The Minkowski dimension of suitable weak solutions of the 3D co-rotational Beris-Edwards system

Abstract

In this paper, we study the possible singular points of suitable weak solutions to the 3D co-rotational Beris-Edwards system. Inspired by the work of He et al. (J. Nonlinear Sci. 29:2681–2698, 2019) and Wang et al. (Nonlinearity 32:4817–4833, 2019) for Navier–Stokes equations, we established a new partial regularity criteria for co-rotational Beris-Edwards system. As an application, we prove the known Minkowski dimension of the potential interior singular set of suitable weak solutions of the co-rotational Beris-Edwards system is $\frac{7}{6}(\approx 1.167)$.

1 Introduction

Liquid crystal is a state of matter capable of flow, but its molecules may be oriented in a crystal-like way. There are numerous attempts to formulate continuum theories describing the behavior of liquid crystals flow, see Stewart’s monograph [31] for example. Commonly, in literature the liquid crystals are categorized by three sub-families, namely the nematics, the cholesterics and the smectics. The nematic liquid crystal appears to be the most common one, where the molecules do not exhibit any positional order, but they have long-range orientational order. For more physical and chemical backward, the readers are referred to [4] and the references therein.

The Oseen-Frank theory [16], the Ericksen-Leslie model [8, 21, 22], and the Beris-Edwards model [2, 25, 26] are three mathematical models for capturing the continuum mechanics of nematic liquid crystals. In the present paper, we use one of the most comprehensive descriptions of nematics, the Beris-Edwards model, describes the hydrodynamic motion of nematic liquid crystals, which couples a forced Navier-Stokes equation for the fluid velocity $\textbf{u}$ with a dissipative parabolic system of $\textbf{Q}$-tensors modeling nematic liquid crystal orientation fields. Recall that the configuration space of $\textbf{Q}$-tensors is the set of traceless, symmetric $3\times 3$-matrices, i.e.,

$$\begin{aligned} S_0^{(3)}=\{\textbf{Q}\in \mathbb {R}^{3\times 3}:\textbf{Q}=\textbf{Q}^{\intercal },\textrm{tr}\textbf{Q}=0\}. \end{aligned}$$

Landau and De Gennes proposed the energy functional in terms of $\textbf{Q}$-tensor consisting of the elastic energy and the bulk energy, one of the wildly accepted simplified form is the following

$$\begin{aligned} E(\textbf{Q}):=\int \limits _{\mathbb {R}^3}\frac{L}{2}|\nabla \textbf{Q}|^2+F(\textbf{Q})\textrm{d}x=\int \limits _{\mathbb {R}^3}\frac{L}{2}|\nabla \textbf{Q}|^2+\frac{a}{2}|\textbf{Q}|^2-\frac{b}{3}\textrm{tr}(\textbf{Q}^3)+\frac{c}{4}|\textbf{Q}|^4\textrm{d}x, \end{aligned}$$

where $F(\textbf{Q})$ is the bulk energy of Landau-De Gennes, $a,b,c>0$ are temperature dependent material constants and $L>0$ denotes the elasticity constant. We denote $H=H(\textbf{Q})$ is the first order variation of the Landau-De Gennes potential functional $E(\textbf{Q})$, i.e.,

$$\begin{aligned} H(\textbf{Q}):=\frac{\delta E(\textbf{Q})}{\delta Q}=L\Delta \textbf{Q}-f(\textbf{Q}) \end{aligned}$$

with

$$\begin{aligned} f(\textbf{Q})=\mathscr {L}[\nabla F(\textbf{Q})]=\mathscr {L}[a\textbf{Q}-b\textbf{Q}^2+c\textbf{Q}\textrm{tr}(\textbf{Q}^2)]=a\textbf{Q} -b\left[ \textbf{Q}^2-\frac{\textrm{tr}(\textbf{Q}^2)}{3}\mathbb {I}_3\right] +c\textbf{Q}\textrm{tr}(\textbf{Q}^2). \end{aligned}$$

Here, $\mathscr {L}[A]$ denotes the projection onto the space of traceless matrices, namely, $\mathscr {L}[A]=A-\frac{1}{3}\textrm{tr}[A]\mathbb {I}_3$ and $\mathbb {I}_3$ denote the $3\times 3$ identity matrix.

With the notations as above, the so-called Beris-Edwards system proposed by Beris and Edwards, which one can find in the physics literature, for instance, in [6], reads as

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\textbf{Q}+\textbf{u}\cdot \nabla \textbf{Q}-S(\nabla \textbf{u},\textbf{Q})=\Gamma H,\\ \partial _t\textbf{u}+\textbf{u}\cdot \nabla \textbf{u}+\nabla P=\mu \Delta \textbf{u}+\textrm{div} (\tau (\textbf{Q})+\sigma (\textbf{Q})),~~~~\text {in}~~~~\mathbb {R}^3\times (0,\infty )\\ \textrm{div}\textbf{u}=0,\\ (\textbf{u}(x,t),\textbf{Q}(x,t))|_{t=0}=(\textbf{u}_0(x),\textbf{Q}_0(x)). \end{array}\right. } \end{aligned}$$

(1.1)

Here, $\textbf{u}$ denotes the fluid velocity field, P is the pressure, $\textbf{Q}$-tensor is a symmetric and traceless $3\times 3$-matrix, physically, it can be interpreted as a suitably normalized second-order moment of the probability distribution function describing the orientation of rod-like liquid crystal molecules, see [1] for more details. $\Gamma >0$ is the macroscopic elastic relaxation time parameter and $\mu >0$ is the fluid viscosity constant. The tensor $S(\nabla \textbf{u},\textbf{Q})$ describes how the flow gradient rotates and stretches the order-parameter, $\textbf{Q}$, given by

$$\begin{aligned} S(\nabla \textbf{u},\textbf{Q})=(\xi D(\textbf{u})+W(\textbf{u}))\left( \textbf{Q}+\frac{1}{3}\mathbb {I}_3\right) + \left( \textbf{Q}+\frac{1}{3}\mathbb {I}_3\right) (\xi D(\textbf{u})-W(\textbf{u}))-2\xi \left( \textbf{Q}+\frac{1}{3}\mathbb {I}_3\right) \textbf{Q}:\nabla \textbf{u}, \end{aligned}$$

where $D(\textbf{u})=\frac{\nabla \textbf{u}+\nabla ^{\top }\textbf{u}}{2}$, and $W(\textbf{u})=\frac{\nabla \textbf{u}-\nabla ^{\top }\textbf{u}}{2}$ is the symmetric and antisymmetric part of the velocity gradient tensor $\nabla \textbf{u}$ respectively, and scalar parameter $\xi \in \mathbb {R}$ denotes rotational parameter measuring the ratio between the aligning and tumbling effects to $\textbf{Q}$ by the fluid velocity field. $\tau (\textbf{Q})$ is the symmetric part of the additional stress tensor given by

$$\begin{aligned} \tau (\textbf{Q})=2\xi H:\textbf{Q}\left( \textbf{Q}+\frac{1}{3}\mathbb {I}_3\right) -\xi \left( H\left( \textbf{Q}+\frac{1}{3}\mathbb {I}_3\right) + \left( \textbf{Q}+\frac{1}{3}\mathbb {I}_3\right) H\right] -L\nabla \textbf{Q}\odot \nabla \textbf{Q}, \end{aligned}$$

and $\sigma (\textbf{Q})$ is the antisymmetric part of the additional stress tensor:

$$\begin{aligned} \sigma (\textbf{Q})=[\textbf{Q},H]=\textbf{Q}H-H\textbf{Q}, \end{aligned}$$

where the notation [A, B] be defined as $[A,B]=AB-BA$. Notice that $f(\textbf{Q})$ is isotropic function of $\textbf{Q}$, thus, we have $[\textbf{Q},f(\textbf{Q})]=0$ so that $\sigma (\textbf{Q})=[\textbf{Q},L\Delta \textbf{Q}-f(\textbf{Q})]=L[\textbf{Q},\Delta \textbf{Q}].$

In this paper, we will focus on the co-rotational Beris-Edwards system (1.1), i.e.,

$$\begin{aligned} \xi =0, \end{aligned}$$

which means that the molecules only tumble in a shear flow and do not align. Since the exact values of $L, \Gamma , \mu$ do not play roles in our analysis, we will assume $L=\Gamma =\mu =1$ for simplicity. Hence, the system (1.1) reduces to the following form:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\textbf{Q}+\textbf{u}\cdot \nabla \textbf{Q}-[W(\textbf{u}),\textbf{Q}]=\Delta \textbf{Q}-f(\textbf{Q}),\\ \partial _t\textbf{u}+\textbf{u}\cdot \nabla \textbf{u}+\nabla P=\Delta \textbf{u}-\nabla \textbf{Q}\cdot \Delta \textbf{Q}+\textrm{div}[\textbf{Q},\Delta \textbf{Q}],~~~~\text {in}~~~~\mathbb {R}^3\times (0,\infty )\\ \textrm{div}\textbf{u}=0,\\ (\textbf{u}(x,t),\textbf{Q}(x,t))|_{t=0}=(\textbf{u}_0(x),\textbf{Q}_0(x)). \end{array}\right. } \end{aligned}$$

(1.2)

In [26], the existence of global in time three-dimensional weak solutions and strong regularity and weak-strong uniqueness results in 2D are proved. F. Guill$\mathrm {\acute{e}}$n-Gonz$\mathrm {\acute{a}}$lez and M. $\mathrm {\acute{A}}$. Rodr$\mathrm {\acute{i}}$guez-Bellido [10] show the existence and uniqueness of a local in time weak solution on a bounded domain, and they also give a regularity criterion which yields such solutions to be global in time. Moreover they prove the global existence and uniqueness of a strong solution provided a viscosity large enough. In [11], F. Guill$\mathrm {\acute{e}}$n-Gonz$\mathrm {\acute{a}}$lez and M. $\mathrm {\acute{A}}$. Rodr$\mathrm {\acute{i}}$guez-Bellido prove the existence of global in time weak solutions, an uniqueness criteria and a maximum principle for $\textbf{Q}$. More process one can refer [5]. However, in the case that spatial dimension is three, a large gap remains between the regularity available in the existence results and additional regularity required in the sufficient conditions to guarantee the smoothness of weak solutions. Recently, inspired by [3] and [23] for Navier-Stokes equations, Du, Hu and Wang [7] introduced the suitable weak solution concept for co-rotational Beris-Edwards system (1.2) and proved the global existence of suitable weak solution. Moreover, the authors established the following partial regularity criteria: there exists some $\varepsilon >0$ such that the condition

$$\begin{aligned} \begin{aligned} \frac{1}{r^2}\int \limits _{\mathbb {P}_r(z_0)}|\textbf{u}|^3+|\nabla \textbf{Q}|^3+|P|^{\frac{3}{2}}\textrm{d}x\textrm{d}t\le \varepsilon , \end{aligned} \end{aligned}$$

(1.3)

yields that $z_0=(x_0,t_0)$ is a regular point for $(\textbf{u},\textbf{Q},P)$, where $\mathbb {P}_r(x_0):=B_r(x_0)\times (t_0-r^2,t_0)$ and $B_r(x_0)$ denotes the ball of center $z_0=(x_0,t_0)$ and radius r. Moreover, under the following smallness condition:

$$\begin{aligned} \limsup _{r\rightarrow 0}\frac{1}{r}\int \limits _{\mathbb {P}_r(z_0)}|\nabla \textbf{u}|^2+|\nabla ^2\textbf{Q}|^2\textrm{d}x\textrm{d}t\le \varepsilon , \end{aligned}$$

(1.4)

they obtained the one-dimensional Hausdorff measure of singular set is zero, which extends the results in [22] for simplified Ericksen-Leslie system and in [3] for Navier-Stokes system.

In this paper, we consider partial regularity criteria of suitable weak solutions of system (1.2). To illuminate the motivations of this paper, we shall recall some regularity criteria results of Navier–Stokes system (the order-parameter $\textbf{Q}$ is not taken into account, i.e., $\textbf{Q}=0$). First, let us recall a point (x, t) singular if $\textbf{u}$ or $\textbf{Q}$ is not $L^{\infty }_{loc}$ in any neighborhood of (x, t); the remaining points, where $\textbf{u}$ and $\textbf{Q}$ are locally essentially bounded, will be called regular points. In 1970 s, Scheffer [28,29,30] considered the potential space-times singular points set of solutions to the Navier-Stokes equations by introducing the suitable weak solutions and proved that the Hausdorff dimension of the singular set of suitable weak solutions of the 3D Navier–Stokes equations is at most $\frac{5}{3}$. Later, in the celebrated work [3], Caffarelli, Kohn and Nirenberg proved the one-dimensional Hausdorff measure of the possible singular set of suitable weak solutions to the 3D Navier–Stokes equations is zero. A new short proof of Caffarelli-Koch-Nirenberg theorem by an indirect argument was given by Lin [23]. In 2007, some different type partial regularity criteria were obtained by Gustafson, Kang and Tsai [14]. By viewing the total pressure $P+\frac{|\textbf{u}|^2}{2}$ as a signed distribution belonging to a certain negative order Sobolev space in local energy inequality, Guevara and Phuc [13] proved that there exists an absolutely positive constant $\varepsilon >0$ such that if $\textbf{u}$ is a suitable weak solution in $\mathbb {P}_{\rho }(z_0)$ and satisfies

$$\begin{aligned} \rho ^{-\frac{3}{2-\sigma }}\int \limits _{t_0-\rho ^2}^{t_0}\Vert |\textbf{u}|^2\Vert ^{\frac{2}{2-\sigma }}_{H^{-\sigma }(B_{\rho }(x_0))}+\Vert P\Vert ^{\frac{2}{2-\sigma }}_{H^{-\sigma }(B_{\rho }(x_0))}\textrm{d}t<\varepsilon ~~\mathrm {for~some}~\sigma \in [0,1], \end{aligned}$$

(1.5)

then $z_0=(x_0,t_0)$ is a regular point. Here, $H^{-\sigma }(B_{\rho }(x_0))$ $(\sigma \in \mathbb {R})$ is the dual space of space of functions f(x) in the $H^{\sigma }_0(\mathbb {R}^3)$ such that ${\textbf{supp}} f(x)\subset \overline{B_{\rho }(x_0)}$, where $H_0^{\sigma }(\mathbb {R}^3)$ is the homogeneous Sobolev space on $\mathbb {R}^3$. Recently, He, Wang and Zhou [15] show the following partial regularity: there exists a $\varepsilon >0$ such that if $\textbf{u}$ be a suitable weak solution in $\mathbb {P}_1(z_0)$ and satisfies

$$\begin{aligned} \begin{aligned} \Vert \textbf{u}\Vert _{L^{p,q}(\mathbb {P}_1(z_0))}+\Vert P\Vert _{L^1(P_1(z_0))}<\varepsilon ,~~~1\le \frac{2}{p}+\frac{3}{q}<2,~1\le p,~q\le \infty , \end{aligned} \end{aligned}$$

(1.6)

then $\textbf{u}\in L^{\infty }(\mathbb {P}_{\frac{1}{2}}(z_0))$. An application, they showed that the Minkowski dimension ( another important notion measuring lower dimensional set) of the potential singular sets to the Navier–Stokes system is bounded by $\frac{2400}{1903}(\approx 1.261)$. The concept of Minkowski dimension is more restrictive than the concept of Hausdorff dimension since it is based on coverings of sets by balls of equal rather than variable radius. In terms of the Navier–Stokes system, Kukavica [18] proved that the Minkowski dimension of the singular points is less than or equal to $\frac{135}{82} (\approx 1.646)$. Later, Koh and Yang [17] showed that the Minkowski dimension is bounded by $\frac{95}{63} ( \approx 1.508)$. Wang and Wu [33] improved the Minkowski dimension to $\frac{360}{277} (\approx 1.300)$. Recently, Wang and Yang [32] refined the bound to $\frac{7}{6}(\approx 1.167)$. See also [15, 19, 27] for some more progress. Inspired of the results for Navier–Stokes system, one of our main objectives is to prove the parabolic fractal dimension of the singular points of co-rotational Beris-Edwards (1.2) is less than or equal to $\frac{7}{6}(\approx 1.167)$. To this end, we shall derive a new partial regularity criterion for suitable weak solutions by modifying the arguments in [32]. our result can be interpreted as a generalizing to the case of the co-rotational Beris-Edwards system. Before we state the main theorems of this paper, we present the definition of Minkowski dimension, more details one can refer to [9] and the references therein.

