1 Introduction

In this paper, we consider the following 3D incompressible micropolar fluid equations

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t} u-\Delta u+(u \cdot \nabla ) u+\nabla P-\nabla \times \omega =0, \\ \partial _{t} \omega -\Delta \omega -\nabla \nabla \cdot \omega +2 \omega +(u \cdot \nabla ) \omega -\nabla \times u=0, \\ \nabla \cdot u=0, \\ u(x, 0)=u_{0}(x),\quad \omega (x, 0)=\omega _{0}(x), \end{array}\right. \end{aligned}$$
(1.1)

where \(u=u(x, t) \in \mathbb {R}^{3}, \omega =\omega (x, t) \in \mathbb {R}^{3}\) and \(P=P(x, t)\) denote the unknown velocity of the fluid, the micro-rotational velocity of the fluid particles and the unknown scalar pressure of the fluid at the point \((x, t) \in \mathbb {R}^{3} \times (0, T)\), respectively, while \(u_{0}, \omega _{0}\) are given initial data satisfying \(\nabla \cdot u=0\) in the sense of distributions.

This model for micropolar fluid flows proposed by Eringen [9] enables to consider some physical phenomena that cannot be treated by the classical Navier–Stokes equations for the viscous incompressible fluids, such as the motion of animal blood, muddy fluids, liquid crystals and dilute aqueous polymer solutions and colloidal suspensions. For more physical background of micropolar fluid flows, we refer the reader to [14] and the references therein.

Mathematically, there is a large literature on the global existence, stability and large time behaviors of solutions to the micropolar fluid equations (see, e.g., [6, 14, 22, 23] and the references therein). However, the global regularity for the weak solutions of the 3D micropolar fluid equations is still an open problem. Therefore, it is of interest to consider the conditional regularity of the weak solutions under certain growth conditions on velocity or pressure. More precisely, Yuan in [29] obtained several scaling invariant regularity criteria

$$\begin{aligned} \left\{ \begin{array}{l} (A)\,\, u(t, x) \in L^{q}\left( (0, T) ; L^{p, \infty }\left( \mathbb {R}^{3}\right) \right) ,\quad \frac{2}{q}+\frac{3}{p} \le 1, \quad 3<p \le \infty ;\\ (B)\,\, \nabla u(t, x) \in L^{q}\left( (0, T) ; L^{p, \infty }\left( \mathbb {R}^{3}\right) \right) , \quad \frac{2}{q}+\frac{3}{p} \le 2, \quad \frac{3}{2}<p \le \infty ;\\ (C)\,\, P(t, x) \in L^{q}\left( (0, T) ; L^{p, \infty }\left( \mathbb {R}^{3}\right) \right) , \quad \frac{2}{q}+\frac{3}{p} \le 2, \quad \frac{3}{2}<p \le \infty ;\\ (D)\,\, \nabla P(t, x) \in L^{q}\left( (0, T) ; L^{p, \infty }\left( \mathbb {R}^{3}\right) \right) , \quad \frac{2}{q}+\frac{3}{p} \le 3,\quad 1<p \le \infty . \end{array}\right. \end{aligned}$$
(1.2)

Velocity/vorticity/pressure regularity criteria for micropolar fluid equations and magneto-micropolar fluid equations (see, e.g., [10, 11, 13, 21, 26] and the references therein) attracted during the last years the attention of many researchers.

In this paper, we are concerned with another type of regularity criterion with geometric significant. A well-known result on the geometrical regularity condition is given by Constantin and Fefferman [7], and they showed if the vorticity direction does not change too rapidly in the regions with high vorticity magnitude, then a weak solution becomes strong solution. This result has been improved by Beirão da Veiga and Berselli [8]. There is extensive literature on the geometric-type regularity conditions, see [1, 3, 4, 16, 25], as well as the references cited therein. In addition, the role of eigenvalues and eigenvectors of the symmetrized gradient of velocity in the regularity problem of Navier–Stokes equations (micro-rotation effects are neglected in (1.1), i.e., \(\omega =0\)) was first proven by Neustupa and Penel [17,18,19,20], and more recently using different methods by Miller [15]. Precisely, they showed that the following condition in terms of the positive part of the intermediate eigenvalue of the strain matrix \(\left( S=S_{i j}=\frac{1}{2}\left( \frac{\partial u_{j}}{\partial x_{i}}+\frac{\partial u_{i}}{\partial x_{j}}\right) \right) \)

