1. Introduction
We are interested in studying the global regularity criteria for the local smooth solutions of the following nonlinear dissipative system in 
$\mathbb {R}^{3}$:
\begin{equation} \begin{cases} \partial_t u+(u\cdot\nabla)u-\Delta u+\nabla p=\Delta\psi\nabla\psi, \\ \nabla\cdot u=0,\\ \partial_t v+(u\cdot\nabla)v=\nabla\cdot(\nabla v-v\nabla\psi),\\ \partial_t w+(u\cdot\nabla)w=\nabla\cdot(\nabla w+w\nabla\psi),\\ \Delta\psi=v-w,\\ (u,v,w)|_{t=0}=(u_{0},v_{0},w_{0}), \end{cases}\end{equation}
where 
$u(x,t)$ is a vector in 
$\mathbb {R}^{3}$, 
$p(x,t), v(x,t), w(x,t)$ and 
$\psi (x,t)$ are scalars. The first and the second equations of the system (1.1) are the conservation equations of the incompressible flow. 
$u$ denotes the velocity filed, 
$p$ denotes the pressure and 
$\psi$ denotes the electrostatic potential caused by the net charged particles. The third and the fourth equations of the system (1.1), which are the Nernst–Planck equations modified by the convective terms 
$u\cdot \nabla v$ and 
$u\cdot \nabla w$, model the balance between diffusion and convective transport of charge densities by flow and electric fields. 
$v$ and 
$w$ are the densities of the negative and positive charged particles. They are coupled by the Poisson equation, the fifth equation of system (1.1).
System (1.1) was first proposed by Rubinstein [Reference Rubinstein22], which is capable of describing electro-chemical and fluid-mechanical transport throughout the cellular environment (see, e.g., [Reference Bazant, Thornton and Ajdari1, Reference Jerome10, Reference Jerome and Sacco11, Reference Lin13, Reference Ryham, Liu and Zikatanov23, Reference Schmuck24] and the references cited therein). Well-posedness of the system (1.1) has been studied by Jerome [Reference Jerome10] based on Kato's semigroup framework. Zhao et al. [Reference Deng, Zhao and Cui7, Reference Zhao, Deng and Cui34, Reference Zhao, Deng and Cui35] established the global existence of strong solutions with small initial data and the local existence of strong solutions with any initial data in the various suitable spaces. However, the global existence of local smooth solutions to the system (1.1) with arbitrary initial data is still unknown. In the absence of a well-posedness theory, the development of blowup/non-blowup theory is of major importance for both theoretical and practical purposes.
When the charge transport is neglected, i.e., 
$v=w=\psi =0$, the system (1.1) reduces to the incompressible Navier–Stokes equations. Much effort has been devoted to establish the global existence and uniqueness of smooth solutions to the Navier–Stokes equations. Different criteria for the regularity of weak solutions have been proposed. The classical Prodi–Serrin condition [Reference Escauriaza and Seregin8, Reference Prodi21, Reference Serrin25] says that if
\begin{equation} u \in L^{q}\left(0, T ; L^{p}\left(\mathbb{R}^{3}\right)\right) \quad \text{with} \ \frac{2}{q}+\frac{3}{p}=1 \quad \text{and}\quad 3\leq p\leq\infty, \end{equation}
then the weak solution 
$u$ is regular on 
$(0,T]$. There are also versions of Serrin type condition involving 
$\nabla u$. For example, Beirãoda Veiga [Reference Da Veiga6] showed that any Leray–Hopf weak solution satisfies
\begin{equation} \nabla u \in L^{q}\left(0, T ; L^{p}\left(\mathbb{R}^{3}\right)\right)\quad \text{with} \ \frac{2}{q}+\frac{3}{p}=2 \quad \text{and}\quad \frac{3}{2}< p \leqslant \infty \end{equation}
is actually smooth. In the past decades, there are many refinements of (1.2) and (1.3) in terms of partial components of velocity appeared. For readers interested in this topic, please refer to [Reference Chemin and Zhang4, Reference Chemin, Zhang and Zhang5, Reference Guo, Caggio and Skalák9, Reference Penel and Pokorny20, Reference Zhang28–Reference Zhang, Yao and Li31] for recent progresses. In this paper, we are concerned with global regularity criteria involving only the middle eigenvalue of the strain tensor. Generally, we see that an eigenvector represents a ‘direction’, and an eigenvalue represents the extent of ‘stretch’ or ‘compression’ in that direction. In fluid dynamics, compression and stretching will bring various interesting phenomena. In particular, Neustupa–Penel [Reference Neustupa and Penel17] first formulated sufficient conditions for the regularity of a so-called suitable weak solution 
$(u, p)$ in a sub-domain 
$D$ of the time-space cylinder 
$Q_T=(\Omega \times (0, T))$ by means of requirements on one of the eigenvalues of strain tensor. Later, for the generalized impermeability boundary conditions and the Navier–Slip boundary condition, in [Reference Neustupa and Penel18, Reference Neustupa and Penel19], they showed that the similar condition holds, then the solution 
$u$ is regular on 
$(0, T]$. In [Reference Chae3], Chae proved that the dynamical behaviours of 
$L^{2}$ norm of vorticity are controlled completely by the second largest eigenvalue 
$\lambda ^{+}_2$ of the deformation tensor for the 3D incompressible Euler equations. In particular, they infer that the regularity/singularity of the 3D Navier–Stokes equations are controlled by the integral
\begin{equation} \int_{0}^{t}\left\|\lambda_{2}^{+}(s)\right\|_{L^{\infty}} \,\textrm{d} s, \end{equation}
where 
$\lambda ^{+}_2(x)=\{\lambda _2(x), 0\}$. Recently, Miller [Reference Miller16] used a different method from Neustupa and Penel's to show that the following condition on the middle eigenvalue of strain tensor
\begin{equation} \lambda^{+}_2 \in L^{q}\left(0, T ; L^{p}\left(\mathbb{R}^{3}\right)\right), \quad \frac{2}{q}+\frac{3}{p}=2,\quad \frac{3}{2}< p \leq \infty \end{equation}implies the smoothness of the solution. More recently, the author in [Reference Wu27] extended the above regularity criteria to the Multiplier space and Besov space.
For the system (1.1), after a careful exploration of the nonlinear coupling between the velocity field and charge transport, some regularity criteria for solutions to (1.1) that only involve the velocity field in the three-dimensional case, the Serrin type blowup criteria (1.2)–(1.3) and the Beale-Kato-Majda type criteria
\begin{equation} \int^{T}_{0}\|\nabla\times u(t)\|_{L^{\infty}} \,\textrm{d}t<\infty \end{equation}for local smooth solutions have been established by Zhao and Bai [Reference Zhao and Bai33], which coincide with the results for conventional Navier–Stokes equations. Later, Zhao and Bai's work has been improved by some other authors (see, e.g., [Reference Liu14, Reference Liu and Zhao15, Reference Wu26, Reference Zhao32]).
