criterion of weak solutions to the 3D micropolar fluids in terms of the pressure. Before stating the main result, let us first recall the definition of weak solutions the 3D micropolar fluids (1.1)–(1.4) (see [19, 23]). Define

$$\begin{aligned}&\mathbf {H}:= \text { closure of } C^{\infty }_{0}(\mathbb {R}^{3})\cap \{u:{\text {div}} u=0\} \text { in } L^{2}(\mathbb {R}^{3}),\\&\mathbf {J}:= \text { closure of } C^{\infty }_{0}(\mathbb {R}^{3})\cap \{u:{\text {div}} u=0\} \text { in } H^{1}(\mathbb {R}^{3}). \end{aligned}$$

Definition 1.1

(weak solutions). Let $0<T<\infty$, $u_{0}\in \mathbf {H}$ and $\omega _{0}\in L^{2}(\mathbb {R}^{3})$. A measurable function $(u,\omega )$ on $\mathbb {R}^{3}\times (0,T)$ is called a weak solution of system (1.1)–(1.4) on (0, T) if $(u,\omega )$ satisfies the following properties

• (i) $u\in L^{\infty }(0,T; \mathbf {H})\cap L^{2}(0,T; \mathbf {J})$ and $\omega \in L^{\infty }(0,T;L^{2}(\mathbb {R}^{3}))\cap L^{2}(0,T;H^{1}(\mathbb {R}^{3}))$;

• (ii) $(u,\omega )$ verifies (1.1)–(1.4) in the sense of distribution, i.e.,

  $$\begin{aligned}&\int _{0}^{T}\int _{\mathbb {R}^{3}}(\partial _{t}\phi +(u\cdot \nabla )\phi )u\text {d}x\text {d}t +\chi \int _{0}^{T}\int _{\mathbb {R}^{3}}\nabla \times \omega \phi \text {d}x\text {d}t+\int _{\mathbb {R}^{3}}u_{0}\phi (x,0)\text {d}x\\&\quad =(\mu +\chi )\int _{0}^{T}\int _{\mathbb {R}^{3}}\nabla u:\nabla \phi \text {d}x\text {d}t;\\&\int _{0}^{T}\int _{\mathbb {R}^{3}}(\partial _{t}\varphi +(u\cdot \nabla )\varphi )\omega \text {d}x\text {d}t +\chi \int _{0}^{T}\int _{\mathbb {R}^{3}}\nabla \times u\varphi \text {d}x\text {d}t+\int _{\mathbb {R}^{3}}\omega _{0}\varphi (x,0)\text {d}x\text {d}t\\&\quad =\int _{0}^{T}\int _{\mathbb {R}^{3}} (\gamma \nabla \omega :\nabla \varphi +\kappa {\text {div}}\omega {\text {div}}\varphi )\text {d}x\text {d}t+2\chi \int _{0}^{T}\int _{\mathbb {R}^{3}}\omega \varphi \text {d}x\text {d}t, \end{aligned}$$

  for all $\phi ,\varphi \in C_{0}^{\infty }(\mathbb {R}^{3}\times [0,T))$ with ${\text {div}}\phi =0$. ${\text {div}} u=0$ in distribution sense, i.e.,

  $$\begin{aligned} \int _{0}^{T}\int _{\mathbb {R}^{3}}u\cdot \nabla \phi \text {d}x\text {d}t=0, \end{aligned}$$

  for all $\phi \in C_{0}^{\infty }(\mathbb {R}^{3}\times [0,T))$.

• (iii) $(u,\omega )$ satisfies the energy inequality, i.e.,

  $$\begin{aligned}&\Vert u(t)\Vert _{L^{2}}^{2}+\Vert \omega (t)\Vert _{L^{2}}^{2}+2(\mu +\chi )\int _{\varepsilon }^{t}\Vert \nabla u(\tau )\Vert _{L^{2}}^{2}\text {d}\tau +2\gamma \int _{\varepsilon }^{t}\Vert \nabla \omega (\tau )\Vert _{L^{2}}^{2}\text {d}\tau \nonumber \\&+2\kappa \int _{\varepsilon }^{t}\Vert {\text {div}}\omega \Vert _{L^{2}}^{2}\text {d}\tau +2\chi \int _{\varepsilon }^{t}\Vert \omega \Vert _{L^{2}}^{2}\text {d}\tau \le \Vert u(\varepsilon )\Vert _{L^{2}}^{2}+\Vert \omega (\varepsilon )\Vert _{L^{2}}^{2} \end{aligned}$$

  (1.10)

  for $0\le \varepsilon \le t\le T$.