$$\begin{aligned} \lambda ^{+}_2 \in L^{q}\left( 0, T ; L^{p}\left( \mathbb {R}^{3}\right) \right) , \quad \frac{2}{q}+\frac{3}{p}=2,\quad \frac{3}{2}<p \le \infty \end{aligned}$$
(1.3)

implies the smoothness of the solution. More recently, the second author in [24] extended above regularity criteria to the Multiplier space and Besov space; namely, if one of the following conditions holds

$$\begin{aligned}{} & {} \int _{0}^{T}\frac{\Vert \lambda ^+_2(t)\Vert ^{\frac{2}{2-r}}_{\dot{X}_{r}}}{1+\ln \left( e+\Vert \lambda _2(t)\Vert _{L^2}\right) }dt<\infty , \quad \text {with}\quad 0\le r\le 1, \end{aligned}$$
(1.4)
$$\begin{aligned}{} & {} \lambda ^+_2 \in L^{q}\left( 0, T; \dot{B}_{\theta , \infty }^{-3\left( \frac{1}{p}-\frac{1}{\theta } \right) }\right) , \quad \text {with}\quad \frac{2}{q}+\frac{3}{p}=2, \quad p \in \left( \frac{3}{2}, \infty \right) , \end{aligned}$$
(1.5)

where \(\theta \in [p, \frac{3p}{3-p}) \text{ if } p \in (\frac{3}{2},3]\,\text {and}\,\theta =\infty \,\text {if}\,p \in (3, \infty )\), then the weak solution u is actually smooth on interval (0, T]. Later on, the second author in different paper [27, 28] refined/extended the conditions (1.3)–(1.5). We also want to mention a very interesting regularity result proved by Houamed in [12], and he investigates the same question under some conditions on one component of the vorticity and unidirectional derivative of one component of the velocity in some critical Besov spaces of the form \(L_{T}^{p}\left( \dot{B}_{2, \infty }^{\alpha , \frac{2}{p}-\frac{1}{2}-\alpha }\right) \) or \(L_{T}^{p}\left( \dot{B}_{q, \infty }^{\frac{2}{p}+\frac{3}{q}-2}\right) \).

Motivated by recent studies [12, 15, 24], the aim of the present paper is to extend and improve eigenvalue regularity criteria to the micropolar fluid equations (1.1). Moreover, we will consider it in some anisotropic Besov spaces of the form \(L^p((B_{2,\infty }^{\alpha })_h(B_{2,\infty }^{s_p-\alpha })_v )\), for \(\alpha \in [0,s_p]\), or \(L^p(\mathcal {B}_{q,p})\), where

$$\begin{aligned} s_p \overset{\hbox {def}}{=} \frac{2}{p}-\frac{1}{2},\quad \text {and}\quad \mathcal {B}_{q,p}\overset{\hbox {def}}{=} \dot{B}^{\frac{3}{q}+\frac{2}{p}-2}_{q,\infty }. \end{aligned}$$
(1.6)

To state the main results, let us first recall the definition of the weak solution of the 3D micropolar fluid equations (1.1).

Definition 1.1

Let \((u_{0}, \omega _{0}) \in L^{2}(\mathbb {R}^{3})\) and suppose that \(\nabla \cdot u_{0}=0\). A measurable function \((u(x, t), \omega (x, t))\) is called a weak solution to the 3D micropolar flows equations (1.1) on (0, T) if \((u, \omega )\) satisfies three properties:

  1. (1)

    \((u, \omega ) \in L^{\infty }((0, T); L^{2}(\mathbb {R}^{3})) \cap L^{2}((0, T); H^{1}(\mathbb {R}^{3}))\) for all \(T>0\);

  2. (2)

    \((u(x, t), \omega (x, t))\) verifies (1.1) in the sense of distribution;

  3. (3)

    For all \(0 \le t \le T\), it holds:

    $$\begin{aligned}&\Vert u(\cdot , t)\Vert _{L^{2}}^{2}+\Vert \omega (\cdot , t)\Vert _{L^{2}}^{2}+2 \int _{0}^{t}\left( \Vert \nabla u(\cdot , \tau )\Vert _{L^{2}}^{2}+\Vert \nabla \omega (\cdot , \tau )\Vert _{L^{2}}^{2}+\Vert \nabla \cdot \omega (\cdot , \tau )\Vert _{L^{2}}^{2}\right) \hbox {d}\tau \\&\quad \le \left\| u_{0}\right\| _{L^{2}}^{2}+\left\| \omega _{0}\right\| _{L^{2}}^{2}. \end{aligned}$$

