A remark on regularity of liquid crystal equations in critical Lorentz spaces

Abstract

The regularity for the 3-D nematic liquid crystal equations is considered in this paper, it is proved that the Leray–Hopf weak solutions (u, d) is in fact smooth, if the velocity field $u\in L^\infty (0,T;L^{3,\infty }_x(\mathbb {R}^3))$ satisfies some addition local small condition

$$\begin{aligned} r^{-3}\left| \left\{ x\in B_r(x_0): |u(x,t_0)|>\varepsilon r^{-1}\right\} \right| \le \varepsilon , \end{aligned}$$

which is inspired by the papers [2, 35].

1 Introduction

The three-dimensional incompressible liquid crystals system are the following coupled equations

$$\begin{aligned} \left\{ \begin{array}{l} {u_t-\Delta u+u\cdot \nabla u +\nabla p=-\mathrm {div}(\nabla d \odot \nabla d),}\\ {\mathrm {div}\ u=0,}\\ {d_t+u\cdot \nabla d-\Delta d=-f(d),}\\ \end{array} \right. \end{aligned}$$

(1.1)

in the domain $Q_T\equiv \mathbb {R}^3\times (0, T)$. Here, the unknowns $u=(u_1, u_2, u_3)$ is the velocity field, p is the scalar pressure and $d=(d_1, d_2, d_3)$ is the optical molecule direction after penalization, and $f(d)=\frac{1}{\sigma ^2}(|d|^2-1)d$, $\nabla d \odot \nabla d$ is a symmetric tensor with its component $(\nabla d \odot \nabla d)_{ij}$ is given by $\partial _i d \cdot \partial _j d$. And the initial conditions are

$$\begin{aligned} u(x,0)=u_0(x), \quad \mathrm {div}(u_0)=0, \quad d(x,0)=d_0(x), \end{aligned}$$

(1.2)

with $|d_0|=1.$

System (1.1) is the simplified system of the original Ericksen–Leslie system of variable length for the flow of liquid crystals, that is the Ginzburg–Landau energy $\int _{\Omega } (\frac{1}{2} |\nabla d |^2 +\frac{(1-|d|^2)^2}{4\sigma ^2})$. For this system, Lin and Liu [19,20,21] first proved a global existence of weak solutions under $L^2$ data and regularity result of the suitable weak solution under the C-K-N condition. Other results to liquid crystals equations refer to [8,9,10,11, 15, 16, 18, 22, 23, 28].

Let us now recall the notion of a suitable weak solution of liquid crystals equations.

Definition 1.1

([20]) a triple (u, d, p) is called a suitable weak solution of (1.1) in $\mathbb {R}^3\times (0,T)$ if the following conditions hold:

• (1). the weak solution (u, d) satisfies system (1.1) in the distribution sense;

• (2). the solution (u, d) satisfy the energy inequality, i.e.,

  $$\begin{aligned}&\Vert u\Vert _{L^{2,\infty }(\mathbb {R}^3\times (0,T))}^2+\Vert \nabla d\Vert _{L^{2,\infty }(\mathbb {R}^3\times (0,T))}^2\\&\quad +\Vert \nabla ^2d\Vert _{L^2(\mathbb {R}^3\times (0,T)))}^2 +\Vert \nabla u\Vert _{L^2(\mathbb {R}^3\times (0,T)))}^2\le c_0; \end{aligned}$$

• (3). the press $p\in L^{\frac{3}{2}}_{loc}(\mathbb {R}^3\times (0,T))$;

• (4). the triple (u, d, p) satisfy the modified generalized local energy inequality, for a.e. $t\in (0,T)$ and for all $\phi \in C_{0}^{\infty }(\mathbb {R}^3\times (0,T))$ with $\phi \ge 0,$

  $$\begin{aligned}&\int _{\mathbb {R}^3\times \{t\}}(|u|^2+|\nabla d|^2)\phi \mathrm{d}x +2\int _{0}^t\int _{\mathbb {R}^3} (|\nabla u|^2+|\nabla ^2 d|^2)\phi \\&\quad \le \int _{0}^t\int _{\mathbb {R}^3} (|u|^2+|\nabla d|^2)(\phi _t+\Delta \phi )\\&\qquad +\int _0^t\int _{\mathbb {R}^3} (|u|^2+|\nabla d|^2+2p)(u\cdot \nabla \phi )\\&\qquad +2\int _0^t\int _{\mathbb {R}^3} ((u\cdot \nabla ) d\odot \nabla d)\cdot \nabla \phi -2\int _0^t\int _{\mathbb {R}^3}\nabla _x f(d)\cdot \nabla \mathrm{d} \phi . \end{aligned}$$

Denote for $z=(x,t)\in \mathbb {R}^{3}\times \mathbb {R}_{+}$ the standard notations

$$\begin{aligned} B_{r}(x)=\{y\in \mathbb {R}^{3}:|y-x|<r\}, \quad Q_{r}(z)=B_{r}(x)\times (t-r^{2},t), \end{aligned}$$

to be the Euclidean ball and parabolic cylinder. For $z=(0,0)$, we simply write them as $B_{r}$ and $Q_{r}$.

Now, we mention some relevant results on regularity. For 3D Navier–Stokes system, the study of partial regularity was originated by Scheffer in a series of papers [31, 32] and [33]. The notion of suitable weak solutions was introduced in a celebrated paper [1] by Caffarelli, Kohn and Nirenberg. It is proved that, for any suitable weak solution (u, p),  there is an open subset in which the velocity field u is Hölder continuous, and the complement of it has zero 1-D Hausdorff measure. Latter Lin [17] gave a simpler proof for the CKN theorem, Lin’s method was used extensively by Seregin’s papers for example in [3, 12]. These results are based on some regularity criterions when certain dimensionless quantities are small. For liquid crystal system (1.1), Lin and Liu[20] proved the following theorem.

Theorem(A). Let the triple (u, d, p) be a suitable weak solution to system (1.1). There exist a small constant $\epsilon _{0}>0$, such that if

$$\begin{aligned} \int _{Q_{1}}|u|^{3}+|\nabla d|^{3}+|p|^{\frac{3}{2}}<\epsilon _{0}, \end{aligned}$$

then u and d are smooth in $\overline{Q_{\frac{1}{2}}}$. In particular, for any $z_0$ if

$$\begin{aligned} \frac{1}{r^2}\int _{Q_r(z_0)}|u|^3+|\nabla d|^3<\epsilon _0, \quad \text {for all}\,\, 0<r\le 1, \end{aligned}$$

then $z_0$ is a regular point.

We can drop the pressure p by Wolf’ method of local suitable weak solutions, the proof to see Sect. 5.

Proposition 1.2

Let the triple (u, d, p) be a suitable weak solution to system (1.1). There exist a small constant $\varepsilon _{0}>0$, such that if

$$\begin{aligned} \int _{Q_{1}}|u|^{3}+|\nabla d|^{3}<\varepsilon _{0}, \end{aligned}$$

then u and d are smooth in $\overline{Q_{\frac{1}{2}}}$.

There is another type of regularity criterion called the Ladyzhenskaya–Prodi–Serrin condition (to see [13, 30, 36]). For system (1.1), [24] showed that if

$$\begin{aligned} ||u||_{L^{p,q}(Q)}=\left(\int ^{0}_{-1}\left(\int _{B}|u|^{p}\right)^{\frac{q}{p}}\right)^{\frac{1}{q}}<\infty , \end{aligned}$$

where $\frac{3}{p}+\frac{2}{q}=1$ and $(p,q)\ne (3,\infty )$, then the weak solution (u, d) is regular in $Q=B\times (-1,0)$, Serrin’s method [36] and Struwe’s method [38] dealing with Navier–Stokes equations are applied. For the Borderline case where $(p,q)=(3,\infty )$, [28] showed that the weak solution (u, d) is smooth in $\mathbb {R}^{3}\times (0,T]$ when $u\in L^{\infty }(0,T;L^{3}(\mathbb {R}^{3}))$(also to see [25]). The $(3,\infty )$ case requires a technique utilizing the backward uniqueness of heat operator and unique continuation through spatial boundary, which was used to deal with the Navier–Stokes equations in [3]. For general Lorenz space $L^{3,q}(\mathbb {R}^3), 3<q<\infty$ case, we prove in [26] if $u\in L^\infty (0,T; L^{3,q}(\mathbb {R}^3))$ then (u, d) is smooth. For N-S equations we refer to [29]. For $L^{3,\infty }(\mathbb {R}^3)$ case, Choe, Wolf and Yang [2] prove that if $u\in L^\infty (0,T; L^{3,\infty }(\mathbb {R}^3))$ and an additional local small condition

$$\begin{aligned} \frac{1}{r^3}|\{x\in B_r(x_0)|\,|u(\cdot ,t_0)|>\frac{\epsilon }{r}\}|\le \epsilon , \end{aligned}$$

(1.3)

then weak solution u of Navier–Stokes equations is smooth on $Q_{\epsilon r}(z_0)$. Also to see Seregin [35].

In this paper, we shall established the regularity of weak solutions of liquid crystals equations in Lorentz space $L^{3,\infty }$, the technicality is different from Navier–Stokes equations, the main technique is to deal with the new norms. Our main results can be stated as following.

Theorem 1.3

Assume (u, d, p) is a weak solution of (1.1) with $u\in L^\infty (0,T; L^{3,\infty }(\mathbb {R}^n))$. There exists a positive constant $\varepsilon$ small with following property. If $z_0=(x_0,t_0)\in Q_T$ and $R>0$ such that $Q_R(z_0)\subset Q_T$ and (d, p) satisfy

$$\begin{aligned} C(\nabla d,R, z_0)+D(p, R, z_0)\le N, \end{aligned}$$

for some $0< r\le R/2,$

$$\begin{aligned} r^{-3}\left| \left\{ x\in B_r(x_0): |u(x,t_0)|>\varepsilon r^{-1}\right\} \right| \le \varepsilon . \end{aligned}$$

Then (u, d) is smooth in $Q_{\varepsilon r}(z_0).$ Here $C(\nabla d,R,z_0)$ and $D(p, R,z_0)$ are dimensionless quantities in Sect. 2.

Theorem 1.4

Let (u, d) be a suitable weak solution to the liquid crystal equations in $Q_T$ with $u\in L^\infty (0,T; L^{3,\infty }(\mathbb {R}^n))$. Then there exist at most finite number $\mathcal {N}$ of singular points at any singular time t.