Definition 1.1

(The Minkowski dimension) Let N(S, r) represent the minimum number of parabolic cylinders $\mathbb {P}_r(z_0)$ required to cover the bounded set $\Omega \subset \mathbb {R}^3\times (0,\infty )$. Then, the Minkowski dimension of the set $\Omega$ is defined as

$$\begin{aligned} \textrm{dim}_f(\Omega )=\limsup _{r\rightarrow 0}\frac{\textrm{log}N(S,r)}{-\textrm{log}r}. \end{aligned}$$

Our main results are as follows.

Theorem 1.1

There exists an $\varepsilon >0$ such that if tripe $(\textbf{u}, \textbf{Q},P)$ is a suitable weak solution of the co-rotational Beris-Edwards system (1.2) in $\mathbb {P}_{1}(z_0)$ and satisfies

$$\begin{aligned} \sup _{t\in [t_0-1,t_0]}\int \limits _{B_1(x_0)} |\textbf{u}|^2+|\nabla \textbf{Q}|^2\textrm{d}x+\int \limits _{\mathbb {P}_1(z_0)}|\nabla \textbf{u}|^2+|\nabla ^2\textbf{Q}|^2 \textrm{d}x\textrm{d}t+\int \limits _{\mathbb {P}_1(z_0)}|P|\textrm{d}x\textrm{d}t\le \varepsilon , \end{aligned}$$

(1.7)

then $(\textbf{u},\textbf{Q})$ is regular at $z_0$.

This Theorem implies the following partial regularity result.

Theorem 1.2

Let the triple $(\textbf{u},\textbf{Q},P)$ be a suitable weak solution to the 3D co-rotational Beris-Edwards system (1.2) in $\mathbb {P}_1(z_0)$. There exists an absolute positive constant $\varepsilon >0$ such that if the triple $(\textbf{u},\textbf{Q},P)$ satisfies

$$\begin{aligned} \begin{aligned} \Vert \textbf{u}\Vert _{L^{p,q}(\mathbb {P}_1(z_0))}+\Vert \nabla \textbf{Q}\Vert _{L^{p,q}(\mathbb {P}_1(z_0))} +\Vert \textbf{P}\Vert _{L^1(P_1(z_0))}<\varepsilon , \end{aligned} \end{aligned}$$

(1.8)

where $1\le \frac{2}{p}+\frac{3}{q}<2,~1\le p,q\le \infty$. Then, $\textbf{u},\nabla \textbf{Q}\in L^{\infty }(\mathbb {P}_{\frac{1}{2}}(z_0))$.

Using the regularity results above, we can prove the following partial regularity theorem:

Theorem 1.3

Suppose that $(\textbf{u},\textbf{Q},P)$ is a suitable weak solution to the co-rotational Beris-Edwards system (1.2). Then for each $\gamma <\frac{1}{2}$, there exist positive numbers $\varepsilon _1$ and $r_1<1$ such that $z_0=(x_0,t_0)$ is a regular point if for some $r<r_1$,

$$\begin{aligned} \begin{aligned} \int \limits _{\mathbb {P}_{r}(z_0)}|\nabla \textbf{u}|^2+|\nabla ^2\textbf{Q}|^2+|\textbf{u}|^{\frac{10}{3}} +|\nabla \textbf{Q}|^{\frac{10}{3}}+|P-\overline{P}_{B_{r}(x_0)}| +|\nabla P|^{\frac{5}{4}}\textrm{d}x\textrm{d}t<r^{\frac{5}{3}-\gamma }\varepsilon _1. \end{aligned} \end{aligned}$$

(1.9)

Applying the new regularity result above, we can improve the bound of the Minkowski dimension of singular points to system (1.2) by the following theorem:

Theorem 1.4

For any $T>0$, let S be the potential singular set of $(\textbf{u},\textbf{Q},\textbf{P})$, then for any compact set $\mathscr {K}\subset \mathbb {R}^3\times (0,T)$, the Minkowski dimension of $S\cap \mathscr {K}$ in (1.2) is at most $\frac{7}{6}$.

Remark 1.1

Our starting point is the following $\varepsilon$-regularity criterion:

$$\begin{aligned} \begin{aligned} \Vert \nabla \textbf{u}\Vert _{L^2(\mathbb {P}_1(z_0))}+\Vert \nabla ^2\textbf{Q}\Vert _{L^2(\mathbb {P}_1(z_0))} +\Vert \textbf{u}\Vert _{L^2(\mathbb {P}_1(z_0))}+\Vert \nabla \textbf{Q}\Vert _{L^2(\mathbb {P}_1(z_0))} +\Vert \nabla \textbf{P}\Vert _{L^{1,\frac{3}{2}}(\mathbb {P}_1(z_0))}<\varepsilon , \end{aligned} \end{aligned}$$

(1.10)

which is derived from

$$\begin{aligned} \begin{aligned} \Vert \textbf{u}\Vert _{L^{2,6}(\mathbb {P}_1(z_0))}+\Vert \nabla \textbf{Q}\Vert _{L^{2,6}(\mathbb {P}_1(z_0))}+ \Vert \nabla \textbf{P}\Vert _{L^{1,\frac{3}{2}}(\mathbb {P}_1(z_0))}<\varepsilon . \end{aligned} \end{aligned}$$

(1.11)

It is worth noting that we get (1.11) by Theorem 1.2 and the Poincar$\acute{e}$-Sobolev inequality.

We end this section by giving some notations. Throughout this paper, we denote

$$\begin{aligned}{} & {} B_r(x_0):=B(x_0,r)=\{x\in \mathbb {R}^3:|x-x_0|\le r\},~\mathbb {P}_r(z_0):=B(x_0,r)\times (t_0-r^2,t_0), \\{} & {} B_r:=B(0,r),~\mathbb {P}_r:=B(0,r)\times (-r^2,0). \end{aligned}$$

The classical Sobolev norm $\Vert \cdot \Vert _{H^s}$ is defined as $\Vert v\Vert ^2_{H^s}=\int \limits _{\mathbb {R}^n}(1+|\xi |)^{2s}|\hat{v}(\xi )|^2\textrm{d}\xi ,~s\in \mathbb {R}$. We denote by $\dot{H}^s$ homogenous Sobolev spaces with the norm $\Vert v\Vert ^2_{\dot{H}^s}=\int \limits _{\mathbb {R}^n}|\xi |^{2s}|\hat{v}(\xi )|^2\textrm{d}\xi$. For simplicity, we write $\Vert v\Vert _{L^{p,q}(Q_r(z_0))}:=\Vert v\Vert _{L^p(t_0-r^2,t_0;L^q(B_r(x_0)))}~\text {and} ~\Vert v\Vert _{L^{p}(Q_r(z_0))}:=\Vert v\Vert _{L^{p,p}(Q_r(z_0))}$ where $p,~q\in [1,\infty ]$. Denote the average of f on the set $\Omega$ by $\bar{f}_{\Omega }$. we shall use the notation $A\lesssim B$ if there is a generic positive constant C such that $|A|\lesssim C|B|$.

2 Preliminaries

First, we begin with the definitions of the suitable weak solutions of the co-rotational Beris-Edwards system (1.2).

Definition 2.1

$({\textbf {Suitable weak solution}})$ We say $(\textbf{u},\textbf{Q},P)$ is called a suitable weak solution to the co-rotational Beris-Edwards system (1.2) provided the following conditions are satisfied:

• (i) $\textbf{u}\in L^{\infty }(0,T;L^2(\mathbb {R}^3))\cap L^2(0,T;H^1(\mathbb {R}^3))$, $P \in L^{\frac{3}{2}}((0,T)\times \mathbb {R}^3)$, and $\textbf{Q}\in L^{\infty }(0,T;H^1(\mathbb {R}^3))\cap L^2(0,T;H^2(\mathbb {R}^3))$;

• (ii) $(\textbf{u},\textbf{Q},P)$ satisfies (1.2) in $\mathbb {R}^3\times [0,T]$ in the sense of distribution;

• (iii) for any $0\le \phi \in C_0^{\infty }(\mathbb {R}^3\times [0,T])$, $(\textbf{u},\textbf{Q},P)$ satisfies the following local energy inequality:

  $$\begin{aligned}&\int \limits _{\mathbb {R}^3}(|\textbf{u}|^2+|\nabla \textbf{Q}|^2)\phi (x,t)\textrm{d}x+2\int \limits _0^T\int \limits _{\mathbb {R}^3} (|\nabla \textbf{u}|^2+|\Delta \textbf{Q}|^2)\phi (x,t)\textrm{d}x\textrm{d}t \nonumber \\&\le \int \limits _0^T\int \limits _{\mathbb {R}^3}(|\textbf{u}|^2+|\nabla \textbf{Q}|^2)(\partial _t\phi +\Delta \phi )(x,t)\textrm{d}x \textrm{d}t\nonumber \\&\quad +\int \limits _0^T\int \limits _{\mathbb {R}^3}[(|\textbf{u}|^2+2P)\textbf{u}\cdot \nabla \phi +2\nabla \textbf{Q}\odot \nabla \textbf{Q}:\textbf{u}\otimes \nabla \phi ](x,t)\textrm{d}x\textrm{d}t\nonumber \\&\quad +2\int \limits _0^T\int \limits _{\mathbb {R}^3}(\nabla \textbf{Q}\odot \nabla \textbf{Q}-|\nabla \textbf{Q}|^2\mathbb {I}_3):\nabla ^2\phi (x,t)\textrm{d}x \textrm{d}t\nonumber \\&\quad -2\int \limits _0^T\int \limits _{\mathbb {R}^3}[\textbf{Q},\Delta \textbf{Q}]:\textbf{u}\otimes \nabla \phi (x,t)\textrm{d}x\textrm{d}t\nonumber \\&\quad -2\int \limits _0^T\int \limits _{\mathbb {R}^3}[[W(\textbf{u}),\textbf{Q}]:(\nabla \textbf{Q}\nabla \phi )+\nabla f(\textbf{Q})\cdot \nabla \textbf{Q}\phi ](x,t)\textrm{d}x\textrm{d}t. \end{aligned}$$

  (2.1)

Definition 2.2

(Scaled functionals) In the light of the natural scaling property of system (1.2), we introduce the following dimensionless quantities:

$$\begin{aligned} A_{z_0}(r)=\frac{1}{r}\sup _{t_0-r^2\le t\le t_0}\int \limits _{B_r(x_0)}(|\textbf{u}(t)|^2+|\nabla \textbf{Q}(t)|^2)\textrm{d}x,~~~ B_{z_0}(r)=\frac{1}{r}\int \limits _{\mathbb {P}_r(z_0)}(|\nabla \textbf{u}|^2+|\nabla ^2\textbf{Q}|^2)\textrm{d}x\textrm{d}t, \end{aligned}$$

$$\begin{aligned} C_{p,z_0}(r)=\frac{1}{r^{5-p}}\int \limits _{\mathbb {P}_r(z_0)}(|\textbf{u}|^p+|\nabla \textbf{Q}|^p)\textrm{d}x\textrm{d}t, ~~~D_{z_0}(r)=\frac{1}{r^{3}}\int \limits _{\mathbb {P}_r(z_0)}|P|\textrm{d}x\textrm{d}t, \end{aligned}$$

$$\begin{aligned} E_{p,z_0}(r)=\frac{1}{r^{5-2p}}\int \limits _{\mathbb {P}_r(z_0)}|P-\overline{P}_{B_r(x_0)}| ^p\textrm{d}x\textrm{d}t,~~D_{1,\frac{3}{2},z_0}(r)=\frac{1}{r}\int \limits _{t_0-r^2}^{t_0} \Bigg (\int \limits _{B_{r}(x_0)}|\nabla P|^{\frac{3}{2}}\textrm{d}x\Bigg )^{\frac{2}{3}}\textrm{d}t. \end{aligned}$$

We recall the following interpolation Lemma, whose proof can be found in [3].

Lemma 2.1

For $\textbf{f}\in H^1(\mathbb {R}^3)$,

$$\begin{aligned} \begin{aligned} \int \limits _{B_r(x_0)}|\textbf{f}|^q\textrm{d}x\lesssim \left( \int \limits _{B_r(x_0)}|\nabla \textbf{f}|^2\textrm{d}x\right) ^{\frac{q}{2}-a}\left( \int \limits _{B_r(x_0)}|\textbf{f}|^2\textrm{d}x\right) ^a+r^{3(1-\frac{q}{2})}\left( \int \limits _{B_r(x_0)}|\textbf{f}|^2\textrm{d}x\right) ^{\frac{q}{2}} \end{aligned} \end{aligned}$$

(2.2)

for every $B_r(x_0)\subset \mathbb {R}^3$, $2\le q\le 6$, $a=\frac{3}{2}(1-\frac{q}{6})$.