We will prove

Theorem 1.1

Let \(T > 0\) and the initial data \((u_0,\omega _0)\in H^{1}(\mathbb {R}^3)\) with \(\nabla \cdot u_{0}=0\) in the sense of distributions. Assume that \((u, \omega )\) is a weak solution to the 3D micropolar fluid equations (1.1) on (0, T). If the positive part of the intermediate eigenvalue of the strain matrix \((i.e.,\lambda ^+_{2}=\max \{\lambda _2, 0\})\) satisfies assumption

$$\begin{aligned} \int _0^T \Vert \lambda ^{+}_2(\cdot ,t)\Vert _{\dot{B}^{\alpha , s_p- \alpha }_{2,\infty }}^p dt < \infty ,\quad \forall \, p \in [2,4],\quad \forall \, \alpha \in \left[ 0, \frac{2}{p}- \frac{1}{2}\right] \end{aligned}$$
(1.7)

or

$$\begin{aligned} \int _0^T \Vert \lambda ^{+}_2(\cdot ,t)\Vert _{\mathcal {B}_{q_1,p_1}}^{p_1} dt < \infty , q_1 \in [3,\infty ),\frac{3}{q_1}+ \frac{2}{p_1} \in (1,2), \end{aligned}$$
(1.8)

then the weak solution \((u,\omega )\) remains smooth on \(\mathbb {R}^{3}\times (0, T]\).

Remark 1.1

All the spaces stated in Theorem 1.1 are scaling invariant spaces based on \(dim\,(\lambda _2)=dim\,(\nabla u)\). From this point of view, eigenvalue regularity criterion (1.7) and (1.8) are optimal.

Remark 1.2

In the case \(p=4\) in (1.7), \(\alpha \) is necessary zero, and this means that the anisotropic space above is nothing but \(L^4_T(\dot{B}^0_{2,\infty })\), which is still larger than \(L^4_T(L^2)\). Therefore, eigenvalue regularity criterion (1.7) is obviously an improvement of (1.3).

Remark 1.3

It is not clear for us if Theorem 1.1 can be further extended to the MHD equations, maybe, it is possible, but this would require much more delicate analysis, and likely somewhat different techniques. Compared with the MHD equations, micropolar flow equations have a better equation structure; that is, we can establish the \(H^1\) estimate of the velocity field only relying on the basic energy estimates.

2 Preliminaries

In this section, we recall some definitions and give several lemmas, which will be used in proof of Theorem 1.1.

Let us first recall some notions of the Littlewood–Paley theory, the definition of anisotropic Besov spaces.

Let \((\psi ,\varphi )\) be a couple of smooth functions with value in [0, 1] satisfying:

$$\begin{aligned} \begin{aligned}&\text {Supp } \psi \subset \left\{ \xi \in \mathbb {R}: |\xi | \le \frac{4}{3}\right\} , \quad{} & {} \text {Supp } \varphi \subset \left\{ \xi \in \mathbb {R}:\frac{3}{4} \le |\xi | \le \frac{8}{3} \right\} \\&\psi (\xi ) + \sum _{q\in \mathbb {N}} \varphi (2^{-q}\xi ) = 1 \quad \forall \xi \in \mathbb {R}, \quad{} & {} \sum _{q\in \mathbb {Z}} \varphi (2^{-q}\xi ) = 1 \quad \forall \xi \in \mathbb {R}\backslash \{0\}. \end{aligned} \end{aligned}$$

Let a be a tempered distribution, \(\hat{a}=\mathcal {F}(a)\) its Fourier transform and \(\mathcal {F}^{-1}\) denote the inverse of \(\mathcal {F}\). We define the homogeneous dyadic blocks \(\Delta _q\) by setting

$$\begin{aligned} \begin{aligned}&\Delta ^v_q a= \mathcal {F}^{-1}\left( \varphi (2^{-q}|\xi _3| \hat{a})\right) ,\quad \forall \; q\in \mathbb {Z}, \quad{} & {} \Delta ^h_j a= \mathcal {F}^{-1}\left( \varphi (2^{-j}|\xi _h| \hat{a})\right) ,\quad \forall \; j\in \mathbb {Z}, \\&S_q^v =\sum _{q^{\prime }< q} \Delta _{q^{\prime } }^v, \quad \forall q \in \mathbb Z, \quad{} & {} S_j^h= \sum _{j^{\prime }< j} \Delta _{j^{\prime } }^h, \quad \forall j \in \mathbb Z. \end{aligned} \end{aligned}$$