2 Notations and preliminaries

Let us recall the scaling property of (1.1). Denote

$$\begin{aligned} u_{\lambda }(x,t)=\lambda u(\lambda x,\lambda ^{2}t),\quad p_{\lambda }(x,t)=\lambda ^{2} p(\lambda x,\lambda ^{2}t),\quad d_{\lambda }(x,t)=d(\lambda x,\lambda ^{2}t). \end{aligned}$$

If (u, p, d) is a solution in $\mathbb {R}^{3}\times (0,T)$, then obviously $(u_{\lambda },p_{\lambda },d_{\lambda })$ is a solution to the following equations

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{\lambda t}-\Delta u_{\lambda }+u_{\lambda }\cdot \nabla u_{\lambda }+\nabla p_{\lambda }=-\nabla \cdot (\nabla d_{\lambda }\odot \nabla d_{\lambda }) &{}\mathrm{in}\quad \mathbb {R}^{3}\times (0,\lambda ^{2}T), \\ \nabla \cdot u_{\lambda }=0 &{}\mathrm{in}\quad \mathbb {R}^{3}\times (0,\lambda ^{2}T), \\ d_{\lambda t}-\Delta d_{\lambda }+u_{\lambda }\cdot \nabla d_{\lambda }=-{\lambda }^{2}f(d_{\lambda }) &{}\mathrm{in}\quad \mathbb {R}^{3}\times (0,\lambda ^{2}T). \end{array}\right. } \end{aligned}$$

Thus, the scaling dimension of corresponding quantities are $\dim u=-1$, $\dim p=-2$, and $\dim d=0$ (we assign x with dimension 1 and t with 2). There are some useful dimensionless quantities and we list them here, let $z_0=(x_0,t_0),$

$$\begin{aligned} A(u,r,z_0)=\, & {} \sup _{t_0-r^{2}<t<t_0}r^{-1}\int _{B_{r}(x_0)}|u|^{2}\mathrm{d}x, \quad E(u,r, z_0)=r^{-1}\int _{Q_{r}(z_0)}|\nabla u|^{2}\mathrm{d}x\mathrm{d}t,\\ C(u,r, z_0)=\, & {} r^{-\frac{16}{7}}\int _{t_0-r^2}^{t_0}\Vert u\Vert ^4_{L^{\frac{14}{5}}(B_r(x_0))}\mathrm{d}t, \quad K(u,r, z_0)=r^{-3}\int _{Q_{r}(z_0)}|u|^{2}\mathrm{d}x\mathrm{d}t,\\ C_1(u,r, z_0)=\, & {} r^{-2}\int _{Q_{r}(z_0)}| u|^{3}\mathrm{d}x\mathrm{d}t,\quad D_1(p, r, z_0)=r^{-2}\int _{Q_{r}(z_0)}|p|^{\frac{3}{2}}\mathrm{d}x\mathrm{d}t,\\ D(p,z_0,r)=\, & {} r^{-\frac{16}{7}}\int _{t_0-r^2}^{t_0}\Vert p\Vert ^{2}_{L^{\frac{7}{5}}(B_r(x_0))}\mathrm{d}t. \end{aligned}$$

Similarly, we denote these notations for $\nabla d$:

$$\begin{aligned} A(\nabla d,r, z_0)=\, & {} \sup _{t_0-r^{2}<t<t_0}r^{-1}\int _{B_{r}(x_0)}|\nabla d|^{2}\mathrm{d}x, \quad E(\nabla d,r, z_0)=r^{-1}\int _{Q_{r}(z_0)}|\nabla ^{2} d|^{2}\mathrm{d}x\mathrm{d}t,\\ C(\nabla d,r, z_0)=\, & {} r^{-\frac{16}{7}}\int _{t_0-r^2}^{t_0}\Vert \nabla d\Vert ^{4}_{L^{\frac{14}{5}}(B_r(x_0))}\mathrm{d}t, \quad K(\nabla d,r,z_0)=r^{-3}\int _{Q_{r}(z_0)}|\nabla d|^{2}\mathrm{d}x\mathrm{d}t,\\ C_1(\nabla d,r, z_0)=\, & {} r^{-2}\int _{Q_{r}(z_0)}| \nabla d|^{3}\mathrm{d}x\mathrm{d}t. \end{aligned}$$

For simplicity when $z_0=(0,0)$, we write $A(u,r)=A(u,r,(0,0))$ , and write $A(r)\equiv A(u,\nabla d,r)=A(u,r)+A(\nabla d,r)$, and the meaning of E(r), C(r), K(r) are alike.

Next, we write down several facts about Lorentz spaces. We say a locally integrable function $f\in L^{p,q}(\Omega )$, if the quasi-norm below is bounded

$$\begin{aligned} ||f||_{L^{p,q}(\Omega )}&=\left( p\int _{0}^{\infty }\alpha ^{q}d_{f,\Omega }(\alpha )^{\frac{q}{p}}\frac{d\alpha }{\alpha }\right) ^{\frac{1}{q}}, \quad q<\infty ,\nonumber \\ ||f||_{L^{p,\infty }(\Omega )}&=\sup _{\alpha >0}\alpha d_{f,\Omega }(\alpha )^{\frac{1}{p}}, \quad q=\infty , \end{aligned}$$

(2.1)

where

$$\begin{aligned} d_{f,\Omega }(\alpha )=|\{x\in \Omega :|f(x)|>\alpha \}|. \end{aligned}$$

A basic fact for such spaces is

$$\begin{aligned} L^{p,q_{1}}\subset L^{p,p}=L^{p}\subset L^{p,q_{2}}\subset L^{p,\infty }=L^{p}_{w}, \end{aligned}$$

(2.2)

where $0<q_{1}<p<q_{2}<\infty$, and $L^{p}_{w}$ is the weak-$L^{p}$ space. If $|\Omega |$ is finite then $L^{p,q}(\Omega )\subset L^r(\Omega )$ for all $0<q\le \infty$ and $0<r<p$,

$$\begin{aligned} \Vert g\Vert _{L^r(\Omega )}\le |\Omega |^{\frac{1}{r}-\frac{1}{p}}\Vert g\Vert _{L^{p,q}(\Omega )}. \end{aligned}$$

(2.3)

Lemma 2.1

Let (u, d, p) be a weak solution to the liquid crystal Eq. (1.1) in $Q=\Omega \times (a,b)$. Let $z_0=(x_0,t_0)$ and let $\rho >0$ be such that $Q_\rho (z_0)\subset Q.$ For every $r<(0,\frac{\rho }{4}]$, we have

$$\begin{aligned} D_1(p,r,z_0)\le &\, {} c\left( \frac{\rho }{r}\right) ^\frac{3}{2}\left[ A(u,\rho , z_0)^\frac{3}{4}E(u,\rho ,z_0)^\frac{3}{4}+A(\nabla d, \rho , z_0)^\frac{3}{4}E(\nabla d, \rho , z_0)^\frac{3}{4}\right] \nonumber \\&\,+c\frac{r}{\rho }D(p,z_0,\rho )^\frac{3}{4}, \end{aligned}$$

(2.4)

$$\begin{aligned} D(p, r, z_0)\le\, & {} c\,\left[ \left( \frac{r}{\rho }\right) ^2D(p,\rho , z_0)+\left( \frac{\rho }{r}\right) ^{\frac{16}{7}} C(u,\nabla d, \rho , z_0)\right] . \end{aligned}$$

(2.5)

Proof

Let $z_0=(0,0)$, we decompose p so that

$$\begin{aligned} p=p_1+p_2, \end{aligned}$$

where $p_1$ satisfies in $B_\rho$ for a.e. $t\in [-\rho ^{2},0]$, in the weak sense,

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta p_1=-\mathrm {div\,div}\,(u\otimes u-[u\otimes u]_{B_\rho }))-\mathrm {div\,div}\,(\nabla d \otimes \nabla d -[\nabla d\otimes \nabla d]_{B_\rho })\\ p_1|_{\partial B_\rho }=0. \end{array}\right. } \end{aligned}$$

(2.6)

And $p_2$ is a harmonic function in $B_\rho$, i.e.,

$$\begin{aligned} \Delta p_2=0. \end{aligned}$$

Regarding $p_1$, by theory of Laplace operator and Calderón–Zygmund theorem, we have

$$\begin{aligned}&\left( \int _{B_\rho }|p_1|^{\frac{3}{2}}dx\right) ^{\frac{2}{3}}\\&\quad \le c\left( \int _{B_\rho }|u\otimes u-[u\otimes u]_{B_\rho }|^{\frac{3}{2}}+|\nabla d\otimes \nabla d-[\nabla d\otimes \nabla d]_{B_\rho }|^{\frac{3}{2}})dx\right) ^{\frac{2}{3}}\\&\quad \le c\int _{B_\rho }|\nabla u||u|+|\nabla ^2 d||\nabla d|\\&\quad \le c\left[ \Vert \nabla u\Vert _{L^2(B_\rho )}\Vert u\Vert _{L^2(B_\rho )}+\Vert \nabla ^2 d\Vert _{L^2(B_\rho )}\Vert \nabla d\Vert _{L^2(B_\rho )}\right] ,\\&\int _{Q_\rho }|p_1|^{\frac{3}{2}}\le c\rho ^2\left[ A^{\frac{3}{4}}(u, \rho )E^{\frac{3}{4}}(u,\rho )+ A^{\frac{3}{4}}(\nabla d, \rho )E^{\frac{3}{4}}(\nabla d,\rho )\right] . \end{aligned}$$

For $x\in B_{\frac{\rho }{2}}$,

$$\begin{aligned} |p_2(x,t)|\le c -\int _{B_\rho }|p_2|\le c \left( -\int _{B_\rho }|p_2|^l\right) ^{\frac{1}{l}}, \quad l>1, \end{aligned}$$

i.e., for $r\le \frac{\rho }{2},$

$$\begin{aligned} \frac{1}{r^2}\int _{Q_r}|p_2|^{\frac{3}{2}}\le\, & {} \frac{c\, r}{\rho ^{\frac{45}{14}}}\int _{-\rho ^2}^0\Vert p_2\Vert _{L^{\frac{7}{5}}(B_\rho )}^{3/2}.\\ D_1(p, r)\le &\,\, {} \frac{1}{r^2}\int _{Q_\rho }|p_1|^{\frac{3}{2}}+\frac{1}{r^2}\int _{Q_r}|p_2|^{\frac{3}{2}}\\\le\, & {} c(\frac{\rho }{r})^2\left[ A^{\frac{3}{4}}(u, \rho )E^{\frac{3}{4}}(u,\rho )+ A^{\frac{3}{4}}(B, \rho )E^{\frac{3}{4}}(\nabla d,\rho )\right] \\&+\,\frac{c\, r}{\rho ^{\frac{45}{14}}}\int _{-\rho ^2}^0\left( \Vert p\Vert _{L^{\frac{7}{5}}(B_\rho )}^{3/2}+ \Vert p_1\Vert _{L^{\frac{7}{5}}(B_\rho )}^{3/2}\right) . \end{aligned}$$

Now,

$$\begin{aligned}&\frac{r}{\rho ^{\frac{45}{14}}}\int _{-\rho ^2}^0\Vert p_1\Vert _{L^{\frac{7}{5}}(B_\rho )}^{3/2}\le \frac{r}{\rho }D_1(p_1, \rho ),\\&\frac{ r}{\rho ^{\frac{45}{14}}}\int _{-\rho ^2}^0\Vert p\Vert _{L^{\frac{7}{5}}(B_\rho )}^{3/2}\le \frac{r}{\rho }D(p,\rho )^{\frac{3}{4}}, \end{aligned}$$

so that

$$\begin{aligned} D_1(p, r)\le c\left(\frac{\rho }{r}\right)^2\left[ A^{\frac{3}{4}}(u, \rho )E^{\frac{3}{4}}(u,\rho )+ A^{\frac{3}{4}}(\nabla d, \rho )E^{\frac{3}{4}}(\nabla d,\rho )\right] +c\frac{r}{\rho }D(p,\rho )^{\frac{3}{4}}. \end{aligned}$$

On the other hand

$$\begin{aligned} \int _{B_r}|p_1|^{\frac{7}{5}}\le c\int _{B_r}|u|^{\frac{14}{5}}+|\nabla d|^{\frac{14}{5}}, \end{aligned}$$

we have

$$\begin{aligned} D(p_1, r)\le c\,\left( \frac{\rho }{r}\right) ^{\frac{16}{7}} C(u, \nabla d, \rho ). \end{aligned}$$

For $x\in B_{\frac{\rho }{2}}$,

$$\begin{aligned} |p_2(x,t)|\le c\left( -\int _{B_\rho }|p_2|^{\frac{7}{5}}dx\right) ^{\frac{5}{7}}, \end{aligned}$$

we have

$$\begin{aligned} D(p_2,r)=\, & {} r^{-\frac{16}{7}}\int _{-r^2}^0\Vert p_2\Vert _{L^{\frac{7}{5}}(B_r)}^2\le c\left( \frac{r}{\rho }\right) ^2D(p_2,\rho )\\\le &\, {} c\left( \frac{r}{\rho }\right) ^2\left[ D(p,\rho )+D(p_1,\rho )\right] , \end{aligned}$$

therefore,

$$\begin{aligned} D(p, r)\le\, & {} c\left[ D(p_1, r)+D(p_2,r)\right] \\\le\, & {} c\,\left( \frac{\rho }{r}\right) ^{\frac{16}{7}} C(u,\nabla d, \rho )+ c\,D(p_2,r)\\\le \,& {} c\,\left[ \left( \frac{r}{\rho }\right) ^2D(p, \rho )+\left( \frac{\rho }{r}\right) ^{\frac{16}{7}} C(u,\nabla d, \rho )\right] . \end{aligned}$$

$\square$

We use following analysis Lemma 2.2 which can be found in ([6],  Lemma 6.1) to prove local estimate Lemma 2.3.