Inspired by the above mentioned works on global regularity criteria of Navier–Stokes equations and system (1.1), the purpose of the present paper is to generalize regularity criteria (1.5) for classical Navier–Stokes equations to the system (1.1), moreover, we will consider it in more general anisotropic Lorentz space. Our strategy is to combine the idea of Miller in [Reference Miller16] and the generalized Sobolev inequality (see lemma 2.5):
\begin{align*} &\left\| \left\| \left\|f\right\|_{L_{x_{1}}^{{2p}/({p-2}),2}}\right\|_{L_{x_{2}}^{{2 q}/({q-2}),2}} \right\|_{L_{x_{3}}^{{2r}/({r-2}),2}}\\ &\quad \leq C\left\|\partial_{1} f\right\|_{L^{2}}^{{1}/{p}}\left\|\partial_{2} f\right\|_{L^{2}}^{{1}/{q}}\left\|\partial_{3} f\right\|_{L^{2}}^{{1}/{r}}\|f\|_{L^{2}}^{1-\left({1}/{p}+{1}/{q}+{1}/{r}\right)}, \end{align*}
for every 
$f \in C_{0}^{\infty }(\mathbb {R}^{3})$, where 
$2 < p, q, r \leq \infty , 1-({1}/{p}+{1}/{q}+{1}/{r}) \geq 0$. In particular, the generalized Sobolev inequality is a strengthening of a similar result in [Reference Guo, Caggio and Skalák9, Reference Liu and Zhao15]. The result shows that the Serrin type criteria via only middle eigenvalue of strain tensor control the breakdown of local smooth solutions of the system (1.1) in the framework of anisotropic Lorentz space. In other words, this reveals that eigenvalue of strain tensor plays a more dominant role than the density functions of charged particles in studying regularity for solutions 
$(u, v, w)$ of system (1.1) (for some geometrical interpretation see remark 1.4). More precisely, we will prove
Theorem 1.1 Let 
$(u,v,w)$ be a local smooth solution to the system (1.1) with initial data 
$u_{0}\in H^{3}(\mathbb {R}^{3})$ and 
$\nabla \cdot u_{0}=0$, 
$(v_{0},w_{0})\in L^{1}(\mathbb {R}^{3})\cap H^{2}(\mathbb {R}^{3})$ and 
$v_{0},w_{0}\geq 0$. Suppose that the middle eigenvalue of strain tensor satisfies
\begin{equation} \int^{T}_{0}\left\|\left\|\left\|\lambda^{+}_2(t)\right\|_{L^{p, \infty}_{x_1}}\right\|_{L^{q, \infty}_{x_2}}\right\|^{{2}/{2-\left({1}/{p}+{1}/{q}+{1}/{r}\right)}}_{L^{r, \infty}_{x_3}} \,\textrm{d}t<\infty, \end{equation}
where 
$2 < p, q, r \leq \infty$ with 
$1-({1}/{p}+{1}/{q}+{1}/{r}) \geq 0$, then the solution 
$(u,v,w)$ can be extended past the time 
$t=T$.
Remark 1.2 Condition (1.7) in theorem 1.1 can be regarded as an Serrin type blowup criteria only involving the middle eigenvalue of strain tensor, whereas our global regularity criteria is on an eigenvalue of the strain matrix, which scales like 
$\nabla \otimes u$. From this point of view, our result is optimal, unfortunately, we require 
$p, q, r$ to be >2 instead of 
${3}/{2}$.
Remark 1.3 While 
$L^{p}(\mathbb {R}^{3})\hookrightarrow L^{p,\infty }(\mathbb {R}^{3})$, clearly 
$L^{p,\infty }$ is a larger space than 
$L^{p}$, hence, regularity criteria (1.7) are much more better. Notice that when we fix 
$p = q = r$ in condition (1.7), it naturally covers the result (1.4) and (1.5) as stated in [Reference Chae3, Reference Miller16]. In particular, we want to mention that there are some global regularity criteria that have been considered in anisotropic Lebesgue space, however, as far as we know, there is still no corresponding result in anisotropic Lorentz space.
Remark 1.4 We give the remark to explain how the eigenvalues of strain tensor affect fluid behaviour. The strain tensor 
$S(x)$ is symmetric, so, it has a diagonal representation in an appropriately chosen orthogonal cartesian coordinate system (which differs from point to point). The diagonal entries coincide with the eigenvalues and the corresponding eigenvectors can be chosen so that they coincide with the directional vectors of the coordinate axes. Each eigenvalue provides the rate of deformation of the cube (for the whole space case, we may regard the boundary values as zero) in the direction of the corresponding eigenvector. If the eigenvalue is positive, the cube is stretched in the corresponding direction (the large is the eigenvalue, the larger is the stretching) and if the eigenvalue is negative, the cube is compressed in the corresponding direction. Since the sum of the eigenvalues is zero (due to the condition of incompressibility), the stretching in some direction must be compensated by compressing in another direction. Thus, in three-dimensional case, the cube can be typically stretched in two directions and compressed in one direction or the cube is stretched in one direction and compressed in two directions. Therefore, our result describes how these two types of deformation influence the global regularity for system (1.1).
2. Preliminaries
In this section, we give some lemmas and introduce some definitions of function spaces, which will be used in the proof of our main result.
Given a measurable function 
$f : \mathbb {R}^{n} \rightarrow \mathbb {R}$ on 
$X$, its distribution function 
$d_f$ defined on 
$[0, \infty )$ as follows:
\[ d_f(\alpha)=\mu(\{x\in X: |f(x)|>\alpha\}). \]
We now define its decreasing rearrangement 
$f^{*}:[0, \infty ) \rightarrow [0, \infty ]$ as
\[ f^{*}(t) =\inf \left\{\alpha: d_{f}(\alpha) \leq t\right\}, \]
with the convention that 
$\inf \emptyset = \infty$. The point of this definition is that 
$f$ and 
$f^{*}$ have the same distribution function,
\[ d_{f^{*}}(\alpha)=d_{f}(\alpha), \]
but 
$f^{*}$ is a positive non-increasing scalar function.