Now, our main result read as follows:

Theorem 1.2

Let $T>0$ be a given time, and $(u_{0},\omega _{0})\in H^{1}(\mathbb {R}^{3})\cap L^{3}(\mathbb {R}^{3})$ with ${\text {div}}u_{0}=0$. Assume the pair $(u,\omega )$ is a weak solution to system (1.1)–(1.4) on $\mathbb {R}^{3}\times (0,T)$. Suppose that the corresponding pressure P satisfies

$$\begin{aligned} \partial _{3}P\in L^{\beta }(0,T;L^{\alpha }(\mathbb {R}^{3}))\text { with }\frac{2}{\beta }+\frac{3}{\alpha }\le 2,\frac{3}{2}\le \alpha<\infty , 1< \beta \le \infty . \end{aligned}$$

(1.11)

Then the solution $(u,\omega )$ is regular on $\mathbb {R}^{3}\times [0,T]$.

Remark 1.3

It is a difficult problem to prove the regularity of the weak solutions to the 3D micropolar equations by adding the Cao and Titi’s type condition (1.7). We hope we can overcome this problem in the near future.

We shall give the proof of our main result in the following section. Later on, we denote by C a universal positive constant, whose value may depends on T, $\mu ,\chi ,\gamma ,\kappa$ and the initial data ($u_{0},d_{0}$), and may change from line to line, and we use $\Vert \cdot \Vert _{L^{p}}$ to denote the norm of the Lebesgue space $L^{p}$.

2 Proof of Theorem 1.2

In this section, we give the proof of Theorem 1.2. In order to prove Theorem 1.2, we need to quote the following lemma from [1] (Chapter 4) (see also [4]), which will play an important role in our discussion.

Lemma 2.1

Let $\delta ,\eta ,\lambda$ and $\zeta$ be four numbers satisfying

$$\begin{aligned} 1\le \delta ,\eta ,\lambda ,\zeta <\infty ,\quad \quad \frac{1}{\eta }+\frac{1}{\lambda }+\frac{1}{\zeta }>1\quad \text { and }\quad 1+\frac{3}{\delta }=\frac{1}{\eta }+\frac{1}{\lambda }+\frac{1}{\zeta }. \end{aligned}$$

Assume that $\phi (x)=\phi (x_{1},x_{2},x_{3})$ with $\partial _{1}\phi \in L^{\eta }(\mathbb {R}^{3})$, $\partial _{2}\phi \in L^{\lambda }(\mathbb {R}^{3})$ and $\partial _{3}\phi \in L^{\zeta }(\mathbb {R}^{3})$. Then, there exists a constant $C=C(\eta ,\lambda ,\zeta )$ such that

$$\begin{aligned} \Vert \phi \Vert _{L^{\delta }}\le C \Vert \partial _{1}\phi \Vert _{L^{\eta }}^{\frac{1}{3}}\Vert \partial _{2}\phi \Vert _{L^{\lambda }}^{\frac{1}{3}}\Vert \partial _{3}\phi \Vert _{L^{\zeta }}^{\frac{1}{3}}. \end{aligned}$$

In particular, when $\eta =\lambda =2$ and $1\le \zeta <\infty$, there exists a constant $C=C(\zeta )$ such that

$$\begin{aligned} \Vert \phi \Vert _{L^{3\zeta }}\le C \Vert \partial _{1}\phi \Vert _{L^{2}}^{\frac{1}{3}}\Vert \partial _{2}\phi \Vert _{L^{2}}^{\frac{1}{3}}\Vert \partial _{3}\phi \Vert _{L^{\zeta }}^{\frac{1}{3}}, \end{aligned}$$

which holds for any $\phi$ with $\partial _{1}\phi ,\partial _{2}\phi \in L^{2}(\mathbb {R}^{3})$ and $\partial _{3}\phi \in L^{\zeta }(\mathbb {R}^{3})$.