Moreover, in all the situations, i.e., for \(\Delta ,S\) with the same index of direction (horizontal or vertical) it holds:

$$\begin{aligned}&\Delta _m\Delta _{m^{\prime }} a =0 \quad \text {if} \; |m-m^{\prime }| \ge 2 \\&\Delta _m\big (S_{m^{\prime }-1}a\Delta _{m^{\prime }} a\big ) =0 \quad \text {if} \; |m-m^{\prime }| \ge 5 \\&\Delta _m\sum _{i\in \{0,1,-1 \}}\sum _{m^{\prime }\in \mathbb {Z}}\left( \Delta _{m^{\prime }+i}a\Delta _{m^{\prime }} a\right) =\Delta _m\sum _{i\in \{0,1,-1 \}}\sum _{m^{\prime }\ge m-5}\left( \Delta _{m^{\prime }+i}a\Delta _{m^{\prime }} a\right) , \end{aligned}$$

We should recall the so-called Bony decomposition (see [2])

$$\begin{aligned} \begin{aligned}&ab= T_a(b) + T_b(a) + R(a,b),{} & {} \\&T_a(b)= \sum _{q\in Z} S_{q-1} a \Delta _q b, \quad{} & {} R(a,b)=\sum _{i\in \{0,1,-1 \}}\sum _{q\in Z} \Delta _{q+i} a \Delta _q b. \end{aligned} \end{aligned}$$

It is also useful sometimes to use the following version

$$\begin{aligned} ab= \widetilde{T}_ab + T_ba, \end{aligned}$$

where

$$\begin{aligned} \widetilde{T}_ab = \sum _{q\in \mathbb {Z}} S_{q+2}a \Delta _q b. \end{aligned}$$

Here again all the situations may be considered; however, particular cases must be precised by using the adequate notations. For instance, if we consider the version for the vertical variable, we have to add the exponent \(^{v}\) in all the operators \(T_a,T_b,R,S_q\) and \(\Delta _q\).

Next, we recall the definition of the anisotropic Besov spaces. See [5] for more details.

Definition 2.1

Let s, t be two real numbers and let \(p_1,p_2,q_1,q_2\) be in \([1,+\infty ]\), we define the space \((\dot{B}^t_{p_1,q_1})_h(\dot{B}^s_{p_2,q_2})_v\) as the space of tempered distributions u such that

$$\begin{aligned} \Vert u \Vert _{\left( \dot{B}^t_{p_1,q_1}\right) _h\left( \dot{B}^s_{p_2,q_2}\right) _v} \Vert { 2^{kt}2^{js} \Vert {\Delta _k^h \Delta _j^v u}\Vert _{L^{p_1}_hL^{p_2}_v}}\Vert _{\ell _k^{q_1}\left( \mathbb {Z};\ell _j^{q_2}(\mathbb {Z})\right) } < \infty . \end{aligned}$$

In the situation where \(q_1=q_2=q\) and \(p_1=p_2=p\), we use the notation \(\dot{B}_{p,q}^{t,s}(\dot{B}^t_{p,q})_h(\dot{B}^s_{p,q})_v\). If \(p=q=2\), then this last space is equivalent to \(\dot{H}^{t,s}\). More precisely, we have:

$$\begin{aligned} \Vert a\Vert _{\dot{B}_{2,2}^{t,s}}^2 \approx \Vert a\Vert _{\dot{H}^{t,s}}^2 \int _{\mathbb {R}^3}|\xi _h|^{2t} |\xi _v|^{2s} |\hat{a}(\xi )|^2 d\xi , \end{aligned}$$

while the proofs of Theorem 1.1 are essentially based on the following ones.

