1 Introduction and the main result

In this paper, we consider the 3D magneto-micropolar fluid equations studied by Galdi and Rionero [5]:

$$\begin{array}{@{}rcl@{}} {}\left\{ \begin{array}{lllll} \partial_{t} u+u\cdot \nabla u-(\mu+\chi)\triangle u-b\cdot \nabla b +\nabla (p+b^{2})-\chi\nabla\times \omega=0,\\ \partial_{t} \omega-\gamma \triangle \omega -\kappa \nabla \text{div}\omega +2\chi \omega +u\cdot \nabla \omega -\chi\nabla \times u=0,\\ \partial_{t} b-\nu \triangle b+u\cdot \nabla b-b\cdot \nabla u=0,\\ \nabla\cdot u=\nabla \cdot b=0,\\ u(x,0)=u_{0}(x),\omega(0,x)=\omega_{0}(x),b(0,x)=b_{0}(x).\vspace*{-13pt} \end{array} \right.\\ \end{array} $$
(1.1)

Here u=u(x,t),b=b(x,t),ω=ω(x,t) represent the velocity field, the magnetic field and the micro-rotational velocity respectively; p denotes the hydrodynamic pressure; μ>0 is the kinematic viscosity, χ>0 is the vortex viscosity, κ>0 and γ>0 are the spin viscosities, 1/ν (with ν>0) is the magnetic Reynold; while u 0, b 0, ω 0 are the corresponding initial data with ∇⋅u 0=∇⋅b 0=0.

The global weak solution to system (1.1) is established by Rojas-Medar and Boldrini [10], while the local strong solutions are given by Rojas-Medar [9]. However, whether or not the local strong solutions can exist globally is still an open problem. Thus regularity criteria appears. Notice that:

  • (1) Yuan [14] first established the following fundamental regularity criterion in terms of the velocity or its gradient

    $$\begin{array}{@{}rcl@{}} {\kern-1.7pc}u\in L^{p}(0,T;L^{q}(\mathbb{R}^{3})),\ \ \ \frac{2}{p}+\frac{3}{q}=1, \ \ \ 3<q\leqslant \infty \end{array} $$
    (1.2)

    and

    $$\begin{array}{@{}rcl@{}} {\kern-1.7pc}\nabla u\in L^{p}(0,T;L^{q}(\mathbb{R}^{3})),\ \ \ \frac{2}{p}+\frac{3}{q}=2, \ \ \ \frac{3}{2}<q\leqslant \infty. \end{array} $$
    (1.3)

    Then Gala [4] extended it to the Morrey–Campanato spaces, Zhang et al [16] improved it to some more general Triebel–Lizorkin spaces.

  • (2) When ω=b=0, system (1.1) is just the classical Naiver–Stokes equations. Serrin [11], Prodi [8] and Beirão da Veiga [1] proved regularity if some scaling-invariant norm of u or ∇u is bounded.

  • (3) When ω=0, system (1.1) is then the 3D MHD equations, He and Xin [6], and Zhou [19] gave criteria similar to the case for Navier–Stokes equations.

  • (4) When b=0, system (1.1) is reduced to the micropolar fluid equations, Yuan [13] gave some criteria in Lorentz spaces.

For later developments, see [2; 3; 15; 18] and references cited therein. Recently, Vasseur [12] proved that if

$$\begin{array}{@{}rcl@{}} \text{div}\frac{u}{\left|u\right|}\in L^{p}(0,T;L^{q}(\mathbb{R}^{3})),\ \ \ \frac{2}{p}+\frac{3}{q}\leqslant \frac{1}{2},\ \ \ p\geqslant 4,\ \ \ q\geqslant 6, \end{array} $$
(1.4)

then the solutions to the Navier–Stokes equations are smooth. Later, Luo [7] extended (1.4) to the MHD equations, but involves the magnetic field also. We now extend the result of Vasseur [12] and Luo [7] to system (1.1). The main result is the following:

Theorem 1.1.