By using the above Lemma 2.1, we now turn to give the proof of Theorem 1.2.

Proof of Theorem1.2 Since the initial data $(u_{0},\omega _{0})\in H^{1}(\mathbb {R}^{3})\cap L^{3}(\mathbb {R}^{3})$ and ${\text {div}} u_{0}=0$, there exists a unique local strong solution $(u,\omega )$ of the 3D micropolar equations on (0, T) (see [19, 23]); thus the proof of Theorem 1.2 is reduced to establishing regular estimates uniformly on (0, T), and then the local strong solution $(u,\omega )$ can be continuously extended to the time $t=T$ argue by standard continuation process (see, e.g., [7]). Therefore, in what follows, we may as well assume that the solution $(u,\omega )$ is sufficiently smooth on (0, T).

Now, we first multiply both sides of Eq. (1.1) by |u|u, and integrate with respect to x over $\mathbb {R}^{3}$. After suitable integration by parts, we obtain

$$\begin{aligned}&\frac{1}{3}\frac{\mathrm{d}}{\mathrm{d}t}\Vert u(\cdot ,t)\Vert _{L^{3}}^{3}+(\mu +\chi )\Vert |u|^{\frac{1}{2}}|\nabla u|(\cdot ,t)\Vert _{L^{2}}^{2}+\frac{4}{9}(\mu +\chi )\Vert \nabla |u|^{\frac{3}{2}}(\cdot ,t)\Vert _{L^{2}}^{2}\nonumber \\&\quad = -\int _{\mathbb {R}^{3}}\nabla P\cdot |u|u\text {d}x+\chi \int _{\mathbb {R}^{3}}\nabla \times \omega \cdot |u|u \text {d}x, \end{aligned}$$

(2.1)

where we have used the following identities due to the divergence-free condition:

$$\begin{aligned} \int _{\mathbb {R}^{3}} (u\cdot \nabla u)\cdot |u|u\text {d}x&=\frac{1}{3}\int _{\mathbb {R}^{3}}u\cdot \nabla |u|^{3}\text {d}x=0;\\ \int _{\mathbb {R}^{3}}(\Delta u) \cdot |u|u\text {d}x&=-\int _{\mathbb {R}^{3}}|\nabla u|^{2}|u|\text {d}x-\int _{\mathbb {R}^{3}}|\nabla |u||^{2}|u|\text {d}x\\&= -\int _{\mathbb {R}^{3}}|\nabla u|^{2}|u|\text {d}x-\frac{4}{9}\int _{\mathbb {R}^{3}}|\nabla |u|^{\frac{3}{2}}|^{2}\text {d}x. \end{aligned}$$

Similarly, multiply both sides of Eq. (1.2) by $|\omega |\omega$, and integrate with respect to x over $\mathbb {R}^{3}$. After suitable integration by parts, it follows that

$$\begin{aligned}&\frac{1}{3}\frac{\mathrm{d}}{\mathrm{d}t}\Vert \omega (\cdot ,t)\Vert _{L^{3}}^{3}+(\gamma +\frac{\kappa }{2})\Vert |\omega |^{\frac{1}{2}}|\nabla \omega |(\cdot ,t)\Vert _{L^{2}}^{2}+\frac{4}{9}(\gamma +\kappa )\Vert \nabla |\omega |^{\frac{3}{2}}(\cdot ,t)\Vert _{L^{2}}^{2}\nonumber \\&\qquad +\frac{\kappa }{2}\Vert |\nabla \times \omega ||\omega |^{\frac{1}{2}}(\cdot ,t)\Vert _{L^{2}}^{2}\nonumber \\&\qquad +2\chi \int _{\mathbb {R}^{3}}|\omega |^{3}\text {d}x\le \chi \int _{\mathbb {R}^{3}}\nabla \times u\cdot |\omega |\omega \text {d}x, \end{aligned}$$