Lemma 2.1

[12] For all \(p\in [2,4]\), for all \(\alpha \in [0,s_p]\), where \(s_p=\frac{2}{p}-\frac{1}{2}\), we have:

$$\begin{aligned} \big | \big ( fg | g \big )_{L^2} \big | \le \frac{1}{10} \Vert {g}\Vert _{\dot{H}^1(\mathbb {R}^3)}^2+ C \Vert {f}\Vert _{ \dot{B}^{\alpha ,s_p- \alpha }_{2,\infty }}^p \Vert {g}\Vert _{L^2(\mathbb {R}^3)}^2. \end{aligned}$$

Lemma 2.2

[12] For any \(p,q\in [1,\infty ]\) satisfying \(\frac{3}{q}+ \frac{2}{p} \in (1,2)\) we have

$$\begin{aligned} \big | \big (fg | g \big )_{L^2} \big | \le \frac{1}{10} \Vert {g}\Vert _{\dot{H}^1(\mathbb {R}^3)}^2+ C \Vert {f}\Vert _{\mathcal {B}_{q,p}}^p \Vert {g}\Vert _{L^2(\mathbb {R}^3)}^2. \end{aligned}$$

Lemma 2.3

[15] For all \(-\frac{3}{2}<\alpha <\frac{3}{2}\) and for all u divergence free in the sense that \(\xi \cdot \hat{u}(\xi )=0\) almost everywhere,

$$\begin{aligned} \Vert S\Vert _{\dot{H}^{\alpha }}^{2}=\Vert A\Vert _{\dot{H}^{\alpha }}^{2}=\frac{1}{2}\Vert \Omega \Vert _{\dot{H}^{\alpha }}^{2}=\frac{1}{2}\Vert \nabla \otimes u\Vert _{\dot{H}^{\alpha }}^{2}, \end{aligned}$$

where symmetric part \(S=S_{i j}=\frac{1}{2}\left( \frac{\partial u_{j}}{\partial x_{i}}+\frac{\partial u_{i}}{\partial x_{j}}\right) \), which we refer to as the strain tensor, anti-symmetric part \(A=A_{i j}=\frac{1}{2}\left( \frac{\partial u_{j}}{\partial x_{i}}-\frac{\partial u_{i}}{\partial x_{j}}\right) \), \(\Omega =\nabla \times u\).

Lemma 2.4

[15] Suppose \(S \in L^{\infty }([0, T]; L^{2}(\mathbb {R}^{3})) \cap L^{2}([0, T]: \dot{H}^{\alpha }\left( \mathbb {R}^{3}\right) )\) is a local strong solution to the fractional Navier–Stokes strain equation and S(x) has eigenvalues \(\lambda _1(x) \le \lambda _2(x) \le \lambda _3(x)\). Define

$$\begin{aligned} \lambda _{2}^{+}(x)=\max \left\{ \lambda _{2}(x), 0\right\} , \end{aligned}$$

then

$$\begin{aligned} -{\textrm{det}}(S) \le \frac{1}{2}|S|^{2} \lambda _{2}^{+}. \end{aligned}$$

3 The Proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1. First, we assume that the condition (1.7) holds, and then, we will divided into two steps for proof of Theorem 1.1. In the first step, we establish the bounds of \(\Vert \nabla u\Vert _{L^2}\) and then derive the bounds of \(\Vert \nabla \omega \Vert _{L^2}\) in the second one.

In [14], Lukaszewicz proved the global existence of weak solutions to the 3D micropolar fluid motion equations by the Galerkin method. Therefore, it is not difficult to verify the corresponding weak solutions \((u, \omega )\) satisfying the energy-type inequality:

$$\begin{aligned} \Vert u\Vert _{L^2}^{2}+\Vert w\Vert _{L^2}^{2}+\int _{0}^{T}\left( \Vert \nabla u(t)\Vert _{L^2}^{2}+\Vert \nabla w(t)\Vert _{L^2}^{2}\right) \hbox {d}t \le \left\| u_{0}\right\| _{L^2}^{2}+\left\| w_{0}\right\| _{L^2}^{2}, \quad \forall \, T>0. \end{aligned}$$

Taking \(\nabla \times \) on the first equation of (1.1), one has

$$\begin{aligned} \partial _{t} \Omega +(u \cdot \nabla ) \Omega -\Delta \Omega = S\Omega +\nabla \times \nabla \times \omega , \end{aligned}$$
(3.1)

taking the operator \(\nabla _{\textrm{sym}}\) \(\left( \hbox {i.e.}, S=\nabla _{s y m}(u)_{i j}=\frac{1}{2}\left( \frac{\partial u_{j}}{\partial x_{i}}+\frac{\partial u_{i}}{\partial x_{j}}\right) \right) \) to the first equation of (1.1) to obtain

$$\begin{aligned} \partial _{t} S+(u \cdot \nabla ) S-\Delta S+S^{2}+\frac{1}{4} \Omega \otimes \Omega -\frac{1}{4}|\Omega |^{2} I_{3}+\hbox {Hess}(P)=\nabla _{\textrm{sym}}(\nabla \times \omega ), \end{aligned}$$
(3.2)

where more details can refer to [15]. Multiplying (3.1) by \(\Omega \) and integrating over \(\mathbb {R}^3\), we have