Definition 2.1 Let 
$(p,q)\in [1,\infty ]^{2}$, the Lorentz space 
$L^{p,q}(\mathbb {R}^{3})$ consists of all measurable functions 
$f$ for which the quantity
\[ \|f\|_{L^{p,q}}:= \begin{cases} \left(\displaystyle\int_{0}^{\infty}[t^{{1}/{p}} f^{*}(t)]^{q} \frac{\mathrm{d} t}{t}\right)^{{1}/{q}} & q<\infty , \\ \sup\limits _{0< t<\infty} t^{{1}/{p}} f^{*}(t) & q=\infty \end{cases} \]is finite.
In order to lead the following definition involving anisotropic Lorentz space, we denote 
$f=f(x_1, x_2, x_3)$ be a measurable function defined on 
$\mathbb {R}^{3}$, 
$f^{*}(t)=f^{*_{1}, *_{2}, *_{3}}(t_{1}, t_{2}, t_{3})$.
Definition 2.2 [Reference Bekmaganbetov K and Toleugazy2]
Let multi indexes 
$p = (p_1, p_2, p_3), q = (q_1, q_2, q_3)$ be such that if 
$0 < p_i < \infty$, then 
$0 < q_i \leq \infty$, and if 
$p_i = \infty$, then 
$q_i = \infty$ for every 
$i = 1, 2, 3$. An anisotropic Lorentz space 
$L^{p_1,q_1}(\mathbb {R}_{x_{1}} ; L^{p_2, q_2}(\mathbb {R}_{x_{2}} ; L^{p_3, q_3}(\mathbb {R}_{x_{3}})))$ is the set of functions for which the following norm is finite:
\begin{align*} &\left\|\left\| \left\| f\right\|_{L_{x_{1}}^{p_1,q_1}}\right\|_{L_{x_{2}}^{p_2,q_2}} \right\|_{L_{x_{3}}^{p_3,q_3}}\\ &\quad :=\left(\int^{\infty}_{0}\left(\int^{\infty}_{0}\left(\int^{\infty}_{0}[t_1^{{1}/{p_1}}t_2^{{1}/{p_2}} t_3^{{1}/{p_3}} f^{*_1, *_2, *_3}(t_1, t_2, t_3)]^{q_1} \frac{\mathrm{d} t_1}{t_1}\right)^{{q_2}/{q_1}} \frac{\mathrm{d} t_2}{t_2}\right)^{{q_3}/{q_2}}\frac{\mathrm{d} t_3}{t_3}\right)^{{1}/{q_3}}. \end{align*}Lemma 2.3 Young's inequality for Lorentz spaces [Reference Lemarie-Rieusset12]
Let 1 
$< p<\infty , 1 \leq q \leq \infty$ and 
${1}/{p^{\prime }}+{1}/{p}=1, {1}/{q^{\prime }}+{1}/{q^{\prime }}=1 .$ Suppose as well that 
$1< p_{1}< p^{\prime }$ and 
$q^{\prime } \leq q \leq \infty .$ If 
${1}/{p_{2}}+1={1}/{p}+{1}/{p_{1}}$ and 
${1}/{q_{2}}={1}/{q}+{1}/{q_{1}},$ then the convolution operator,
\[ *: L^{p, q}\left(\mathbb{R}^{n}\right) \times L^{p_{1}, q_{1}}\left(\mathbb{R}^{n}\right) \mapsto L^{p_{2}, q_{2}}\left(\mathbb{R}^{n}\right) \]is a bounded bilinear operator.
Lemma 2.4 For 
$2< p<\infty$, there exists a constant 
$C=C(p)$ such that 
$f\in \dot {H}^{{1}/{p}}(\mathbb {R})$, then 
$f\in L^{{2p}/({p-2}),2}(\mathbb {R})$ and
\begin{equation} \|f\|_{L^{{2p}/({p-2}),2}}\leq C\|f\|_{\dot{H}^{{1}/{p}}}. \end{equation}Proof. We first make the pointwise definition, 
$\gamma (\xi )=|\xi |^{s} \hat {f}(\xi )$; since 
$f \in \dot {H}^{{1}/{p}}(\mathbb {R})$, 
$\gamma \in L^{2}(\mathbb {R})$. If we set 
$g=\mathcal {F}^{-1} \gamma$, then 
$g \in L^{2}(\mathbb {R})$ and 
$\|g\|_{L^{2}}=\|\gamma \|_{L^{2}}=\|f\|_{\dot {H}^{{1}/{p}}}$. Now,
\[ \hat{f}(\xi)=\frac{|\xi|^{{1}/{p}} \hat{f}(\xi)}{|\xi|^{{1}/{p}}}=\hat{g}(\xi)|\xi|^{-{1}/{p}}. \]
Combining the fact that if 
$P_{\alpha }(x)=|x|^{-\alpha },$ then 
$\widehat {P_{\alpha }(\xi )}=C_{ \alpha } P_{1-\alpha }(\xi )$. Thus we obtain 
$f=g * C_{1-{1}/{p}}^{-1} P_{1-{1}/{p}}$. The function 
$P_{1-{1}/{p}}=|x|^{-({p-1})/{p}}\in L^{{p}/({p-1}), \infty }(\mathbb {R})$ but not in 
$L^{{p}/({p-1})}(\mathbb {R})$. Applying lemma 2.3, we find that
\begin{align} \left\|f\right\|_{L^{{2p}/({p-2}),2}}&=\left\|g * C_{1-{1}/{p}}^{{-}1} P_{1-{1}/{p}}\right\|_{L^{{2p}/({p-2}),2}}\nonumber\\ &\leq C\left\|g\right\|_{L^{2}}\left\||x|^{-({p-1})/{p}}\right\|_{L^{{p}/({p-1}), \infty}} \leq C\left\|f\right\|_{\dot{H}^{{1}/{p}}}. \end{align}Lemma 2.5 There exists a positive constant C such that
\begin{align} &\left\| \left\| \left\|f\right\|_{L_{x_{1}}^{{2 p}/({p-2}),2}}\right\|_{L_{x_{2}}^{{2 q}/({q-2}),2}} \right\|_{L_{x_{3}}^{{2 r}/({r-2}),2}}\notag\\ &\quad \leq C\left\|\partial_{1} f\right\|_{L^{2}}^{{1}/{p}}\left\|\partial_{2} f\right\|_{L^{2}}^{{1}/{q}}\left\|\partial_{3} f\right\|_{L^{2}}^{{1}/{r}}\|f\|_{L^{2}}^{1-\left({1}/{p}+{1}/{q}+{1}/{r}\right)}, \end{align}
for every 
$f \in C_{0}^{\infty }(\mathbb {R}^{3})$, where 
$2 < p, q, r \leq \infty , 1-({1}/{p}+{1}/{q}+{1}/{r}) \geq 0$.