(2.2)

where we have used the fact that $\nabla {\text {div}} \omega =\nabla \times (\nabla \times \omega )+ \Delta \omega$ implies

$$\begin{aligned}&-\int _{\mathbb {R}^{3}}\nabla {\text {div}}\omega \cdot |\omega |\omega \text {d}x = -\int _{\mathbb {R}^{3}}(\nabla \times (\nabla \times \omega )+\Delta \omega )\cdot |\omega |\omega \text {d}x \\&\quad = \int _{\mathbb {R}^{3}} |\nabla \times \omega |^{2} |\omega |\text {d}x +\int _{\mathbb {R}^{3}} \nabla \times \omega \cdot \nabla |\omega |\times \omega \text {d}x +\int _{\mathbb {R}^{3}}|\nabla \omega |^{2}|\omega |\text {d}x\\&\qquad +\frac{2}{3}\int _{\mathbb {R}^{3}}|\nabla |\omega |^{\frac{3}{2}}|^{2}\text {d}x\\&\quad \ge \int _{\mathbb {R}^{3}} |\nabla \times \omega |^{2} |\omega |\text {d}x -\frac{1}{2}\int _{\mathbb {R}^{3}} (|\nabla \times \omega |^{2}|\omega |+|\nabla |\omega ||^{2}|\omega |)\text {d}x\\&\qquad +\int _{\mathbb {R}^{3}}|\nabla \omega |^{2}|\omega |\text {d}x +\frac{4}{9}\int _{\mathbb {R}^{3}}|\nabla |\omega |^{\frac{3}{2}}|^{2}\text {d}x\\&\quad =\frac{1}{2}\int _{\mathbb {R}^{3}} |\nabla \times \omega |^{2} |\omega |\text {d}x+\frac{1}{2}\int _{\mathbb {R}^{3}}|\nabla \omega |^{2}|\omega |\text {d}x +\frac{4}{9}\int _{\mathbb {R}^{3}}|\nabla |\omega |^{\frac{3}{2}}|^{2}\text {d}x. \end{aligned}$$

Combining (2.1) and (2.2) together, it follows that

$$\begin{aligned}&\frac{1}{3}\frac{\mathrm{d}}{\mathrm{d}t}(\Vert u(\cdot ,t)\Vert _{L^{3}}^{3}+\Vert \omega (\cdot ,t)\Vert _{L^{3}}^{3})+(\mu +\chi )\Vert |u|^{\frac{1}{2}}|\nabla u|(\cdot ,t)\Vert _{L^{2}}^{2}\nonumber \\&\qquad +\frac{4}{9}(\mu +\chi )\Vert \nabla |u|^{\frac{3}{2}}(\cdot ,t)\Vert _{L^{2}}^{2}\nonumber \\&\qquad +(\gamma +\frac{\kappa }{2})\Vert |\omega |^{\frac{1}{2}}|\nabla \omega |(\cdot ,t)\Vert _{L^{2}}^{2}+\frac{4}{9}(\gamma +\kappa )\Vert \nabla |\omega |^{\frac{3}{2}}(\cdot ,t)\Vert _{L^{2}}^{2}\nonumber \\&\qquad +\frac{\kappa }{2}\Vert |\nabla \times \omega ||\omega |^{\frac{1}{2}}(\cdot ,t)\Vert _{L^{2}}^{2}+2\chi \Vert \omega (\cdot ,t)\Vert _{L^{3}}^{3}\nonumber \\&\quad \le \left| \int _{\mathbb {R}^{3}}Pu\cdot \nabla |u|\text {d}x\right| +\chi \int _{\mathbb {R}^{3}}|\omega ||u||\nabla u|\text {d}x+\chi \int _{\mathbb {R}^{3}}|u||\omega ||\nabla \omega |\text {d}x\nonumber \\&\quad \equiv I+I_{2}+I_{3}, \end{aligned}$$