Let \(u_{0},\omega _{0},b_{0}\in H^{1}(\mathbb {R}^{3})\) with ∇⋅u 0 =∇⋅b 0 =0 in the sense of distributions. Suppose that (u,ω,b) is a strong solution to ( 1.1 ) in (0,T) such that

$$\begin{array}{@{}rcl@{}} u,\ \omega,\ b\in C((0,T);H^{1}(\mathbb{R}^{3}))\cap C((0,T);H^{2}(\mathbb{R}^{3})) \end{array} $$

and ∇⋅u=∇⋅b=0. If additionally,

$$\begin{array}{@{}rcl@{}} \text{div}\frac{u}{\left|u\right|}\in L^{p}(0,T;L^{q}(\mathbb{R}^{3})),\! \ \ \frac{2}{p}+\frac{3}{q}\leqslant \frac{1}{2},\! \ \ 4\leqslant p<\infty,\ \ \ 6\leqslant q\leqslant \infty\\ \end{array} $$
(1.5)

and

$$\begin{array}{@{}rcl@{}} b\in L^{r}(0,T;L^{s}(\mathbb{R}^{3})),\ \ \ \frac{2}{r}+\frac{3}{s}\leqslant 1,\ \ \ 2\leqslant r<\infty,\ \ \ 3\leqslant s\leqslant \infty, \end{array} $$
(1.6)

then the solution can be extended smoothly beyond t=T.

Remark 1.1.

Theorem 1.1 shows that it is enough to control the rate of change in the direction of the velocity and the norm of b to get full regularity of the solutions. Notice that we add no conditions on the micro-rotational velocity ω.

Remark 1.2.

Our theorem covers the results of Vasseur [12] and Luo [7] for Navier–Stokes and MHD equations, respectively. Observe that the condition (1.6) is a scaling-invariant, but (1.5) is not. Whether or not the 1/2 in (1.5) can increase to 1 is our future work.

Before giving a proof, let us first recall the definition of weak solutions to system (1.1).

DEFINITION 1.1

Let \(u_{0},\omega _{0},b_{0}\in L^{2}(\mathbb {R}^{3})\) with ∇⋅u 0=∇⋅b 0=0. A triple (u,ω,b) of measurable functions on \(\mathbb {R}^{3}\times (0,T)\) is said to be a weak solution of system (1.1) if

  • (1) \(u,\omega ,b\in L^{\infty }(0,T;L^{2})\cap L^{2}(0,T;H^{1})\) with ∇⋅u=∇⋅b=0;

  • (2) System (1.1) holds in the sense of distributions.

Remark 1.3.

Testing (1.1) 1, (1.1) 2, (1.1) 3 by u,ω,b respectively, after suitable integration by parts, one has the energy inequality:

$$\begin{array}{@{}rcl@{}} & &{}\left\|\left(u(t),\omega(t),b(t)\right)\right\|_{L^{2}}^{2} +2(\mu+\chi) {{\int}_{0}^{t}} \left\|\nabla u(s)\right\|_{L^{2}}^{2}\mathrm{d} s\\ & &{\kern.3pc}+2\gamma {{\int}_{0}^{t}} \left\|\nabla \omega(s)\right\|_{L^{2}}^{2}\mathrm{d} s +2\nu {{\int}_{0}^{t}} \left\|\nabla b(s)\right\|_{L^{2}}^{2}\mathrm{d}s\\ & & {\kern.3pc}+2{\chi{\int}_{0}^{t}} \left\|\omega(s)\right\|_{L^{2}}^{2}\mathrm{d}s \leqslant \left\|\left(u_{0},\omega_{0},b_{0}\right)\right\|_{L^{2}}^{2}. \end{array} $$
(1.7)

Throughout the proof in the next section, we shall frequently use the following interpolation inequality (see [17]):

$$\begin{array}{@{}rcl@{}} \left\|u\right\|_{p,q}\leqslant C \left\|u\right\|_{\infty,2}^{\frac{3}{q}-\frac{1}{2}} \left\|\nabla u\right\|_{2,2}^{\frac{3}{2}-\frac{3}{q}} \leqslant C\left(\left\|u\right\|_{\infty,2}+\left\|\nabla u\right\|_{2,2}\right), \end{array} $$
(1.8)

for (p,q) satisfying

$$\begin{array}{@{}rcl@{}} \frac{2}{p}+\frac{3}{q}\geqslant \frac{3}{2},\ \ \ 2\leqslant q\leqslant 6. \end{array} $$

In this paper, we shall use standard notations for Lebesgue space \(L^{q}(\mathbb {R}^{3})\) endowed with the norm ∥⋅∥ q , and anisotropic Lebesgue space \(L^{p}(I; L^{q}(\mathbb {R}^{3}))\) endowed with the norm ∥⋅∥ p,q . Here \(I\subset \mathbb {R}^{+}\) is an interval. A constant C (C=C(∗,∗,…) which depends on the parameters) may differ from line to line.