Proof. Let 
$\Lambda ^{p}_1$ be the Fourier multiplier defined as
\[ \mathcal{F}_{1}\left(\Lambda_{1}^{p} f\right)\left(\xi_{1}, x_{2}, x_{3}\right)=\left|\xi_{1}\right|^{p} \mathcal{F}_{1} f\left(\xi_{1}, x_{2}, x_{3}\right) \]with
\[ \mathcal{F}_{1} f\left(\xi_{1}, x_{2}, x_{3}\right)=\int_{\mathbb{R}} \textrm{e}^{-\textrm{i} \xi_{1} x_{1}} f\left(x_{1}, x_{2}, x_{3}\right) \mathrm{d} x_{1}, \]
$\Lambda ^{p}_2$ and 
$\Lambda ^{p}_3$ can be defined analogously. Then by the Sobolev's embedding theorem (lemma 2.4), the Minkowski's inequality and the Hölder's inequality to obtain
\begin{align} &\left\| \left\| \left\|f\right\|_{L_{x_{1}}^{{2 p}/({p-2}),2}}\right\|_{L_{x_{2}}^{{2 q}/({q-2}),2}} \right\|_{L_{x_{3}}^{{2 r}/({r-2}),2}}\nonumber\\ &\quad \leq C \left\|\left\|\left\|\Lambda_{1}^{{1}/{p}} f\right\|_{L^{2}_{x_1}}\right\|_{L_{x_{2}}^{{2 q}/({q-2}),2}} \right\|_{L_{x_{3}}^{{2 r}/({r-2}),2}}\leq \left\|\left\|\left\|\Lambda_{1}^{{1}/{p}} f\right\|_{L_{x_{2}}^{{2 q}/({q-2}),2}}\right\|_{L^{2}_{x_1}} \right\|_{L_{x_{3}}^{{2 r}/({r-2}),2}}\nonumber\\ &\quad\leq C\left\|\left\|\Lambda_{2}^{{1}/{q}}\Lambda_{1}^{{1}/{p}} f\right\|_{L^{2}_{x_1, x_2}} \right\|_{L_{x_{3}}^{{2 r}/({r-2}),2}}\leq C \left\|\left\|\Lambda_{2}^{{1}/{q}} \Lambda_{1}^{{1}/{p}} f\right\|_{L_{x_{3}}^{{2 r}/({r-2}),2}} \right\|_{L^{2}_{x_1, x_2}}\nonumber\\ &\quad\leq C\left\|\Lambda_{3}^{{1}/{r}}\Lambda_{2}^{{1}/{q}}\Lambda_{1}^{{1}/{p}} f\right\|_{L^{2}}. \end{align}Combining the Fourier–Plancherel formula and the Hölder's inequality, we have
\begin{align} &C\left\|\Lambda_{3}^{{1}/{r}} \Lambda_{2}^{{1}/{q}} \Lambda_{1}^{{1}/{p}} f\right\|_{L^{2}} \nonumber\\ &\quad \leq C\left(\int_{\mathbb{R}^{3}}\left|\xi_{1}\right|^{{2}/{p}}\left|\xi_{2}\right|^{{2}/{q}} \left|\xi_{3}\right|^{{2}/{r}}\left|\mathcal{F} f\left(\xi_{1}, \xi_{2}, \xi_{3}\right)\right|^{2} \, \mathrm{d} \xi_{1} \,\mathrm{d} \xi_{2} \,\mathrm{d} \xi_{3}\right)^{{1}/{2}}\nonumber\\ &\quad =C\left(\int_{\mathbb{R}^{3}}\left|\xi_{1}\right|^{{2}/{p}}|\mathcal{F} f(\xi)|^{{2}/{p}} \left|\xi_{2}\right|^{{2}/{q}}| \mathcal{F} f(\xi)|^{{2}/{q}}\left|\xi_{3}\right|^{{2}/{r}}\right.\nonumber\\ &\qquad \times \left.\vphantom{\int_{\mathbb{R}^{3}}}|\mathcal{F} f(\xi)|^{{2}/{r}}| \mathcal{F} f(\xi)|^{2-({2}/{p}+{2}/{q}+{2}/{r})} \,\mathrm{d} \xi_{1} \, \mathrm{d} \xi_{2} \,\mathrm{d} \xi_{3}\right)^{{1}/{2}}\nonumber\\ &\quad\leq C\|\mathcal{F} f\|_{L^{2}}^{1-{1}/{p}-{1}/{q}-{1}/{r}} \left(\int_{\mathbb{R}^{3}}\left|\xi_{1}\right|^{2}|\mathcal{F} f|^{2} \,\mathrm{d} \xi\right)^{{1}/{2 p}}\nonumber\\ &\qquad\times \left(\int_{\mathbb{R}^{3}}\left|\xi_{2}\right|^{2}|\mathcal{F} f|^{2} \,\mathrm{d} \xi\right)^{{1}/{2 q}} \left(\int_{\mathbb{R}^{3}}\left|\xi_{3}\right|^{2}|\mathcal{F} f|^{2} \,\mathrm{d} \xi\right)^{{1}/{2 r}}\nonumber\\ &\quad\leq C\left\|\partial_{1} f\right\|_{L^{2}}^{{1}/{p}}\left\|\partial_{2} f\right\|_{L^{2}}^{{1}/{q}} \left\|\partial_{3} f\right\|_{L^{2}}^{{1}/{r}}\|f\|_{L^{2}}^{1-({1}/{p}+{1}/{q}+{1}/{r})}. \end{align}Remark 2.6 In fact, since 
$L^{{2 p}/({p-2}),2}\hookrightarrow L^{{2 p}/({p-2}),{2 p}/({p-2})}$ for 
$2< p<\infty$, we have similar result for estimate (2.3) in anisotropic Lebesgue space, for more detail refer to [Reference Liu and Zhao15]. However, we should point out that lemma 2.5 that holds in Lorentz space mainly depends on the Sobolev's embedding in lemma 2.4.
Lemma 2.7 [Reference Miller16]
For all 
$-{3}/{2}<\alpha <{3}/{2}$ and for all 
$u$ divergence free in the sense that 
$\xi \cdot \hat {u}(\xi )=0$ almost everywhere,
\begin{equation} \|S\|_{\dot{H}^{\alpha}}^{2}=\|A\|_{\dot{H}^{\alpha}}^{2}=\frac{1}{2}\|\omega\|_{\dot{H}^{\alpha}}^{2}=\frac{1}{2}\|\nabla \otimes u\|_{\dot{H}^{\alpha}}^{2}, \end{equation}
where symmetric part 
$S=S_{i j}={1}/{2}({\partial u_{j}}/{\partial x_{i}}+{\partial u_{i}}/{\partial x_{j}})$, which we refer to as the strain tensor, anti-symmetric part 
$A=A_{i j}={1}/{2}({\partial u_{j}}/{\partial x_{i}}-{\partial u_{i}}/{\partial x_{j}})$, 
$\omega =\nabla \times u$.