Next, we give some auxiliary lemmas which are helpful in the proof of Theorems 1.1–1.3.

Lemma 2.2

For $0<r\le \frac{1}{8}\rho$, there exists an absolute constant C independent of r and $\rho$ such that

$$\begin{aligned} \begin{aligned}&D_{1,\frac{3}{2},z_0}(r)\lesssim \bigg (\frac{\rho }{r}\bigg )B_{z_0}(\rho )+\bigg (\frac{r}{\rho }\bigg )D_{\frac{5}{4},z_0}(\rho ). \end{aligned} \end{aligned}$$

(2.3)

$$\begin{aligned} \begin{aligned} r^{-\frac{3}{2}}\int \limits _{t_0-r^2}^{t_0}\Vert P\Vert _{L^2(B_r(x_0))}\textrm{d}t \lesssim \bigg (\frac{r}{\rho }\bigg )\rho ^{-3}\int \limits _{t_0-\rho ^2}^{t_0}\Vert P\Vert _{L^1(B_{\rho }(x_0))}\textrm{d}t +\bigg (\frac{\rho }{r}\bigg )^{\frac{3}{2}}(A_{z_0}(\rho )+B_{z_0}(\rho )). \end{aligned} \end{aligned}$$

(2.4)

Proof

Let

$$\begin{aligned} P(x,t):=P_1(x,t)+P_2(x,t), \end{aligned}$$

where $P_1(x,t):=R_iR_j[(\textbf{U}_{i,j}+\textbf{D}_{i,j}-\frac{1}{2}\widetilde{\textbf{D}}_{i,j})\chi _{B_{\frac{\rho }{2}}}]$. Here, $R_i=\partial _i(-\Delta )^{-\frac{1}{2}},i=1,2,3$, is the i-th Riesz transform, $\textbf{U}_{i,j}=(\textbf{u}_i-(\overline{\textbf{u}_i})_{B_{\frac{\rho }{2}}(x_0)}) (\textbf{u}_j-(\overline{\textbf{u}_j})_{B_{\frac{\rho }{2}}(x_0)}),~ \textbf{D}_{i,j}=\partial _i\textbf{Q}_{\alpha \beta } \partial _j\textbf{Q}_{\alpha \beta }$, $\widetilde{\textbf{D}}_{i,j}=\partial _i\textbf{Q}_{\alpha \beta } \partial _j\textbf{Q}_{\alpha \beta }\delta _{ij}$, $\alpha ,\beta =1,2,3$. Note that, for any $\phi \in C_0^{\infty }(B_{\frac{\rho }{2}}(x_0))$, we have

$$\begin{aligned} \begin{aligned} -\int \limits _{B_{\frac{\rho }{2}}(x_0)}P_1\Delta \phi \textrm{d}x&=\int \limits _{B_{\frac{\rho }{2}}(x_0)} \Big (\textbf{U}_{i,j}+\textbf{D}_{i,j}-\frac{1}{2}\widetilde{\textbf{D}}_{i,j}\Big )\partial _{ij}\phi \textrm{d}x\\ {}&=\int \limits _{B_{\frac{\rho }{2}}(x_0)} \Big (\textbf{u}_i\textbf{u}_j+\partial _i\textbf{Q}_{\alpha \beta }\partial _j\textbf{Q}_{\alpha \beta }-\frac{1}{2}\partial _i\textbf{Q}_{\alpha \beta }\partial _j\textbf{Q}_{\alpha \beta }\delta _{ij}\Big )\partial _{ij}\phi \textrm{d}x, \end{aligned} \end{aligned}$$

(2.5)

where we used the facts that $-R_iR_j(\Delta \phi )=\partial _{ij}\phi$, $\nabla \cdot \textbf{u}=0$ and $\textrm{div}^2[\textbf{Q},\Delta \textbf{Q}]=0$. For the proof of $\textrm{div}^2[\textbf{Q},\Delta \textbf{Q}]=0$, we can refer to Lemma 2.3 in [7]. Thus, as P also solves

$$\begin{aligned} -\Delta P=\nabla \cdot \nabla \cdot \Big (\textbf{u}\otimes \textbf{u}+\nabla \textbf{Q} \otimes \nabla \textbf{Q}-\frac{1}{2}|\nabla \textbf{Q}|^2\mathbb {I}_3\Big ) \end{aligned}$$

in the distributional sense, we know that $P_2$ is harmonic in $B_{\frac{\rho }{2}}(x_0)$ for a.e. t. Then for $r\in (0,\frac{\rho }{8}]$, it holds that

where we used the notation . Hence,

$$\begin{aligned} \begin{aligned}&\int \limits _{B_r(x_0)}|P-\overline{P}_{B_r(x_0)}|^2\textrm{d}x\lesssim \int \limits _{B_{\frac{\rho }{2}}(x_0)}|P_1|^2\textrm{d}x+\frac{r^5}{\rho ^8} \Vert P_2-\overline{P}_{2,B_{\frac{\rho }{2}}(x_0)}\Vert _{L^1(B_{\frac{\rho }{2}}(x_0))}\\&\lesssim \int \limits _{B_{\frac{\rho }{2}}(x_0)}|P_1|^2\textrm{d}x+\frac{r^5}{\rho ^8} \left( \Vert P_1\Vert _{L^1(B_{\frac{\rho }{2}}(x_0))}+\Vert P-\overline{P}_{B_{\frac{\rho }{2}}(x_0)}\Vert _{L^1(B_{\frac{\rho }{2}}(x_0))}\right) \\&\lesssim \int \limits _{B_{\frac{\rho }{2}}(x_0)}|P_1|^2\textrm{d}x+\frac{r^5}{\rho ^8} \Vert P-\overline{P}_{B_{\frac{\rho }{2}}(x_0)}\Vert _{L^1(B_{\frac{\rho }{2}}(x_0))}. \end{aligned} \end{aligned}$$

(2.6)

On the other hand, by the Calder$\mathrm {\acute{o}}$n-Zygmund theorem, Sobolev inequality and Lemma 2.1 we get

$$\begin{aligned}{} & {} \int \limits _{B_{\frac{\rho }{2}}(x_0)}|P_1|^2\textrm{d}x\lesssim \int \limits _{B_{\frac{\rho }{2}}(x_0)} |\textbf{u}-(\overline{\textbf{u}})_{B_{\frac{\rho }{2}}(x_0)}|^4+ |\nabla \textbf{Q}|^4\textrm{d}x \nonumber \\ {}\lesssim & {} \left( \int \limits _{B_{\frac{\rho }{2}}(x_0)}|\nabla \textbf{u}|^2\textrm{d}x\right) ^{\frac{3}{2}} \left( \int \limits _{B_{\frac{\rho }{2}}(x_0)}|\textbf{u}|^2\textrm{d}x\right) ^{\frac{1}{2}} \nonumber \\{} & {} \;+ \left( \int \limits _{B_{\frac{\rho }{2}}(x_0)}|\nabla ^2\textbf{Q}|^2\textrm{d}x\right) ^{\frac{3}{2}} \left( \int \limits _{B_{\frac{\rho }{2}}(x_0)}|\nabla \textbf{Q}|^2\textrm{d}x\right) ^{\frac{1}{2}}+\frac{1}{\rho ^3}\left( \int \limits _{B_r(x_0)}|\nabla \textbf{Q}|^2\textrm{d}x\right) ^2. \end{aligned}$$

(2.7)

Combining (2.6) and (2.7), we get

$$\begin{aligned}{} & {} \Vert P-\overline{P}_{B_r(x_0)}\Vert _{L^2(B_r(x_0))}\nonumber \\{} & {} \lesssim \frac{r^{\frac{5}{2}}}{\rho ^4}\Vert P-\overline{P}_{B_{\frac{\rho }{2}}(x_0)}\Vert _{L^1(B_{\rho }(x_0))}+ \left( \int \limits _{B_{\frac{\rho }{2}}(x_0)}|\nabla \textbf{u}|^2\textrm{d}x\right) ^{\frac{3}{4}} \left( \int \limits _{B_{\frac{\rho }{2}}(x_0)}|\textbf{u}|^2\textrm{d}x\right) ^{\frac{1}{4}} \nonumber \\{} & {} \quad \;+C \left( \int \limits _{B_{\frac{\rho }{2}}(x_0)}|\nabla ^2\textbf{Q}|^2\textrm{d}x\right) ^{\frac{3}{4}} \left( \int \limits _{B_{\frac{\rho }{2}}(x_0)}|\nabla \textbf{Q}|^2\textrm{d}x\right) ^{\frac{1}{4}}+\frac{1}{\rho ^{\frac{3}{2}}}\int \limits _{B_r(x_0)}|\nabla \textbf{Q}|^2\textrm{d}x. \end{aligned}$$

(2.8)

Integrating over $t\in (t_0-r^2,t_0)$ and using Hölder’s inequality and Young’s inequality we obtain (2.4).

To prove (2.3), without loss of generality, we assume that $z_0=0$. Fix a smooth function $\phi$ supported in $B_{\frac{\rho }{2}}$ and with value 1 on the ball $B_{\frac{3}{8}\rho }$. Moreover, there holds $0\le \phi \le 1$ and $|\nabla \phi |^2+|\nabla ^2\phi |\le C \rho ^{-2}$. Using the divergence-free condition one has

$$\begin{aligned} \partial _i\partial _i(\partial _kP\phi )=-\phi \partial _i\partial _j \Big [\partial _k\textbf{U}_{i,j}+\partial _k\textbf{D}_{i,j}-\frac{1}{2}\partial _k\widetilde{\textbf{D}}_{i,j}\Big ]+2\partial _i\phi \partial _i\partial _kP+ \partial _kP\partial _i\partial _i\phi , \end{aligned}$$

then for $x\in B_{\frac{3}{8}\rho }$, we can use the Green function representation:

$$\begin{aligned} \begin{aligned}&\partial _kP(x)=\mathscr {G}*\Big \{-\phi \partial _i\partial _j\Big [\partial _k\textbf{U}_{i,j} +\partial _k\textbf{D}_{i,j}-\frac{1}{2}\partial _k\widetilde{\textbf{D}}_{i,j}\Big ]+2\partial _i\phi \partial _i\partial _kP +\partial _kP\partial _i\partial _i\phi \Big \}\\ {}&\quad \quad \quad =-\partial _i\partial _j\mathscr {G}*\Big (\phi \Big [\partial _k\textbf{U}_{i,j} +\partial _k\textbf{D}_{i,j}-\frac{1}{2}\partial _k\widetilde{\textbf{D}}_{i,j}\Big ]\Big )+2\partial _i\mathscr {G}*\Big (\partial _j\phi \Big [\partial _k\textbf{U}_{i,j} +\partial _k\textbf{D}_{i,j}-\frac{1}{2}\partial _k\widetilde{\textbf{D}}_{i,j}\Big ]\Big )\\ {}&\quad \quad \quad \quad -\mathscr {G}*\Big (\partial _i\partial _j\phi \Big [\partial _k\textbf{U}_{i,j} +\partial _k\textbf{D}_{i,j}-\frac{1}{2}\partial _k\widetilde{\textbf{D}}_{i,j}\Big ]\Big )+2\partial _k\partial _i\mathscr {G}*(\partial _i\phi P) \\&\quad \quad \quad \quad +2\partial _i\mathscr {G}*(\partial _k\partial _i\phi P)+ \partial _k\mathscr {G}*(\partial _i\partial _i\phi P)+ \mathscr {G}*(\partial _i\partial _i\partial _k\phi P) \\ {}&\quad \quad \quad =:\partial _kP_3(x)+\partial _kP_4(x)+\partial _kP_5(x), \end{aligned} \end{aligned}$$

(2.9)

where

$$\begin{aligned}{} & {} \partial _kP_3(x)=-\partial _i\partial _j\mathscr {G}*\Big (\phi \Big [\partial _k\textbf{U}_{i,j} +\partial _k\textbf{D}_{i,j}-\frac{1}{2}\partial _k\widetilde{\textbf{D}}_{i,j}\Big ]\Big ), \\ {}{} & {} \partial _kP_4(x)=2\partial _i\mathscr {G}*\Big (\partial _j\phi \Big [\partial _k\textbf{U}_{i,j} +\partial _k\textbf{D}_{i,j}-\frac{1}{2}\partial _k\widetilde{\textbf{D}}_{i,j}\Big ]\Big ) -\mathscr {G}*\Big (\partial _i\partial _j\phi \Big [\partial _k\textbf{U}_{i,j} +\partial _k\textbf{D}_{i,j}-\frac{1}{2}\partial _k\widetilde{\textbf{D}}_{i,j}\Big ]\Big ), \\ {}{} & {} \partial _kP_5(x)=2\partial _k\partial _i\mathscr {G}*(\partial _i\phi P) +2\partial _i\mathscr {G}*(\partial _k\partial _i\phi P)+ \partial _k\mathscr {G}*(\partial _i\partial _i\phi P)+ \mathscr {G}*(\partial _i\partial _i\partial _k\phi P). \end{aligned}$$

and $\mathscr {G}$ is the standard normalized fundamental solution of the Laplace equation. Since $\phi (x)=1$ on $B_{\frac{\rho }{4}}$, we know that there is no singularity in $\partial _k P_4$ and $\partial _kP_5$. Thus, we get

$$\begin{aligned}{} & {} |\partial _kP_4(x)|\le \rho ^{-3}\int \limits _{B_{\frac{\rho }{2}}}|\partial _k\textbf{U}_{i,j} +\partial _k\textbf{D}_{i,j}-\frac{1}{2}\partial _k\widetilde{\textbf{D}}_{i,j}|\textrm{d}x, \\{} & {} |\partial _kP_5(x) |\le \rho ^{-4}\int \limits _{B_{\frac{\rho }{2}}}|P|\textrm{d}x. \end{aligned}$$

By Hölder’s inequality yields

$$\begin{aligned} \Vert \partial _kP_4(x)\Vert _{L^{\frac{3}{2}}(B_r)}\le C {\bigg (\frac{r}{\rho }\bigg )}^2 \Vert \partial _k{\textbf {U}}_{i,j} +\partial _k{\textbf {D}}_{i,j}-\frac{1}{2}\partial _k\widetilde{{\textbf {D}}}_{i,j}\Vert _{L^{\frac{3}{2}} (B_{\frac{\rho }{4}} )}, \end{aligned}$$

$$\begin{aligned} \begin{aligned} \Vert \partial _kP_5(x)\Vert _{L^{\frac{3}{2}}(B_r)}\le C \big (\frac{r}{\rho }\big )^2 \Vert P\Vert _{L^{\frac{15}{7}} (B_{\frac{\rho }{4}})}. \end{aligned} \end{aligned}$$