(2.3)

where we have use the following identities

$$\begin{aligned}&\int _{\mathbb {R}^{3}}\nabla P\cdot |u|u\text {d}x=-\int _{\mathbb {R}^{3}}P u\cdot \nabla |u|\text {d}x;\\&\int _{\mathbb {R}^{3}}\nabla \times \omega \cdot |u|u\text {d}x=-\int _{\mathbb {R}^{3}}|u|\omega \cdot \nabla \times u\text {d}x-\int _{\mathbb {R}^{3}}\omega \cdot \nabla |u|\times u\text {d}x;\\&\int _{\mathbb {R}^{3}}\nabla \times u\cdot |\omega |\omega \text {d}x=-\int _{\mathbb {R}^{3}}|\omega |u\cdot \nabla \omega \text {d}x -\int _{\mathbb {R}^{3}}u\cdot \nabla |\omega |\times \omega \text {d}x \end{aligned}$$

and the facts that $|\nabla \times u|\le |\nabla u|$ and $|\nabla |u||\le |\nabla u|$. Then we shall estimate the above terms $I_{1}$, $I_{2}$ and $I_{3}$ one by one. For $I_{2}$, by using the Hölder inequality and the Young inequality, one has

$$\begin{aligned} I_{2}&\le \chi \Vert |\omega ||u|^{\frac{1}{2}}\Vert _{L^{2}}\Vert |u|^{\frac{1}{2}}|\nabla u|\Vert _{L^{2}}\nonumber \\&\le \frac{(\mu +\chi )}{4}\Vert |u|^{\frac{1}{2}}|\nabla u|\Vert _{L^{2}}^{2}+C\Vert |\omega ||u|^{\frac{1}{2}}\Vert _{L^{2}}^{2}\nonumber \\&\le \frac{(\mu +\chi )}{4}\Vert |u|^{\frac{1}{2}}|\nabla u|\Vert _{L^{2}}^{2}+C\Vert \omega \Vert _{L^{3}}^{2}\Vert |u|^{\frac{1}{2}}\Vert _{L^{\frac{3}{2}}}^{2}\nonumber \\&\le \frac{(\mu +\chi )}{4}\Vert |u|^{\frac{1}{2}}|\nabla u|\Vert _{L^{2}}^{2}+C\Vert \omega \Vert _{L^{3}}^{2}\Vert u\Vert _{L^{3}}\nonumber \\&\le \frac{(\mu +\chi )}{4}\Vert |u|^{\frac{1}{2}}|\nabla u|\Vert _{L^{2}}^{2}+C(\frac{2}{3}\Vert \omega \Vert _{L^{3}}^{3}+\frac{1}{3}\Vert u\Vert _{L^{3}}^{3}). \end{aligned}$$

(2.4)

Similarly, one can estimate $I_{3}$ as

$$\begin{aligned} I_{3}\le&\Vert |u||\omega |^{\frac{1}{2}}\Vert _{L^{2}}\Vert |\omega |^{\frac{1}{2}}|\nabla \omega |\Vert _{L^{2}}\nonumber \\ \le&\frac{\gamma }{2}\Vert |\omega |^{\frac{1}{2}}|\nabla \omega |\Vert _{L^{2}}^{2}+C(\Vert \omega \Vert _{L^{3}}^{3}+\Vert u\Vert _{L^{3}}^{3}). \end{aligned}$$

(2.5)

To estimate the term $I_{1}$, let us first take the gradient on (1.1) and use the facts ${\text {div}} u=0$ and ${\text {div}}(\nabla \times \omega )=0$ yield

$$\begin{aligned} -\Delta P= \nabla \cdot (u\cdot \nabla u)=\sum _{i,j=1}^{3}\partial _{i}\partial _{j}(u_{i}u_{j}). \end{aligned}$$

By using the Calderon–Zygmund inequality, it is easy to obtain that there exists a absolute positive constant C such that

$$\begin{aligned} \Vert P\Vert _{L^{r}}\le C \Vert u\Vert _{L^{2r}}^{2} \text { for }1<r<\infty . \end{aligned}$$

(2.6)

Taking $\nabla {\text {div}}$ on both sides of equation (1.1), it follows that

$$\begin{aligned} -\Delta (\nabla P)=\nabla {\text {div}} (u\cdot \nabla u)=\sum _{i,j=1}^{3}\partial _{i}\partial _{j}\nabla (u_{i}u_{j}), \end{aligned}$$

where we have used the facts ${\text {div}}u=0$ and ${\text {div}}(\nabla \times \omega )=0$ again. The Calderon–Zygmund inequality implies that

$$\begin{aligned} \Vert \nabla P\Vert _{L^{s}}\le C\Vert |u||\nabla u|\Vert _{L^{s}} \text { for all } 1<s<\infty . \end{aligned}$$