Lemma 2.8 [Reference Miller16]
Suppose 
$S \in L^{\infty }([0, T] ; L^{2}(\mathbb {R}^{3})) \cap L^{2}([0, T] : \dot {H}^{1}(\mathbb {R}^{3}))$ is a local smooth solution to the Navier–Stokes strain equation and 
$S(x)$ has eigenvalues 
$\lambda _1(x) \leq \lambda _2(x) \leq \lambda _3(x)$. Define
\[ \lambda_{2}^{+}(x)=\max \left\{\lambda_{2}(x), 0\right\}, \]then
\[ -\operatorname{det}(S) \leq \frac{1}{2}|S|^{2} \lambda_{2}^{+}. \]3. The proof of theorem 1.1
This section is devoted to the proof of theorem 1.1. Based on the Beale-Kato-Majda type criteria (1.6), due to 
$H^{2}(\mathbb {R}^{3})\hookrightarrow L^{\infty }(\mathbb {R}^{3})$, it suffices to present the bound
\[ \int^{T}_{0}\|u(t)\|_{H^{3}} \,\textrm{d}t<\infty \]under condition (1.7).
We first recall from [Reference Schmuck24, Reference Zhao and Bai33] that the fundamental energy inequalities have been established, i.e., for any 
$0< t< T$,
\begin{align} &\|v\|^{2}_{L^{2}}+\|w\|^{2}_{L^{2}}+2\int^{t}_{0}\|\nabla v(s)\|^{2}_{L^{2}}+\|\nabla w(s)\|^{2}_{L^{2}}\,\textrm{d}s\leq\|v_{0}\|^{2}_{L^{2}}+\|w_{0}\|^{2}_{L^{2}}, \end{align}
\begin{align} &\|u\|^{2}_{L^{2}}+\|\nabla\psi\|^{2}_{L^{2}}+2\int^{t}_{0}(\|\nabla u(s)\|^{2}_{L^{2}}+\|\Delta \psi(s)\|^{2}_{L^{2}}) \,\textrm{d}s\leq C, \end{align}
where 
$C$ is a constant depending only on 
$\left \|u_{0}\right \|_{L^{2}}$, 
$\left \|v_{0}\right \|_{L^{1} \cap L^{2}}$ and 
$\left \|w_{0}\right \|_{L^{1} \cap L^{2}}$. Next, we derive the 
$H^{1}$ estimate for the local smooth solution. First, we consider the 
$H^{1}$ estimate of velocity 
$u$. Taking 
$\nabla \times$ on the first equation of (1.1), we see that
\begin{equation} \partial_{t} \omega+(u \cdot \nabla) \omega-\Delta \omega-S\omega=\nabla\times(\Delta\psi\nabla\psi), \end{equation}
taking the operator 
$\nabla _{sym}$ 
$(\textrm {i.e.},\ S=\nabla _{s y m}(u)_{i j}={1}/{2}({\partial u_{j}}/{\partial x_{i}}+{\partial u_{i}}/{\partial x_{j}}))$ to the first equation of (1.1) to obtain
\begin{align} \partial_{t} S+(u \cdot \nabla) S-\Delta S+S^{2}+\frac{1}{4} \omega \otimes \omega-\frac{1}{4}|\omega|^{2} I_{3}+H e s s(p)=\nabla_{sym}(\Delta\psi\nabla\psi), \end{align}
where for more details readers can refer to [Reference Miller16]. Multiplying (3.3) by 
$\omega$ and integrating over 
$\mathbb {R}^{3}$ to get
\begin{align} \frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\|\omega\|_{L^{2}}^{2}+\|\nabla \omega\|_{L^{2}}^{2}&= \int_{\mathbb{R}^{3}} S\omega\cdot\omega \,\textrm{d}x+\int_{\mathbb{R}^{3}} \nabla\times( \Delta\psi\nabla\psi)\cdot\omega \,\textrm{d}x\nonumber\\ &=\int_{\mathbb{R}^{3}} S\omega\cdot\omega \,\textrm{d}x-\int_{\mathbb{R}^{3}} (\Delta\psi\nabla\psi)\cdot\Delta u \,\textrm{d}x. \end{align}
Multiplying (3.4) by 
$S$ and integrating by parts in 
$\mathbb {R}^{3}$ to obtain
\begin{align} \frac{1}{2} \frac{\textrm{d}}{\textrm{d} t}\|S\|_{L^{2}}^{2}+\|\nabla S\|_{L^{2}}^{2}&={-} \int_{\mathbb{R}^{3}} S^{2}\cdot S\,\textrm{d}x-\frac{1}{4}\int_{\mathbb{R}^{3}} \omega \otimes \omega\cdot S \,\textrm{d}x - \int_{\mathbb{R}^{3}} \operatorname{Hess}(p)\cdot S \,\textrm{d}x\nonumber\\ &\quad +\frac{1}{4}\int_{\mathbb{R}^{3}} |\omega|^{2} I_{3}\cdot S\,\textrm{d}x+ \int_{\mathbb{R}^{3}}\nabla_{sym}(\Delta\psi\nabla\psi)\cdot S \,\textrm{d}x\nonumber\\ &={-} \int_{\mathbb{R}^{3}} \operatorname{tr}\left(S^{3}\right)\textrm{d}x-\frac{1}{4}\int_{\mathbb{R}^{3}} \omega \otimes \omega\cdot S \,\textrm{d}x\notag\\ &\quad -\frac{1}{2}\int_{\mathbb{R}^{3}} (\Delta\psi\nabla\psi)\cdot\Delta u \,\textrm{d}x, \end{align}where we have used the following facts
\begin{align*} &\left\langle|\omega|^{2} I_{3}, S\right\rangle_{L^{2}}= 0, \end{align*}
\begin{align*} &\langle\mathrm{Hess}(p), S\rangle_{L^{2}}= 0 , \end{align*}
\begin{align*} &\left\langle S^{2}, S\right\rangle_{L^{2}}=\int_{\mathbb{R}^{3}} \operatorname{tr}\left(S^{3}\right), \end{align*}