(2.10)

Using the Hölder’s inequality and the Poincar$\mathrm {\acute{e}}$-Sobolev inequality, we see that

$$\begin{aligned} \begin{aligned}&\Vert \partial _k\textbf{U}_{i,j}+\partial _k\textbf{D}_{i,j}-\frac{1}{2}\partial _k\widetilde{\textbf{D}}_{i,j}\Vert _{L^{1,\frac{3}{2}}\big (\mathbb {P}_{\frac{\rho }{2}}\big )}\\&\lesssim \Vert \textbf{u}-\overline{\textbf{u}}_{B_{\frac{\rho }{2}}}\Vert _{L^{2,6}\big (\mathbb {P}_{\frac{\rho }{2}}\big )} \Vert \nabla \textbf{u}\Vert _{L^{2}\big (\mathbb {P}_{\frac{\rho }{2}}\big )}+ \Vert \nabla \textbf{Q}-\overline{\nabla \textbf{Q}}_{B_{\frac{\rho }{2}}}\Vert _{L^{2,6}\big (\mathbb {P}_{\frac{\rho }{2}}\big )} \Vert \nabla ^2\textbf{Q}\Vert _{L^{2}\big (\mathbb {P}_{\frac{\rho }{2}}\big )}+\Vert \nabla ^2\textbf{Q}\Vert _{L^2\big (\mathbb {P}_{\frac{\rho }{2}}\big )}\\ {}&\lesssim \Vert \nabla \textbf{u}\Vert _{L^{2}\big (\mathbb {P}_{\rho }\big )}^2+\Vert \nabla ^2\textbf{Q}\Vert _{L^{2}\big (\mathbb {P}_{\rho }\big )}^2, \end{aligned} \end{aligned}$$

(2.11)

where we have used the fact that

$$\begin{aligned} \nabla (\nabla \textbf{Q}\otimes \nabla \textbf{Q})&=\nabla \Big [\Big (\nabla \textbf{Q}-\overline{\nabla \textbf{Q}}_{B_{\frac{\rho }{2}}}\Big )\otimes \Big (\nabla \textbf{Q}-\overline{\nabla \textbf{Q}}_{B_{\frac{\rho }{2}}}\Big ) \end{aligned}$$

(2.12)

$$\begin{aligned}&\quad \qquad +\overline{\nabla \textbf{Q}}_{B_{\frac{\rho }{2}}}\otimes \Big (\nabla \textbf{Q}-\overline{\nabla \textbf{Q}}_{B_{\frac{\rho }{2}}}\Big )+\Big (\nabla \textbf{Q}-\overline{\nabla \textbf{Q}}_{B_{\frac{\rho }{2}}}\Big )\otimes \overline{\nabla \textbf{Q}}_{B_{\frac{\rho }{2}}}\Big ]. \end{aligned}$$

(2.13)

Thus,

$$\begin{aligned} \begin{aligned} \Vert \partial _kP_4(x)\Vert _{L^{1,\frac{3}{2}}(\mathbb {P}_{r})}\lesssim \big (\frac{r}{\rho }\big )^2 \left( \Vert \nabla \textbf{u}\Vert _{L^{2}\big (\mathbb {P}_{\frac{\rho }{2}}\big )}^2 +\Vert \nabla ^2\textbf{Q}\Vert _{L^{2}\big (\mathbb {P}_{\frac{\rho }{2}}\big )}^2\right) . \end{aligned} \end{aligned}$$

(2.14)

Notice that $P-\overline{P}_{\frac{\rho }{2}}$ also satisfies (2.9), we derive from (2.10) that

$$\begin{aligned} \begin{aligned} \Vert \partial _kP_5(x)\Vert _{L^{1,\frac{3}{2}}\big (\mathbb {P}_{r}\big )}\lesssim \big (\frac{r}{\rho }\big )^2\Vert P- \overline{P}_{B_{\frac{\rho }{2}}}\Vert _{L^{\frac{5}{4},\frac{15}{7}}\big (\mathbb {P}_{\frac{\rho }{2}}\big )} \lesssim \big (\frac{r}{\rho }\big )^2\Vert \nabla P\Vert _{L^{\frac{5}{4}}\big (\mathbb {P}_{\frac{\rho }{2}}\big )}. \end{aligned} \end{aligned}$$

(2.15)

According to the Calder$\mathrm {\acute{o}}$n-Zygmund theorem and (2.11), we know that

$$\begin{aligned} \begin{aligned} \Vert \partial _kP_3(x)\Vert _{L^{1,\frac{3}{2}}(\mathbb {P}_{r})}\lesssim \Vert \partial _k\textbf{U}_{i,j}+\partial _k\textbf{D}_{i,j}-\frac{1}{2}\widetilde{\textbf{D}}_{i,j}\Vert _{L^{1,\frac{3}{2}}\big (\mathbb {P}_{\frac{\rho }{2}}\big )} \lesssim \Vert \nabla \textbf{u}\Vert _{L^{2}(\mathbb {P}_{\rho })}^2+\Vert \nabla ^2\textbf{Q}\Vert _{L^{2}(\mathbb {P}_{\rho })}^2. \end{aligned} \end{aligned}$$

(2.16)

Combining the estimates (2.14), (2.15) and (2.16) yields

$$\begin{aligned} \begin{aligned} \Vert \partial _kP(x)\Vert _{L^{1,\frac{3}{2}}(\mathbb {P}_{r})}&\lesssim \Vert \partial _kP_3(x)\Vert _{L^{1,\frac{3}{2}}(\mathbb {P}_{r})} +\Vert \partial _kP_4(x)\Vert _{L^{1,\frac{3}{2}}(\mathbb {P}_{r})}+ \Vert \partial _kP_5(x)\Vert _{L^{1,\frac{3}{2}}(\mathbb {P}_{r})}\\ {}&\lesssim \Vert \nabla \textbf{u}\Vert _{L^{2}(\mathbb {P}_{\rho })}^2+\Vert \nabla ^2\textbf{d}\Vert _{L^{2}(\mathbb {P}_{\rho })}^2 +\big (\frac{r}{\rho }\big )^2\Vert \partial _kP(x)\Vert _{L^{\frac{5}{4}}(\mathbb {P}_{\rho })}, \end{aligned} \end{aligned}$$

(2.17)

this is $D_{1,\frac{3}{2}}(r)\le C (\frac{\rho }{r})B(\rho )+(\frac{r}{\rho })D_{\frac{5}{4}}(\rho )$. Thus, the proof of Lemma 2.2 is completed after a translation.

$\square$

Lemma 2.3

(Cf. Lemma 2.2 of [15] and Lemma 5.1 of [20]) Let $1<\frac{2}{p}+\frac{3}{q}<2$, $1\le q\le \infty$. There is an absolute constant C such that

$$\begin{aligned} \begin{aligned} \Vert \textbf{u}\Vert ^3_{L^3(\mathbb {P}_{\rho })}\lesssim \rho ^{\frac{3(\alpha -1)}{2}}\Vert \textbf{u}\Vert ^{\alpha }_{L^{p,q}(\mathbb {P}_{\rho })} \left( \Vert \textbf{u}\Vert ^2_{L^{\infty ,2}(\mathbb {P}_{\rho })}+\Vert \nabla \textbf{u}\Vert ^2_{L^2(\mathbb {P}_{\rho })}\right) ^{\frac{3-\alpha }{2}}; \end{aligned} \end{aligned}$$

(2.18)

$$\begin{aligned} \begin{aligned} \Vert \nabla \textbf{Q}\Vert ^3_{L^3(\mathbb {P}_{\rho })}\lesssim \rho ^{\frac{3(\alpha -1)}{2}}\Vert \nabla \textbf{Q}\Vert ^{\alpha }_{L^{p,q}(\mathbb {P}_{\rho })} \left( \Vert \nabla \textbf{Q}\Vert ^2_{L^{\infty ,2}(\mathbb {P}_{\rho })}+\Vert \nabla ^2\textbf{Q}\Vert ^2_{L^2(\mathbb {P}_{\rho })}\right) ^{\frac{3-\alpha }{2}}, \end{aligned} \end{aligned}$$

(2.19)

where $\alpha =\frac{2}{\frac{3}{p}+\frac{2}{q}}>1$.

$$\begin{aligned} \begin{aligned} C_{3,z_0}(r)\le C\bigg (\frac{\rho }{r}\bigg )^3A_{z_0}^{\frac{3}{4}}(\rho )B_{z_0}(\rho )+C\bigg (\frac{r}{\rho }\bigg )^3A_{z_0}^{\frac{3}{2}}(\rho ). \end{aligned} \end{aligned}$$

(2.20)

Remark 2.1

The proof of (2.19) and (2.20) can be shown by the exactly the same method as that of [15] and [20] respectively, thus we omit the detail here.

Lemma 2.4

(Cf. Lemma V.3.1 of [12]) Let I(s) be a bounded nonnegative function in the interval [r, R]. Assume that for every $\sigma , \rho \in [r,R]$ and $\sigma <\rho$ we have

$$\begin{aligned} I(\sigma )\le A_1(\rho -\sigma )^{-\alpha _1}+A_2(\rho -\sigma )^{-\alpha _2}+A_3+{\mathcalligra {I}}I(\rho ), \end{aligned}$$

for some nonnegative constants $A_1,A_2,A_3$, nonnegative exponents $\alpha _1\ge \alpha _2$ and a parameter $\xi \in [0,1)$. Then, there holds

$$\begin{aligned} I(r)\le C(\alpha _1,\xi )[A_1(R-r)^{-\alpha _1}+A_2(R-r)^{-\alpha _2}+A_3]. \end{aligned}$$

Next, let us recall the following partial regularity criteria for suitable weak solutions to the co-rotational Beris-Edwards system (1.2), which have proved in [24].

Lemma 2.5

Let $(\textbf{u},\textbf{Q},P)$ be a suitable weak solution of the 3d co-rotational Beris-Edwards system (1.2) and $z_0\in \mathbb {R}^3\times (0,T)$. There exists $\varepsilon _*>0$ such that for some positive constant $0<r_*\le 1$, $\mathbb {P}_{r_*}(z_0)\subset \mathbb {R}^3\times (0,T)$ and

$$\begin{aligned} \begin{aligned} \sup _{0< r< r_*}A(\textbf{u},\nabla \textbf{Q};r)\le \varepsilon _*, \end{aligned} \end{aligned}$$

(2.21)

then $(\textbf{u},\nabla \textbf{Q})$ is regular at $z_0$.

3 The proof of Theorem 1.1

According to Lemma 2.5, it is enough to show that

$$\begin{aligned} \sup _{0<r<\frac{1}{4}}[A_z(r)+B_z(r)]\le C \varepsilon ^{\frac{1}{2}}\le \varepsilon _*, \end{aligned}$$

for every $z\in \mathbb {P}_{\frac{1}{2}}$. Without lose of generality, it suffices to consider the $z=(0,0)\in \mathbb {R}^3\times (0,\infty )$. Let $\theta \in (0,\frac{1}{4}]$ be determined later and define $r_n=\theta ^n,~n\in \mathbb {N}$. Hence, we only to prove that

$$\begin{aligned} \begin{aligned} A(\theta ^n)+B(\theta ^n)\le C\varepsilon ^{\frac{1}{2}}, \end{aligned} \end{aligned}$$

(3.1)

for a fixed $\theta \in (0,\frac{1}{4}]$ and for all $n=1,2,\cdots$. Due to hypothesis, inequality (3.1) holds in the case $n=1$ provided $\varepsilon _0$ is sufficiently small. Using inductive argument, Suppose now that it holds for $n=1,\cdots ,m-1$ with $m\ge 2$. Let $\phi _m=\chi \psi _m$, where $0\le \chi \le 1$ is a smooth cutoff function which equals 1 on $\mathbb {P}_{\theta ^2}$ and vanishes in $\mathbb {R}^3\times (-\infty ,0){\setminus } \mathbb {P}_{\theta }$, and $\psi _m$ is given by

$$\begin{aligned} \psi _m(x,t)=(r_m^2-t)^{-\frac{3}{2}}e^{-\frac{|x|^2}{4(r_m^2-t)}},~~~t<r_m^2. \end{aligned}$$

Then it is easy to see that $\phi _m\ge 0$, $(\partial _t+\Delta )\phi _m=0$ in $\mathbb {P}_{\theta ^2}$, and

$$\begin{aligned}&|(\partial _t+\Delta )\phi _m|\le C~~~\text {on}~~\mathbb {P}_{\theta }, \end{aligned}$$

(3.2)

$$\begin{aligned}&2^{-\frac{3}{2}}r_m^{-3}\le \phi _m\le r_m^{-3},~~~|\nabla \phi _m|\le C r_m^{-4} ~\text {on}~\mathbb {P}_{r_m},~~m\ge 2, \end{aligned}$$

(3.3)

$$\begin{aligned}&\phi _m\le C r_k^{-3},~~|\nabla \phi _m|\le C r_k^{-4}~~\text {on}~~\mathbb {P}_{r_k-1}\setminus \mathbb {P}_{r_k},~~1<k\le m. \end{aligned}$$

(3.4)

Using $\phi _m$ as a test function in the generalized energy inequality and combining (3.2), (3.3), we find that

$$\begin{aligned} \begin{aligned} A(r_m)+B(r_m)\le C (I+II+III+IV+V+VI), \end{aligned} \end{aligned}$$

(3.5)

where

$$\begin{aligned}{} & {} I=r_m^2\int \limits _{\mathbb {P}_{\theta }}|\textbf{u}|^2+|\nabla \textbf{Q}|^2\textrm{d}x\textrm{d}t,~II=r_m^2\int \limits _{\mathbb {P}_{\theta }}(|\textbf{u}|^3+|\nabla \textbf{Q}|^3)|\nabla \phi _m|\textrm{d}x\textrm{d}t, \\{} & {} III=r_m^2\left| \int \limits _{\mathbb {P}_{\theta }}P(\textbf{u}\cdot \nabla \phi _m)\textrm{d}x\textrm{d}t\right| ,~IV=r_m^2\int \limits _{\mathbb {P}_{\theta }}|\nabla \textbf{Q}|^2|\nabla ^2\phi _m|\textrm{d}x\textrm{d}t,\\{} & {} V=r_m^2\int \limits _{\mathbb {P}_{\theta }}(|\Delta \textbf{Q}||\textbf{u}|+|\nabla \textbf{u}||\nabla \textbf{Q}|)|\nabla \phi _m|\textrm{d}x\textrm{d}t,~VI=r_m^2\int \limits _{\mathbb {P}_{\theta }}|\nabla \textbf{Q}|^2\phi _m\textrm{d}x\textrm{d}t. \end{aligned}$$