Lemma 2.2

Let I(s) be a bounded nonnegative function in the interval $[R_1,R_2].$ Suppose that for any s,  $\rho \in [R_1,R_2]$ and $s<\rho$, the following yields

$$\begin{aligned} I(s)\le \left[ a_1(\rho -s)^{-\alpha }+a_2(\rho -s)^{-\beta }+a_3(\rho -s)^{-\gamma }+ a_4\right] +\theta I(\rho ), \end{aligned}$$

with $\alpha>\beta>\gamma >0$, $a_i>0 , i=1,2,3, 4$ and $\theta \in [0,1)$. Then,

$$\begin{aligned} I(R_1)\le c(\alpha ,\beta ,\gamma )[a_1(R_2-R_1)^{-\alpha }+a_2(R_2-R_1)^{-\beta }+a_3(R_2-R_1)^{-\gamma }+a_4]. \end{aligned}$$

Lemma 2.3

Let (u, d, p) be a suitable weak solution to the liquid crystal Eq. (1.1) in $Q=\Omega \times (a,b).$ Assume that $z_0=(x_0,t_0)$ and $1\ge r>0$ with $Q_r(z_0)\subset Q$. Then the following holds:

$$\begin{aligned}&A(u,\nabla d, r/2, z_0)+E(u,\nabla d, r/2, z_0)\nonumber \\&\quad \le c\left[ C(u, \nabla d, r, z_0)^{\frac{1}{2}}+C(u, r, z_0)^{\frac{1}{2}}C( \nabla d, r, z_0)^{\frac{1}{2}}+C(u,r,z_0)\right. \nonumber \\&\qquad +\left. D(p, r, z_0)^\frac{7}{10}C(u, r, z_0)^\frac{3}{20}\right] . \end{aligned}$$

(2.7)

Proof

Let $r/2\le s<\rho \le r<1,$ and $Q_r\subset Q_1\equiv Q.$ Choosing test function $\phi (x,t)=\eta _1(x)\eta _2(t)$ with $\eta _1\in C^\infty _0(B_\rho (x_0)),$ $0\le \eta _1\le 1$ in $\mathbb {R}^3,$ $\eta _1\equiv 1$ on $B_s(x_0),$ and $|\nabla ^\alpha \eta _1|\le \frac{C}{(\rho -s)^{|\alpha |}},$ for all multi-index $\alpha ,$ with $|\alpha |\le 3.$ And $\eta _2\in C^\infty _0(t_0-\rho ^2,t_0+\rho ^2),$ $0\le \eta _2\le 1$ in $\mathbb {R},$ $\eta _2(t)\equiv 1$ for $t\in [t_0-s^2,t_0+s^2],$ with $|\eta _2'(t)|\le \frac{C}{\rho ^2-s^2}\le \frac{C}{r(\rho -s)}.$ From the local energy inequality we have

$$\begin{aligned}&\int _{\Omega }(|u|^2+|\nabla d|^2)\phi \mathrm{d}x+2\int _{a}^{t}\int _{\Omega }(|\nabla u|^2+|\nabla ^2 d|^2)\phi \mathrm{d}x\mathrm{d}s\nonumber \\&\quad \le c\int _{t_0-\rho ^2}^{t_0}\Vert |u|^2+|\nabla d|^2\Vert _{W^{-1,2}(B_\rho (x_0))}\Vert \nabla (\phi _t+\Delta \phi )\Vert _{L^2(B_\rho (x_0))}\mathrm{d}t \nonumber \\&\qquad +c\int _{t_0-\rho ^2}^{t_0}\Vert |u|^2\Vert _{W^{-1,2}(B_\rho (x_0))}\Vert \nabla (u\cdot \nabla \phi )\Vert _{L^2(B_\rho (x_0))}\mathrm{d}t\nonumber \\&\qquad +c\int _{t_0-\rho ^2}^{t_0}\Vert |u||\nabla d|\Vert _{W^{-1,2}(B_\rho (x_0))}\Vert |\nabla ^2d||\nabla \phi |+|\nabla d||\nabla ^2\phi |\Vert _{L^2(B_\rho (x_0))}\mathrm{d}t\nonumber \\&\qquad +c\int _{t_0-\rho ^2}^{t_0}\int _{B_\rho }\,|p\,u\cdot \nabla \phi | \mathrm{d}x\mathrm{d}t + c\int _{t_0-\rho ^2}^{t_0}\int _{B_\rho }\,|\nabla d|^2\phi \nonumber \\&\quad =J_1+J_2+J_3+J_4+J_5. \end{aligned}$$

(2.8)

Here, we rewrite the term

$$\begin{aligned} \int _{t_0-\rho ^2}^{t_0}\int _{B_\rho }|\nabla d|^2 \, u\cdot \nabla \phi \mathrm{d}x\mathrm{d}t= \int _{t_0-\rho ^2}^{t_0}\int _{B_\rho }(u\otimes \nabla d):(\nabla \phi \otimes \nabla d) \mathrm{d}x\mathrm{d}t. \end{aligned}$$

Denote

$$\begin{aligned} I(s)= & {} \sup _{t_0-s^2\le t\le t_0}\int _{B_s(x_0)}|\nabla d|^2\mathrm{d}x+\sup _{t_0-s^2\le t\le t_0}\int _{B_s(x_0)}|u|^2\mathrm{d}x\\&+\int _{t_0-s^2}^{t_0}\int _{B_s(x_0)}|\nabla ^2 d|^2\mathrm{d}x\mathrm{d}t+\int _{t_0-s^2}^{t_0}\int _{B_s(x_0)}|\nabla u|^2\mathrm{d}x\mathrm{d}t\\=\, & {} sA(\nabla d,s,z_0)+sA(u,s,z_0)+sE(\nabla d,s,z_0)+sE(u,s,z_0)\\=\, & {} I_1(\nabla d,s)+I_1(u,s)+I_2(\nabla d,s)+I_2(u,s), \end{aligned}$$

and

$$\begin{aligned} I(u,s)=I_1(u,s)+I_2(u,s), \quad I(\nabla d,s)=I_1(\nabla d,s)+I_2(\nabla d,s). \end{aligned}$$

Estimate $J_1,$ $J_2,$ $J_3, J_4$ and $J_5$, respectively, as the following:

$$\begin{aligned} J_1\le\, & {} \frac{c\rho ^{3/2}}{(\rho -s)^3}\int _{t_0-\rho ^2}^{t_0}\Vert |u|^2+|\nabla d|^2\Vert _{W^{-1,2}(B_\rho )}\mathrm{d}t\\\le\, & {} \frac{c\rho ^{\frac{5}{2}}}{(\rho -s)^3}\left[ \int _{t_0-\rho ^2}^{t_0}\Vert |u|^2+|\nabla d|^2\Vert _{W^{-1,2}(B_\rho )}^2\mathrm{d}t\right] ^{\frac{1}{2}}, \end{aligned}$$

by Young’s inequality, we get

$$\begin{aligned} J_2\le \,& {} c\int _{t_0-\rho ^2}^{t_0}\left[ \Vert |u|^2\Vert _{W^{-1,2}(B_\rho )}\left( \frac{\Vert \nabla u\Vert _{L^2(B_\rho )}}{\rho -s}+\frac{\Vert u\Vert _{L^2(B_\rho )}}{(\rho -s)^2}\right) \right] \mathrm{d}t\\\le\, & {} \frac{c}{\rho -s}\left[ \int _{t_0-\rho ^2}^{t_0}\Vert |u|^2\Vert ^2_{W^{-1,2}(B_\rho )}\mathrm{d}t\right] ^\frac{1}{2}I_2(v,\rho )^{\frac{1}{2}}\\&+\frac{c\rho }{(\rho -s)^2}\left[ \int _{t_0-\rho ^2}^{t_0}\Vert |u|^2\Vert ^2_{W^{-1,2}(B_\rho )}\mathrm{d}t\right] ^\frac{1}{2}I_1(u,\rho )^{\frac{1}{2}}\\\le\, & {} \frac{1}{4}I(u,\rho )+\left[ \frac{c}{(\rho -s)^2}+\frac{c\rho ^2}{(\rho -s)^4}\right] \int _{t_0-\rho ^2}^{t_0}\Vert |u|^2\Vert ^2_{W^{-1,2}(B_\rho )}\mathrm{d}t, \end{aligned}$$

similarly, we have

$$\begin{aligned} J_3\le\, & {} c\int _{t_0-\rho ^2}^{t_0}\left[ \Vert |u||\nabla d|\Vert _{W^{-1,2}(B_\rho )}\left( \frac{\Vert \nabla ^2 d\Vert _{L^2(B_\rho )}}{\rho -s}+\frac{\Vert \nabla d\Vert _{L^2(B_\rho )}}{(\rho -s)^2}\right) \right] \mathrm{d}t\\\le\, & {} \frac{1}{4}I(\nabla d,\rho )+\left[ \frac{c}{(\rho -s)^2}+\frac{c\rho ^2}{(\rho -s)^4}\right] \int _{t_0-\rho ^2}^{t_0}\Vert |u\nabla d|\Vert ^2_{W^{-1,2}(B_\rho )}\mathrm{d}t. \end{aligned}$$