2 Proof of Theorem 1.1

By decreasing p or r if necessary, we may assume that

$$\begin{array}{@{}rcl@{}} \frac{2}{p}+\frac{3}{q}=\frac{1}{2},\ \ \ \frac{2}{r}+\frac{3}{s}=1. \end{array} $$

For an ε>0 to be chosen sufficiently small (see (2.12)), choose t 1∈(0,T) such that

$$\begin{array}{@{}rcl@{}} \left\|\text{div}\frac{u}{\left|u\right|}\right\|_{p,q}<\varepsilon \end{array} $$
(2.1)

and

$$\begin{array}{@{}rcl@{}} \left\|b\right\|_{r,s}<\varepsilon. \end{array} $$
(2.2)

Hereafter, the integrals are over \(\mathbb {R}^{3}\times (t_{1}, T )\).

Utilizing the regularity criteria (1.2), we complete the proof of Theorem 1.1 provided

$$\begin{array}{@{}rcl@{}} u\in L^{8}(t_{1}, T;L^{4}(\mathbb{R}^{3})). \end{array} $$
(2.3)

To this end, denote by

$$\begin{array}{@{}rcl@{}} I=\||u|^{2}\|_{\infty,2}+\|\nabla |u|^{2}\|_{2,2} + \| |b |^{2} \|_{\infty,2}+ \|\nabla |b |^{2}\|_{2,2}. \end{array} $$
(2.4)

Multiplying (1.1) 1,(1.1) 3 by \(\left |u\right |^{2}u, \left |b\right |^{2}b\) respectively and integrating over \(\mathbb {R}^{3}\times (t_{1}, T )\), we find that

$$\begin{array}{@{}rcl@{}} &&{}\frac{1}{4} \| | u |^{2} \|_{\infty,2}^{2} +\frac{\mu+\chi}{2} \|\nabla | u |^{2} \|_{2,2}^{2} +(\mu+\chi)\left\|\left|u\right|\left|\nabla u\right|\right\|_{2,2}^{2}\\ &&{\kern1pt}={\int}_{t_{1}}^ T {\int}_{\mathbb{R}^{3}} {}-b\cdot{} \nabla(\left|u\right|^{2}u)\cdot b {\kern-1.2pt}+{}({\kern-1.2pt}p{\kern-1.2pt}+{}\left|b\right|^{2}{})u\cdot{}\nabla \left|u\right|^{2} {}\\ &&{\kern12pt}+\,\chi \omega{}\cdot{}\nabla{}\times{}({}\left|u\right|^{2}u{})\mathrm{d}x\mathrm{d}t, \end{array} $$
(2.5)

as well as

$$\begin{array}{@{}rcl@{}} &&{}\frac{1}{4} \| |b |^{2} \|_{\infty,2}^{2} +\frac{\nu}{2} \|\nabla |b |^{2} \|_{2,2}^{2} +\nu\left\|\left|b\right|\left|\nabla b\right|\right\|_{2,2}^{2}\\ &&{\kern2pt}={\int}_{t_{1}}^ T {\int}_{\mathbb{R}^{3}} -b\cdot \nabla (\left|b\right|^{2}b)\cdot u \,\mathrm{d}x\mathrm{d}t. \end{array} $$
(2.6)

Using (2.5) and (2.6), notice that (see [12])

$$\begin{array}{@{}rcl@{}} \text{div}\frac{u}{\left|u\right|}=-\frac{u}{\left|u\right|^{2}}\nabla u \end{array} $$

and (see [13])

$$\begin{array}{@{}rcl@{}} \left|\nabla \right|{u}\|\leqslant \left|{\nabla u}\right|. \end{array} $$