By the hypothesis, we have

$$\begin{aligned} I\le r_m^2\varepsilon \le \varepsilon ^{\frac{3}{4}}, \end{aligned}$$

From the above properties of test function $\phi _m$, we have

$$\begin{aligned} \begin{aligned} II&=r_m^2\sum _{k=1}^{m-1}\int \limits _{\mathbb {P}_{r_k}\setminus \mathbb {P}_{r_{k+1}}}\left( |\textbf{u}|^3+|\nabla \textbf{Q}|^3\right) |\nabla \phi _m|\textrm{d}x\textrm{d}t+ r_m^2\int \limits _{\mathbb {P}_{r_m}}\left( |\textbf{u}|^3+|\nabla \textbf{Q}|^3\right) |\nabla \phi _m|\textrm{d}x\textrm{d}t\\ {}&\lesssim r_m^2\sum _{k=1}^{m-1}r_k^{-4}\int \limits _{\mathbb {P}_{r_k}}\left( |\textbf{u}|^3+|\nabla \textbf{d}|^3\right) \textrm{d}x\textrm{d}t. \end{aligned} \end{aligned}$$

(3.6)

Thus by (2.20) and inductive hypothesis, it follows that

$$\begin{aligned} II\lesssim r_m^2 \sum _{k=1}^{m-1}r_k^{-2}\varepsilon ^{\frac{3}{4}}\le C\varepsilon ^{\frac{3}{4}}. \end{aligned}$$

As for the term III, we write

$$\begin{aligned} \phi _m=\chi _1\phi _m=\sum _{k=1}^{m-1}(\chi _k-\chi _{k+1})\phi _m+\chi _m\phi _m, \end{aligned}$$

where $\chi _k, k=1,2,\cdots ,m$, is a smooth cutoff function such that $0\le \chi _k\le 1,~\chi _k=1$ in $\mathbb {P}_{\frac{7k}{8}}, \chi _k=0$ in $\mathbb {R}^3\times (-\infty ,0){\setminus } \mathbb {P}_{r_k}$, and $|\nabla \chi _k|\le \frac{C}{r_k}$. Then

$$\begin{aligned} \begin{aligned} III&\lesssim r_m^2\left| \sum _{k=1}^{m-1}\int \limits _{\mathbb {P}_{r_k}}P\textbf{u}\cdot \nabla [(\chi _k-\chi _{k+1})\phi _m] \textrm{d}x\textrm{d}t\right| +r_m^2\left| \int \limits _{\mathbb {P}_{r_m}}P\textbf{u}\cdot \nabla (\chi _m\phi _m)\textrm{d}x\textrm{d}t \right| \\&\lesssim r_m^2\sum _{k=2}^mr_k^4\int \limits _{\mathbb {P}_{r_k}}|P\textbf{u}|\textrm{d}x\textrm{d}t+\theta ^{-2} \int \limits _{\mathbb {P}_{\theta }}|P\textbf{u}|\textrm{d}x\textrm{d}t\\ {}&\lesssim r_m^2\sum _{k=2}^mr_k^{-4}\int \limits _{-r_k^2}^0\Vert P\Vert _{L^2(B_{r_k})}\Vert \textbf{u}\Vert _{L^2(B_{r_k})}\textrm{d}t +\theta ^{-2}\int \limits _{-\theta ^2}^0\Vert P\Vert _{L^2(B_{\theta })}\Vert \textbf{u}\Vert _{L^2(B_{\theta })}\textrm{d}t. \end{aligned} \end{aligned}$$

(3.7)

By inductive hypothesis, this gives

$$\begin{aligned} \begin{aligned} III\lesssim r_m^2\sum _{k=2}^m r_k^{-2}\varepsilon ^{\frac{1}{4}}r_k^{-\frac{3}{2}}\int \limits _{-r_k^2}^0 \Vert P\Vert _{L^2(B_{r_k})}\textrm{d}t+ \varepsilon ^{\frac{1}{2}}\theta ^{-\frac{3}{2}}\int \limits _{-\theta ^2}^0\Vert P\Vert _{L^2(B_{\theta })}\textrm{d}t. \end{aligned} \end{aligned}$$

(3.8)

We now let $U(r_k)=r_k^{-\frac{3}{2}}\int \limits _{-r_k^2}^0\Vert P\Vert _{L^2(B_{r_k})}\textrm{d}t$. By (2.4) and H$\mathrm {\ddot{o}}$lder’s inequality, for $2\le k\le m$ we know

$$\begin{aligned} U(r_k)\lesssim \theta U(r_{k-1})+\theta ^{-\frac{3}{2}}A(r_{k-1})^{\frac{1}{4}}B(r_{k-1})^{\frac{3}{4}}, \end{aligned}$$

Choosing $\theta =\frac{1}{2C}$ and iterating this inequality we obtain

$$\begin{aligned} U(r_k)=\Big (\frac{1}{2}\Big )^{k-1}U(r_1)+C\theta ^{-\frac{3}{2}}\sum _{l=1}^{k-1}\Big (\frac{1}{2}\Big )^{l-1} A(r_{k-l})^{\frac{1}{4}}B(r_{k-l})^{\frac{3}{4}}.\end{aligned}$$

Then by inductive hypothesis we get

$$\begin{aligned} U(r_k)\le U(r_1)+C\sum _{l=1}^{k=1}\Big (\frac{1}{2}\Big )^{l-1}\varepsilon ^{\frac{1}{2}}\le \theta ^{-\frac{3}{2}}\int \limits _{-\theta ^2}^0\Vert P\Vert _{L^2(B_{\theta })}\textrm{d}t +C\varepsilon ^{\frac{1}{2}},\end{aligned}$$

Combining this with (3.8), we arrive at

$$\begin{aligned} III\le C\varepsilon ^{\frac{1}{4}}\theta ^{-\frac{3}{2}}\int \limits _{-\theta ^2}^{0}\Vert P\Vert _{L^2(B_{\theta })} \textrm{d}t+C\varepsilon ^{\frac{3}{4}}, \end{aligned}$$

which by (2.4) gives

$$\begin{aligned} III\le C \varepsilon ^{\frac{1}{4}}[D(2\theta )+A(2\theta )^{\frac{1}{4}}B(2\theta )^{\frac{3}{4}}]+ C\varepsilon ^{\frac{3}{4}}\le C (\varepsilon ^{\frac{5}{4}}+\varepsilon ^{\frac{3}{4}})\le C \varepsilon ^{\frac{3}{4}}. \end{aligned}$$

$$\begin{aligned} \begin{aligned}&IV=r_m^2\sum _{k=1}^{m-1}\int \limits _{\mathbb {P}_{r_k}\setminus \mathbb {P}_{r_{k+1}}}|\nabla \textbf{Q}|^2|\nabla ^2\phi _m|\textrm{d}x\textrm{d}t+ r_m^2\int \limits _{\mathbb {P}_{r_m}}|\nabla \textbf{Q}|^2|\nabla ^2\phi _m|\textrm{d}x\textrm{d}t\\ {}&\quad \quad \lesssim r_m^2\sum _{k=1}^{m-1}r_k^{-5}\int \limits _{\mathbb {P}_{r_k}}|\nabla \textbf{Q}|^2\textrm{d}x\textrm{d}t\lesssim r_m^2\sum _{k=1}^{m-1}r_k^{-3}\int \limits _{B_{r_k}}|\nabla \textbf{Q}|^2\textrm{d}x\lesssim r_m^2\sum _{k=1}^{m-1}r_k^{-2} \varepsilon ^{\frac{1}{2}}\le C\varepsilon ^{\frac{1}{2}}. \end{aligned} \end{aligned}$$

(3.9)

$$\begin{aligned} \begin{aligned}&V\lesssim r_m\int \limits _{\mathbb {P}_{\theta }}(|\textbf{u}|^2+|\nabla \textbf{Q}|^2)|\nabla \phi |\textrm{d}x\textrm{d}t+r_m^3\int \limits _{\mathbb {P}_{\theta }}(|\Delta \textbf{Q}|^2+|\nabla \textbf{u}|^2)|\nabla \phi |\textrm{d}x\textrm{d}t\\ {}&\quad \lesssim r_m\sum _{k=1}^{m-1}\int \limits _{\mathbb {P}_{r_k}\backslash \mathbb {P}_{r_k+1}}(|\textbf{u}|^2+|\nabla \textbf{Q}|^2)|\nabla \phi _m|\textrm{d}x\textrm{d}t+r_m\int \limits _{\mathbb {P}_{r_m}}(|\textbf{u}|^2+|\nabla \textbf{Q}|^2)|\nabla \phi _m|\textrm{d}x\textrm{d}t\\ {}&\quad \quad +r_m^3\sum _{k=1}^{m-1}\int \limits _{\mathbb {P}_{r_k}\backslash \mathbb {P}_{r_k+1}}(|\nabla \textbf{u}|^2+|\nabla ^2\textbf{Q}|^2)|\nabla \phi _m|\textrm{d}x\textrm{d}t+r_m^3\int \limits _{\mathbb {P}_{r_m}}(|\nabla \textbf{u}|^2+|\nabla ^2\textbf{Q}|^2)|\nabla \phi _m|\textrm{d}x\textrm{d}t\\ {}&\quad \lesssim r_m\sum _{k=1}^{m-1}r_k^{-2}\int \limits _{B_{r_k}}(|\textbf{u}|^2+|\nabla \textbf{Q}|^2)\textrm{d}x+r_m^3\sum _{k=1}^{m-1}r_k^{-4}\int \limits _{\mathbb {P}_{r_k}}(|\nabla \textbf{u}|^2+|\nabla ^2\textbf{Q}|^2)\textrm{d}x\textrm{d}t\\ {}&\quad \le Cr_m\sum _{k=1}^{m-1}r_k^{-1}\varepsilon ^{\frac{3}{4}}+r_m^3\sum _{k=1}^{m-1}r_k^{-3}\varepsilon ^{\frac{3}{4}}\lesssim C\varepsilon ^{\frac{1}{2}}, \end{aligned} \end{aligned}$$

(3.10)

where we have used the Young’s inequality $\textbf{a}\cdot \textbf{b}\le \frac{C}{r_m}\textbf{a}^2+r_m\textbf{b}^2$.

$$\begin{aligned} \begin{aligned} VI&=r_m^2\sum _{k=1}^{m-1}\int \limits _{\mathbb {P}_{r_k}\backslash \mathbb {P}_{r_k+1}}|\nabla \textbf{Q}|^2\phi _m\textrm{d}x\textrm{d}t+r_m^2\int \limits _{\mathbb {P}_{r_m}}|\nabla \textbf{Q}|^2|\phi _m|\textrm{d}x\textrm{d}t\\ {}&\lesssim r_m^2\sum _{k=1}^{m-1}r_k^{-3}\int \limits _{\mathbb {P}_{r_k}}|\nabla \textbf{Q}|^2\textrm{d}x\textrm{d}t \le C \varepsilon ^{\frac{3}{4}}. \end{aligned} \end{aligned}$$

(3.11)

Combining (3.5) and the estimates for I–VI, we obtain

$$\begin{aligned} A(r_m)+B(r_m)\le C \varepsilon ^{\frac{3}{4}}\le C\varepsilon ^{\frac{1}{2}}\le \varepsilon _* \end{aligned}$$

provided $\varepsilon$ is small enough. This proves (3.1) and the proof of Theorem 1.1 is complete.         $\square$

4 The proof of Theorem 1.2

For simplicity, we assume that $z_0=(0,0)\in \mathbb {R}^3\times (0,\infty )$. Let $0<\frac{R}{2}\le r<\frac{3r+\rho }{4}<\frac{r+\rho }{2}<\rho \le R$ and $\eta (x,t)$ be nonnegative smooth function supported in $\mathbb {P}_{\frac{r+\rho }{2}}$ such that $\eta (x,t)\equiv 1$ on $\mathbb {P}_{\frac{3r+\rho }{4}}$, $|\nabla \eta |\le \frac{C}{\rho -r}$ and $|\nabla ^2\eta |+|\partial _t\eta |\le \frac{C}{(\rho -r)^2}$. By means of H$\mathrm {\ddot{o}}$lder’s inequality, we have

$$\begin{aligned} \begin{aligned}&\int \limits _{-T}^t\int \limits _{\mathbb {R}^3}(\nabla \textbf{Q}\otimes \nabla \textbf{Q}-|\nabla \textbf{Q}|^2\mathbb {I}_3):\nabla ^2\eta \textrm{d}x\textrm{d}t+ \int \limits _{-T}^t\int \limits _{\mathbb {R}^3}\nabla f(\textbf{Q})\cdot \nabla \textbf{Q} \eta \textrm{d}x\textrm{d}t\\ {}&\quad + \int \limits _{-T}^t\int \limits _{\mathbb {R}^3}(|\textbf{u}|^2+|\nabla \textbf{Q}|^2)(\partial _s\eta +\Delta \eta )\textrm{d}x \textrm{d}t\\ {}&\le \frac{C}{(\rho -r)^2}\int \limits _{\mathbb {P}_{\frac{r+\rho }{2}}}(|\textbf{u}|^2+|\nabla \textbf{Q}|^2)\textrm{d}x \textrm{d}t\\ {}&\le \frac{C\rho ^{\frac{5}{3}}}{(\rho -r)^2}\left( \int \limits _{\mathbb {P}_{\rho }}|\textbf{u}|^3+|\nabla \textbf{Q}|^3\textrm{d}x \textrm{d}t\right) ^{\frac{2}{3}}=:M_1. \end{aligned} \end{aligned}$$

(4.1)

Thanks to $\partial _i\partial _iP=-\partial _i\partial _j[\textbf{u}_i\textbf{u}_j +\partial _i\textbf{Q}_{\alpha \beta }\partial _j\textbf{Q}_{\alpha \beta }-\frac{1}{2}\partial _i\textbf{Q}_{\alpha \beta }\partial _j\textbf{Q}_{\alpha \beta }]$ and Leibniz’s formula, we know that

$$\begin{aligned} \partial _i\partial _i(P\phi )=-\phi \partial _i\partial _j\Big [\textbf{u}_i\textbf{u}_j +\partial _i\textbf{Q}\partial _j\textbf{Q}-\frac{1}{2}\partial _i\textbf{Q}_{\alpha \beta }\partial _j\textbf{Q}_{\alpha \beta }\Big ]+2\partial _i\phi \partial _iP +P\partial _i\partial _i\phi .\end{aligned}$$