(2.7)

Then, we have

By letting $0\le \theta \le \frac{1}{4}$ such that $\frac{7}{3}+\frac{8\theta }{3}\le 3$, and by using the Young inequality, it follows that

$$\begin{aligned} \Vert u\Vert _{L^{3}}^{\frac{7}{3}+\frac{8\theta }{3}}\le C(\Vert u\Vert _{L^{3}}^{3}+1). \end{aligned}$$

Setting $\beta =\frac{2(1-\theta )}{1+2\theta }$, then the restrictions $0\le \theta \le \frac{1}{4}$, $\frac{1-\theta }{r}+\frac{2\theta }{3}=\frac{1}{3}$ and $1+\frac{3}{r}=\frac{1}{\alpha }+\frac{5}{3}$ imply that $0\le \theta =\frac{\alpha -3}{4\alpha -3}\le \frac{1}{4}$, i.e., $\alpha \ge 3$, and $\frac{2}{\beta }+\frac{3}{\alpha }= 2$. Hence, we found that

$$\begin{aligned} |I_{1}|\le \frac{(\mu +\chi )}{4}\Vert \nabla |u|^{\frac{3}{2}}\Vert _{L^{2}}^{2}+C(\Vert \partial _{3}P\Vert _{L^{\alpha }}^{\beta }+ \Vert \nabla u\Vert _{L^{2}}^{2})(\Vert u\Vert _{L^{3}}^{3}+1), \end{aligned}$$

(2.8)

for all $\alpha \ge 3$ and $\frac{2}{\beta }+\frac{3}{\alpha }=2$. On the other hand, under the assumption $\frac{3}{2}\le \alpha \le 3$, one can estimate $I_{1}$ as follows

Notice that the condition $\frac{3}{2}\le \alpha \le 3$ implies that $2\le \frac{4\alpha }{3(\alpha -1)}\le 4<6$. Hence, by using the Gagliardo–Nirenberg inequality, it follows that

$$\begin{aligned} \Vert |u|^{\frac{3}{2}}\Vert _{L^{\frac{4\alpha }{3(\alpha -1)}}}\le \Vert |u|^{\frac{3}{2}}\Vert _{L^{2}}^{\frac{9(\alpha -1)}{4\alpha }-\frac{1}{2}}\Vert \nabla |u|^{\frac{3}{2}}\Vert _{L^{2}}^{\frac{3}{2}-\frac{9(\alpha -1)}{4\alpha }}. \end{aligned}$$

Thus

$$\begin{aligned} |I_{1}|\le&\Vert \partial _{3} P\Vert _{L^{\alpha }}^{\frac{1}{3}}\Vert \nabla u\Vert _{L^{2}}^{\frac{2}{3}}\Vert |u|^{\frac{3}{2}}\Vert _{L^{2}}^{\frac{\alpha -1}{\alpha }-\frac{2}{9}} \Vert \nabla |u|^{\frac{3}{2}}\Vert _{L^{2}}^{\frac{2}{3}-\frac{\alpha -1}{\alpha }} \Vert u\Vert _{L^{3}}^{\frac{1}{2}}\Vert \nabla |u|^{\frac{3}{2}}\Vert _{L^{2}}\nonumber \\ =&\Vert \partial _{3} P\Vert _{L^{\alpha }}^{\frac{1}{3}}\Vert \nabla u\Vert _{L^{2}}^{\frac{2}{3}} \Vert u\Vert _{L^{3}}^{\frac{3(\alpha -1)}{2\alpha }+\frac{1}{6}}\Vert \nabla |u|^{\frac{3}{2}} \Vert _{L^{2}}^{\frac{5}{3}-\frac{\alpha -1}{\alpha }}\nonumber \\ \le&\frac{\mu +\chi }{3} \Vert \nabla |u|^{\frac{3}{2}}\Vert _{L^{2}}^{2} +C[\Vert \partial _{3}P\Vert _{L^{\alpha }}^{\frac{1}{3}}\Vert \nabla u\Vert _{L^{2}}^{\frac{2}{3}}\Vert u\Vert _{L^{3}}^{\frac{3(\alpha -2)}{2\alpha }+\frac{1}{6}}]^{\frac{6\alpha }{4\alpha -3}}\qquad \\&\text { by the Young inequality} \nonumber \\ =&\frac{\mu +\chi }{3} \Vert \nabla |u|^{\frac{3}{2}}\Vert _{L^{2}}^{2} +C\Vert \partial _{3}P\Vert _{L^{\alpha }}^{\frac{2\alpha }{4\alpha -3}}\Vert \nabla u\Vert _{L^{2}}^{\frac{4\alpha }{4\alpha -3}}\Vert u\Vert _{L^{3}}^{\frac{9(\alpha -1)}{4\alpha -3}+\frac{\alpha }{4\alpha -3}} \nonumber \\ \le&\frac{\mu +\chi }{3} \Vert \nabla |u|^{\frac{3}{2}}\Vert _{L^{2}}^{2} +C(\Vert \partial _{3}P\Vert _{L^{\alpha }}^{\frac{2\alpha }{2\alpha -3}}+\Vert \nabla u\Vert _{L^{2}}^{2})\Vert u\Vert _{L^{3}}^{\frac{10\alpha -9)}{4\alpha -3}}. \qquad \\&\text { by the Young inequality} \end{aligned}$$

