Logarithmically improved blow-up criterion for the nematic liquid crystal system with zero viscosity

Abstract

In this paper, we establish a criterion for the breakdown of local in time classical solutions to the incompressible nematic liquid crystal system with zero viscosity in dimensions three. More precisely, let $T_{*}$ be the maximal existence time of the local classical solution, then $T_{*}<+\infty$ if and only if

$$\begin{aligned} \int \limits _{0}^{T_{*}}\frac{\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}}{\sqrt{1+\ln (e+\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }} +\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }})}}\hbox {d}t=\infty . \end{aligned}$$

The result can be regarded as a corresponding logarithmical blow-up criterion in Huang and Wang (Commun. Partial Differ. Equ. 37:875–884, 2012) for the nematic liquid crystal system with zero viscosity.

1 Introduction

Liquid crystals are a state of the matter that has properties between those of a conventional liquid and those of a solid crystal that are optically anisotropic, even when they are at rest [2]. In the period of 1958 through 1968, Ericksen and Leslie developed the hydrodynamic theory to describe the flow phenomena of nematic liquid crystals (see [3, 12, 13]). After that, Lin [15] proposed a simplified version which still retains most of the interesting mathematical properties (without destroy the basic nonlinear structure) of the original Ericksen–Leslie model for the hydrodynamics of nematic liquid crystals. The Cauchy problem of the three-dimensional simplified version reads as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}u-\nu \Delta u+u\cdot \nabla u+\nabla P=-\lambda \nabla \cdot (\nabla d \odot \nabla d) &{}\text {in}\ \mathbb {R}^{3}\times (0,+\infty ),\\ \mathrm{div } u=0 &{}\text {in}\ \mathbb {R}^{3}\times (0,+\infty ),\\ \partial _{t}d+u\cdot \nabla d=\gamma (\Delta d+|\nabla d|^{2}d) &{}\text {in}\ \mathbb {R}^{3}\times (0,+\infty ),\\ (u,d)|_{t=0}=(u_{0},d_{0}) &{}\text {in}\ \mathbb {R}^{3}, \end{array}\right. } \end{aligned}$$

(1.1)

where $u\in \mathbb {R}^{3}$ denotes the velocity field of the fluid, $d\in \mathbb {S}^{2}$ (the unit sphere in $\mathbb {R}^{3}$) the unit vector field that denotes the macroscopic/continuum nematic liquid crystal molecular orientations, and $P\in \mathbb {R}$ is the pressure arising from the incompressibility condition $\mathrm{div } u=0$, which including both the hydrostatic part and the induced elastic part from the orientation field, and $\nu$, $\lambda$, and $\gamma$ are positive constants which represent viscosity, the competition between kinetic energy and potential energy, and microscopic elastic relaxation time for the molecular orientation field. The term $\nabla d\odot \nabla d$ in the stress tensor represents the anisotropic feature of the system, which is the $3\times 3$ matrix whose $(i,j)$-th entry is given by $\partial _{i}d\cdot \partial _{j}d (1\le i,j\le 3)$. $u_{0}$ and $d_{0}$ are the initial datum of $u$ and $d$, and $u_{0}$ satisfies $\mathrm{div } u_{0}=0$ in distributional sense. It is easy to verify that

$$\begin{aligned} \nabla \cdot (\nabla d\odot \nabla d)=\nabla \left( \frac{|\nabla d|^{2}}{2}\right) +\Delta d\cdot \nabla d. \end{aligned}$$

(1.2)

Note that the system (1.1) is a nonlinear coupling of the conventional Navier–Stokes equations, and the system for the flow of harmonic maps into sphere. If $|\nabla d|^{2}d$ in the third equation of (1.1) is replaced by $\frac{(1-|d|^{2})d}{\varepsilon } \;(\varepsilon \,\hbox {is a positive parameter})$, the system has been studied in a series of papers by Lin [15] and Lin and Liu [16, 17]. More precisely, they proved in [16] the global existence of weak solutions and classical solutions in dimensions two and three, and discussed uniqueness and some stability properties of solutions. In [17], they proved that the one-dimensional space–time Hausdorff measure of the singular set of “suitable” weak solutions is zero. Compared with these results, the studies for the system (1.1) were only started in recent years. Lin et al. [18] established global existence of Leray–Hopf-type weak solutions to the initial boundary value on bounded domain in two space dimensions, see also Hong [7]. The uniqueness of such weak solutions was subsequently obtained by Lin and Wang [19] and Xu and Zhang [26]. The global existence and uniqueness of strong solutions of (1.1) with small initial data were also considered by many authors; we refer the readers to see [6, 14, 24], and the references cited there.

In this paper, we consider the three-dimensional nematic liquid crystal flow system (1.1) with partial viscosity, that is, $\nu =0$, this occurs in the turbulent flow regime when the Reynolds number is very big. The three-dimensional Cauchy problem reads as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}u+u\cdot \nabla u+\nabla P=-\nabla \cdot (\nabla d \odot \nabla d) &{}\text {in}\ \mathbb {R}^{3}\times (0,+\infty ),\\ \mathrm{div } u=0 &{}\text {in}\ \mathbb {R}^{3}\times (0,+\infty ),\\ \partial _{t}d+u\cdot \nabla d=\Delta d+|\nabla d|^{2}d &{}\text {in}\ \mathbb {R}^{3}\times (0,+\infty ),\\ (u,d)|_{t=0}=(u_{0},d_{0}) &{}\text {in}\ \mathbb {R}^{3}. \end{array}\right. } \end{aligned}$$

(1.3)

Since the concrete values of the constants $\lambda$ and $\gamma$ do not play a special role in our discussion, for simplicity, we have already assumed that they all equal to one.