\begin{align*} &2\langle\nabla_{sym}(\Delta\psi\nabla\psi), S\rangle_{L^{2}}=\langle\Delta\psi\nabla\psi, -\Delta u\rangle_{L^{2}}. \end{align*}Due to lemma 2.7, for (3.5), we derive
\begin{equation} \frac{\textrm{d}}{\textrm{d}t}\|S\|_{L^{2}}^{2}+2\|\nabla S\|_{L^{2}}^{2}= \int_{\mathbb{R}^{3}} S\omega\otimes\omega \,\textrm{d}x-\int_{\mathbb{R}^{3}} (\Delta\psi\nabla\psi)\cdot\Delta u \,\textrm{d}x. \end{equation}Combining (3.6) and (3.7), we conclude that
\begin{align} \frac{\textrm{d}}{\textrm{d} t}\|S\|_{L^{2}}^{2}+2\|\nabla S\|_{L^{2}}^{2}& ={-}\frac{4}{3} \int_{\mathbb{R}^{3}} \operatorname{tr}\left(S^{3}\right)-\int_{\mathbb{R}^{3}} (\Delta\psi\nabla\psi)\cdot\Delta u \,\textrm{d}x\nonumber\\ &=I_{1}+I_{2}. \end{align}
Using the interpolation inequality, the energy estimate (3.1)–(3.2) and the Young's inequality, 
$I_{2}$ can be estimated as
\begin{align} I_{2}&\leq\left|\int_{\mathbb{R}^{3}} (\Delta\psi\nabla\psi)\cdot\Delta u \,\textrm{d}x\right|\nonumber\\ &\leq \epsilon\|\Delta u\|^{2}_{L^{2}}+C\|\Delta\psi\|^{2}_{L^{4}}\|\nabla\psi\|^{2}_{L^{4}}\nonumber\\ &\leq \epsilon\|\Delta u\|^{2}_{L^{2}}+C\|(v,w)\|^{2}_{L^{4}}\|\nabla\psi\|^{2}_{L^{4}}\nonumber\\ &\leq \epsilon\|\Delta u\|^{2}_{L^{2}}+C\|(v,w)\|^{{1}/{2}}_{L^{2}}\|(\nabla v,\nabla w)\|^{{3}/{2}}_{L^{2}} \|\nabla\psi\|^{{1}/{2}}_{L^{2}}\|\Delta\psi\|^{{3}/{2}}_{L^{2}}\nonumber\\ &\leq \epsilon\|\Delta u\|^{2}_{L^{2}}+C\|(v,w)\|^{2}_{L^{2}}\|(\nabla v,\nabla w)\|^{2}_{L^{2}} +C\|(v,w)\|^{2}_{L^{2}}\|\nabla\psi\|^{2}_{L^{2}}\nonumber\\ &\leq \epsilon\|\Delta u\|^{2}_{L^{2}}+C (\|(\nabla v,\nabla w)\|^{2}_{L^{2}}+1)\nonumber\\ &\leq 2\epsilon\|\nabla S \|^{2}_{L^{2}}+C (\|(\nabla v,\nabla w)\|^{2}_{L^{2}}+1), \end{align}
here and below, 
$\|(f,g)\|_{L^{p}}$ denotes the norm of 
$\|f\|_{L^{p}}+\|g\|_{L^{p}}$, 
$1\leq p\leq \infty$. For 
$I_1$, since 
$\operatorname {tr}(S)=\nabla \cdot u=0$, by the Hölder's inequality, Young's inequality, lemmas 2.5 and 2.8, we have
\begin{align} I_1&={-}\frac{4}{3} \int_{\mathbb{R}^{3}} \operatorname{tr}\left(S^{3}\right) \textrm{d}x\nonumber\\ &={-}\frac{4}{3} \int_{\mathbb{R}^{3}}\lambda_{1}^{3}+\lambda_{2}^{3}+\lambda_{3}^{3} \,\textrm{d}x \nonumber\\ &={-}\frac{4}{3} \int_{\mathbb{R}^{3}}\lambda_{1}^{3}+\lambda_{2}^{3}+\left(-\lambda_{1}-\lambda_{2}\right)^{3} \,\textrm{d}x \nonumber\\ &={-}4\int_{\mathbb{R}^{3}}\left(-\lambda_{1}-\lambda_{2}\right) \lambda_{1} \lambda_{2} \,\textrm{d}x={-}4\int_{\mathbb{R}^{3}} \lambda_{1} \lambda_{2} \lambda_{3} \,\textrm{d}x\nonumber\\ &={-}4\int_{\mathbb{R}^{3}} \operatorname{det}(S) \,\textrm{d}x\leq 2\int_{\mathbb{R}^{3}}|S|^{2} \lambda_{2}^{+} \,\textrm{d}x\nonumber\\ &\leq 2\left\|\left\|\left\| \lambda_{2}^{+}\right\|_{L^{p, \infty}_{x_1}}\right\|_{L^{q, \infty}_{x_2}}\right\|_{L^{r, \infty}_{x_3}} \left\| \left\| \left\|S\right\|_{L_{x_{1}}^{{2 p}/({p-2}),2}}\right\|_{L_{x_{2}}^{{2 q}/({q-2}),2}} \right\|_{L_{x_{3}}^{{2 r}/({r-2}), 2}}\left\|S\right\|_{L^{2}}\nonumber\\ &\leq C\left\|\left\|\left\|\lambda_{2}^{+}\right\|_{L^{p, \infty}_{x_1}}\right\|_{L^{q, \infty}_{x_2}}\right\|_{L^{r, \infty}_{x_3}} \left\|\partial_{1}S \right\|_{L^{2}}^{{1}/{p}}\left\|\partial_{2} S\right\|_{L^{2}}^{{1}/{q}}\left\|\partial_{3} S\right\|_{L^{2}}^{{1}/{r}} \notag\\ &\quad \times \|S\|_{L^{2}}^{1-({1}/{p}+{1}/{q}+{1}/{r})}\|S\|_{L^{2}}\nonumber\\ &\leq C\left\|\left\|\left\|\lambda_{2}^{+}\right\|_{L^{p, \infty}_{x_1}}\right\|_{L^{q, \infty}_{x_2}}\right\|_{L^{r, \infty}_{x_3}} \left\|\nabla S \right\|_{L^{2}}^{{1}/{p}+{1}/{q}+{1}/{r}}\|S\|_{L^{2}}^{2- ({1}/{p}+{1}/{q}+{1}/{r})}\nonumber\\ &\leq C \left\|\left\|\left\|\lambda_{2}^{+}\right\|_{L^{p,\infty}_{x_1}}\right\|_{L^{q,\infty}_{x_2}}\right\|_{L^{r, \infty}_{x_3}}^{{2}/{2-({1}/{p}+{1}/{q}+{1}/{r})}} \|S\|^{2}_{L^{2}}+\epsilon\|\nabla S\|_{L^{2}}^{2}. \end{align}Inserting estimates (3.10) and (3.9) into (3.8), we deduce that