Thus we have

$$\begin{array}{@{}rcl@{}} & &{\kern-1.8pc}I^{2}+\left\|\left|u\right|\left|\nabla u\right|\right\|_{2,2}^{2} +\left\|\left|b\right|\left|\nabla b\right|\right\|_{2,2}^{2}\\ &&{}\leqslant C{\int}_{t_{1}}^ T {\int}_{\mathbb{R}^{3}} \left|b\right|^{2}\left|u\right|\left(\left|u\right|\left|\nabla u\right|+\left|b\right|\left|\nabla b\right|\right) \mathrm{d} x\mathrm{d} t\\ & & {\kern4pt}+\,C{\int}_{t_{1}}^ T {\int}_{\mathbb{R}^{3}} (p+\left|b\right|^{2} )\left|u\right|^{3}\left|\text{div}\frac{u}{\left|{u}\right|}\right|\mathrm{d}x\mathrm{d} t\\ & & {\kern4pt}+\,C{\int}_{t_{1}}^ T {\int}_{\mathbb{R}^{3}}\left|\omega\right|\left|u\right|^{2}\left|\nabla u\right|\mathrm{d}x\mathrm{d} t \end{array} $$
(2.7)
$$\begin{array}{@{}rcl@{}} &\equiv&I_{1}+I_{2}+I_{3}. \end{array} $$
(2.8)

Here C is a constant depending only on μ,χ,ν.

Using Cauchy–Schwartz inequality, I 1 can be bounded as

$$\begin{array}{@{}rcl@{}} I_{1}&\leqslant& \frac{1}{4}\left\|\left|u\right|\left|\nabla u\right|\right\|_{2,2}^{2} +\frac{1}{2}\left\|\left|b\right|\left|\nabla b\right|\right\|_{2,2}^{2} +C \| |b |^{2} |u \|_{2,2}^{2}. \end{array} $$

By generalized Hölder inequality and (1.8), it follows that

$$\begin{array}{@{}rcl@{}} \| |b |^{2} | u | \|_{2,2}^{2} &\leqslant& \left\|b\right\|_{r,s}^{2}\left\|b\right\|_{a,b}^{2}\left\|u\right\|_{c,d}^{2}\\ &=& \left\|b\right\|_{r,s}^{2} \||b |^{2}\|_{\frac{a}{2},\frac{b}{2}} \||u|^{2}\|_{\frac{c}{2},\frac{d}{2}}\\ &\leqslant& C\varepsilon^{2} I^{2}, \end{array} $$

where

$$\begin{array}{@{}rcl@{}} \left\{ \begin{array}{lll} \displaystyle{\frac{1}{r}+\frac{1}{a}+\frac{1}{c}=\frac{1}{2} =\frac{1}{s}+\frac{1}{b}+\frac{1}{d},}\\ \displaystyle{\frac{2}{a/2}+\frac{3}{b/2}=\frac{3}{2},}\\ \displaystyle{\frac{2}{c/2}+\frac{3}{d/2}=\frac{3}{2},}\\ \end{array} \right. \end{array} $$

and we have used (1.8). In fact, we can choose

$$\begin{array}{@{}rcl@{}} a=c=\frac{4r}{r-2},\quad b=d=\frac{4s}{s-2}, \end{array} $$

where r,s are as in (1.6). Thus

$$\begin{array}{@{}rcl@{}} I_{1}\leqslant \frac{1}{4}\left\|\left|u\right|\left|\nabla u\right|\right\|_{2,2}^{2} +\frac{1}{2}\left\|\left|b\right|\left|\nabla b\right|\right\|_{2,2}^{2}+C\varepsilon I^{2}. \end{array} $$
(2.9)

For I 2, let us first take divergence of (1.1) 1 to see

$$\begin{array}{@{}rcl@{}} -\triangle p=\sum\limits_{i,j=1}^{3} \partial_{ij} (u_{i}u_{j}-b_{i}b_{j}+\delta_{ij}\left|b\right|^{2}), \end{array} $$

thus classical Calder\(\acute {\mathrm {o}}\)n–Zygmund estimates imply

$$\begin{array}{@{}rcl@{}} \left\|p\right\|_{a,b}\leqslant C(\| | u |^{2} \|_{a,b}+ \| |b |^{2} \|_{a,b}), \end{array} $$