For the term $J_4$, using Hölder’s inequality and Sobolev inequality, we have

$$\begin{aligned} J_4\le &\, {} c\int _{t_0-\rho ^2}^{t_0}\Vert p\Vert _{L^{\frac{7}{5}}(B_\rho (x_0))}\Vert u\nabla \phi \Vert _{L^{\frac{7}{2}}(B_\rho (x_0))}\\\le\, & {} c\int _{t_0-\rho ^2}^{t_0}\Vert p\Vert _{L^{\frac{7}{5}}(B_\rho )}\Vert \nabla (u\nabla \phi )\Vert _{L^2(B_\rho )}^{\frac{4}{7}}\Vert u\nabla \phi \Vert _{L^{\frac{9}{4}}(B_\rho )}^{\frac{3}{7}}\\\le\, & {} c\int _{t_0-\rho ^2}^{t_0}\Vert p\Vert _{L^{\frac{7}{5}}(B_\rho )}\left[ \Vert \nabla u\nabla \phi )\Vert _{L^2(B_\rho )}+\Vert u\nabla ^2\phi \Vert _{L^2(B_\rho )}\right] ^{\frac{4}{7}}\Vert u\nabla \phi \Vert _{L^{\frac{9}{4}}(B_\rho )}^{\frac{3}{7}} \\\le\, & {} \frac{c}{\rho -s}\Vert \nabla u\Vert _{L^2(Q_\rho (z_0))}^{\frac{4}{7}}\left[ \int _{t_0-\rho ^2}^{t_0}\Vert p\Vert _{L^{\frac{7}{5}}(B_\rho )}^{\frac{7}{5}}\Vert u\Vert _{L^{\frac{9}{4}}(B_\rho )}^{\frac{3}{5}}\right] ^{\frac{5}{7}}\\&+\frac{c}{(\rho -s)^{\frac{11}{7}}}\sup _t\Vert u\Vert _{L^2(B_\rho )}^{\frac{4}{7}}\int _{t_0-\rho ^2}^{t_0}\Vert p\Vert _{L^{\frac{7}{5}}(B_\rho )} \Vert u\Vert _{L^{\frac{9}{4}}(B_\rho )}^{\frac{3}{7}}\\\le\, & {} \frac{1}{4}I(u,\rho )+ \frac{c}{(\rho -s)^{\frac{7}{5}}}\int _{t_0-\rho ^2}^{t_0}\Vert p\Vert _{L^{\frac{7}{5}}(B_\rho )}^{\frac{7}{5}}\Vert u\Vert _{L^{\frac{9}{4}}(B_\rho )}^{\frac{3}{5}}\\&+ \frac{c}{(\rho -s)^{\frac{11}{5}}}\left[ \int _{t_0-\rho ^2}^{t_0}\Vert p\Vert _{L^{\frac{7}{5}}(B_\rho )} \Vert u\Vert _{L^{\frac{9}{4}}(B_\rho )}^{\frac{3}{7}}\right] ^{\frac{7}{5}}\\\le\, & {} \frac{1}{4}I(u,\rho )+\frac{c\rho ^{\frac{16}{35}}}{(\rho -s)^{\frac{7}{5}}}\left[ \int _{t_0-\rho ^2}^{t_0}\Vert p\Vert _{L^{\frac{7}{5}}(B_\rho )}^{2}\right] ^{\frac{7}{10}}\left[ \int _{t_0-\rho ^2}^{t_0}\Vert u\Vert _{L^{\frac{14}{5}}(B_\rho )}^{4}\right] ^{\frac{3}{20}}\\&+ \frac{c\rho ^{\frac{44}{35}}}{(\rho -s)^{\frac{11}{5}}}\left[ \int _{t_0-\rho ^2}^{t_0}\Vert p\Vert _{L^{\frac{7}{5}}(B_\rho )}^{2}\right] ^{\frac{7}{10}}\left[ \int _{t_0-\rho ^2}^{t_0}\Vert u\Vert _{L^{\frac{14}{5}}(B_\rho )}^{4}\right] ^{\frac{3}{20}}.\\ J_5\le\, & {} c\,\rho ^{\frac{13}{7}}\left[ \int _{t_0-\rho ^2}^{t_0}\Vert \nabla d\Vert _{L^{\frac{14}{5}}(B_\rho )}^4\right] ^{1/2}. \end{aligned}$$

From (2.8), using estimates above with respect to $J_1, J_2, J_3, J_4$ and $J_5$ we get

$$\begin{aligned} I(s)\le\, & {} \frac{1}{2}I(\rho )+\frac{c r^\frac{5}{2}}{(\rho -s)^3}\left[ \int _{t_0-r^2}^{t_0}\Vert |u|^2+|\nabla d|^2\Vert _{W^{-1,2}(B_r)}^2\right] ^{\frac{1}{2}}\nonumber \\&+\,\left[ \frac{c}{(\rho -s)^2}+\frac{c r^2}{(\rho -s)^4}\right] \int _{t_0-r^2}^{t_0}\left[ \Vert |u\,\nabla d|\Vert ^2_{W^{-1,2}(B_r)}+\Vert |u|^2\Vert ^2_{W^{-1,2}(B_r)}\right] \nonumber \\&+\,\frac{c r^{\frac{16}{35}}}{(\rho -s)^{\frac{7}{5}}}\left[ \int _{t_0-r^2}^{t_0} \Vert p\Vert _{L^{\frac{7}{5}}(B_r)}^{2}\right] ^{\frac{7}{10}}\left[ \int _{t_0-r^2}^{t_0}\Vert u\Vert _{L^{\frac{14}{5}}(B_r)}^{4}\right] ^{\frac{3}{20}}\nonumber \\&+\, \frac{c r^{\frac{44}{35}}}{(\rho -s)^{\frac{11}{5}}}\left[ \int _{t_0-r^2}^{t_0}\Vert p\Vert _{L^{\frac{7}{5}}(B_r)}^{2}\right] ^{\frac{7}{10}}\left[ \int _{t_0-r^2}^{t_0}\Vert u\Vert _{L^{\frac{14}{5}}(B_r)}^{4}\right] ^{\frac{3}{20}}\nonumber \\&+\,c\,\rho ^{\frac{13}{7}}\left[ \int _{t_0-\rho ^2}^{t_0}\Vert \nabla d\Vert _{L^{\frac{14}{5}}(B_\rho )}^4\right] ^{1/2}. \end{aligned}$$

(2.9)

By Lemma 2.2, we have

$$\begin{aligned} I(r/2)\le\, & {} cr^{-\frac{1}{2}}\left[ \int _{t_0-r^2}^{t_0}\Vert |u|^2+|\nabla d|^2\Vert _{W^{-1,2}(B_r)}^2\right] ^{\frac{1}{2}}\\&+cr^{-2}\int _{t_0-r^2}^{t_0}\left( \Vert |u\,\nabla d|\Vert ^2_{W^{-1,2}(B_r)}+\Vert |u|^2\Vert ^2_{W^{-1,2}(B_r)}\right) \mathrm{d}t\\&+\,c r^{-\frac{33}{35}}\left[ \int _{t_0-r^2}^{t_0} \Vert p\Vert _{L^{\frac{7}{5}}(B_r)}^{2}\right] ^{\frac{7}{10}}\left[ \int _{t_0-r^2}^{t_0}\Vert u\Vert _{L^{\frac{14}{5}}(B_r)}^{4}\right] ^{\frac{3}{20}}\\&+\, c\,r^{\frac{13}{7}}\left[ \int _{t_0-r^2}^{t_0}\Vert \nabla d\Vert _{L^{\frac{14}{5}}(B_r)}^4\right] ^{1/2}. \end{aligned}$$

Finally, we get

$$\begin{aligned}&A(u,\nabla d,r/2,z_0)+E(u,\nabla d,r/2,z_0)\nonumber \\&\quad \le c r^{-\frac{3}{2}}\left[ \int _{t_0-r^2}^{t_0}\Vert |u|^2+|\nabla d|^2\Vert _{W^{-1,2}(B_r)}^2\right] ^{1/2}\nonumber \\&\qquad + c r^{-3}\int _{t_0-r^2}^{t_0}\left( \Vert |u\,\nabla d|\Vert ^2_{W^{-1,2}(B_r)}+\Vert |u|^2\Vert ^2_{W^{-1,2}(B_r)}\right) \nonumber \\&\qquad + cr^{-\frac{68}{35}}\left[ \int _{t_0-r^2}^{t_0} \Vert p\Vert _{L^{\frac{7}{5}}(B_r)}^{2}\right] ^{\frac{7}{10}}\left[ \int _{t_0-r^2}^{t_0}\Vert u\Vert _{L^{\frac{14}{5}}(B_r)}^{4}\right] ^{\frac{3}{20}}\nonumber \\&\qquad + c\,r^{\frac{6}{7}}\left[ \int _{t_0-r^2}^{t_0}\Vert \nabla d\Vert _{L^{\frac{14}{5}}(B_r)}^4\right] ^{1/2}. \end{aligned}$$

(2.10)

For $f\in L^{7/5}(B_r(x_0))$ and $\varphi \in C^\infty _0(B_r(x_0))$, we have

$$\begin{aligned} \left| \int _{B_r(x_0)}\varphi f(x)dx\right|&\le c\int _{B_r(x_0)}\left[ \int _{B_r(x_0)}\frac{|\nabla \varphi (y)|}{|x-y|^2}\mathrm{d}y\right] |f(x)|\mathrm{d}x\\&= c\int _{B_r(x_0)}|\nabla \varphi (y)|\left[ \int _{B_r(x_0)}\frac{|f(x)|}{|x-y|^2}\mathrm{d}x\right] \mathrm{d}y\\&\le c\Vert \nabla \varphi \Vert _{L^2(B_r(x_0))}\Vert \mathbf {I}_1(\chi _{B_r(x_0)}|f|)\Vert _{L^2(B_r(x_0))}. \end{aligned}$$

where $\mathbf {I}_1$ is the first order Riesz’s potential defined by

$$\begin{aligned} \mathbf {I}_1(\mu )(x)=c\int _{\mathbb {R}^3}\frac{d\mu (y)}{|x-y|^2},\quad x\in \mathbb {R}^3. \end{aligned}$$

By using Hardy–Littlewood–Sobolev inequality, we have

$$\begin{aligned} \Vert f\Vert _{W^{-1,2}(B_r(x_0))}\le & {} \Vert \mathbf {I}_1(\chi _{B_r(x_0)}|f|)\Vert _{L^2(B_r(x_0))}\nonumber \\\le & {} c\Vert f\Vert _{L^{\frac{6}{5}}(B_r(x_0))}\le cr^{\frac{5}{14}}\Vert f\Vert _{L^{\frac{7}{5}}(B_r(x_0))}. \end{aligned}$$

(2.11)

Applying (2.11) with $f=|y|^2+|\nabla d|^2,$ and $f= |u||\nabla d|,$ we obtain

$$\begin{aligned}&r^{-3}\int _{t_0-r^2}^{t_0}\Vert |u|^2+|\nabla d|^2\Vert _{W^{-1,2}(B_r)}^2\\&\quad \le r^{\frac{-16}{7}}\int _{t_0-r^2}^{t_0}\left( \Vert u\Vert _{L^{\frac{14}{5}}(B_r)}^4 +\Vert \nabla d\Vert _{L^{\frac{14}{5}}(B_r)}^4\right) \\&\quad = C(u,\nabla d, r, z_0),\\&r^{-3}\int _{t_0-r^2}^{t_0}\left( \Vert |u\,\nabla d|\Vert ^2_{W^{-1,2}(B_r)}+\Vert |u|^2\Vert ^2_{W^{-1,2}(B_r)}\right) \\&\quad \le \left[ C(u,r, z_0)^{\frac{1}{2}} C(\nabla d, r, z_0)^{\frac{1}{2}}+C(u, r, z_0)\right] . \end{aligned}$$

$\square$

We need the bounded estimates for $C(\nabla d,r)$ and D(p, r) with the help of the bounded of C(u, r).