Local existence of smooth solutions of the system (1.3) has been announced in [8]. For a given unit vector $\bar{d}\in \mathbb {S}^{2}$ and $s>0$, we set

$$\begin{aligned} H^{s}_{\bar{d}}(\mathbb {R}^{3}, \mathbb {S}^{2}):=\Big \{d: d-\bar{d}\in H^{s}(\mathbb {R}^{3}, \mathbb {R}^{3}), |d|=1\ \text {a.e.}\ \text {in} \ \mathbb {R}^{3}\Big \}. \end{aligned}$$

It follows from [8, 25] that if the initial velocity $u_{0}\in H^{s}(\mathbb {R}^{3},\mathbb {R}^{3})$ with $\mathrm{div }u_{0}=0$ and $d_{0}\in H^{s+1}_{\bar{d}}(\mathbb {R}^{3}, \mathbb {S}^{2})$ for $s\ge 3$, then there exists a positive time $T_{*}$ depending only on $\Vert (u_{0}, \nabla d_{0})\Vert _{H^{s}}$ such that the system (1.3) has a unique smooth solution $(u, d)$ in $\mathbb {R}^{3}\times [0,T_{*})$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} u\in C\left( [0,T], H^{s}\left( \mathbb {R}^{3},\mathbb {R}^{3}\right) \right) \cap C^{1}\left( [0,T], H^{s-1}\left( \mathbb {R}^{3}, \mathbb {R}^{3}\right) \right) , \\ d\in C\left( [0,T], H^{s+1}_{\bar{d}}\left( \mathbb {R}^{3}, \mathbb {S}^{2}\right) \right) \cap C^{1}\left( [0,T], H^{s-1}_{\bar{d}}\left( \mathbb {R}^{3}, \mathbb {S}^{2}\right) \right) . \end{array}\right. } \end{aligned}$$

(1.4)

for any $0<T<T_{*}$. However, whether this smooth solution of (1.3) on $[0,T_{*})$ will lead to a singularity at $t=T_{*}$ is an outstanding open problem. For the well-known Euler equations [taking the orientation field $d\equiv 1$ in the system (1.3)], Beale et al. [1] in their pioneering work showed that, by using the logarithmic Sobolev inequality, if the smooth solution $u$ blows up at the time $t=T_{*}$, then

$$\begin{aligned} \int \limits _{0}^{T_{*}}\Vert \nabla \times u\left( \cdot , t\right) \Vert _{L^{\infty }}\;\hbox {d}t=\infty . \end{aligned}$$

(1.5)

Later, Kozono and Taniuchi [11] and Konozo et al. [10] refined the BKM-type criterion (1.5) to

$$\begin{aligned} \int \limits _{0}^{T_{*}}\Vert \nabla \times u(\cdot , t)\Vert _{BMO}\;\hbox {d}t=\infty \ \ \text {and}\ \ \int \limits _{0}^{T_{*}}\Vert \nabla \times u(\cdot , t)\Vert _{\dot{B}^{0}_{\infty , \infty }}\;\hbox {d}t=\infty , \end{aligned}$$

(1.6)

respectively, where $BMO$ is the space of Bounded Mean Oscillation and $\dot{B}^{0}_{\infty , \infty }$ is the homogeneous Besov space.

When the velocity field $u$ is identically vanishing, the system (1.3) becomes to the heat flow of harmonic maps. In [23], Wang established a Serrin-type regularity criterion, which implies that if the solution $d$ blows up at time $T_{*}$, then

$$\begin{aligned} \sup _{0\le t< T_{*}}\Vert \nabla d(\cdot , t)\Vert _{L^{n}}=\infty . \end{aligned}$$

(1.7)

Motivated by the conditions (1.5) and (1.7), Huang and Wang [8] established a BKM-type blow-up criterion for the system (1.1); more precisely, by using the logarithmic Sobolev inequality, they characterized the first finite singular time $T_{*}$ as that if $T_{*}<\infty$, then

$$\begin{aligned} \int \limits _{0}^{T_{*}}\big (\Vert \nabla \times u(\cdot , t)\Vert _{L^{\infty }}+\Vert \nabla d(\cdot , t)\Vert _{L^{\infty }}^{2}\big )\;\hbox {d}t=\infty . \end{aligned}$$

(1.8)

This result also holds for the system (1.3), and the condition (1.8) can be improved by

$$\begin{aligned} \int \limits _{0}^{T_{*}}\big (\Vert \nabla \times u(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d(\cdot , t)\Vert _{L^{\infty }}^{2}\big )\;\hbox {d}t=\infty , \end{aligned}$$

(1.9)

for details, see Wang and Wang [25].

Inspired by the above works on blow-up criteria of the Euler equations and the heat flow of harmonic maps, the purpose of this paper is to establish logarithmically improved blow-up criterion for the system (1.3) in terms of the homogeneous Besov space. We observe here that, for the full system (1.1), we have already proved in [20] that the smooth solution $(u,d)$ of (1.1) blows up at the time $T_{*}$ if and only if

$$\begin{aligned} \int \limits _{0}^{T_{*}}\frac{\Vert \nabla \times u(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}}{\sqrt{1+\text {ln}(e+\Vert \nabla \times u(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }} +\Vert \nabla d(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }})}}\text {d}t=\infty . \end{aligned}$$

The methods used in [20] cannot be applied to the system (1.3) because we used the viscosity term $\nu \Delta u$ to control some bad terms when we establish the higher derivative estimates of solution $(u,d)$. Compared with [20], the additional difficulties arise when we deal with the higher derivative estimates of the external force term $\nabla \cdot (\nabla d\odot \nabla d)$ and the convective term $u\cdot \nabla d$. We emphasize that the proof of our main result uses in a fundamental way the algebraical structure of the system (1.3).