Noticing that for $\frac{3}{2}\le \alpha \le 3$ implies that $\frac{10\alpha -9}{4\alpha -3}\le 3$, then by using the Young inequality, one has

$$\begin{aligned} \Vert u\Vert _{L^{3}}^{\frac{10\alpha -9}{4\alpha -3}}\le C( \Vert u\Vert _{L^{3}}^{3}+1). \end{aligned}$$

Letting $\beta =\frac{2\alpha }{2\alpha -3}$, it follows that

$$\begin{aligned} I_{1}\le \frac{\mu +\chi }{3} \Vert \nabla |u|^{\frac{3}{2}}\Vert _{L^{2}}^{2} +C(\Vert \partial _{3}P\Vert _{L^{\alpha }}^{\beta }+\Vert \nabla u\Vert _{L^{2}}^{2})(\Vert u\Vert _{L^{3}}^{3}+1), \end{aligned}$$

(2.9)

for $\frac{3}{2}\le \alpha \le 3$ and $\frac{2}{\beta }+\frac{3}{\alpha }=2$. Combining (2.8) and (2.9) together, we found that for all $\frac{3}{2}\le \alpha <\infty$ and $\frac{2}{\beta }+\frac{3}{\alpha }=2$, there holds

$$\begin{aligned} I_{1}\le \frac{\mu +\chi }{4} \Vert |u|^{\frac{1}{2}}|\nabla u|\Vert _{L^{2}}^{2}+\frac{\mu +\chi }{3} \Vert \nabla |u|^{\frac{3}{2}}\Vert _{L^{2}}^{2} +C(\Vert \partial _{3}P\Vert _{L^{\alpha }}^{\beta }+\Vert \nabla u\Vert _{L^{2}}^{2})(\Vert u\Vert _{L^{3}}^{3}+1). \end{aligned}$$

(2.10)

Inserting the estimates (2.4), (2.5) and (2.10) into (2.3), one has

$$\begin{aligned}&\frac{1}{3}\frac{\mathrm{d}}{\mathrm{d}t}(\Vert u(\cdot ,t)\Vert _{L^{3}}^{3}+\Vert \omega (\cdot ,t)\Vert _{L^{3}}^{3}) +\frac{\mu +\chi }{2}\Vert |u|^{\frac{1}{2}}|\nabla u|(\cdot ,t)\Vert _{L^{2}}^{2}\\&\qquad +\frac{\mu +\chi }{3}\Vert \nabla |u|^{\frac{3}{2}}(\cdot ,t)\Vert _{L^{2}}^{2}\nonumber \\&\qquad +(\gamma +\frac{\kappa }{2})\Vert |\omega |^{\frac{1}{2}}|\nabla \omega |(\cdot ,t)\Vert _{L^{2}}^{2}+\frac{4}{9}(\gamma +\kappa )\Vert \nabla |\omega |^{\frac{3}{2}}(\cdot ,t)\Vert _{L^{2}}^{2} \\&\qquad +\frac{\kappa }{2}\Vert |\nabla \times \omega ||\omega |^{\frac{1}{2}}(\cdot ,t)\Vert _{L^{2}}^{2}+2\chi \Vert \omega (\cdot ,t)\Vert _{L^{3}}^{3}\nonumber \\&\quad \le C(\Vert \partial _{3}P\Vert _{L^{\alpha }}^{\beta }+\Vert \nabla u\Vert _{L^{2}}^{2})(\Vert u\Vert _{L^{3}}^{3}+\Vert \omega \Vert _{L^{3}}^{3}+1), \end{aligned}$$