Thus, for any $x\in B_{\frac{r+3\rho }{4}}$,

$$\begin{aligned} \begin{aligned} P(x)&=\mathscr {G}*\Big \{-\eta \partial _i\partial _j\Big [\textbf{u}_i\textbf{u}_j +\partial _i\textbf{Q}_{\alpha \beta }\partial _j\textbf{Q}_{\alpha \beta }-\frac{1}{2}\partial _i\textbf{Q}_{\alpha \beta }\partial _j\textbf{Q}_{\alpha \beta }\Big ]+2\partial _i\eta \partial _iP +P\partial _i\partial _i\eta \Big \}\\ {}&=-\partial _i\partial _j\mathscr {G}*\Big (\eta \Big [\textbf{u}_i\textbf{u}_j +\partial _i\textbf{Q}_{\alpha \beta }\partial _j\textbf{Q}_{\alpha \beta }-\frac{1}{2}\partial _i\textbf{Q}_{\alpha \beta }\partial _j\textbf{Q}_{\alpha \beta }\Big ]\Big )\\ {}&\quad +2\partial _i\mathscr {G}*\Big (\partial _j\eta \Big [\textbf{u}_i\textbf{u}_j +\partial _i\textbf{Q}_{\alpha \beta }\partial _j\textbf{Q}_{\alpha \beta }-\frac{1}{2}\partial _i\textbf{Q}_{\alpha \beta }\partial _j\textbf{Q}_{\alpha \beta }\Big ]\Big )\\ {}&\quad -\mathscr {G}*\Big (\partial _i\partial _j\eta \Big [\textbf{u}_i\textbf{u}_j +\partial _i\textbf{Q}_{\alpha \beta }\partial _j\textbf{Q}_{\alpha \beta }-\frac{1}{2}\partial _i\textbf{Q}_{\alpha \beta }\partial _{j}\textbf{Q}_{\alpha \beta }\Big ]\Big )+2\partial _i\mathscr {G}*(\partial _i\eta P) -\mathscr {G}*(\partial _i\partial _i\eta P)\\ {}&=:\tilde{P}_1(x)+\tilde{P}_2(x)+\tilde{P}_3(x), \end{aligned} \end{aligned}$$

(4.2)

where $\mathscr {G}$ denote the standard normalized fundamental solution of Laplace equation and

$$\begin{aligned}{} & {} \tilde{P}_1=-\partial _i\partial _j\mathscr {G}*(\eta [\textbf{u}_i\textbf{u}_j +\partial _i\textbf{Q}_{\alpha \beta }\partial _j\textbf{Q}_{\alpha \beta }-\frac{1}{2}\partial _i\textbf{Q}_{\alpha \beta }\partial _j\textbf{Q}_{\alpha \beta }]), \\{} & {} \begin{aligned} \tilde{P}_2&=2\partial _i\mathscr {G}*\Big (\partial _j\eta \Big [\textbf{u}_i\textbf{u}_j +\partial _i\textbf{Q}_{\alpha \beta }\partial _j\textbf{Q}_{\alpha \beta }-\frac{1}{2}\partial _i\textbf{Q}_{\alpha \beta }\partial _j\textbf{Q}_{\alpha \beta }\Big ]\Big )\\ {}&\quad -\mathscr {G}*\Big (\partial _i\partial _j\eta \Big [\textbf{u}_i\textbf{u}_j +\partial _i\textbf{Q}_{\alpha \beta }\partial _j\textbf{Q}_{\alpha \beta }-\frac{1}{2}\partial _i\textbf{Q}_{\alpha \beta }\partial _j\textbf{Q}_{\alpha \beta }\Big ]\Big ), \end{aligned} \\{} & {} \tilde{P}_3=2\partial _i\mathscr {G}*(\partial _i\eta P) -\mathscr {G}*(\partial _i\partial _i\eta P). \end{aligned}$$

Due to the local energy inequality (2.1) and the decomposition of pressure, we get that

$$\begin{aligned} \begin{aligned}&\int \limits _{B_{\frac{r+\rho }{2}}}(|\textbf{u}|^2+|\nabla \textbf{Q}|^2)\eta \textrm{d}x+2 \int \limits _{\mathbb {P}_{\frac{r+\rho }{2}}}(|\nabla \textbf{u}|^2+|\nabla ^2\textbf{Q}|^2)\eta \textrm{d}x\textrm{d}t\\ {}&\le M_1+M_2+M_3+M_4+M_5+M_6, \end{aligned} \end{aligned}$$

(4.3)

where

$$\begin{aligned}{} & {} M_2=\frac{C}{(\rho -r)}\int \limits _{\mathbb {P}_{\frac{r+\rho }{2}}}|\textbf{u}|^3+|\nabla \textbf{d}|^3\textrm{d}x\textrm{d}t,~~~\\{} & {} M_3=\frac{C}{(\rho -r)}\int \limits _{\mathbb {P}_{\frac{r+\rho }{2}}}\textbf{u}\tilde{P}_1\textrm{d}x\textrm{d}t, \\{} & {} M_4=\frac{C}{(\rho -r)}\int \limits _{\mathbb {P}_{\frac{r+\rho }{2}}}\textbf{u}\tilde{P}_2\textrm{d}x\textrm{d}t,\\{} & {} M_5=\frac{C}{(\rho -r)}\int \limits _{\mathbb {P}_{\frac{r+\rho }{2}}}\textbf{u}\tilde{P}_3\textrm{d}x\textrm{d}t, \\{} & {} M_6=\frac{C}{(\rho -r)}\int \limits _{\mathbb {P}_{\frac{r+\rho }{2}}}|\Delta \textbf{Q}||\textbf{u}|+|\nabla \textbf{u}||\nabla \textbf{Q}|\textrm{d}x\textrm{d}t. \end{aligned}$$

By the H$\mathrm {\ddot{o}}$lder inequality, Young’s inequality and Calder$\mathrm {\acute{o}}$n-Zygmund theorem yields

$$\begin{aligned} \begin{aligned}&M_3\le \frac{C}{(\rho -r)}\Vert \tilde{P}_1\Vert _{L^{\frac{3}{2}}\bigg (\mathbb {P}_{\frac{r+\rho }{2}}\bigg )} \Vert \textbf{u}\Vert _{L^3\bigg (\mathbb {P}_{\frac{r+\rho }{2}}\bigg )}\\ {}&\le \frac{C}{(\rho -r)}\left( \Vert \textbf{u}\Vert ^2_{L^3(\mathbb {P}_{\rho })}+ \Vert \nabla \textbf{Q}\Vert ^2_{L^3(\mathbb {P}_{\rho })}\right) \Vert \textbf{u}\Vert _{L^3\bigg (\mathbb {P}_{\frac{r+\rho }{2}}\bigg )} \\ {}&\le \frac{C}{(\rho -r)}\left( \Vert \textbf{u}\Vert ^3_{L^3(\mathbb {P}_{\rho })}+\Vert \nabla \textbf{Q}\Vert ^3_{L^3(\mathbb {P}_{\rho })}\right) . \end{aligned} \end{aligned}$$

(4.4)

Noting that $\tilde{P}_2$ and $\tilde{P}_3$ have not singularity in $B_{\frac{r+\rho }{2}}$ because of the property of the cutoff function. Hence, a straightforward computation we have

$$\begin{aligned} \Vert \tilde{P}_2\Vert _{L^{\frac{3}{2}}\bigg (\mathbb {P}_{\frac{r+\rho }{2}}\bigg )}\le \frac{C\rho ^3}{(\rho -r)^3} \left( \Vert \textbf{u}\Vert ^2_{L^3(\mathbb {P}_{\rho })}+\Vert \nabla \textbf{Q}\Vert ^2_{L^3(\mathbb {P}_{\rho })}\right) ,~~ \Vert \tilde{P}_3\Vert _{L^{1,2}\bigg (\mathbb {P}_{\frac{r+\rho }{2}}\bigg )}\le \frac{C\rho ^{\frac{3}{2}}}{(\rho -r)^3} \Vert P\Vert _{L^1(\mathbb {P}_{\rho })}.\end{aligned}$$

This yields

$$\begin{aligned} \begin{aligned} M_4&\le \frac{C}{(\rho -r)}\Vert \tilde{P}_2\Vert _{L^{\frac{3}{2}}\bigg (\mathbb {P}_{\frac{r+\rho }{2}}\bigg )} \Vert \textbf{u}\Vert _{L^3\bigg (\mathbb {P}_{\frac{r+\rho }{2}}\bigg )}\\ {}&\le \frac{C\rho ^3}{(\rho -r)^4}\left( \Vert \textbf{u}\Vert ^2_{L^3(\mathbb {P}_{\rho })}+ \Vert \nabla \textbf{Q}\Vert ^2_{L^3(\mathbb {P}_{\rho })}\right) \Vert \textbf{u}\Vert _{L^3\bigg (\mathbb {P}_{\frac{r+\rho }{2}}\bigg )} \\ {}&\le \frac{C\rho ^3}{(\rho -r)^4}\left( \Vert \textbf{u}\Vert ^3_{L^3(\mathbb {P}_{\rho })}+\Vert \nabla \textbf{Q}\Vert ^3_{L^3(\mathbb {P}_{\rho })}\right) . \end{aligned} \end{aligned}$$

(4.5)

$$\begin{aligned} \begin{aligned} M_5\le \frac{C}{(\rho -r)}\Vert \tilde{P}_3\Vert _{L^{1,2}\bigg (\mathbb {P}_{\frac{r+\rho }{2}}\bigg )} \Vert \textbf{u}\Vert _{L^{\infty ,2}\bigg (\mathbb {P}_{\frac{r+\rho }{2}}\bigg )}\le \frac{C\rho ^{\frac{3}{2}}}{(\rho -r)^4} \Vert P\Vert _{L^1(\mathbb {P}_{\rho })}\Vert \textbf{u}\Vert _{L^{\infty ,2}(\mathbb {P}_{\rho })}. \end{aligned} \end{aligned}$$

(4.6)

Plugging (2.18) and (2.19) into (4.1), (4.5) and (4.6) respectively, using the Young’s inequality, we get

$$\begin{aligned} \begin{aligned} M_1\le&\frac{C\rho ^{3+\frac{2}{\alpha }}}{(\rho -r)^{\frac{6}{\alpha }}}\left( \Vert \textbf{u}\Vert ^2_{L^{p,q}(\mathbb {P}_{\rho })}+\Vert \nabla \textbf{Q}\Vert ^2_{L^{p,q}(\mathbb {P}_{\rho })}\right) \\ {}&+\frac{1}{8}\left( \Vert \textbf{u}\Vert ^2_{L^{\infty ,2}(\mathbb {P}_{\rho })}+\Vert \nabla \textbf{Q}\Vert ^2_{L^{\infty ,2}(\mathbb {P}_{\rho })} +\Vert \nabla \textbf{u}\Vert ^2_{L^2(\mathbb {P}_{\rho })} +\Vert \nabla ^2\textbf{Q}\Vert ^2_{L^2(\mathbb {P}_{\rho })}\right) ; \end{aligned} \end{aligned}$$

(4.7)

$$\begin{aligned} \begin{aligned} M_2+M_3\le&\frac{C\rho ^3}{(\rho -r)^{\frac{2}{\alpha -1}}}\left( \Vert \textbf{u}\Vert ^{\frac{2\alpha }{\alpha -1}}_{L^{p,q}(\mathbb {P}_{\rho })} +\Vert \nabla \textbf{Q}\Vert ^{\frac{2\alpha }{\alpha -1}}_{L^{p,q}(\mathbb {P}_{\rho })}\right) \\ {}&+\frac{1}{8}\left( \Vert \textbf{u}\Vert ^2_{L^{\infty ,2}(\mathbb {P}_{\rho })}+\Vert \nabla \textbf{Q}\Vert ^2_{L^{\infty ,2}(\mathbb {P}_{\rho })} +\Vert \nabla \textbf{u}\Vert ^2_{L^2(\mathbb {P}_{\rho })} +\Vert \nabla ^2\textbf{Q}\Vert ^2_{L^2(\mathbb {P}_{\rho })}\right) ; \end{aligned} \end{aligned}$$

(4.8)

$$\begin{aligned} \begin{aligned} M_4\le&\frac{C\rho ^{\frac{3(\alpha +1)}{\alpha -1}}}{(\rho -r)^{\frac{8}{\alpha -1}}}\left( \Vert \textbf{u}\Vert ^{\frac{2\alpha }{\alpha -1}}_{L^{p,q}(\mathbb {P}_{\rho })} +\Vert \nabla \textbf{Q}\Vert ^{\frac{2\alpha }{\alpha -1}}_{L^{p,q}(\mathbb {P}_{\rho })}\right) \\ {}&+\frac{1}{8}\left( \Vert \textbf{u}\Vert ^2_{L^{\infty ,2}(\mathbb {P}_{\rho })}+\Vert \nabla \textbf{Q}\Vert ^2_{L^{\infty ,2}(\mathbb {P}_{\rho })} +\Vert \nabla \textbf{u}\Vert ^2_{L^2(\mathbb {P}_{\rho })} +\Vert \nabla ^2\textbf{Q}\Vert ^2_{L^2(\mathbb {P}_{\rho })}\right) ; \end{aligned} \end{aligned}$$

(4.9)

For term $M_5$, utilizing the Young’s inequality again, we get

$$\begin{aligned} \begin{aligned} M_5\le \frac{C\rho ^3}{(\rho -r)^8}\Vert P\Vert ^2_{L^1(\mathbb {P}_{\rho })}+\frac{1}{8} \left( \Vert \textbf{u}\Vert ^2_{L^{\infty ,2}(\mathbb {P}_{\rho })}+\Vert \nabla \textbf{Q}\Vert ^2_{L^{\infty ,2}(\mathbb {P}_{\rho })}\right) . \end{aligned} \end{aligned}$$

(4.10)