Now, we state our main result as follows:

Theorem 1.1

Let $u_{0}\in H^{3}(\mathbb {R}^{3},\mathbb {R}^{3})$ with $\mathrm{div }u_{0}=0$ and $d_{0}\in H^{4}_{\bar{d}}(\mathbb {R}^{3},\mathbb {S}^{2})$, and let $T_{*}>0$ be the maximal existence time such that the system (1.3) has a unique solution $(u,d)$ satisfying (1.4) for any $0<T<T_{*}$. If $T_{*}<+\infty$, then

$$\begin{aligned} \int \limits _{0}^{T_{*}}\frac{\Vert \nabla u(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}}{\sqrt{1+\ln (e+\Vert \nabla u(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }} +\Vert \nabla d(\cdot ,t)\Vert _{\dot{B}^{0}_{\infty ,\infty }})}}\hbox {d}t=\infty . \end{aligned}$$

(1.10)

In particular, it holds that

$$\begin{aligned} \limsup _{t\rightarrow T_{*}}\left( \Vert \nabla u(\cdot ,t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d(\cdot ,t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}\right) =\infty . \end{aligned}$$

By the Biot–Savart law, we have the following two representations:

$$\begin{aligned} \frac{\partial u}{\partial x_{j}}=R_{j}(R\times \omega ) \; \; \text {for}\; \; j=1,2,3 \; \; \text {and}\; \; \frac{\partial u^{i}}{\partial x_{j}}=R_{j}\Big (\sum _{k=1}^{3}R_{k} \mathrm{Def }u_{ki}\Big )\; \; \text {for}\; \; i, j=1,2,3, \end{aligned}$$

where $\omega =\nabla \times u=\left( \frac{\partial u^{k}}{\partial x_{j}}-\frac{\partial u^{j}}{\partial x_{k}}\right) _{1\le j, k\le 3}$ and $\mathrm{Def }u_{ki}=\left( \frac{\partial u^{k}}{\partial x_{i}}+\frac{\partial u^{i}}{\partial x_{k}}\right) _{1\le k, i\le 3}$ are the vorticity and the deformation tensor of the velocity field, respectively; $R=(R_{1}, R_{2}, R_{3})$ and $R_{j}=\frac{\partial }{\partial x_{j}}(-\Delta )^{-\frac{1}{2}}\; (j=1,2,3)$ denotes the Riesz transforms. Since the Riesz transforms are bounded in $\dot{B}^{0}_{\infty ,\infty }(\mathbb {R}^{3})$, we can get the following result as a corollary:

Corollary 1.2

Let $u_{0}\in H^{3}(\mathbb {R}^{3},\mathbb {R}^{3})$ with $\mathrm{div }u_{0}=0$ and $d_{0}\in H^{4}_{\bar{d}}(\mathbb {R}^{3},\mathbb {S}^{2})$, and let $T_{*}>0$ be the maximal existence time such that the system (1.3) has a unique solution $(u,d)$ satisfying (1.4) for any $0<T<T_{*}$. If $T_{*}<+\infty$, then

$$\begin{aligned} \int \limits _{0}^{T_{*}}\frac{\Vert \omega (\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}}{\sqrt{1+\ln \left( e+\Vert \omega (\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }} +\Vert \nabla d(\cdot ,t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}\right) }}\mathrm{d}t=\infty \end{aligned}$$

and

$$\begin{aligned} \int \limits _{0}^{T_{*}}\frac{\Vert \mathrm{Def }u(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}}{\sqrt{1+\ln \left( e+\Vert \mathrm{Def }u(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }} +\Vert \nabla d(\cdot ,t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}\right) }}\mathrm{d}t=\infty . \end{aligned}$$

Due to $L^{\infty }(\mathbb {R}^{3})\subset BMO\subset \dot{B}^{0}_{\infty ,\infty }(\mathbb {R}^{3})$, thus Corollary 1.2 can be regarded as an extension of the blow-up criterion in [25].

This paper is written as follows. In Sect. 2, we collect some crucial tools used in the proof of Theorem 1.1. In Sect. 3, we present the proof of Theorem 1.1. Throughout the paper, $C$ denotes a constant and may change from line to line; $\Vert \cdot \Vert _{X}$ stands for the norm of the Banach space $X$.

2 Preliminaries

In this section, we collect some analytic tools used later in the proof of Theorem 1.1. We denote by $H^{s}_{p}(\mathbb {R}^{3})$ with $s>0$ and $1<p<\infty$ the usual Sobolev space $\{f\in L^{p}(\mathbb {R}^{3}):\ \nabla ^{s}f\in L^{p}(\mathbb {R}^{3})\},\; BMO$ the space of the bounded mean oscillation, and $\dot{B}^{s}_{p,q}(\mathbb {R}^{3})$ with $s\in \mathbb {R}$ and $1\le p, q\le \infty$ the homogeneous Besov space.

Lemma 2.1

Let $1\le p<q<\infty$. Then, there exists a constant $C$ depending only on $p$ and $q$ such that

$$\begin{aligned} \Vert f\Vert _{L^{q}}\le C\Vert f\Vert _{BMO}^{1-\frac{p}{q}}\Vert f\Vert _{L^{p}}^{\frac{p}{q}} \end{aligned}$$

(2.1)

for all $f\in BMO\cap L^{p}(\mathbb {R}^{3})$.

Proof

This is an Exercise 7.4.1 in [4], see p. 554, we omit the proof here.$\square$

Lemma 2.2

For all $f\in H^{s-1}(\mathbb {R}^{3})$ with $s>\frac{5}{2}$, we have

$$\begin{aligned} \Vert f\Vert _{BMO}\le C\big (1+\Vert f\Vert _{\dot{B}^{0}_{\infty , \infty }}\ln ^{\frac{1}{2}}\left( e+\Vert f\Vert _{H^{s-1}}\right) \big ). \end{aligned}$$

(2.2)

Proof

By Theorem 2.1 in [10], we know that

$$\begin{aligned} \Vert f\Vert _{\dot{B}^{0}_{\infty ,2}}\le C\big (1+\Vert f\Vert _{\dot{B}^{0}_{\infty , \infty }}\ln ^{\frac{1}{2}}(e+\Vert f\Vert _{H^{s-1}})\big ), \end{aligned}$$

which combining the Sobolev embedding result $\dot{B}^{0}_{\infty ,2}(\mathbb {R}^{3})\hookrightarrow BMO$ implies (2.2) immediately. $\square$