Lemma 2.4

Suppose that (u, d, p) is a suitable weak solution in $Q_1(z_0)=B_1(x_0)\times (t_0-1,t_0)$. Let

$$\begin{aligned} C(u,r, z_0)\le M\quad \text {for any }0<r\le 1, \end{aligned}$$

for $M>0$. Then for every $0<r<1/4,$ we have the following estimates:

$$\begin{aligned}&A(u,\nabla d, r/2, z_0)+E(u,\nabla d,r/2, z_0)+C(\nabla d,r/2, z_0)+D(p,r/2,z_0)\\&\quad \le c(M,C(\nabla d, 1/2, z_0 ), D(p,1/2, z_0));\\&C_1(u,\nabla d, r, z_0)+D_1(p, r, z_0)\le c(M, D(p,1/2, z_0), C(\nabla d, 1/2, z_0)). \end{aligned}$$

Proof

Without loss of generality, we consider $z_0=(0,0).$ It is easy to see that

$$\begin{aligned} C(\nabla d,r/2)\le c\left[ E(\nabla d,r)^\frac{5}{7}C(\nabla d,r)^\frac{2}{7}+C(\nabla d,r)^\frac{9}{14}\right] . \end{aligned}$$

(2.12)

Combining with (2.7) and (2.12) we obtain

$$\begin{aligned} C(\nabla d,r/2)\le \,& {} c(M)\,\left[ 1+ C(\nabla d,r)^{\frac{1}{2}}+D(p,r)^{\frac{7}{10}}\right] ^{\frac{5}{7}}C(\nabla d,r)^{\frac{2}{7}} \nonumber \\&+c\,C(\nabla d,r)^{\frac{9}{14}}. \end{aligned}$$

(2.13)

Let $r=\theta \rho$ with $\theta \le \frac{1}{4}.$ From (2.5) (2.13), we have

$$\begin{aligned}&C(\nabla d,r)+D(p,r)^\frac{5}{6}\nonumber \\&\quad \le c(M)\theta ^{-\frac{32}{49}}\,\left[ 1+\theta ^{-\frac{8}{7}} C(\nabla d,\rho )^{\frac{1}{2}}+\theta ^{-\frac{8}{5}}D(p,\rho )^{\frac{7}{10}}\right] ^{\frac{5}{7}}C(\nabla d,\rho )^{\frac{2}{7}}\nonumber \\&\qquad +c\theta ^{-\frac{72}{49}}\,C(\nabla d,\rho )^{\frac{9}{14}} +c\theta ^\frac{5}{3}D(p,\rho )^\frac{5}{6}+c\theta ^{-\frac{40}{21}}C(\nabla d,\rho )^\frac{5}{6}+c(M,\theta ) \end{aligned}$$

Using Young’s inequality, we have

$$\begin{aligned} C(\nabla d,r)+D(p,r)^\frac{5}{6}\le \eta C(\nabla d,\rho )+(\eta +c\theta ^\frac{5}{3})D(p,\rho )^\frac{5}{6}+c(\theta ,\eta ,M) \end{aligned}$$

Set $F(r)=C(\nabla d,r)+D(p,r)^\frac{5}{6}.$ Choose $\eta >0$ and $\theta >0$ small enough, we have

$$\begin{aligned} F(r)\le \frac{1}{2}F(\rho )+c. \end{aligned}$$

By the standard iterating argument

$$\begin{aligned} C(\nabla d,r)+D(p,r)^\frac{5}{6}\le c\left( M, D(p,1/2), C(\nabla d,1/2)\right) , \quad r\in (0,\frac{1}{4}]. \end{aligned}$$

So that for $0<r\le 1/4,$

$$\begin{aligned} A(u,\nabla d,r)+E(u,\nabla d,r)+C(\nabla d,r)+D(p,r)\le c(M,C(\nabla d,1/2),D(p,1/2)). \end{aligned}$$

The estimates of $C_1(u,\nabla d,r)$ and $D_1(p,r)$ are immediate results. $\square$

3 Proof of Theorem 1.3

We shall prove the following Proposition 3.1, Theorem 1.3 is an immediate result.

Proposition 3.1

Let (u, d, p) a weak solution of (1.1) with

$$\begin{aligned} \Vert u\Vert _{L^\infty (0, T; L^{3,\infty }(\mathbb {R}^3))}\le M, \end{aligned}$$

for $z_0=(x_0,t_0)$ and $R>0$ such that $Q_R(z_0)\subset Q_T$, (d, p) satisfy

$$\begin{aligned} C(\nabla d,R,z_0)+D(p, R, z_0)\le N. \end{aligned}$$

there exists a positive number $\varepsilon (M, N)<\frac{1}{4}$ such that if for some $0<r\le R/2$,

$$\begin{aligned} r^{-3}\left| \left\{ x\in B_r(x_0): |u(x,t_0)|>\varepsilon r^{-1}\right\} \right| \le \varepsilon , \end{aligned}$$

(3.1)

then there exists $\rho \in [2r\varepsilon , r]$ such that

$$\begin{aligned} \frac{1}{\rho ^2}\int _{Q_\rho (z_0)}|u|^3+|\nabla d|^3<\varepsilon _0 \end{aligned}$$

(3.2)

where $\varepsilon _0$ is the same number in Proposition 1.2.

Proof

Let (u, d, p) be a weak solution of (1.1) with $u\in L^\infty (0,T; L^{3,\infty }(\mathbb {R}^3)$. We assume

$$\begin{aligned} \sup _{0<r<T}\Vert u(t)\Vert _{L^{3,\infty }(\mathbb {R}^3)}\le M. \end{aligned}$$

(3.3)

Note that $|d|\le 1$, we have (see [28])

$$\begin{aligned} ||\nabla d||_{L^{4}_{x}}\le c||\nabla d||^{\frac{1}{2}}_{\dot{B}^{-1}_{\infty ,\infty }}||\nabla d||^{\frac{1}{2}}_{\dot{H}^{1}}\le c\Vert d\Vert _{L^\infty }^{1/2}||\nabla d||^{\frac{1}{2}}_{\dot{H}^{1}}. \end{aligned}$$

Also since the real interpolation $L^{4}=[L^{6,\infty },L^{3,\infty }]_{\frac{1}{2},4}$ holds, then

$$\begin{aligned} ||u||_{L^{4}_{x}}\le c||u||^{\frac{1}{2}}_{L^{6,\infty }_{x}}||u||^{\frac{1}{2}}_{L^{3,\infty }_{x}}\le c||\nabla u||^{\frac{1}{2}}_{L^{2}_{x}}M^{\frac{1}{2}}. \end{aligned}$$

From energy inequality and estimates above we get

$$\begin{aligned} \Vert (u, \nabla d)\Vert _{L^4(Q_T)}\le c(M, c_0), \end{aligned}$$

(3.4)

which yields (u, d, p) is a local suitable weak solution of (1.1) and $u\in C([0,T]; L^2(\mathbb {R}^3))$.

We use a contradiction argument for $z_0=(0,0)$ and $R=1$. Fixed $N, M>0$ if the assertion of the proposition were false, then there would exist $\epsilon _k \downarrow 0$, and suitable weak solutions $(u_k,d_k, p_k)$ of (1.1) and $r_k\le 1/2$ such that

$$\begin{aligned}&\Vert u_k\Vert _{L^\infty (-1,0; L^{3,\infty }(\mathbb {R}^3))}\le M; \end{aligned}$$

(3.5)

$$\begin{aligned}&C(\nabla d_k, 1)+D(p_k, 1)\le N;\end{aligned}$$

(3.6)

$$\begin{aligned}&r_k^{-3}\left| \left\{ x\in B_{r_k}(0): |u_k(x,0)|>\epsilon _k r_k^{-1}\right\} \right| \le \epsilon _k; \end{aligned}$$

(3.7)

and for all $\rho \in [2r_k\epsilon _k, r_k]$,

$$\begin{aligned} \frac{1}{\rho ^2}\int _{Q_\rho (0)}|u_k|^3+|\nabla d_k|^3>\varepsilon _0/2. \end{aligned}$$

(3.8)

Since for $0<r\le 1$

$$\begin{aligned} C(u_k, r)= & {} r^{-16/7}\int _{-r^2}^0|B_{r}|^{2/21}\Vert u_k\Vert _{L^{3,\infty }(B_{r})}^4\\\le & {} c\, \Vert u_k\Vert _{L^\infty (-1,0; L^{3,\infty }(B_1))}^4\le M^4, \end{aligned}$$

combining the estimate and (3.6) with Lemma 2.4, we get, for $0<r\le 1/2$,

$$\begin{aligned} A(u_k,\nabla d_k, r)+ E(u_k,\nabla d_k, r)+ C(\nabla d_k, r) + D(p_k, r)\le c(M,N). \end{aligned}$$

Similarly, for any $z_0\in Q_{1/2}$ and $0<r\le 1/2$, we have

$$\begin{aligned} A(u_k,\nabla d_k, r, z_0)+ E(u_k,\nabla d_k, r, z_0)+ C(\nabla d_k, r, z_0) + D(p_k, r, z_0)\le c(M,N). \end{aligned}$$

Define, for $(x,t)\in Q_{r_k^{-1}}$,

$$\begin{aligned} {\left\{ \begin{array}{ll} U_{k}(x,t) = r_{k}u_k(r_{k}x,r_{k}^{2}t), \\ D_{k}(x,t) = d_k(r_{k}x,r_{k}^{2}t), \\ P_{k}(x,t) = r_{k}^{2}p_k(r_{k}x,r_{k}^{2}t). \end{array}\right. } \end{aligned}$$

(3.9)

Obviously, $(U_k, D_k, P_k)$ are weak solutions to system (1.1) with the right side of (1.1)$_{3}$ replaced by $r_{k}^{2}f(D_{k})$ in $Q_{r_k^{-1}}$. Now, for $a>0,$ and $a r_k\le 1/2,$

$$\begin{aligned} \Vert U_k\Vert _{L^{\infty }(-r_k^{-1},0; L^{3,\infty }(B_{r_k^{-1}}))}=\, & {} \Vert u_k\Vert _{L^\infty (-1, 0; L^{3,\infty }(B_1))}\le M,\\ C(\nabla D_k, a)+D(P_k, a)=\, & {} C(\nabla d_k, ar_k)+D(p_k, ar_k)\le N,\\ C(U_k, a)= \,& {} C(u_k, ar_k)\\\le\, & {} (ar_k)^{-16/7}\int _{-(ar_k)^2}^0|B_{ar_k}|^{2/21}\Vert u_k\Vert _{L^{3,\infty }(B_{ar_k})}^4\\\le\, & {} c\, \Vert u_k\Vert _{L^\infty (-1,0; L^{3,\infty }(B_1))}^4\le M^4. \end{aligned}$$

We have by Lemma 2.4 again

$$\begin{aligned}&A(U_k,\nabla D_k, a)+E(U_k,\nabla D_k,a)+D(P_k,a)\nonumber \\&\quad +C_1(U_k,\nabla D_k, a)+D_1(P_k,a)\le c(M,N). \end{aligned}$$

(3.10)

So that

$$\begin{aligned} \Vert U_k\Vert _{L^4(Q_a)}^4\le\, & {} c\,\int _{-a^2}^0\Vert U_k\Vert _{L^{3,\infty }(B_a)}^2\Vert U_k\Vert _{L^6(B_a)}^{2}\nonumber \\\le\, & {} c\,\int _{-a^2}^0\Vert U_k\Vert _{L^{3,\infty }(B_{a})}^{2}\Vert \nabla U_k\Vert _{L^2(B_{a})}^{2}+c\,\int _{-a^2}^0|B_a|\Vert U_k\Vert _{L^{3,\infty }(B_{a})}^{2}\Vert U_k\Vert _{L^2(B_{a})}^2\nonumber \\\le\, & {} c\,a M^2\left( A(U_k, a)+E(U_k, a)\right) ,\\ \Vert \nabla D_k\Vert _{L^4(Q_a)}^4\le \,& {} a\, c(M,N)\left( A(\nabla D_k, a)+E(\nabla D_k, a)\right) . \end{aligned}$$