Similarly, applying the Hölder’s inequality and the Young’s inequality we have

$$\begin{aligned} \begin{aligned} M_6&\le \frac{C}{(\rho -r)}\Vert \Delta \textbf{Q}\Vert _{L^2(\mathbb {P}_{\rho })}\Vert \textbf{u}\Vert _{L^2(\mathbb {P}_{\rho })}+\frac{C}{(\rho -r)}\Vert \nabla \textbf{u}\Vert _{L^2(\mathbb {P}_{\rho })}\Vert \nabla \textbf{Q}\Vert _{L^2(\mathbb {P}_{\rho })}\\ {}&\le \frac{C}{(\rho -r)^2}\int \limits _{\mathbb {P}_{\rho }}|\textbf{u}|^2+|\nabla \textbf{Q}|^2\textrm{d}x\textrm{d}t+\frac{1}{8}(\Vert \nabla \textbf{u}\Vert ^2_{L^2(\mathbb {P}_{\rho })}+\Vert \nabla ^2\textbf{Q}\Vert ^2_{L^2(\mathbb {P}_{\rho })})\\ {}&\le M_1+\frac{1}{8}(\Vert \nabla \textbf{u}\Vert ^2_{L^2(\mathbb {P}_{\rho })}+\Vert \nabla ^2\textbf{Q}\Vert ^2_{L^2(\mathbb {P}_{\rho })})\\ {}&\le \frac{C\rho ^{3+\frac{2}{\alpha }}}{(\rho -r)^{\frac{6}{\alpha }}}\left( \Vert \textbf{u}\Vert ^2_{L^{p,q}(\mathbb {P}_{\rho })}+\Vert \nabla \textbf{Q}\Vert ^2_{L^{p,q}(\mathbb {P}_{\rho })}\right) \\ {}&\quad +\frac{1}{4}\left( \Vert \textbf{u}\Vert ^2_{L^{\infty ,2}(\mathbb {P}_{\rho })}+\Vert \nabla \textbf{Q}\Vert ^2_{L^{\infty ,2}(\mathbb {P}_{\rho })} +\Vert \nabla \textbf{u}\Vert ^2_{L^2(\mathbb {P}_{\rho })} +\Vert \nabla ^2\textbf{Q}\Vert ^2_{L^2(\mathbb {P}_{\rho })}\right) ; \end{aligned} \end{aligned}$$

(4.11)

Collecting all the above estimates, we know that

$$\begin{aligned} \begin{aligned}&\Vert \textbf{u}\Vert ^2_{L^{\infty ,2}(\mathbb {P}_{r})}+\Vert \nabla \textbf{Q}\Vert ^2_{L^{\infty ,2}(\mathbb {P}_{r})} +\Vert \nabla \textbf{u}\Vert ^2_{L^2(\mathbb {P}_{r})} +\Vert \nabla ^2\textbf{Q}\Vert ^2_{L^2(\mathbb {P}_{r})}\\ {}&\le C \frac{\rho ^{\frac{3\alpha +2}{\alpha }}}{(\rho -r)^{\frac{6}{\alpha }}}\left( \Vert \textbf{u}\Vert ^2_{L^{p,q}(\mathbb {P}_{\rho })} +\Vert \nabla \textbf{Q}\Vert ^2_{L^{p,q}(\mathbb {P}_{\rho })}\right) \\ {}&\quad +C\frac{\rho ^3}{(\rho -r)^{\frac{2}{\alpha -1}}} \left( \Vert \textbf{u}\Vert ^{\frac{2\alpha }{\alpha -1}}_{L^{p,q}(\mathbb {P}_{\rho })} +\Vert \nabla \textbf{d}\Vert ^{\frac{2\alpha }{\alpha -1}}_{L^{p,q}(\mathbb {P}_{\rho })}\right) \\ {}&\quad +C\frac{\rho ^{\frac{3(\alpha +1)}{\alpha -1}}}{(\rho -r)^{\frac{8}{\alpha -1}}} \left( \Vert \textbf{u}\Vert ^{\frac{2\alpha }{\alpha -1}}_{L^{p,q}(\mathbb {P}_{\rho })} +\Vert \nabla \textbf{d}\Vert ^{\frac{2\alpha }{\alpha -1}}_{L^{p,q}(\mathbb {P}_{\rho })}\right) +C \frac{\rho ^3}{(\rho -r)^8}\Vert P\Vert ^2_{L^1(\mathbb {P}_{\rho })}\\ {}&\quad +\frac{3}{4}\left( \Vert \textbf{u}\Vert ^2_{L^{\infty ,2}(\mathbb {P}_{\rho })}+\Vert \nabla \textbf{Q}\Vert ^2_{L^{\infty ,2}(\mathbb {P}_{\rho })} +\Vert \nabla \textbf{u}\Vert ^2_{L^2(\mathbb {P}_{\rho })} +\Vert \nabla ^2\textbf{Q}\Vert ^2_{L^2(\mathbb {P}_{\rho })}\right) . \end{aligned} \end{aligned}$$

(4.12)

Using iteration Lemma 2.4 we conclude that

$$\begin{aligned} \begin{aligned}&\Vert \textbf{u}\Vert ^2_{L^{\infty ,2}(\mathbb {P}_{\frac{R}{2}})}+\Vert \nabla \textbf{Q}\Vert ^2_{L^{\infty ,2}(\mathbb {P}_{\frac{R}{2}})} +\Vert \nabla \textbf{u}\Vert ^2_{L^2(\mathbb {P}_{\frac{R}{2}})} +\Vert \nabla ^2\textbf{Q}\Vert ^2_{L^2(\mathbb {P}_{\frac{R}{2}})}\\ {}&\le C R^{\frac{3\alpha -4}{\alpha }}\left( \Vert \textbf{u}\Vert ^2_{L^{p,q}(\mathbb {P}_{R})} +\Vert \nabla \textbf{Q}\Vert ^2_{L^{p,q}(\mathbb {P}_{R})}\right) \\ {}&\quad +CR^{\frac{3\alpha -5}{\alpha -1}} \left( \Vert \textbf{u}\Vert ^{\frac{2\alpha }{\alpha -1}}_{L^{p,q}(\mathbb {P}_{R})} +\Vert \nabla \textbf{Q}\Vert ^{\frac{2\alpha }{\alpha -1}}_{L^{p,q}(\mathbb {P}_{R})}\right) +CR^{-6}\Vert P\Vert ^2_{L^1(\mathbb {P}_{\rho })}. \end{aligned} \end{aligned}$$

(4.13)

Due to the regularity assumption and the Theorem 1.1, we complete the proof of Theorem 1.2. $\square$

5 The proof of Theorem 1.3

Assume that for some fixed $2\rho <\rho _0\le 1$,

$$\begin{aligned} \begin{aligned} \int \limits _{\mathbb {P}_{2\rho }(z_0)}|\nabla \textbf{u}|^2+|\textbf{u}|^{\frac{10}{3}} +|\nabla ^2\textbf{Q}|^2+|\nabla \textbf{Q}|^{\frac{10}{3}} +|P-\overline{P}_{B_{2\rho }(x_0)}|^{\frac{5}{3}}+|\nabla P|^{\frac{5}{4}} \textrm{d}x\textrm{d}t<(2\rho )^{\frac{5}{3}-\gamma }\varepsilon _1. \end{aligned} \end{aligned}$$

(5.1)

Taking $\phi (x,t)$, which is a smooth positive function supported in $\mathbb {P}_{2\rho }(z_0)$ and with value 1 on the $\mathbb {P}_{\rho }(z_0)$, and satisfy property:

$$\begin{aligned} |\nabla ^2\phi |+|\nabla \phi |^2+|\partial _t\phi |\lesssim \frac{1}{\rho ^2}~~~~\textrm{on}~~~\mathbb {P}_{2\rho }(z_0). \end{aligned}$$

(5.2)

From the local energy inequality (1.2), (5.2) and incompressible condition we obtain that

$$\begin{aligned} \begin{aligned}&\sup _{t_0-\rho ^2\le t<t_0}\int \limits _{B_{\rho }(x_0)}(|\textbf{u}|^2+|\nabla \textbf{Q}|^2)\textrm{d}x +\int \limits _{\mathbb {P}_{\rho }(z_0)}(|\nabla \textbf{u}|^2+|\nabla ^2\textbf{Q}|^2)\textrm{d}x\textrm{d}t\\ {}&\le \left( \int \limits _{\mathbb {P}_{2\rho }(z_0)}|\textbf{u}|^{\frac{10}{3}}+|\nabla \textbf{Q}|^{\frac{10}{3}} \textrm{d}x\textrm{d}t\right) ^{\frac{3}{5}}+\int \limits _{\mathbb {P}_{2\rho }(z_0)} (|\textbf{u}|^2+|\nabla \textbf{Q}|^2)\textbf{u}\cdot \nabla \phi \textrm{d}x\textrm{d}t \\ {}&\quad +\int \limits _{\mathbb {P}_{2\rho }(z_0)}(P-\overline{P}_{B_{2\rho }(x_0)})\textbf{u}\cdot \nabla \phi \textrm{d}x\textrm{d}t+ \int \limits _{\mathbb {P}_{2\rho }(z_0)}(|\Delta \textbf{Q}||\textbf{u}|+|\nabla \textbf{u}||\nabla \textbf{Q}|)|\nabla \phi |\textrm{d}x\textrm{d}t. \end{aligned} \end{aligned}$$

(5.3)

Using the incompressible condition, H$\mathrm {\ddot{o}}$lder’s inequality, and the Gagliardo-Nirenberg inequality, we know that

$$\begin{aligned}&\int \limits _{\mathbb {P}_{2\rho }(z_0)}(|\textbf{u}|^2+|\nabla \textbf{Q}|^2)\textbf{u}\cdot \nabla \phi \textrm{d}x\textrm{d}t \nonumber \\&\lesssim \rho ^{-1}(\Vert |\textbf{u}|^2-\overline{|\textbf{u}|^2}_{B_{2\rho }(x_0)}\Vert _{L^{\frac{10}{7},\frac{15}{8}}(\mathbb {P}_{2\rho }(z_0))} +\Vert |\nabla \textbf{Q}|^2-\overline{|\nabla \textbf{Q}|^2}_{B_{2\rho }(x_0)}\Vert _{L^{\frac{10}{7},\frac{15}{8}}(\mathbb {P}_{2\rho })}) \Vert \textbf{u}\Vert _{L^{\frac{10}{3},\frac{15}{7}}(\mathbb {P}_{2\rho })}\nonumber \\ {}&\lesssim \rho ^{-1}\Vert |\textbf{u}|^2-\overline{|\textbf{u}|^2}_{B_{2\rho }(x_0)} \Vert ^{\frac{1}{2}}_{L^{\frac{5}{3}}(\mathbb {P}_{2\rho }(z_0))}\Vert |\textbf{u}|^2-\overline{|\textbf{u}|^2}_{B_{2\rho }(x_0)} \Vert ^{\frac{1}{2}}_{L^{\frac{5}{4},\frac{15}{7}}(\mathbb {P}_{2\rho }(z_0))} \Vert \textbf{u}\Vert _{L^{\frac{10}{3},\frac{15}{7}}(\mathbb {P}_{2\rho }(z_0))}\nonumber \\ {}&\quad +\rho ^{-1}\Vert |\nabla \textbf{Q}|^2-\overline{|\nabla \textbf{Q}|^2}_{B_{2\rho }(x_0)} \Vert ^{\frac{1}{2}}_{L^{\frac{5}{3}}(\mathbb {P}_{2\rho }(z_0))}\Vert |\nabla \textbf{Q}|^2-\overline{|\nabla \textbf{Q}|^2}_{B_{2\rho }(x_0)} \Vert ^{\frac{1}{2}}_{L^{\frac{5}{4},\frac{15}{7}}(\mathbb {P}_{2\rho }(z_0))} \Vert \textbf{u}\Vert _{L^{\frac{10}{3},\frac{15}{7}}(\mathbb {P}_{2\rho }(z_0))}\nonumber \\ {}&\lesssim \rho ^{-1} \Vert |\textbf{u}|^2 \Vert ^{\frac{1}{2}}_{L^{\frac{5}{3}}(\mathbb {P}_{2\rho }(z_0))}\Vert \textbf{u}\cdot \nabla \textbf{u} \Vert ^{\frac{1}{2}}_{L^{\frac{5}{4}}(\mathbb {P}_{2\rho }(z_0))} \Vert \textbf{u}\Vert _{L^{\frac{10}{3},\frac{15}{7}}(\mathbb {P}_{2\rho }(z_0))}\nonumber \\ {}&\quad + \rho ^{-1}\Vert |\nabla \textbf{Q}|^2 \Vert ^{\frac{1}{2}}_{L^{\frac{5}{3}}(\mathbb {P}_{2\rho }(z_0))}\Vert \nabla \textbf{Q}\cdot \nabla ^2\textbf{Q} \Vert ^{\frac{1}{2}}_{L^{\frac{5}{4}}(\mathbb {P}_{2\rho }(z_0))} \Vert \textbf{u}\Vert _{L^{\frac{10}{3},\frac{15}{7}}(\mathbb {P}_{2\rho }(z_0))}\nonumber \\&\lesssim \rho ^{-\frac{1}{2}}\Vert \textbf{u}\Vert ^{\frac{5}{2}}_{L^{\frac{10}{3}}(\mathbb {P}_{2\rho }(z_0))}\Vert \nabla \textbf{u} \Vert ^{\frac{1}{2}}_{L^2(\mathbb {P}_{2\rho }(z_0))}+\rho ^{-\frac{1}{2}} \Vert \nabla \textbf{Q}\Vert ^{\frac{3}{2}}_{L^{\frac{10}{3}}(\mathbb {P}_{2\rho }(z_0))} \Vert \nabla ^2\textbf{Q}\Vert ^{\frac{1}{2}}_{L^{2}(\mathbb {P}_{2\rho }(z_0))} \Vert \textbf{u}\Vert _{L^{\frac{10}{3}}(\mathbb {P}_{2\rho }(z_0))}. \end{aligned}$$

(5.4)

And

$$\begin{aligned} \begin{aligned}&\int \limits _{\mathbb {P}_{2\rho }(z_0)}(P-\overline{P}_{B_{2\rho }(x_0)})\textbf{u}\cdot \nabla \phi \textrm{d}x\textrm{d}t\\ {}&\le \lesssim \rho ^{-1}\Vert P-\overline{P}_{B_{2\rho }(x_0)}\Vert _{L^{\frac{10}{7},\frac{15}{8}}(\mathbb {P}_{2\rho }(z_0))} \Vert \textbf{u}\Vert _{L^{\frac{10}{3},\frac{15}{7}}(\mathbb {P}_{2\rho }(z_0))}\\ {}&\lesssim \rho ^{-\frac{1}{2}} \Vert P-\overline{P}_{B_{2\rho }(x_0)}\Vert ^{\frac{1}{2}}_{L^{\frac{5}{3}}(\mathbb {P}_{2\rho }(z_0))} \Vert \nabla P\Vert ^{\frac{1}{2}}_{L^{\frac{5}{4}}(\mathbb {P}_{2\rho }(z_0))} \Vert \textbf{u}\Vert _{L^{\frac{10}{3}}(\mathbb {P}_{2\rho }(z_0))}. \end{aligned} \end{aligned}$$