Lemma 2.3

Let $1<p<\infty ,\; s_{1}\ge 1$ and $s_{2}\ge 1$. Then, we have

$$\begin{aligned} \Vert \Lambda ^{s_{1}}f\Lambda ^{s_{2}}g\Vert _{L^{p}} \le C\big (\Vert f\Vert _{BMO}\Vert \Lambda ^{s_{1}+s_{2}}g\Vert _{L^{p}}+\Vert g\Vert _{BMO} \Vert \Lambda ^{s_{1}+s_{2}}f\Vert _{L^{p}}\big ) \end{aligned}$$

(2.3)

for all $f,g \in BMO\cap H^{s_{1}+s_{2}}_{p}(\mathbb {R}^{3})$ with the constant $C$ depending only on $p, s_{1}$ and $s_{2}$. Here, we denote by $\Lambda =(-\Delta )^{\frac{1}{2}}$.

Proof

This is Lemma 2.1 in [11]. $\square$

Lemma 2.4

Let $1<p<q<\infty$, and let $s=\beta (\frac{q}{p}-1)>0$. Then, there exists a constant $C$ depending only on $\beta ,\; p$ and $q$ such that the estimate

$$\begin{aligned} \Vert f\Vert _{L^{q}}\le C\Vert \Lambda ^{s}f\Vert _{L^{p}}^{\frac{p}{q}}\Vert f \Vert _{\dot{B}^{-\beta }_{\infty ,\infty }}^{1-\frac{p}{q}} \end{aligned}$$

(2.4)

holds for all $f\in \dot{H}^{s}_{p}(\mathbb {R}^{3}) \cap \dot{B}^{-\beta }_{\infty ,\infty }(\mathbb {R}^{3})$. In particular, by taking $s=\beta =1, p=2$ and $q=4$, (2.4) implies that

$$\begin{aligned} \Vert f\Vert _{L^{4}}\le C\Vert \Lambda f\Vert _{L^{2}}^{\frac{1}{2}}\Vert f\Vert _{ \dot{B}^{-1}_{\infty ,\infty }}^{\frac{1}{2}}. \end{aligned}$$

(2.5)

Proof

This is Theorem 2.2 in [5], which is originally from [22]. $\square$

Lemma 2.5

For $s>1$, we have

$$\begin{aligned} \Vert \Lambda ^{s}(fg)-f\Lambda ^{s}g\Vert _{L^{p}}\le C\big (\Vert \Lambda f\Vert _{L^{p_{1}}}\Vert \Lambda ^{s-1}g\Vert _{L^{q_{1}}}+ \Vert \Lambda ^{s}f\Vert _{L^{p_{2}}}\Vert g\Vert _{L^{q_{2}}}\big ) \end{aligned}$$

(2.6)

with $1<p, q_{1}, p_{2}<\infty$ such that $\frac{1}{p}= \frac{1}{p_{1}}+ \frac{1}{q_{1}}=\frac{1}{p_{2}}+\frac{1}{q_{2}}$.

Proof

This is Lemma X1 in Appendix of [9].$\square$

3 The proof of Theorem 1.1

We prove Theorem 1.1 by contradiction. Suppose that (1.10) is not true. Then, there exists a positive constant $M$ such that

$$\begin{aligned} \int \limits _{0}^{T_{*}}\frac{\Vert \nabla u(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }} +\Vert \nabla d(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}}{\sqrt{1+\ln \left( e+\Vert \nabla u(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }} +\Vert \nabla d(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}\right) }}\hbox {d}t\le M<\infty . \end{aligned}$$

(3.1)

In the sequel, we will show that if assumption (3.1) holds, then

$$\begin{aligned} \lim _{t\rightarrow T_{*}^{-}}(\Vert \nabla \Delta u(\cdot ,t)\Vert _{L^2}^{2} +\Vert \Delta ^{2} d(\cdot , t)\Vert _{L^2}^{2})\le C, \end{aligned}$$

(3.2)

where $C$ is a constant depending only on $u_{0}, d_{0}, T_{*}$ and $M$. The estimate (3.2) is enough to ensure the extension of smooth solution ($u, d$) beyond the time $T_{*}$. That is to say, $[0,T_{*})$ is not a maximal existence interval, which leads to the contradiction. We split the proof of (3.2) into the following three steps.

Step 1 (Energy inequalities) Multiplying the first equation of (1.3) by $u$, integrating over $\mathbb {R}^{3}$, after integration by parts, we see that

$$\begin{aligned} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\Vert u\Vert _{L^{2}}^{2}=-\int \limits _{\mathbb {R}^{3}}\Delta d\cdot \nabla d\cdot u\hbox {d}x, \end{aligned}$$

(3.3)

where we have used the facts (1.2) and $\mathrm{div }u=0$ which imply that

$$\begin{aligned} \int \limits _{\mathbb {R}^{3}}\nabla \left( \frac{|\nabla d|^{2}}{2}\right) \cdot u\hbox {d}x=-\int \limits _{\mathbb {R}^{3}}\left( \frac{|\nabla d|^{2}}{2}\right) \mathrm{div }u\hbox {d}x=0. \end{aligned}$$

Multiplying the third equation of (1.3) by $-\Delta d$, integrating over $\mathbb {R}^{3}$, and using the facts $|d|=1$ and $|\nabla d|^{2}=-d\cdot \Delta d$, one obtains that

$$\begin{aligned} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\Vert \nabla d\Vert _{L^{2}}^{2}+\Vert \Delta d\Vert _{L^{2}}^{2}-\int \limits _{\mathbb {R}^{3}}u\cdot \nabla d\cdot \Delta d\hbox {d}x&=-\int \limits _{\mathbb {R}^{3}}|\nabla d|^{2}d\cdot \Delta d \hbox {d}x\nonumber \\&=\int \limits _{\mathbb {R}^{3}}|d\cdot \Delta d|^{2}\hbox {d}x\le \int \limits _{\mathbb {R}^{3}}|\Delta d|^{2}\hbox {d}x. \end{aligned}$$