invoking again the generalized Hölder inequality and (1.8),

$$\begin{array}{@{}rcl@{}} I_{2}&\leqslant& C \|p + | b |^{2} \|_{a,b} \| | u |^{3} \|_{c,d} \| \text{div}\frac{u}{\left|u\right|}\|_{p,q}\\ &\leqslant& C\varepsilon \left( \left\|u\right\|_{a_{1},b_{1}}\left\|u\right\|_{3c,3d}+\left\|b\right\|_{a_{1},b_{1}}\left\|b\right\|_{3c,3d} \right) \left\|u\right\|^{3}_{3c,3d}\\ &=& C\varepsilon \left\|u\right\|_{a_{1},b_{1}} \| | u |^{2} \|_{\frac{3c}{2},\frac{3d}{2}}^{2} +C\varepsilon \left\|b\right\|_{a_{1},b_{1}} \| | b |^{2} \|_{\frac{3c}{2},\frac{3d}{2}}^{\frac{1}{2}} \| | u |^{2} \|_{\frac{3c}{2},\frac{3d}{2}}^{\frac{3}{2}}\\ &\leqslant& C\varepsilon I^{2}, \end{array} $$
(2.10)

where

$$\begin{array}{@{}rcl@{}} \left\{ \begin{array}{lll} \displaystyle{ \frac{1}{a}+\frac{1}{c}+\frac{1}{p} =1= \frac{1}{b}+\frac{1}{d}+\frac{1}{q},}\\ \displaystyle{ \frac{1}{a}=\frac{1}{a_{1}}+\frac{1}{3c},\ \ \ \frac{1}{b}=\frac{1}{b_{1}}+\frac{1}{3d},}\\ \displaystyle{ \frac{2}{a_{1}}+\frac{3}{b_{1}}=\frac{3}{2},\ \ \ \frac{2}{3c/2}+\frac{3}{3d/2}=\frac{3}{2}.} \end{array} \right. \end{array} $$

In fact, we can choose

$$\begin{array}{@{}rcl@{}} a=c=\frac{2}{3}a_{1}=\frac{2p}{p-1},\quad b=d=\frac{2}{3}b_{1}=\frac{2q}{q-1}, \end{array} $$

where p,q are as in (1.5).

Finally, using (1.8), I 3 is treated as

$$\begin{array}{@{}rcl@{}} I_{3}&\leqslant& \frac{1}{4}\left\|\left|u\right|\left|\nabla u\right|\right\|_{2,2}^{2} +C\left\|\left|\omega\right|\left|u\right|\right\|_{2,2}^{2}\\ &\leqslant& \frac{1}{4}\left\|\left|u\right|\left|\nabla u\right|\right\|_{2,2}^{2} +C\left\|\omega\right\|_{2,4}^{2}\left\|u\right\|_{\infty,4}^{2}\\ &\leqslant& \frac{1}{4}\left\|\left|u\right|\left|\nabla u\right|\right\|_{2,2}^{2} +CI\\ &\leqslant& \frac{1}{4}\left\|\left|u\right|\left|\nabla u\right|\right\|_{2,2}^{2} +C\varepsilon I^{2}+C. \end{array} $$
(2.11)

Combining the estimates for I 1,I 2,I 3, i.e. (2.9), (2.10), (2.11), and substituting into (2.7), we find

$$\begin{array}{@{}rcl@{}} & &{}I^{2}+\frac{1}{2}\left\|\left|u\right|\left|\nabla u\right|\right\|_{2,2}^{2} +\frac{1}{2}\left\|\left|b\right|\left|\nabla b\right|\right\|_{2,2}^{2}\leqslant 3C\varepsilon I^{2}+C, \end{array} $$

where C is the generic constant appearing in (2.9), (2.10) and (2.11). Thus, we see that

$$\begin{array}{@{}rcl@{}} I\leqslant \sqrt{2C}<\infty, \end{array} $$

provided

$$\begin{array}{@{}rcl@{}} \varepsilon=\frac{1}{6C}. \end{array} $$
(2.12)

Consequently, by (2.4), we have

$$\begin{array}{@{}rcl@{}} u\in L^{\infty}(t_{1}, T , L^{4}(\mathbb{R}^{3}))\subset L^{8}(t_{1}, T ,L^{4}(\mathbb{R}^{3})). \end{array} $$

The proof is completed.