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\Vert \Omega (t)\Vert _{2}^{2}+\Vert \nabla \Omega (t)\Vert _{2}^{2}&=\int _{\mathbb {R}^3} S\Omega \cdot \Omega \textrm{d} x+ \int _{\mathbb {R}^{3}} \nabla \times \nabla \times \omega \cdot \Omega \textrm{d} x\nonumber \\&=\int _{\mathbb {R}^3} S\Omega \cdot \Omega \textrm{d} x- \int _{\mathbb {R}^{3}} \nabla \times \omega \cdot \Delta u \textrm{d} x. \end{aligned}$$
(3.3)

Testing (3.2) by S and integrating by parts in \(\mathbb {R}^3\) to obtain

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\Vert S(t)\Vert _{L^{2}}^{2}+\Vert \nabla S(t)\Vert _{L^{2}}^{2}&=- \int _{\mathbb {R}^3} S^{2}\cdot S\hbox {d}x-\frac{1}{4}\int _{\mathbb {R}^3} \Omega \otimes \Omega \cdot S \textrm{d}x - \int _{\mathbb {R}^3} \text {Hess}(P)\cdot S \textrm{d}x\nonumber \\&\quad +\frac{1}{4}\int _{\mathbb {R}^3} |\Omega |^{2} I_{3}\cdot S\textrm{d}x+\int _{\mathbb {R}^3}\nabla _{\textrm{sym}}(\nabla \times \omega )\cdot S\textrm{d}x\nonumber \\&=- \int _{\mathbb {R}^{3}} \text {tr}\left( S^{3}\right) \textrm{d}x-\frac{1}{4}\int _{\mathbb {R}^3} \Omega \otimes \Omega \cdot S \textrm{d}x-\frac{1}{2}\int _{\mathbb {R}^3} \nabla \times \omega \cdot \Delta u \textrm{d}x, \end{aligned}$$
(3.4)

where we have used the following facts

$$\begin{aligned} \left\langle |\omega |^{2} I_{3}, S\right\rangle _{L^2}= & {} 0,\\ \langle \textrm{Hess}(p), S\rangle _{L^2}= & {} 0,\\ \left\langle S^{2}, S\right\rangle _{L^2}= & {} \int _{\mathbb {R}^{3}} \text {tr}\left( S^{3}\right) . \end{aligned}$$

Due to Lemma 2.3, for (3.3), we derive

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t}\Vert S(t)\Vert _{L^{2}}^{2}+2\Vert \nabla S(t)\Vert _{L^{2}}^{2}=\int _{\mathbb {R}^3} S\Omega \cdot \Omega \textrm{d} x- \int _{\mathbb {R}^{3}} \nabla \times \omega \cdot \Delta u \textrm{d} x. \end{aligned}$$
(3.5)

Combining (3.4) and (3.5), we have

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t}\Vert S(t)\Vert _{L^{2}}^{2}+2\Vert \nabla S(t)\Vert _{L^{2}}^{2}=-\frac{4}{3} \int _{\mathbb {R}^{3}} \text {tr}\left( S^{3}\right) \textrm{d} x- \int _{\mathbb {R}^{3}} \nabla \times \omega \cdot \Delta u \textrm{d}x. \end{aligned}$$
(3.6)