Thus, the $L^p$ estimate holds for $(U_k, D_k, P_k)$ in $Q_a$, for any $a>0$,

$$\begin{aligned}&\int _{Q_{a}}|U_k|^4+ |\nabla D_k|^4+|\partial _{t}U_k|^{\frac{4}{3}}+ |\partial _{t}\nabla D_k|^{\frac{4}{3}}+|\nabla ^{2}U_k|^{\frac{4}{3}}+|\nabla ^{2}\nabla D_k|^{\frac{4}{3}} +|\nabla P_k|^{\frac{4}{3}}\nonumber \\&\quad \le c_2(a,M, N). \end{aligned}$$

(3.11)

By Aubin–Lion’s lemma, there exists a triplet (v, e, q) such that

$$\begin{aligned} {\left\{ \begin{array}{ll} U_k\rightarrow v, \quad &{}\text {in }L^3(Q_a),\\ \nabla D_k\rightarrow \nabla e, \quad &{}\text {in }L^3(Q_a),\\ P_k\rightharpoonup q, \quad &{}\text {in }L^\frac{3}{2}(Q_a), \end{array}\right. } \end{aligned}$$

(3.12)

and

$$\begin{aligned} U_k\rightarrow v,\quad \nabla D_k\rightarrow \nabla e \quad \text {in }C([-a^2,0]; L^{\frac{4}{3}}(B_a)). \end{aligned}$$

Using estimates above, the limit function (v, e, q) satisfy, in the sense of suitable weak solutions on $\mathbb {R}^3\times (-\infty ,0)$,

$$\begin{aligned} {\left\{ \begin{array}{ll} v_{t}-\Delta v+v\cdot \nabla v+\nabla q=-\nabla \cdot (\nabla e\odot \nabla e), \\ \nabla \cdot v=0, \\ e_{t}-\Delta e+v\cdot \nabla e=0. \end{array}\right. } \end{aligned}$$

(3.13)

From (3.7) and (3.8), we get

$$\begin{aligned} \left| \left\{ x\in B(0): |U_k(x,0)|>\epsilon _k\right\} \right| <\epsilon _k, \end{aligned}$$

(3.14)

and for $\rho \in [2\epsilon _k,1]$

$$\begin{aligned} \frac{1}{\rho ^2}\int _{Q_{\rho }}|U_{k}|^{3}+|\nabla D_{k}|^{3}>\varepsilon _0/2. \end{aligned}$$

(3.15)

Taking limit we get

$$\begin{aligned} v(\cdot , 0)=0, \quad \text {in}\, B_1(0), \end{aligned}$$

(3.16)

and for $\rho \in (0, 1]$

$$\begin{aligned} \frac{1}{\rho ^2}\int _{Q_{\rho }}|v|^{3}+|\nabla e|^{3}\ge \varepsilon _0/2. \end{aligned}$$

(3.17)

The crucial point here is a reduction to backward uniqueness for the heat operator with lower order terms as [3]. Set

$$\begin{aligned} v_k=\rho _kv(\rho _kx,\rho _k^2t), \quad e_k=e(\rho _kx,\rho _k^2t),\quad q_k=\rho _k^2q(\rho _kx,\rho _k^2t). \end{aligned}$$

Then $(v_k, e_k, q_k)$ satisfy (3.13), and similar to (3.10) for any $a>0$

$$\begin{aligned} A(v_k,\nabla e_k, a)+ & {} E(v_k,\nabla e_k,a)+D(q_k,a)\nonumber \\+ & {} C_1(v_k,\nabla e_k, a)+D_1(q_k,a)\le c(M,N). \end{aligned}$$

(3.18)

As before there exists a triplet $(\widetilde{v},\widetilde{e},\widetilde{q})$ is suitable weak solution of (3.13) such that

$$\begin{aligned} {\left\{ \begin{array}{ll} v_k\rightarrow \widetilde{v}, \quad &{}\text {in }L^3(Q_a),\\ \nabla e_k\rightarrow \nabla \widetilde{e}, \quad &{}\text {in }L^3(Q_a),\\ q_k\rightharpoonup \widetilde{q}, \quad &{}\text {in }L^\frac{3}{2}(Q_a), \end{array}\right. } \end{aligned}$$

(3.19)

and

$$\begin{aligned} v_k\rightarrow \widetilde{v},\quad \nabla e_k\rightarrow \nabla \widetilde{e} \quad \text {in }C([-a^2,0]; L^{\frac{4}{3}}(B_a)). \end{aligned}$$

From (3.16) we get

$$\begin{aligned} \widetilde{v}(\cdot , 0)=0, \quad \text {in}\,\mathbb {R}^3, \end{aligned}$$

(3.20)

from (3.17) and take $\rho =\rho _k$

$$\begin{aligned} \int _{Q_{1}}|\widetilde{v}|^{3}+|\nabla \widetilde{e}|^{3}\ge \varepsilon _0/2. \end{aligned}$$

(3.21)

On the other hand, for fixed $z_0$ and $0<R\le \frac{1}{2r_k}$, as (3.18)

$$\begin{aligned}&A(v_k,\nabla e_k, R, z_0)+E(v_k,\nabla e_k,R, z_0)+D(q_k,R, z_0)\nonumber \\&\quad +C_1(v_k,\nabla e_k, R, z_0)+D_1(q_k,R, z_0)\le c(M,N). \end{aligned}$$

(3.22)

By the Fubini theorem, we have,

$$\begin{aligned}&\left| \left\{ (x,t)\in \mathbb {R}^3\times (-T,0): \, |\widetilde{v}(x,t)|>\gamma \right\} \right| \\&\quad =\int _{-T}^0d_{\widetilde{v}(t)}(\gamma )\mathrm{d}t\le \gamma ^{-3}M^3T. \end{aligned}$$

Hence, for any $\eta >0,$ there exists a $B_R$ such that

$$\begin{aligned} \left| \left\{ (x,t)\in (\mathbb {R}^3\backslash B_R)\times (-T,0): \, |\widetilde{v}(x,t)|>\gamma \right\} \right| <\eta . \end{aligned}$$

Let $Q_1(z_0)\subset (\mathbb {R}^3\backslash B_R)\times (-T,0]$, by (3.22), we have $A(\widetilde{v},\nabla \widetilde{e},\theta ,z_0)+E(\widetilde{v},\nabla \widetilde{e},\theta ,z_0)\le C(M,N)$ for any $0<\theta \le 1.$ Thus, by the interpolation inequality we have

$$\begin{aligned} \theta ^{-5/3}\int _{Q_\theta (z_0)}|\widetilde{v}|^{10/3}+|\nabla \widetilde{e}|^{10/3}\mathrm{d}x\mathrm{d}t\le c(M,N). \end{aligned}$$

Thus

$$\begin{aligned} C_1(\widetilde{v},1, z_0)&\le \gamma ^3|Q_1(z_0)|+\mathop {\iint }\limits _{Q_1(z_0)\cap \{|\widetilde{v}|>\gamma \}}|\widetilde{v}|^3dxdt\\&\le c\gamma ^3+\Vert \widetilde{v}\Vert _{L^{10/3}(Q_1(z_0))}\left| Q_1(z_0)\cap \left\{ |\widetilde{v}|>\gamma \right\} \right| ^{1/10}\\&\le c(\gamma ^3+\eta ^\frac{1}{10}). \end{aligned}$$

For any $\epsilon >0$ we choose $\gamma$ and $\eta$ such that $C_1(\widetilde{v},1,z_0)<\epsilon$.

It is easy to see that, by [28] (to see Lemma 3.2),

$$\begin{aligned} K(\nabla \widetilde{e},\theta , z_{0})\le \,& {} c\theta ^{-3}C_1(\widetilde{v},1,z_0)^{\frac{2}{3}}\left[ A(\nabla \widetilde{e}, 1,z_0) +E(\nabla \widetilde{e}, 1,z_0)\right] +\theta ^{2} K(\nabla \widetilde{e}, 1,z_0))\nonumber \\\le\, & {} c(M,N) (\theta ^{-3}\epsilon ^{2/3}+\theta ^2) \end{aligned}$$

(3.23)

Utilizing (3.23) and Hölder’s inequality we have

$$\begin{aligned} C_1(\nabla \widetilde{e},\theta , z_{0})\equiv \,& {} \theta ^{-2}\int _{Q_{\theta }(z_{0})}|\nabla \widetilde{e}|^{3}\nonumber \\\le \,& {} \left[ \theta ^{-\frac{5}{3}}\int _{Q_{\theta }(z_{0})}|\nabla \widetilde{e}|^{\frac{10}{3}}\right] ^{\frac{3}{4}}\left[ \theta ^{-3}\int _{Q_{\theta }(z_{0})}|\nabla \widetilde{e}|^{2}\right] ^{\frac{1}{4}}\nonumber \\\le &\, {} c(M, N) K(\nabla \widetilde{e},\theta ;z_{0})^{\frac{1}{4}}\nonumber \\\le\, & {} c(M, N)(\theta ^{-3/4}\epsilon ^{1/6}+\theta ^{1/2}). \end{aligned}$$

(3.24)

Thus

$$\begin{aligned} C_1(\widetilde{v},\nabla \widetilde{e}, \theta , z_0)\le & {} \theta ^{-2}\epsilon + c(M, N)(\theta ^{-3/4}\epsilon ^{1/6}+\theta ^{1/2}). \end{aligned}$$

First we take $\theta$ such that $c(M,N)\theta ^{1/2}\le \varepsilon _0/2,$ then take $\epsilon$ such that $\theta ^{-2}\epsilon +c(M, N)\theta ^{-3/4}\epsilon ^{1/6}\le \varepsilon _0/2$, i.e.,

$$\begin{aligned} C_1(\widetilde{v},\nabla \widetilde{e}, \theta , z_0)\le \varepsilon _0, \end{aligned}$$

which implies that $z_0$ is a regular point by Proposition 1.2, therefor $(\widetilde{v},\widetilde{e},\widetilde{q})$ are smooth and their derivatives are bounded in $(\mathbb {R}^{3}\setminus B_{2R})\times (-T/2,0)$. Next, we show

$$\begin{aligned} \nabla \widetilde{e}(\cdot ,0)=0. \end{aligned}$$

(3.25)

For any B(y) and $\phi \in C_0^\infty (B(y)),$ since e is Hölder continuous (to see following Lemma 3.2), we have

$$\begin{aligned}&\left| \int _{B(y)}\nabla \widetilde{e}(x,0)\phi dx\right| \\&\quad \le \int _{B(y)}|\nabla \widetilde{e}(x,0)-\nabla e_{k}(x,0)|\mathrm{d}x+\left| \int _{B(y)}\nabla e_{k}\phi \right| \\&\quad \le c\,\Vert \nabla \widetilde{e}(x,0)-\nabla e_{k}(x,0)\Vert _{L^{4/3}(B(y))}+c\,r_{k}^{-3}\int _{B_{r_{k}}(r_{k}y)}|e(x,0)-e(0,0)|\mathrm{d}x\\&\quad \le o(1)+c\, r_{k}^{-3}r_{k}^{3}r_{k}^{\alpha }\\&\quad \le o(1). \end{aligned}$$

The backward uniqueness theorem of parabolic equations [3], we conclude

$$\begin{aligned} \widetilde{e}(x,t)=0,\quad \text {in }\mathbb {R}^3\backslash \bar{B}_{2R}(0)\times (-T/2,0]. \end{aligned}$$

Using unique continuation theorem of parabolic equation in the bounded domain again [3], we conclude that

$$\begin{aligned} \widetilde{e}(x,t)=0\quad \text {in }\mathbb {R}^3\times (-T/2,0). \end{aligned}$$

Thus, $\widetilde{v}$ satisfies Navier–Stokes equations in $\mathbb {R}^3\times (-T/2,0)$

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}\partial _t\widetilde{v}-\Delta \widetilde{v}+\widetilde{v}\cdot \nabla \widetilde{v}+\nabla \widetilde{q}=0,\\ &{}\mathrm {div} \widetilde{v}=0.\end{array}\right. } \end{aligned}$$

Using (3.20) and backward uniqueness of heat operator again [3], we get

$$\begin{aligned} \widetilde{v}(x,t)=0\quad \text {in }\mathbb {R}^3\times (-T/2,0), \end{aligned}$$

which is a contradiction with (3.21).$\square$

We need following lemma.