(3.4)

Adding the above estimates (3.3) and (3.4) together, we have

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\big (\Vert u\Vert _{L^{2}}^{2}+\Vert \nabla d\Vert _{L^{2}}^{2}\big )\le 0, \end{aligned}$$

(3.5)

and it follows that

$$\begin{aligned} \sup _{t\ge 0}\big (\Vert u(\cdot , t)\Vert _{L^{2}}^{2}+ \Vert \nabla d(\cdot , t)\Vert _{L^{2}}^{2}\big )\le \Vert u_{0}\Vert _{L^{2}}^{2}+ \Vert \nabla d_{0}\Vert _{L^{2}}^{2}. \end{aligned}$$

(3.6)

Step 2 (Estimates for the gradient of $(u,\nabla d)$) Multiplying the first equation of (1.3) by $-\Delta u$, integrating over $\mathbb {R}^{3}$, and using $\mathrm{div } u=0$, one obtains that

$$\begin{aligned} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\Vert \nabla u\Vert _{L^{2}}^{2}&=\int \limits _{\mathbb {R}^{3}}u \cdot \nabla u\cdot \Delta u\hbox {d}x+\int \limits _{\mathbb {R}^{3}}\nabla \hbox {d}\cdot \Delta \hbox {d}\cdot \Delta u\hbox {d}x\nonumber \\&=-\int \limits _{\mathbb {R}^{3}}\nabla u\cdot \nabla u\cdot \nabla u\hbox {d}x +\int \limits _{\mathbb {R}^{3}}\nabla d\cdot \Delta d\cdot \Delta u\hbox {d}x. \end{aligned}$$

(3.7)

Taking $\Delta$ on the third equation of (1.3), multiplying the resulting equation by $\Delta d$, and integrating over $\mathbb {R}^{3}$, and using $\mathrm{div } u=0$ again, we see that

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

(3.8)

By adding the above estimates (3.7) and (3.8), note that the second term in the right-hand side of (3.7) and the first term in the right-hand side of (3.8) can be canceled, we obtain

$$\begin{aligned} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\big (\Vert \nabla u\Vert _{L^{2}}^{2}+\Vert \Delta d\Vert _{L^{2}}^{2}\big )+\Vert \nabla \Delta d\Vert _{L^{2}}^{2}&=-\int \limits _{\mathbb {R}^{3}}\nabla u\cdot \nabla u\cdot \nabla u\hbox {d}x-2\int \limits _{\mathbb {R}^{3}}\nabla u\cdot \nabla \nabla d\cdot \Delta d \hbox {d}x\nonumber \\&\quad +\,\int \limits _{\mathbb {R}^{3}}\Delta (d|\nabla d|^{2})\cdot \Delta d \hbox {d}x\nonumber \\&=I_{1}+I_{2}+I_{3}. \end{aligned}$$

(3.9)

Now, we estimate the terms $I_{i} \;(i=1,2,3)$ one by one. For $I_{1}$, applying the Hölder inequality and Lemma 2.1 with $q=4$ and $p=2$, we have

$$\begin{aligned} I_{1}&\le |\int \limits _{\mathbb {R}^{3}}\nabla u\cdot \nabla u\cdot \nabla u\hbox {d}x|\le C\Vert \nabla u\Vert _{L^{4}}^{2}\Vert \nabla u\Vert _{L^{2}}\nonumber \\&\le C\Vert \nabla u\Vert _{BMO}\Vert \nabla u\Vert _{L^{2}}^{2}. \end{aligned}$$

(3.10)

For $I_{2}$, by using the Hölder inequality and Lemma 2.4 with $f=\nabla ^{2} d$, we obtain

$$\begin{aligned} I_{2}&\le 2|\int \limits _{\mathbb {R}^{3}}\nabla u\cdot \nabla \nabla d\cdot \Delta d \hbox {d}x|\le C\Vert \nabla u\Vert _{L^{2}}\Vert \nabla ^{2} d\Vert _{L^{4}}\Vert \Delta d\Vert _{L^{4}}\nonumber \\&\le C\Vert \nabla u\Vert _{L^{2}}\Vert \nabla ^{2} d\Vert _{L^{4}}^{2} \le C\Vert \nabla u\Vert _{L^{2}}\Vert \nabla ^{2} d\Vert _{\dot{B}^{-1}_{\infty ,\infty }}\Vert \nabla \Delta d\Vert _{L^{2}}\nonumber \\&\le C\Vert \nabla u\Vert _{L^{2}}\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}\Vert \nabla \Delta d\Vert _{L^{2}}\nonumber \\&\le \frac{1}{4}\Vert \nabla \Delta d\Vert _{L^{2}}^{2}+C\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}\Vert \nabla u\Vert _{L^{2}}^{2}, \end{aligned}$$