\begin{align} \frac{\textrm{d}}{\textrm{d} t}\|S\|_{L^{2}}^{2}+\|\nabla S\|_{L^{2}}^{2}&\leq C \left\|\left\|\left\|\lambda_{2}^{+}\right\|_{L^{p,\infty}_{x_1}}\right\|_{L^{q,\infty}_{x_2}}\right\|_{L^{r, \infty}_{x_3}}^{{2}/{2-({1}/{p}+{1}/{q}+{1}/{r})}} \|S\|^{2}_{L^{2}}\nonumber\\ &\quad +C \left(\|\left(\nabla v,\nabla w\right)\|^{2}_{L^{2}}+1\right). \end{align}
Therefore Gronwall's inequality implies that for all 
$0< t\leq T$,
\begin{align} \sup\limits_{0< t\leq T}\|S\|^{2}_{L^{2}}&\leq \left(\| S_{0}\|^{2}_{L^{2}}+C\int^{T}_{0}\|(\nabla v,\nabla w)\|^{2}_{L^{2}}+1\,\textrm{d}t\right)\nonumber\\ &\quad\times\exp C\int^{T}_{0} \left\|\left\|\left\|\lambda_{2}^{+}\right\|_{L^{p,\infty}_{x_1}}\right\|_{L^{q,\infty}_{x_2}}\right\|_{ L^{r,\infty}_{x_3}}^{{2}/{2-({1}/{p}+{1}/{q}+{1}/{r})}}\,\textrm{d}t, \end{align}
where we used the energy equality (3.1) to control the term involving the integral of 
$\int ^{T}_{0}\|(\nabla v,\nabla w)\|^{2}_{L^{2}}\,\textrm {d}t$. Combining (3.12) and lemma 2.7, we have
\[ u\in L^{\infty}(0,T;H^{1}(\mathbb{R}^{3}))\cap L^{2}(0,T;H^{2}(\mathbb{R}^{3})). \] Now, we have uniform estimate of velocity 
$u$ at hand, then we estimate the 
$H^{1}$-norm of 
$v$ and 
$w$. Multiplying 
$\Delta v$ to the third equation of system (1.1), after integrating over 
$\mathbb {R}^{3}$ to see
\begin{equation} \frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\|\nabla v\|^{2}_{L^{2}}+\|\Delta v\|^{2}_{L^{2}}={-}\int_{\mathbb{R}^{3}}(u\cdot\nabla) v\cdot\Delta v \,\textrm{d}x-\int_{\mathbb{R}^{3}}\nabla \cdot(v\cdot\nabla\psi)\Delta v\,\textrm{d}x. \end{equation}
Applying the interpolation inequality, Hölder's inequality and the Young's inequality, 
$-\int _{\mathbb {R}^{3}}(u\cdot \nabla ) v\cdot \Delta v\,\textrm {d}x$ can be bounded by
\begin{align} -\int_{\mathbb{R}^{3}}(u\cdot\nabla) v\cdot\Delta v \,\textrm{d}x&\leq \|(u\cdot\nabla)v\|_{L^{2}}\| \Delta v\|_{L^{2}}\nonumber\\ &\leq\epsilon\| \Delta v\|^{2}_{L^{2}}+C \|(u\cdot\nabla)v\|^{2}_{L^{2}}\nonumber\\ &\leq\epsilon\| \Delta v\|^{2}_{L^{2}}+C \|u\|^{2}_{L^{6}}\|\nabla v\|^{2}_{L^{3}}\nonumber\\ &\leq \epsilon\| \Delta v\|^{2}_{L^{2}}+C \|\nabla u\|^{2}_{L^{2}}\|\nabla v\|_{L^{2}}\|\nabla v\|_{L^{6}}\nonumber\\ &\leq 2\epsilon\|\Delta v\|^{2}_{L^{2}}+C \|\nabla v\|^{2}_{L^{2}}. \end{align}
Similarly, for the term 
$-\int _{\mathbb {R}^{3}}\nabla \cdot (v\cdot \nabla \psi )\Delta v\,\textrm {d}x$, we have
\begin{align} -\int_{\mathbb{R}^{3}}\nabla \cdot(v\cdot\nabla\psi)\Delta v \,\textrm{d}x&={-}\int_{\mathbb{R}^{3}}\nabla v\nabla\psi\cdot\Delta v \,\textrm{d}x-\int_{\mathbb{R}^{3}}v\Delta\psi\cdot\Delta v \,\textrm{d}x \nonumber\\ &\leq\epsilon\|\Delta v\|^{2}_{L^{2}}+C (\|\nabla v\nabla \psi\|^{2}_{L^{2}}+\|v\Delta\psi\|^{2}_{L^{2}})\nonumber\\ &\leq \epsilon\| \Delta v\|^{2}_{L^{2}}+C (\|\nabla v \|^{2}_{L^{3}}\|\nabla \psi\|^{2}_{L^{6}}+\|(v, w)\|^{4}_{L^{4}})\nonumber\\ &\leq \epsilon\| \Delta v\|^{2}_{L^{2}}+C (\|(v, w)\|^{2}_{L^{2}}\|\nabla v\|_{L^{2}}\|\Delta v\|_{L^{2}}\nonumber\\ &\quad +\|(v, w)\|^{{5}/{2}}_{L^{2}}\|(\Delta v ,\Delta w)\|^{{3}/{2}}_{L^{2}})\nonumber\\ &\leq 3\epsilon\| (\Delta v,\Delta w)\|^{2}_{L^{2}}+C (1+\|(\nabla v,\nabla w)\|^{2}_{L^{2}}), \end{align}where the interpolation inequality have been used
\begin{equation} \|f\|_{L^{4}}\leq C\|f\|_{L^{2}}^{{5}/{8}}\|\Delta f\|_{L^{2}}^{{3}/{8}}.\end{equation}Plugging (3.14) and (3.15) into (3.13), we obtain
\begin{equation} \frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\|\nabla v\|^{2}_{L^{2}}+\|\Delta v\|^{2}_{L^{2}}\leq 5\epsilon\| (\Delta v,\Delta w)\|^{2}_{L^{2}}+C (1+\|(\nabla v,\nabla w)\|^{2}_{L^{2}}). \end{equation}
In a similar way, one can bound 
$w$ as