which implies that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}(\Vert u(\cdot ,t)\Vert _{L^{3}}^{3}+\Vert \omega (\cdot ,t)\Vert _{L^{3}}^{3}+1) +(\mu +\chi )\Vert \nabla |u|^{\frac{3}{2}}(\cdot ,t)\Vert _{L^{2}}^{2}+\gamma \Vert \nabla |\omega |^{\frac{3}{2}}(\cdot ,t)\Vert _{L^{2}}^{2}\nonumber \\&\quad \le C(\Vert \partial _{3}P\Vert _{L^{\alpha }}^{\beta }+\Vert \nabla u\Vert _{L^{2}}^{2})(\Vert u\Vert _{L^{3}}^{3}+\Vert \omega \Vert _{L^{3}}^{3}+1), \end{aligned}$$

for all $\frac{3}{2}\le \alpha <\infty$ and $\frac{2}{\beta }+\frac{3}{\alpha }=2$. Then, applying the Gronwall inequality, it follows that

$$\begin{aligned}&\!\sup _{0\le t\le T}\!\!\{\Vert u(\cdot ,t)\Vert _{\!L^{\!3}}^{3}\!+\!\Vert \omega (\cdot ,t)\Vert _{\!L^{\!3}}^{3}\!+\!1\!\} \! +\!(\mu \!+\!\chi )\!\!\int _{0}^{T}\!\! \Vert \nabla |u|^{\frac{3}{2}}(\cdot ,t)\Vert _{\!L^{\!2}}^{2}\text {d}t \nonumber \\&\qquad +\!\gamma \!\!\int _{0}^{T}\!\!\Vert \nabla |\omega |^{\frac{3}{2}}(\cdot ,t)\Vert _{\!L^{\!2}}^{2}\text {d} t\nonumber \\&\quad \le e^{C(\int _{0}^{T}\Vert \partial _{3}P\Vert _{L^{\alpha }}^{\beta }\text {d}t+ \int _{0}^{T}\Vert \nabla u\Vert _{L^{2}}^{2}\text {d}t)}(\Vert u_{0}\Vert _{L^{3}}^{3}+\Vert \omega _{0}\Vert _{L^{3}}^{3}+1),\nonumber \\&\quad \le e^{C(\int _{0}^{T}\Vert \partial _{3}P\Vert _{L^{\alpha }}^{\beta }\text {d}t +\Vert u_{0}\Vert _{L^{2}}^{2})}(\Vert u_{0}\Vert _{L^{3}}^{3}+\Vert \omega _{0}\Vert _{L^{3}}^{3}+1), \end{aligned}$$

(2.11)

where we have used the energy inequality (1.10) in the last inequality. From (2.11), we get by using the assumption (1.11) that

$$\begin{aligned} \Vert u\Vert _{L^{3}(0,T;L^{9})}+\Vert \omega \Vert _{L^{3}(0,T;L^{9})}<\infty . \end{aligned}$$

(2.12)

From (2.12) and the standard Serrin regularity criterion (see, e.g., [22]), we have $(u,\omega )$ smooth on $(0,T)\times \mathbb {R}^{3}$. Then, by using the standard arguments of the continuation of local solutions, it is easy to conclude that the above estimate (2.12) implies that the solution $(u(x,t),\omega (x,t))$ can be extended beyond T. This completes the proof of Theorem 1.2. $\Box$