(5.5)

$$\begin{aligned} \begin{aligned}&\int \limits _{\mathbb {P}_{2\rho }(z_0)}(|\Delta \textbf{Q}||\textbf{u}|+|\nabla \textbf{u}||\nabla \textbf{Q}|)|\nabla \phi |\textrm{d}x\textrm{d}t\\ {}&\lesssim \rho ^{-1}(\Vert \Delta \textbf{Q}\Vert _{L^2(\mathbb {P}_{2\rho }(z_0))}\Vert \textbf{u}\Vert _{L^2(\mathbb {P}_{2\rho }(z_0))}+\Vert \nabla \textbf{u}\Vert _{L^2(\mathbb {P}_{2\rho }(z_0))}\Vert \nabla \textbf{Q}\Vert _{L^2(\mathbb {P}_{2\rho }(z_0))})\\ {}&\lesssim \Vert \Delta \textbf{Q}\Vert _{L^2(\mathbb {P}_{2\rho }(z_0))}\Vert \textbf{u}\Vert _{L^{\frac{10}{3}}(\mathbb {P}_{2\rho }(z_0))}+\Vert \nabla \textbf{u}\Vert _{L^2(\mathbb {P}_{2\rho }(z_0))}\Vert \nabla \textbf{Q}\Vert _{L^{\frac{10}{3}}(\mathbb {P}_{2\rho }(z_0))}. \end{aligned} \end{aligned}$$

(5.6)

Combining (5.3)–(5.6) and using the assumption (5.1) yields

$$\begin{aligned} \begin{aligned}&\sup _{t_0-\rho ^2\le t\le t_0}\int \limits _{B_{\rho }(x_0)}|\textbf{u}|^2+|\nabla \textbf{Q}|^2\textrm{d}x+ \int \limits _{\mathbb {P}_{\rho }(z_0)}|\nabla \textbf{u}|^2+|\nabla ^2\textbf{Q}|^2\textrm{d}x\textrm{d}t\\ {}&\lesssim \Vert \textbf{u}\Vert ^2_{L^{\frac{10}{3}}(\mathbb {P}_{2\rho }(z_0))}+\Vert \nabla \textbf{Q}\Vert ^2_{L^{\frac{10}{3}}(\mathbb {P}_{2\rho }(z_0))} +\rho ^{-\frac{1}{2}}\Vert \textbf{u}\Vert ^{\frac{5}{2}}_{L^{\frac{10}{3}}(\mathbb {P}_{2\rho }(z_0))}\Vert \nabla \textbf{u} \Vert ^{\frac{1}{2}}_{L^2(\mathbb {P}_{2\rho }(z_0))}\\ {}&\quad +\rho ^{-\frac{1}{2}} \Vert \nabla \textbf{Q}\Vert ^{\frac{3}{2}}_{L^{\frac{10}{3}}(\mathbb {P}_{2\rho }(z_0))} \Vert \nabla ^2\textbf{Q}\Vert ^{\frac{1}{2}}_{L^{2}(\mathbb {P}_{2\rho }(z_0))} \Vert \textbf{u}\Vert _{L^{\frac{10}{3}}(\mathbb {P}_{2\rho }(z_0))}\\ {}&\quad +\rho ^{-\frac{1}{2}} \Vert P-\overline{P}_{B_{2\rho }(x_0)}\Vert ^{\frac{1}{2}}_{L^{\frac{5}{3}}(\mathbb {P}_{2\rho }(z_0))} \Vert \nabla P\Vert ^{\frac{1}{2}}_{L^{\frac{5}{4}}(\mathbb {P}_{2\rho }(z_0))} \Vert \textbf{u}\Vert _{L^{\frac{10}{3}}(\mathbb {P}_{2\rho }(z_0))}\\ {}&\quad +\Vert \Delta \textbf{Q}\Vert _{L^2(\mathbb {P}_{2\rho }(z_0))}\Vert \textbf{u}\Vert _{L^{\frac{10}{3}}(\mathbb {P}_{2\rho }(z_0))}+\Vert \nabla \textbf{u}\Vert _{L^2(\mathbb {P}_{2\rho }(z_0))}\Vert \nabla \textbf{Q}\Vert _{L^{\frac{10}{3}}(\mathbb {P}_{2\rho }(z_0))}\\ {}&\lesssim \varepsilon _1^{\frac{3}{5}}\rho ^{1-\frac{3}{5}\gamma }+ \varepsilon _1\rho ^{\frac{7}{6}-\gamma }+\varepsilon _1^{\frac{4}{5}}\rho ^{\frac{4}{3}-\frac{4}{5}\gamma }\lesssim \varepsilon _1^{\frac{3}{5}}\rho ^{\frac{7}{6}-\gamma }, \end{aligned} \end{aligned}$$

(5.7)

where we have need that $\gamma \ge \frac{5}{12}$. Hence, we get

$$\begin{aligned} \begin{aligned} A(r)\lesssim \varepsilon _1^{\frac{3}{5}}\rho ^{\frac{1}{6}-\gamma }. \end{aligned} \end{aligned}$$

(5.8)

Since

$$\begin{aligned} \begin{aligned} \int \limits _{B_r(x_0)}|\textbf{u}|^2\textrm{d}x&\lesssim \int \limits _{B_r(x_0)}|\textbf{u}-\overline{\textbf{u}}_{B_{\rho }(x_0)}|^2\textrm{d}x +\int \limits _{B_r(x_0)}|\overline{\textbf{u}}_{B_{\rho }(x_0)}|^2\textrm{d}x\\ {}&\lesssim r^2\int \limits _{B_{\rho }(x_0)} |\nabla \textbf{u}|^2\textrm{d}x+\frac{r^3}{\rho ^3}\int \limits _{B_{\rho }(x_0)}|\textbf{u}|^2\textrm{d}x, \end{aligned} \end{aligned}$$

(5.9)

where we have used the Poincar$\mathrm {\acute{e}}$ inequality and H$\mathrm {\ddot{o}}$lder’s inequality. Integrating with respect to t from $t_0-r^2$ to $t_0$ and using the H$\mathrm {\ddot{o}}$lder’s inequality again, we have

$$\begin{aligned} \begin{aligned} \int \limits _{\mathbb {P}_r(z_0)}|\textbf{u}|^2\textrm{d}x\textrm{d}t\lesssim r^2\int \limits _{\mathbb {P}_{\rho }(z_0)} |\nabla \textbf{u}|^2\textrm{d}x\textrm{d}t+\frac{r^5}{\rho ^3} \sup _{t_0-\rho ^2\le t\le t_0}\int \limits _{B_{\rho }(x_0)}|\textbf{u}|^2\textrm{d}x. \end{aligned} \end{aligned}$$

(5.10)

Similarly,

$$\begin{aligned} \begin{aligned} \int \limits _{\mathbb {P}_r(z_0)}|\nabla \textbf{Q}|^2\textrm{d}x\textrm{d}t\lesssim r^2\int \limits _{\mathbb {P}_{\rho }(z_0)} |\nabla ^2\textbf{Q}|^2\textrm{d}x\textrm{d}t+\frac{r^5}{\rho ^3} \sup _{t_0-\rho ^2\le t\le t_0}\int \limits _{B_{\rho }(x_0)}|\nabla \textbf{Q}|^2\textrm{d}x. \end{aligned} \end{aligned}$$

(5.11)

Adding (5.10) and (5.11) we get

$$\begin{aligned} \begin{aligned} C_{2,z_0}(r)\lesssim (\frac{\rho }{r})B_{z_0}(\rho )+\bigg (\frac{r}{\rho }\bigg )^2A_{z_0}(\rho ). \end{aligned} \end{aligned}$$

(5.12)

Let $\theta =\rho ^{\beta }<\frac{1}{8}$, then from (5.1), (5.8) and (5.12) we have

$$\begin{aligned} \begin{aligned} C_{2,z_0}(\theta \rho )&\le \theta ^{-1}B_{z_0}(\rho )+\theta ^2A_{z_0}(\rho )\lesssim \varepsilon _1^{\frac{3}{5}}\theta ^{-1} \rho ^{\frac{2}{3}-\gamma }+ \varepsilon _1^{\frac{3}{5}}\theta ^2 \rho ^{\frac{1}{6}-\gamma }\\ {}&\lesssim \varepsilon _1^{\frac{3}{5}} \rho ^{-\beta +\frac{2}{3}-\gamma }+\varepsilon _1^{\frac{3}{5}} \rho ^{2\beta +\frac{1}{6}-\gamma }\lesssim \varepsilon _1^{\frac{3}{5}} \rho ^{\frac{1}{2}-\gamma }\lesssim \varepsilon _1^{\frac{3}{5}}, \end{aligned} \end{aligned}$$

(5.13)

where we let $\beta =\frac{1}{6}$ and $\gamma \le \frac{1}{2}$. From (2.3) assumption (5.1) yields

$$\begin{aligned} \begin{aligned} B_{z_0}(\theta \rho )+D_{1,\frac{3}{2},z_0}(\theta \rho )&\lesssim \theta ^{-1}B_{z_0}(\rho )+\theta D_{\frac{5}{4},z_0}(\rho ) \lesssim \varepsilon _1\theta ^{-1}\rho ^{\frac{2}{3}-\gamma }+\varepsilon _1^{\frac{4}{5}}\theta \rho ^{\frac{1}{3}-\frac{4}{5}\gamma }\\ {}&\lesssim \varepsilon _1^{\frac{4}{5}}\rho ^{-\frac{1}{6}+\frac{2}{3}-\gamma }+ \varepsilon _1^{\frac{4}{5}}\rho ^{\frac{1}{6}+\frac{1}{3}-\frac{4}{5}\gamma }\lesssim \varepsilon _1^{\frac{4}{5}}\rho ^{\frac{1}{2}-\gamma }. \end{aligned} \end{aligned}$$

(5.14)

Combining (5.13) and (5.14), we have

$$\begin{aligned} B_{z_0}(\theta \rho )+C_{2,z_0}(\theta \rho )+D_{1,\frac{3}{2},z_0}(\theta \rho )\lesssim \varepsilon _1^{\frac{3}{5}}<\varepsilon \end{aligned}$$

if $\varepsilon _1$ sufficient small. Hence,

$$\begin{aligned} \Vert \textbf{u}\Vert _{L^{2,6}(\mathbb {P}_{\theta \rho }(z_0))}+\Vert \nabla \textbf{Q}\Vert _{L^{2,6}(\mathbb {P}_{\theta \rho }(z_0))}+ \Vert \nabla P\Vert _{L^{1,\frac{3}{2}}(\mathbb {P}_{\theta \rho }(z_0))}<\varepsilon , \end{aligned}$$

which implies

$$\begin{aligned} \Vert \textbf{u}\Vert _{L^{p,q}(\mathbb {P}_r(z_0))}+\Vert \nabla \textbf{Q}\Vert _{L^{p,q}(\mathbb {P}_r(z_0))} +\Vert P\Vert _{L^1(\mathbb {P}_r(z_0))}<\varepsilon ,~~\Big (1\le \frac{2}{p}+\frac{3}{q}<2\Big ). \end{aligned}$$

Due to Theorem 1.2, we know $z_0=(x_0,t_0)$ is a regular point. Thus the proof of Theorem 1.3 is completed. $\square$

Using Theorem 1.3, we are now in a position to prove Theorem 1.4.

6 The proof of Theorem 1.4

Notice that the Theorem 1.3 implies that if $z_0\in S$ is a interior singular point, then for all sufficiently small $0<\rho <r$, one has

$$\begin{aligned} \begin{aligned} \rho ^{\frac{5}{3}-\gamma }\varepsilon _1\le \int \limits _{\mathbb {P}_{\rho }(z_0)}|\nabla \textbf{u}|^2+|\textbf{u}|^{\frac{10}{3}} +|\nabla ^2\textbf{Q}|^2+|\nabla \textbf{Q}|^{\frac{10}{3}} +|P-\overline{P}_{B_{\rho }(x_0)}|^{\frac{5}{3}}+|\nabla P|^{\frac{5}{4}} \textrm{d}x\textrm{d}t. \end{aligned} \end{aligned}$$

(6.1)

Now, fix $5\rho<r<1$ small enough and consider the covering $\{\mathbb {P}_{\rho }(z):z\in S\}$ of the singular point set S. By the Vitali covering lemma, there is a finite disjoint sub-family $\{\mathbb {P}_{\rho }(z_i)~|~i=1,2,\cdots ,N\}$, such that $S\subset \bigcup _{i=1}^N\mathbb {P}_{5\rho }(z_i)$. Summing the inequality above at $z_i$ for $i=1,2,\cdots ,N$ yields

$$\begin{aligned} \begin{aligned}&N\varepsilon _1\rho ^{\frac{5}{3}-\gamma }\\ {}&\le \sum _{i=1}^N\int \limits _{\mathbb {P}_{z_i}(\rho )}(|\nabla \textbf{u}|^2+ |\nabla ^2\textbf{Q}|^2+|\textbf{u}|^{\frac{10}{3}}+|\nabla \textbf{Q}|^{\frac{10}{3}}+ |P-\overline{P}_{B_{\rho }(x_0)}|^{\frac{5}{3}}+|\nabla P|^{\frac{5}{4}}) \textrm{d}x\textrm{d}t\\ {}&\le \int \limits _{\Omega _T}(|\nabla \textbf{u}|^2+ |\nabla ^2\textbf{Q}|^2+|\textbf{u}|^{\frac{10}{3}}+|\nabla \textbf{Q}|^{\frac{10}{3}}+ |P-\overline{P}_{B_{\rho }(x_0)}|^{\frac{5}{3}}+|\nabla P|^{\frac{5}{4}}) \textrm{d}x\textrm{d}t=:\tilde{C}<\infty . \end{aligned} \end{aligned}$$

(6.2)

Let $N(S\cap \Omega _T;\rho )$ denotes the minimum number of parabolic cylinders $\mathbb {P}_z(r)$ required to cover the set $S\cap \Omega _T$, then one has

$$\begin{aligned} N(S\cap \Omega _T;\rho )\le N\le \tilde{C}\varepsilon _1^{-1}\rho ^{-\frac{5}{3}+\gamma }, \end{aligned}$$

which implies that

$$\begin{aligned} \limsup _{\rho \rightarrow 0}\frac{\textrm{log}N(S\cap \Omega _T;\rho )}{-\textrm{log}\rho }\le \frac{5}{3}-\gamma . \end{aligned}$$

Since $\gamma$ can be arbitrary close to $\frac{1}{2}$, this completes the proof of Theorem 1.4. $\square$