(3.11)

where we have used the fact that the norms $\Vert \Delta d\Vert _{\dot{B}^{-1}_{\infty ,\infty }}$ and $\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}$ are equivalent. For $I_{3}$, after integration by parts, by using the Hölder inequality, the Young inequality and the fact $|d|=1$, we obtain

$$\begin{aligned} I_{3}&\le \left| \int \limits _{\mathbb {R}^{3}}\Delta (d|\nabla d|^{2})\cdot \Delta d \hbox {d}x\right| =\left| \int \limits _{\mathbb {R}^{3}} \nabla (d|\nabla d|^{2})\cdot \nabla \Delta d \hbox {d}x\right| \nonumber \\&\le \left| \int \limits _{\mathbb {R}^{3}}\nabla d|\nabla d|^{2}\cdot \nabla \Delta d \hbox {d}x\right| +\left| \int \limits _{\mathbb {R}^{3}} d\nabla (|\nabla d|^{2})\cdot \nabla \Delta d \hbox {d}x\right| \nonumber \\&\le \left| \int \limits _{\mathbb {R}^{3}}\nabla (\nabla d|\nabla d|^{2})\cdot \Delta d \hbox {d}x\right| +\left| \int \limits _{\mathbb {R}^{3}} d\nabla (|\nabla d|^{2})\cdot \nabla \Delta d \hbox {d}x\right| \nonumber \\&\le C\left| \int \limits _{\mathbb {R}^{3}}|\nabla d|^{2}|\nabla ^{2} d|^{2} \hbox {d}x\right| +C\left| \int \limits _{\mathbb {R}^{3}} |\nabla d||\nabla ^{2}d||\nabla \Delta d| \hbox {d}x\right| \nonumber \\&\le C\Vert \nabla d\Vert _{L^{4}}^{2}\Vert \nabla ^{2} d\Vert _{L^{4}}^{2}+C\Vert \nabla d\Vert _{L^{4}}\Vert \nabla ^{2} d\Vert _{L^{4}}\Vert \nabla \Delta d\Vert _{L^{2}}\nonumber \\&\le \frac{1}{8}\Vert \nabla \Delta d\Vert _{L^{2}}^{2}+C\Vert \nabla d\Vert _{L^{4}}^{2}\Vert \nabla ^{2} d\Vert _{L^{4}}^{2}\nonumber \\&\le \frac{1}{8}\Vert \nabla \Delta d\Vert _{L^{2}}^{2}+C\Vert d\Vert _{L^{\infty }}\Vert \Delta d\Vert _{L^{2}}\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}\Vert \nabla \Delta d\Vert _{L^{2}}\nonumber \\&\le \frac{1}{4}\Vert \nabla \Delta d\Vert _{L^{2}}^{2}+C\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}\Vert \Delta d\Vert _{L^{2}}^{2}, \end{aligned}$$

(3.12)

where we have used Lemma 2.4 with $f=\nabla ^{2} d$ and the following Gagliardo–Nirenberg inequalities with $q=2$:

$$\begin{aligned} \Vert \nabla f\Vert _{L^{2q}}\le \Vert f\Vert _{L^{\infty }}^{\frac{1}{2}}\Vert \Delta f\Vert _{L^{q}}^{\frac{1}{2}}\ \ \text {for}\ \ q>\frac{3}{2}. \end{aligned}$$

(3.13)

Putting the above estimates (3.10)–(3.12) into (3.9), and using Lemma 2.2, one concludes that

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}&\big (\Vert \nabla u\Vert _{L^{2}}^{2}+\Vert \Delta d\Vert _{L^{2}}^{2}\big ) +\Vert \nabla \Delta d\Vert _{L^{2}}^{2}\le C\big (\Vert \nabla u\Vert _{BMO}+\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}\big )\big (\Vert \nabla u\Vert _{L^{2}}^{2} +\Vert \Delta d\Vert _{L^{2}}^{2}\big )\nonumber \\&\le C\big (1+\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}\ln ^{\frac{1}{2}} \left( e+\Vert \nabla \Delta u\Vert _{L^{2}}\right) +\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty , \infty }}^{2}\big )\big (\Vert \nabla u\Vert _{L^{2}}^{2}+\Vert \Delta d\Vert _{L^{2}}^{2} \big )\nonumber \\&\le C\big (1\!+\!\left( \Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}\!+\!\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}\right) \!\ln ^{\frac{1}{2}}\left( e\!+\!\Vert \nabla \!\Delta u\Vert _{L^{2}}\!+\!\Vert \Delta ^{2} d\Vert _{L^{2}}\right) \big )\big (\Vert \nabla u\Vert _{L^{2}}^{2}\!+\!\Vert \!\Delta d\Vert _{L^{2}}^{2}\big )\nonumber \\&\le C\big (\Vert \nabla u\Vert _{L^{2}}^{2}+\Vert \Delta d\Vert _{L^{2}}^{2}\big )+ C\frac{\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}}{\sqrt{1+\ln \left( e+\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}\right) }}\nonumber \\&\quad \times \sqrt{1+\ln \left( e+\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}\right) }\nonumber \\&\quad \ln ^{\frac{1}{2}}\left( e+\Vert \nabla \Delta u\Vert _{L^{2}}+\Vert \Delta ^{2} d\Vert _{L^{2}}\right) \big (\Vert \nabla u\Vert _{L^{2}}^{2}+\Vert \Delta d\Vert _{L^{2}}^{2}\big )\nonumber \\&\le C\big (\Vert \nabla u\Vert _{L^{2}}^{2}+\Vert \Delta d\Vert _{L^{2}}^{2}\big )+ C\frac{\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}}{\sqrt{1+\ln \left( e+\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}\right) }}\nonumber \\&\quad \times \ln \left( e+\Vert \nabla \Delta u\Vert _{L^{2}}+\Vert \Delta ^{2} d\Vert _{L^{2}}\right) \big (\Vert \nabla u\Vert _{L^{2}}^{2}+\Vert \Delta d\Vert _{L^{2}}^{2}\big ), \end{aligned}$$

(3.14)

where we have used the following inequality by applying the basic energy inequality (3.6):

$$\begin{aligned} \sqrt{1+\ln \left( e+\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}\right) }&\le C\sqrt{\ln \left( e+\Vert \nabla u\Vert _{L^{\infty }}+\Vert \nabla d\Vert _{L^{\infty }}\right) }\\&\le C\sqrt{\!\ln \left( e\!+\!\Vert u\Vert _{L^{2}}^{\frac{1}{6}}\Vert \!\nabla \!\Delta u\Vert _{L^{2}}^{\frac{5}{6}} \!+\!\Vert \nabla d\Vert _{L^{2}}^{\frac{1}{2}}\Vert \!\Delta ^{2} d\Vert _{L^{2}}^{\frac{1}{2}}\right) }\\&\le C\sqrt{\ln \left( e+\Vert \nabla \Delta u\Vert _{L^{2}}+\Vert \Delta ^{2} d\Vert _{L^{2}}\right) }. \end{aligned}$$