Lemma 3.2

For $v\in L^\infty (0, T; L^{3,\infty }(\mathbb {R}^3))$, if e satisfies in $\mathbb {R}^3\times (0,T)$

$$\begin{aligned} \partial _t e-\Delta e=-v\cdot \nabla e. \end{aligned}$$

Then e is Hölder continuous.

Proof

From [34], if $v\in L^\infty (0, T; BMO^{-1}(\mathbb {R}^3))$, then e is Hölder continuous. We only prove the following inclusion relationship for $3<p<\infty$,

$$\begin{aligned} L^{3}_{w}(\mathbb {R}^{3})\subset \dot{B}^{-1+\frac{3}{p}}_{p,\infty }(\mathbb {R}^{3})\subset BMO^{-1}(\mathbb {R}^{3}), \end{aligned}$$

where $L^{3}_{w}(\mathbb {R}^{3})=L^{3,\infty }(\mathbb {R}^{3})$ is the weak Lebesgue space, and $\dot{B}^{-1+\frac{3}{p}}_{p,\infty }(\mathbb {R}^{3})$ is a homogeneous Besov space.

The first inclusion is obtained through Sobolev embedding and real interpolation. To be specific, we write weak space $L^{3}_{w}$ a real interpolation

$$\begin{aligned} L^{3}_{w}(\mathbb {R}^{3})=(L^{2}(\mathbb {R}^{3}),L^{p}(\mathbb {R}^{3}))_{\theta ,\infty }, \end{aligned}$$

where $\theta =\frac{p}{3(p-2)}$. Notice we have the following two embedding relations

$$\begin{aligned} L^{2}(\mathbb {R}^{3})=\, & {} F^{0}_{2,2}(\mathbb {R}^{3})=B^{0}_{2,2}(\mathbb {R}^{3})\subset B^{\frac{3}{p}-\frac{3}{2}}_{p,2}(\mathbb {R}^{3}),\\ L^{p}(\mathbb {R}^{3})=\, & {} F^{0}_{p,2}(\mathbb {R}^{3})\subset B^{0}_{p,p}(\mathbb {R}^{3}), \end{aligned}$$

where we have used Littlewood–Paley Theorem to characterize $L^{r}$ by the Triebel–Lizorkin space $F^{0}_{r,2}$ (for $1<r<\infty$). Thus the identity map is bounded:

$$\begin{aligned}&id:L^{2}(\mathbb {R}^{3})\rightarrow B^{\frac{3}{p}-\frac{3}{2}}_{p,2}(\mathbb {R}^{3}),\\&id:L^{p}(\mathbb {R}^{3})\rightarrow B^{0}_{p,p}(\mathbb {R}^{3}). \end{aligned}$$

By real interpolation, and $(1-\theta )\cdot (\frac{3}{p}-\frac{3}{2})+\theta \cdot 0=-1+\frac{3}{p}$, this leads to

$$\begin{aligned} B^{-1+\frac{3}{p}}_{p,\infty }(\mathbb {R}^{3})=(B^{\frac{3}{p}-\frac{3}{2}}_{p,2}(\mathbb {R}^{3}),B^{0}_{p,p}(\mathbb {R}^{3}))_{\theta ,\infty }, \end{aligned}$$

thus the identity map

$$\begin{aligned} id:L^{3}_{w}(\mathbb {R}^{3})\rightarrow B^{-1+\frac{3}{p}}_{p,\infty }(\mathbb {R}^{3})\subset \dot{B}^{-1+\frac{3}{p}}_{p,\infty }(\mathbb {R}^{3}) \end{aligned}$$

is also bounded. The second inclusion is obtained by using the heat kernel characterization of the corresponding spaces, denote $s=-1+\frac{3}{p}$,

$$\begin{aligned}&||f||_{\dot{B}^{s}_{p,\infty }(\mathbb {R}^{3})}=\sup _{t>0}||t^{-\frac{s}{2}}e^{t\Delta }f||_{L^{p}(\mathbb {R}^{3})},\\&||f||_{BMO^{-1}(\mathbb {R}^{3})}=\sup _{x\in \mathbb {R}^{3},R>0}\left[\frac{1}{|B_{R}(x)|}\int _{B_{R}(x)}\int _{0}^{R^{2}}|e^{t\Delta }f|^{2}\mathrm{d}y\mathrm{d}t\right]^{\frac{1}{2}}. \end{aligned}$$

Direct calculation yields:

$$\begin{aligned} ||f||_{BMO^{-1}(\mathbb {R}^{3})}&= \sup _{x\in \mathbb {R}^{3},R>0}\left[\frac{1}{|B_{R}(x)|}\int _{0}^{R^{2}}\int _{B_{R}(x)}|e^{t\Delta }f|^{2}\mathrm{d}y\mathrm{d}\right]^{\frac{1}{2}}\\&\le \sup _{x\in \mathbb {R}^{3},R>0}\left[|B_{R}(x)|^{-\frac{2}{p}}\int _{0}^{R^{2}}(\int _{B_{R}(x)}|e^{t\Delta }f|^{p}dy)^{\frac{2}{p}}\mathrm{d}t\right]^{\frac{1}{2}}\\&\le \sup _{x\in \mathbb {R}^{3},R>0}\left[|B_{R}(x)|^{-\frac{2}{p}}\int _{0}^{R^{2}}||e^{t\Delta }f||_{L^{p}(\mathbb {R}^{3})}^{2}\mathrm{d}t\right]^{\frac{1}{2}}\\&\le \sup _{x\in \mathbb {R}^{3},R>0}\left[|B_{R}(x)|^{-\frac{2}{p}}\int _{0}^{R^{2}}t^{s}||f||^{2}_{\dot{B}^{s}_{p,\infty }(\mathbb {R}^{3})}\mathrm{d}t\right]^{\frac{1}{2}}\\&\le c||f||_{\dot{B}^{s}_{p,\infty }(\mathbb {R}^{3})}. \end{aligned}$$

The proof is thus finished.$\square$

4 Proof of Theorem 1.4

Define for any $r>0$ such that $Q_r(z_0)\subset Q_T,$

$$\begin{aligned} C_2(u, r,z_0)=r^{-1}\int _{Q_r(z_0)}|u|^4,\quad C_2(\nabla d, r,z_0)=r^{-1}\int _{Q_r(z_0)}|\nabla d|^4. \end{aligned}$$

By interpolation inequality we have

$$\begin{aligned} C(\nabla d, R, z_0)\le A(\nabla d, R, z_0)^{6/7} C_2(\nabla d,R,z_0)^{4/7}\le c(c_0, R), \end{aligned}$$

and

$$\begin{aligned} C_2(u, R,z_0) \le c\, M^2\left[ A(u, R, z_0)+E(u,R,z_0)\right] , \end{aligned}$$

where $\Vert u\Vert _{L^\infty (-1,0; L^{3,\infty }(\mathbb {R}^3))}\le M$. On the other hand, by Calderón-Zygmund theorem,

$$\begin{aligned} \Vert p(t)\Vert _{L^{s}(\mathbb {R}^3)}^{s}\le c\,\Vert (u,\nabla d)(t)\Vert _{L^{2s}(\mathbb {R}^3)}^{2s}\quad \text {for}\,\, 1<s<\infty , \end{aligned}$$

and

$$\begin{aligned} (u, \nabla d)\in L^4(0,T; L^3(\mathbb {R}^3)), \end{aligned}$$

which implies

$$\begin{aligned} p\in L^2(0,T; L^{3/2}(\mathbb {R}^3)), \end{aligned}$$

we have

$$\begin{aligned} D(p, R,z_0)\le c(c_0, R). \end{aligned}$$

Since for any $0<r\le R,$

$$\begin{aligned} C(u, r,z_0)\le c\, M^4, \end{aligned}$$

by Lemma 2.4, we have for any $0<r\le R/2$ and $z_0\in \Omega \times (0,T), \Omega \subset \subset \mathbb {R}^3$,

$$\begin{aligned}&A(u,\nabla d, r, z_0)+E(u,\nabla d,r,z_0)+C_1(u, \nabla d, r,z_0)+C_2(u, \nabla d, r,z_0)\nonumber \\&\qquad +C(\nabla d, r, z_0)+D(p, r,z_0)+D_1(p, r,z_0)\nonumber \\&\quad \le c(M, R, c_0)\equiv N. \end{aligned}$$

(4.1)

The number $\varepsilon (M,N)$ of Proposition 3.1 can be determined.