\begin{equation} \frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\|\nabla w\|^{2}_{L^{2}}+\|\Delta w\|^{2}_{L^{2}}\leq 5\epsilon\| (\Delta v,\Delta w)\|^{2}_{L^{2}}+C (1+\|(\nabla v,\nabla w)\|^{2}_{L^{2}}). \end{equation}Therefore then
\begin{equation} \frac{\textrm{d}}{\textrm{d}t}(\|\nabla v\|^{2}_{L^{2}}+\|\nabla w\|^{2}_{L^{2}})+\|\Delta v\|^{2}_{L^{2}}+\|\Delta w\|^{2}_{L^{2}}\leq C (1+\|(\nabla v,\nabla w)\|^{2}_{L^{2}}). \end{equation}That is to say
\[ v, w\in L^{\infty}(0,T;H^{1}(\mathbb{R}^{3}))\cap L^{2}(0,T;H^{2}(\mathbb{R}^{3})). \]
Finally, we derive the 
$H^{2}$ uniform estimate for the field of velocity 
$u$. Taking the operator 
$\Delta$ on both sides of the first equation of system (1.1), then multiplying the resulting equality with 
$\Delta u$ and integrating over 
$\mathbb {R}^{3}$ to obtain
\begin{align} \frac{1}{2}\frac{\textrm{d}}{\textrm{d}t}\|\Delta u\|^{2}_{L^{2}}+\|\nabla\Delta u\|^{2}_{L^{2}}={-}\int_{\mathbb{R}^{3}}\Delta((u\cdot\nabla)u) \cdot\Delta u \,\textrm{d}x+\int_{\mathbb{R}^{3}}\Delta(\Delta \psi\nabla\psi)\cdot\Delta u \,\textrm{d}x. \end{align}
Using the same trick as (3.15) present, it follows from the estimates of velocity 
$u$ that
\begin{align} -\int_{\mathbb{R}^{3}}\Delta((u\cdot\nabla)u) \cdot\Delta u \,\textrm{d}x &\leq\epsilon\| \nabla\Delta u\|^{2}_{L^{2}}+C \|\nabla u\otimes\nabla u\|^{2}_{L^{2}}+C\|(u\cdot\nabla)\nabla u\|^{2}_{L^{2}}\nonumber\\ &\leq\epsilon\| \nabla\Delta u\|^{2}_{L^{2}}+C \|\nabla u\|^{4}_{L^{4}}+C\|u\|^{2}_{L^{6}}\|\Delta u\|^{2}_{L^{3}}\nonumber\\ &\leq \epsilon\|\nabla \Delta u\|^{2}_{L^{2}}+C \|\nabla u\|^{{5}/{2}}_{L^{2}}\|\nabla \Delta u\|^{{3}/{2}}_{L^{2}}\notag\\ &\quad +C\|\nabla u\|^{2}_{L^{2}}\|\Delta u \|_{L^{2}}\|\nabla\Delta u\|_{L^{2}}\nonumber\\ &\leq 2\epsilon\|\nabla \Delta u\|^{2}_{L^{2}}+C (1+\|\Delta u\|^{2}_{L^{2}}) \end{align}and
\begin{align} \int_{\mathbb{R}^{3}}\Delta(\Delta\psi\nabla\psi)\cdot\Delta u \,\textrm{d}x& \leq|\int_{\mathbb{R}^{3}}\nabla((v-w)\nabla\psi)\cdot\nabla\Delta u \,\textrm{d}x|\nonumber\\ &\leq\epsilon\| \nabla\Delta u\|^{2}_{L^{2}}+C\left( \|(\nabla v-\nabla w)\nabla\psi\|^{2}_{L^{2}}+\|(v-w)\nabla^{2} \psi\|^{2}_{L^{2}}\right)\nonumber\\ &\leq\epsilon\| \nabla\Delta u\|^{2}_{L^{2}}+C\left( \|(\nabla v ,\nabla w)\|^{2}_{L^{3}}\|\nabla\psi\|^{2}_{L^{6}}+\|(v,w)\|^{4}_{L^{4}}\right)\nonumber\\ &\leq \epsilon\|\nabla\Delta u\|^{2}_{L^{2}}+\epsilon\|(\Delta v,\Delta w)\|^{2}_{L^{2}}+C (1+\|(\nabla v,\nabla w) \|^{2}_{L^{2}}). \end{align}Inserting (3.21) and (3.22) into (3.20), we get
\begin{align} \frac{\textrm{d}}{\textrm{d}t}\|\Delta u\|^{2}_{L^{2}}+\|\nabla\Delta u\|^{2}_{L^{2}}\leq 2\epsilon\|(\Delta v,\Delta w)\|^{2}_{L^{2}}+C (1+\|(\nabla v,\nabla w)\|^{2}_{L^{2}}+\|\Delta u\|^{2}_{L^{2}}). \end{align}Combining (3.19) and (3.23), we find that
\begin{align*} &\frac{\textrm{d}}{\textrm{d}t}(\|\Delta u\|^{2}_{L^{2}}+\|(\nabla v,\nabla w)\|^{2}_{L^{2}})+(\|\nabla\Delta u\|^{2}_{L^{2}}+\|(\Delta v,\Delta w)\|^{2}_{L^{2}})\nonumber\\ &\quad \leq C (1+\|(\nabla v,\nabla w)\|^{2}_{L^{2}}+\|\Delta u\|^{2}_{L^{2}}). \end{align*}
For all 
$0\leq t\leq T$, by Gronwall's inequality implies
\begin{align*} &u\in L^{\infty}(0,T;H^{2}(\mathbb{R}^{3}))\cap L^{2}(0,T;H^{3}(\mathbb{R}^{3})), \end{align*}
\begin{align*}&v,w\in L^{\infty}(0,T;H^{1}(\mathbb{R}^{3}))\cap L^{2}(0,T;H^{2}(\mathbb{R}^{3})).\end{align*}
Since 
$H^{2}(\mathbb {R}^{3})\hookrightarrow L^{\infty }(\mathbb {R}^{3})$, we have
\[ \int^{T}_{0}\|\nabla\times u(t)\|_{L^{\infty}} \,\textrm{d}t< C . \]Combining the classical Beale-Kato-Majda's type global regularity criteria (1.6), as in [Reference Zhao and Bai33], we complete the proof of theorem 1.1.
Acknowledgments
The author would like to express his sincere gratitude to Prof Jiri Neustupa for many helpful discussions and suggestions, and to the anonymous referee for their extremely thorough and helpful report.