Based on the inequality (3.14), by using the condition (3.1), one obtains that for any small constant $\varepsilon >0$, there exists $T=T(\varepsilon )\in (0,T_{*})$ such that

$$\begin{aligned} \int \limits _{T}^{T_{*}}\frac{\Vert \nabla u(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }} +\Vert \nabla d(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}}{\sqrt{1+\ln (e+\Vert \nabla u(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }} +\Vert \nabla d(\cdot , t)\Vert _{\dot{B}^{0}_{\infty ,\infty }})}}\hbox {d}t\le \varepsilon . \end{aligned}$$

Thus for any $T\le t<T_{*}$, if we set

$$\begin{aligned} W(t):=\sup _{T<\tau \le t} \left( \Vert \nabla \Delta u(\cdot , \tau )\Vert _{L^{2}}^{2}+\Vert \Delta ^{2} d(\cdot , \tau )\Vert _{L^{2}}^{2}\right) . \end{aligned}$$

then we can derive from (3.14) that

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}&\left( \Vert \nabla u\Vert _{L^{2}}^{2}+\Vert \Delta d\Vert _{L^{2}}^{2}\right) +\Vert \nabla \Delta d\Vert _{L^{2}}^{2} \le C\left( \Vert \nabla u\Vert _{L^{2}}^{2}+\Vert \Delta d\Vert _{L^{2}}^{2}\right) \nonumber \\&\quad +\,C\frac{\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}}{\sqrt{1+\ln \left( e+\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}\right) }} \ln (e+W(t))\big (\Vert \nabla u\Vert _{L^{2}}^{2}+\Vert \Delta d\Vert _{L^{2}}^{2}\big ). \end{aligned}$$

(3.15)

By applying Gronwall’s inequality to (3.15) in the interval $[T, t)$, it follows that

$$\begin{aligned} \Vert \nabla u\Vert _{L^{2}}^{2}&+\Vert \Delta d\Vert _{L^{2}}^{2} \le C(\varepsilon )\exp \Big \{\int \limits _{T}^{t}C\mathrm{d}\tau \nonumber \\&\quad +\,C\int \limits _{T}^{t}\frac{\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }} +\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}}{\sqrt{1+\ln \left( e+\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }} +\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}\right) }}\mathrm{d}\tau \ln (e+W(t))\Big \}\nonumber \\&\le C(\varepsilon )\exp \Big \{C\big [(T_{*}-T)+\varepsilon \ln (e+W(t))\big ]\Big \}\nonumber \\&\le C(\varepsilon )\exp \Big \{C\varepsilon (1+\ln (e+W(t)))\Big \}\nonumber \\&\le C(\varepsilon )(e+W(t))^{2C\varepsilon }, \end{aligned}$$

(3.16)

where $C(\varepsilon )=C\{\Vert \nabla u (\cdot ,T(\varepsilon )) \Vert _{L^{2}}^{2}+\Vert \Delta d(\cdot ,T(\varepsilon ))\Vert _{L^{2}}^{2}\}$ is a bounded positive constant depending on $\varepsilon$ which may change from line to line, and $C$ is the positive constant whose value is independent of either $\varepsilon$ or $T$.

Step 3 (Higher energy estimates) In this step, we derive the higher energy estimates. Applying $\nabla \Delta$ on the first equation of (1.3), multiplying $\nabla \Delta u$ and integrating over $\mathbb {R}^{3}$, after integration by parts, one obtains

$$\begin{aligned} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\Vert \nabla \Delta u\Vert _{L^{2}}^{2}&=-\int \limits _{\mathbb {R}^{3}} \nabla \Delta (u\cdot \nabla u)\cdot \nabla \Delta u\hbox {d}x -\int \limits _{\mathbb {R}^{3}}\nabla \Delta (\Delta d\cdot \nabla d)\cdot \nabla \Delta u\hbox {d}x\nonumber \\&=-\int \limits _{\mathbb {R}^{3}}\big [\nabla \Delta (u\cdot \nabla u)-u\cdot \nabla \nabla \Delta u\big ]\cdot \nabla \Delta u\hbox {d}x\nonumber \\&\quad -\,\int \limits _{\mathbb {R}^{3}}\big [\nabla \Delta (\nabla d\cdot \Delta d)-\nabla d\cdot \nabla \Delta ^{2} d\big ]\cdot \nabla \Delta u\hbox {d}x\nonumber \\&\quad -\,\int \limits _{\mathbb {R}^{3}}\nabla d\cdot \nabla \Delta ^{2} d\cdot \nabla \Delta u\hbox {d}x, \end{aligned}$$

(3.17)

where we have used the fact that $\mathrm{div } u=0$ implies that $\int _{\mathbb {R}^{3}}\nabla \Delta \nabla (\frac{|\nabla d|^{2}}{2})\cdot \nabla \Delta u\text {d}x=0$.