Let S be a singular points set of (u, d) at $\left\{ (x,T): x\in \mathbb {R}^3\right\}$. Assume that it contains more than $M^3\varepsilon ^{-4}$ elements. Letting $P=[M^3\varepsilon ^{-4}]+1$, we can find P different singular points $\left\{ (x_k, T): k = 1, 2,\ldots , P\right\}$ of the set S. We can choose $R_0\le R$ such that $B_{R_0}(x_k)\cap B_{R_0}(x_l)=\emptyset , k\ne l,$ and bounded domain $\Omega$ such that $\cup _{k=1}^PB_{R_0}(x_k)\subset \Omega$. According to Proposition 3.1, for all $r\in (0,R_0/2]$, it holds true

$$\begin{aligned} \varepsilon \le \frac{1}{r^3} \left| \left\{ x\in B_r(x_k): |u(x,T)|>\frac{\varepsilon }{r}\right\} \right| \end{aligned}$$

(4.2)

for all $k=1,2,\ldots , P.$ In particular, taking $r=r_0=R_0/2,$ we have

$$\begin{aligned} P\varepsilon\le \,& {} \sum _{k=1}^P\frac{1}{r_0^3} \left| \left\{ x\in B_{r_0}(x_k): |u(x,T)|>\frac{\varepsilon }{r_0}\right\} \right| \\\le\, & {} \frac{1}{r_0^3} \left| \left\{ x\in \cup _{k=1}^P B_{r_0}(x_k): |u(x,T)|>\frac{\varepsilon }{r_0}\right\} \right| \\\le\, & {} \frac{1}{r_0^3} \left| \left\{ x\in \Omega : |u(x,T)|>\frac{\varepsilon }{r_0}\right\} \right| \\\le\, & {} \varepsilon ^{-3}\Vert u(\cdot , T)\Vert _{L^{3,\infty }(\Omega )}^3\\\le & {} \varepsilon ^{-3}M^3, \end{aligned}$$

i.e., $P\le M^3\varepsilon ^{-4}<P,$ which is a contradiction. $\square$

5 Appendix: Proof of Proposition 1.2

According to the $L^p$ theorem of Stokes system in [5], if $\mathbf {f}\in W^{-1,q}(\Omega ;\mathbb {R}^3), 1<q<\infty$, and $\Omega$ is a $C^1$ bounded domain, the following Stokes equations

$$\begin{aligned} {\left\{ \begin{array}{ll}-\Delta v+\nabla p=\mathbf {f}, \\ \mathrm {div}\,v=0, \quad \int _\Omega p=0,\\ v|_{\partial \Omega }=0,\end{array}\right. } \end{aligned}$$

(5.1)

there exists exactly one solution $(v,p)\in W^{1,q}(\Omega )\times L^q(\Omega ),$ and

$$\begin{aligned} \Vert \nabla v\Vert _{L^{q}(\Omega )}+\Vert p\Vert _{L^q(\Omega )}\le c(q)\Vert \mathbf {f}\Vert _{W^{-1,q}(\Omega )}. \end{aligned}$$

(5.2)

Wolf’s the local pressure projection $\mathcal {P}_q$ tell us,

$$\begin{aligned} \mathcal {P}_q: W^{-1,q}(\Omega )\rightarrow W^{-1,q}(\Omega ),\quad \mathcal {P}_q(\mathbf {f})=p. \end{aligned}$$

As in [39], we have following Lemma.

Lemma 5.1

Let (u, d) be a weak solution of (1.1), then for every $C^2$ bounded sub-domain $\Omega$, and any $\phi \in C_0^\infty (\Omega \times (0,T))$, there holds

$$\begin{aligned}&-\int _0^T\int _\Omega (u+\nabla p_h)\cdot \phi _t-\int _0^T\int _\Omega (u\otimes u+\nabla d\odot \nabla d+p_1\mathbf {I}): \nabla \phi \nonumber \\&\quad +\int _0^T\int _\Omega (\nabla u-p_2\mathbf {I}):\nabla \phi =0, \end{aligned}$$

(5.3)

i.e., set $v_\Omega =v:=u+\nabla p_h$,

$$\begin{aligned} \partial _t v+\mathrm {div}\,(u\otimes u)+\nabla p_1+\nabla p_2=\Delta v-\nabla \cdot (\nabla d\odot \nabla d), \end{aligned}$$

(5.4)

where $\mathbf {I}$ is identity matrix, and

$$\begin{aligned} {\left\{ \begin{array}{ll}p_h=-\mathcal {P}_2(u),\\ p_1=-\mathcal {P}_{3/2}(u\otimes u+\nabla d\odot \nabla d),\\ p_2=\mathcal {P}_2(\Delta u). \end{array}\right. } \end{aligned}$$

In addition, following estimates hold for a.e. $t\in (0,T)$

$$\begin{aligned} {\left\{ \begin{array}{ll}\Vert \nabla p_h(t)\Vert _{L^m(\Omega )}\le c\,\Vert u(t)\Vert _{L^m(\Omega )}, \quad 1<m\le 6,\\ \Vert p_1(t)\Vert _{L^{3/2}(\Omega )}\le c\,\Vert u\otimes u+\nabla d\odot \nabla d\Vert _{L^{3/2}(\Omega )},\\ \Vert p_2(t)\Vert _{L^2(\Omega )}\le c\,\Vert \nabla u(t)\Vert _{L^2(\Omega )}. \end{array}\right. } \end{aligned}$$

(5.5)

Here $c>0$ depends on the geometry of $\Omega$ and in (5.5)$_1$ on m only. In particular, if $\Omega$ is the ball $B_R(x_0)$ then c in (5.5)$_1$ depends only on m, while in (5.5)$_2$ and (5.5)$_3$ c is an absolute constant.

Hence, we have local energy inequality, for $\varphi \in C_0^\infty (\Omega \times (0,T))$

$$\begin{aligned}&\int _\Omega (|v(t)|^2+|\nabla d(t)|^2)\varphi +2\int _0^t\int _\Omega (|\nabla v|^2+|\nabla ^2 d|^2)\varphi \nonumber \\&\quad \le \int _0^t\int _\Omega (|v|^2+|\nabla d|^2)(\varphi _t+\Delta \varphi )+\int _0^t\int _\Omega (|u|^2u+|\nabla d|^2v)\cdot \nabla \varphi \nonumber \\&\qquad +\int _0^t\int _\Omega 2(p_1+p_2)v\cdot \nabla \varphi + 2\int _0^t\int _\Omega u^iu^j\partial _i(\partial _j p_h\varphi )\nonumber \\&\qquad +2\int _0^t\int _\Omega (u\cdot \nabla d)\cdot (\nabla d\nabla \varphi )\nonumber \\&\qquad + \int _0^t\int _\Omega 2\nabla ^2p_h:(\nabla d\odot \nabla d)\varphi -|\nabla d|^2\nabla p_h\cdot \nabla \varphi \nonumber \\&\qquad -2\int _0^t\int _\Omega \nabla _xf(d)\nabla d\,\varphi . \end{aligned}$$

(5.6)

Note that the suitable weak solution of (1.1) satisfies the local energy inequality (5.6).

From local energy inequality we can get the Caccioppoli-type estimates

$$\begin{aligned}&\Vert W\Vert _{L^{10/3}(Q_{R/2})}^2+\Vert \nabla W\Vert _{L^2(Q_{R/2})}^2\nonumber \\&\quad \le c\,R^{-1/3}\Vert W\Vert _{L^{3}(Q_R)}^2+ c\,R^{-1}\Vert W\Vert _{L^{3}(Q_R)}^3. \end{aligned}$$

(5.7)

Here, $W=(u, \nabla d)$, and

$$\begin{aligned} \Vert W\Vert _{L^{k}(Q_r)}^2=\, & {} \Vert u\Vert _{L^{k}(Q_r)}^2+\Vert \nabla d\Vert _{L^{k}(Q_r)}^2,\\ \Vert \nabla W\Vert _{L^{2}(Q_r)}^2=\, & {} \Vert \nabla u\Vert _{L^{2}(Q_r)}^2+\Vert \nabla ^2 d\Vert _{L^{2}(Q_r)}^2. \end{aligned}$$

Obviously,

$$\begin{aligned} C_1(W, r, z_0)=C_1(u, \nabla d, r, z_0), \quad E(W,r,z_0)=E(u,\nabla d, r, z_0). \end{aligned}$$

Proposition 1.2 is an immediate result of following lemma and Theorem (A).

Lemma 5.2

Suppose that (u, d) is a local suitable weak solution of (1.1). Then there exist universal constants $\varepsilon ^*>0$ and $\theta \in (0,\frac{1}{4}]$ with following property. For any $\varepsilon \in (0, \varepsilon ^*]$ if

$$\begin{aligned} C_{1}(W, 1, z_0)\le \varepsilon , \end{aligned}$$

then

$$\begin{aligned} C_{1}(W,\theta , z_0)\le \varepsilon . \end{aligned}$$

Proof

We prove by contradiction. Let $\theta \in (0,\frac{1}{4}]$ be a constant to be specified later. Suppose there exist a decreasing sequence $\{\varepsilon _n\}$ converging to 0, and a sequence of pairs of local suitable weak solutions $(u_n, d_n, p_n )$ such that

$$\begin{aligned} C_{1}(W_n, 1, z_0)=\varepsilon _n^3, \end{aligned}$$

(5.8)

and

$$\begin{aligned} C_{1}(W_n, \theta , z_0)>\varepsilon _n^3. \end{aligned}$$

(5.9)

Define $(v_n, e_n, q_n)=(\frac{u_n}{\varepsilon _n}, \frac{d_n}{\varepsilon _n},\frac{p_n}{\varepsilon _n}),$ then they satisfy

$$\begin{aligned}&\partial _t v_n+\varepsilon _n v_n\cdot \nabla v_n+\nabla q_n=\Delta v_n-\varepsilon _n\mathrm {div}\,(\nabla e_n\odot \nabla e_n), \quad \mathrm {div} v_n=0,\\&\partial _t e_n+\varepsilon _n v_n\cdot \nabla e_n-\Delta e_n=-\sigma ^{-2}(|d_n|^2-1))e_n. \end{aligned}$$

Write $w_n=(v_n, \nabla e_n)$, then

$$\begin{aligned} C_{1}(w_n, 1, z_0)=1, \end{aligned}$$

(5.10)

and

$$\begin{aligned} C_{1}(w_n, \theta , z_0)>1. \end{aligned}$$

(5.11)

Using the Caccioppoli estimate (5.7) we conclude

$$\begin{aligned} \Vert w_n\Vert _{L^{10/3}(Q_{1/2}(z_0))}+\Vert \nabla w_n\Vert _{L^2(Q_{1/2}(z_0))}\le c, \end{aligned}$$

(5.12)

which implies

$$\begin{aligned} \Vert w_n\cdot \nabla w_n\Vert _{L^{5/4}(Q_{1/2}(z_0))}\le \Vert \nabla w_n\Vert _{L^2(Q_{1/2}(z_0))}\Vert w_n\Vert _{L^{10/3}(Q_{1/2}(z_0))}\le c. \end{aligned}$$

The coercive estimate for the Stokes system (see, for instance, [27]) with a suitable cutoff function implies

$$\begin{aligned} \int _{Q_{\frac{1}{3}}} |\partial _tw_n|^{\frac{5}{4}}+|\nabla ^2w_n|^{\frac{5}{4}}+|\nabla q_n|^{\frac{5}{4}}+|w_n|^{\frac{5}{4}}\le c, \end{aligned}$$

where the constant c is independent of n. Thanks to the compact embedding theorem and (5.12), there exist $w\in L^{3}(Q_{1/3}(z_0))$ and $q\in L^{\frac{5}{4}}(Q_{1/3}(z_0))$ such that

$$\begin{aligned}&w_n\rightarrow w=(v,\nabla e) \quad \text {in}\,\, L^{3}(Q_{1/3}(z_0)),\\&q_n\rightharpoonup q \quad \text {in}\,\, L^{\frac{5}{4}}(Q_{1/3}(z_0)). \end{aligned}$$

Thus, (v, e, q) satisfy

$$\begin{aligned}&\partial _t v-\Delta v+\nabla q=0, \quad \mathrm {div}\, v=0,\\&\partial _t e-\Delta e=\sigma ^{-2}e. \end{aligned}$$

Moreover

$$\begin{aligned} \Vert w\Vert _{L^{3}(Q_{1/3}(z_0))}+\Vert q\Vert _{L^{\frac{5}{4}}(Q_{3/4}(z_0))}\le c. \end{aligned}$$

By the classical estimate of the Stokes system [37], we get

$$\begin{aligned} \sup _{Q_{1/3}(z_0)}|w|\le c, \end{aligned}$$

which implies that for $0<\theta \le 1/3$

$$\begin{aligned} C_{1}(w,\theta , z_0)\le c\,\theta ^3. \end{aligned}$$

This contradicts (5.11), if we choose $\theta$ sufficiently small. The lemma is proved.$\square$