Since \(\text {tr}(S)=\nabla \cdot u=0\), by Hölder’s inequality, Young’s inequality, and Lemmas 2.1, 2.3 and 2.4, one has

$$\begin{aligned} -\frac{4}{3} \int _{\mathbb {R}^{3}} \text {tr}\left( S^{3}\right) \textrm{d} x&=-\frac{4}{3} \int _{\mathbb {R}^{3}}\lambda _{1}^{3}+\lambda _{2}^{3}+\lambda _{3}^{3} \textrm{d} x \nonumber \\&=-\frac{4}{3} \int _{\mathbb {R}^{3}}\lambda _{1}^{3}+\lambda _{2}^{3}+(-\lambda _{1}-\lambda _{2})^{3} \textrm{d} x\nonumber \\&=-4\int _{\mathbb {R}^{3}}(-\lambda _{1}-\lambda _{2}) \lambda _{1} \lambda _{2} \textrm{d} x=-4\int _{\mathbb {R}^{3}} \lambda _{1} \lambda _{2} \lambda _{3} \textrm{d} x\nonumber \\&=-4\int _{\mathbb {R}^{3}} \text {det}(S) \textrm{d} x\le 2\int _{\mathbb {R}^{3}}|S|^{2} \lambda _{2}^{+} \textrm{d} x\nonumber \\&\le \frac{1}{5} \Vert S\Vert _{\dot{H}^1}^2 + C \Vert \lambda _{2}^{+}\Vert _{\dot{B}^{\alpha , s_p-\alpha }_{2,\infty }}^p \Vert S\Vert _{L^2}^2 \end{aligned}$$
(3.7)

and

$$\begin{aligned} - \int _{\mathbb {R}^{3}} \nabla \times \omega \cdot \Delta u \textrm{d} x&\le \frac{1}{10} \Vert \Delta u\Vert ^2_{L^2} + C \Vert \nabla \omega \Vert _{L^2}^2\nonumber \\&\approx \frac{1}{5} \Vert \nabla S\Vert ^2_{L^2} + C \Vert \nabla \omega \Vert _{L^2}^2. \end{aligned}$$
(3.8)

Putting (3.7) and (3.8) into (3.6), we deduce that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t}\Vert S(t)\Vert _{L^{2}}^{2}+\Vert \nabla S(t)\Vert _{L^{2}}^{2}\le C \Vert \lambda _{2}^{+}\Vert _{\dot{B}^{\alpha , s_p-\alpha }_{2,\infty }}^p \Vert S\Vert _{L^2}^2+C \Vert \nabla \omega \Vert _{L^2}^2. \end{aligned}$$
(3.9)

Gronwall’s lemma leads then to

$$\begin{aligned} \sup \limits _{0< t\le T}\Vert S(t)\Vert ^{2}_{L^{2}}+\int ^T_{0}\Vert \nabla S(t)\Vert _{L^{2}}^{2} \textrm{d}t\le \left( \Vert S_{0}\Vert ^{2}_{L^{2}}+{ CT}\right) \exp C\int ^{T}_{0} \Vert \lambda _{2}^{+}\Vert _{\dot{B}^{\alpha , s_p-\alpha }_{2,\infty }}^p\textrm{d}t. \end{aligned}$$
(3.10)

Hence, it follows from Lemma 2.3 that

$$\begin{aligned} u\in L^{\infty }\left( 0,T;H^{1}(\mathbb {R}^3)\right) \cap L^{2}\left( 0,T;H^{2}(\mathbb {R}^3)\right) . \end{aligned}$$

Next, we derive the estimate of \(\Vert \nabla \omega \Vert _{L^2}\). Testing the second equation of (1.1) by \(-\Delta \omega \), and integrating by parts, we have

$$\begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\Vert \nabla \omega \Vert _{L^2}^{2}+\Vert \Delta \omega \Vert _{L^2}^{2}+\Vert \nabla \cdot (\nabla \omega )\Vert _{L^2}^{2}+\Vert \nabla \omega \Vert _{L^2}^{2}\nonumber \\&\quad =\int _{\mathbb {R}^{3}}(u \cdot \nabla ) \omega \cdot \Delta \omega \textrm{d} x-2 \int _{\mathbb {R}^{3}} \nabla \times u \cdot \Delta \omega \textrm{d}x. \end{aligned}$$
(3.11)

By Hölder’s inequality and Young’s inequality, it yields that

$$\begin{aligned} \left| \int _{\mathbb {R}^{3}}(u \cdot \nabla ) \omega \cdot \Delta \omega \textrm{d} x\right|&=\left| \int _{\mathbb {R}^{3}} u_{i} \partial _{i} \omega _{j} \partial _{kk} \omega _{j}\textrm{d} x\right| =\left| \int _{\mathbb {R}^{3}} \partial _{k} u_{i} \partial _{i} \omega _{j} \partial _{k} \omega _{j}\textrm{d} x\right| \nonumber \\&\le \Vert \nabla u\Vert _{L^2}\Vert \nabla \omega \Vert _{L^4}^{2} \nonumber \\&\le \Vert \nabla u\Vert _{L^2}\Vert \nabla \omega \Vert _{L^{2}}^{\frac{1}{2}}\Vert \Delta \omega \Vert _{L^{2}}^{\frac{3}{2}} \nonumber \\&\le C\Vert \nabla u\Vert _{L^2}^{4}\Vert \nabla \omega \Vert _{2}^{2}+\epsilon \Vert \Delta \omega \Vert _{2}^{2} \end{aligned}$$
(3.12)

and

$$\begin{aligned} \left| -2 \int _{\mathbb {R}^{3}} \nabla \times u \cdot \Delta \omega \textrm{d} x\right| \le C\Vert \nabla u\Vert _{2}^{2}+\epsilon \Vert \Delta \omega \Vert _{2}^{2}. \end{aligned}$$
(3.13)

Inserting (3.12) and (3.13) into (3.11), we have

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t}\Vert \nabla \omega \Vert _{2}^{2}+\Vert \Delta \omega \Vert _{L^2}^{2} \le C\Vert \nabla u\Vert _{2}^{4}\Vert \nabla \omega \Vert _{2}^{2}+C\Vert \nabla u\Vert _{2}^{2}. \end{aligned}$$
(3.14)

With the aid of the Gronwall inequality, we then have the desired boundedness:

$$\begin{aligned} (u, \omega )\in L^{\infty }\left( 0,T;H^{1}(\mathbb {R}^3)\right) \cap L^{2}\left( 0,T;H^{2}(\mathbb {R}^3)\right) . \end{aligned}$$

The estimates of higher-order derivatives can be obtained by an inductive procedure. This finishes the proof of Theorem 1.1 under the eigenvalue criteria (1.7).

Now, we prove Theorem 1.1 under the condition (1.8). The proof of eigenvalue regularity criteria (1.8) does not differ a lot from the previous one. We restart from (3.6), and applying Lemmas 2.2–2.4 yields that

$$\begin{aligned} -\frac{4}{3} \int _{\mathbb {R}^{3}} \text {tr}\left( S^{3}\right) \textrm{d}x&=-4\int _{\mathbb {R}^{3}} \text {det}(S) \textrm{d}x\le 2\int _{\mathbb {R}^{3}}|S|^{2} \lambda _{2}^{+} \textrm{d}x\nonumber \\&\le \frac{1}{5} \Vert {\nabla u}\Vert _{\dot{H}^1}^2 + C \Vert \lambda _{2}^{+}\Vert _{\mathcal {B}_{q_1,p_1} }^{p_1} \Vert S\Vert _{L^2}^2 \end{aligned}$$
(3.15)

and

$$\begin{aligned} - \int _{\mathbb {R}^{3}} \nabla \times \omega \cdot \Delta u \textrm{d} x \le \frac{1}{5} \Vert \nabla S\Vert ^2_{L^2} + C \Vert \nabla \omega \Vert _{L^2}^2. \end{aligned}$$
(3.16)

Similarly, we have

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t}\Vert S(t)\Vert _{L^{2}}^{2}+\Vert \nabla S(t)\Vert _{L^{2}}^{2}\le C \Vert \lambda _{2}^{+}\Vert _{\mathcal {B}_{q_1,p_1} }^{p_1} \Vert S\Vert _{L^2}^2+C \Vert \nabla \omega \Vert _{L^2}^2. \end{aligned}$$
(3.17)

Therefore, Gronwall’s inequality implies that for all \(0<t\le T\),

$$\begin{aligned} \sup \limits _{0< t\le T}\Vert S(t)\Vert ^{2}_{L^{2}}+\int ^T_{0}\Vert \nabla S(t)\Vert _{L^{2}}^{2} \textrm{d}t\le \left( \Vert S_{0}\Vert ^{2}_{L^{2}}+{ CT}\right) \exp C\int ^{T}_{0} \Vert \lambda _{2}^{+}\Vert _{\mathcal {B}_{q_1,p_1} }^{p_1} \textrm{d}t. \end{aligned}$$
(3.18)

And then

$$\begin{aligned} u\in L^{\infty }\left( 0,T;H^{1}(\mathbb {R}^3)\right) \cap L^{2}\left( 0,T;H^{2}(\mathbb {R}^3)\right) . \end{aligned}$$

We thus complete the proof of Theorem 1.1.