Taking $\Delta ^{2}$ on the third equation of (1.3), multiplying $\Delta ^{2}d$ and integrating over $\mathbb {R}^{3}$, one obtains

$$\begin{aligned} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\Vert \Delta ^{2} d\Vert _{L^{2}}^{2}&+\Vert \nabla \Delta ^{2} d\Vert _{L^{2}}^{2} =-\int \limits _{\mathbb {R}^{3}}\Delta ^{2}(u\cdot \nabla d)\cdot \Delta ^{2}d \hbox {d}x +\int \limits _{\mathbb {R}^{3}}\Delta ^{2}(|\nabla d|^{2}d)\cdot \Delta ^{2}d \hbox {d}x\nonumber \\&=\int \limits _{\mathbb {R}^{3}}\nabla \Delta (u\cdot \nabla d)\cdot \nabla \Delta ^{2}d \hbox {d}x +\int \limits _{\mathbb {R}^{3}}\Delta ^{2}(|\nabla d|^{2}d)\cdot \Delta ^{2}d \hbox {d}x\nonumber \\&=\int \limits _{\mathbb {R}^{3}}\big [\nabla \Delta (u\cdot \nabla d)-\nabla d\cdot \nabla \Delta u\big ]\cdot \nabla \Delta ^{2}d \hbox {d}x\nonumber \\&\quad +\,\int \limits _{\mathbb {R}^{3}}\nabla d\cdot \nabla \Delta u\cdot \nabla \Delta ^{2}d \hbox {d}x+\int \limits _{\mathbb {R}^{3}}\Delta ^{2}(|\nabla d|^{2}d)\cdot \Delta ^{2}d \hbox {d}x. \end{aligned}$$

(3.18)

Adding (3.17) and (3.18) together, we see that

$$\begin{aligned} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\big (\Vert \nabla \Delta u\Vert _{L^{2}}^{2}&+\Vert \Delta ^{2} d\Vert _{L^{2}}^{2}\big )+\Vert \nabla \Delta ^{2} d\Vert _{L^{2}}^{2} =-\int \limits _{\mathbb {R}^{3}}\big [\nabla \Delta (u\cdot \nabla u)-u\cdot \nabla \nabla \Delta u\big ]\cdot \nabla \Delta u\hbox {d}x\nonumber \\&\qquad -\int \limits _{\mathbb {R}^{3}}\big [\nabla \Delta (\nabla d\cdot \Delta d)-\nabla d\cdot \nabla \Delta ^{2} d\big ]\cdot \nabla \Delta u\hbox {d}x\nonumber \\&\qquad +\int \limits _{\mathbb {R}^{3}}\big [\nabla \Delta (u\cdot \nabla d)-\nabla d\cdot \nabla \Delta u\big ]\cdot \nabla \Delta ^{2}d \hbox {d}x\nonumber \\&\qquad +\int \limits _{\mathbb {R}^{3}}\Delta ^{2}(|\nabla d|^{2}d)\cdot \Delta ^{2}d \hbox {d}x\nonumber \\&\quad =J_{1}+J_{2}+J_{3}+J_{4}. \end{aligned}$$

(3.19)

The terms $J_{i} \;(i=1,2,3,4)$ can be estimated as follows: For $J_{1}$, by applying the Leibniz’s rule, Lemma 2.3 with $p=2$, and Lemma 2.2 with $f=\nabla u$, we obtain

$$\begin{aligned} J_{1}&\le \Big |\int \limits _{\mathbb {R}^{3}}\big [\nabla \Delta (u\cdot \nabla u)-u\cdot \nabla \nabla \Delta u\big ]\cdot \nabla \Delta u\hbox {d}x\Big |\nonumber \\&\le \Vert \nabla \Delta (u\cdot \nabla u)-u\cdot \nabla \nabla \Delta u\Vert _{L^{2}}\Vert \nabla \Delta u\Vert _{L^{2}}\nonumber \\&\le C\big (\Vert \nabla \Delta u\cdot \nabla u\Vert _{L^{2}}+\Vert \Delta u\cdot \nabla \nabla u\Vert _{L^{2}} +\Vert \nabla u\cdot \nabla \Delta u\Vert _{L^{2}}\big )\Vert \nabla \Delta u\Vert _{L^{2}}\nonumber \\&\le C\Vert \nabla u\Vert _{BMO}\Vert \nabla \Delta u\Vert _{L^{2}}^{2}\nonumber \\&\le C\Big (1+\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}\ln ^{\frac{1}{2}}(e+\Vert \nabla \Delta u\Vert _{L^{2}})\Big ) \Vert \nabla \Delta u\Vert _{L^{2}}^{2}\nonumber \\&\le C\Big (1\!\!+\!\!\frac{\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}}{\sqrt{1\!+\!\ln (e+\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}\!+\!\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }})}}\Big )\ln (e\!+\!\Vert \nabla \Delta u\Vert _{L^{2}}\!+\!\Vert \Delta ^{2} d\Vert _{L^{2}}) \Vert \nabla \Delta u\Vert _{L^{2}}^{2}. \end{aligned}$$

(3.20)

For $J_{2}$, Lemmas 2.4 and 2.5 lead to

$$\begin{aligned} J_{2}&\le \Big |\int \limits _{\mathbb {R}^{3}}\big [\nabla \Delta (\nabla d\cdot \Delta d)-\nabla d\cdot \nabla \Delta ^{2} d\big ]\cdot \nabla \Delta u\hbox {d}x\Big |\nonumber \\&\le C\Vert \Delta d\Vert _{L^{4}}\Vert \Delta ^{2} d\Vert _{L^{4}}\Vert \nabla \Delta u\Vert _{L^{2}}\nonumber \\&\le C\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}\Vert \nabla \Delta ^{2} d\Vert _{L^{2}}\Vert \nabla \Delta u\Vert _{L^{2}}\nonumber \\&\le \frac{1}{6}\Vert \nabla \Delta ^{2} d\Vert _{L^{2}}^{2}+C\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}\Vert \nabla \Delta u\Vert _{L^{2}}^{2}\nonumber \\&\le \frac{1}{6}\Vert \nabla \Delta ^{2} d\Vert _{L^{2}}^{2}+C\frac{\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}}{\sqrt{1+\ln (e+\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }})}}\nonumber \\&\qquad \qquad \qquad \times \ln (e+\Vert \nabla \Delta u\Vert _{L^{2}}+\Vert \Delta ^{2} d\Vert _{L^{2}})\Vert \nabla \Delta u\Vert _{L^{2}}^{2}, \end{aligned}$$

(3.21)

where we have used the following generalized Gagliardo–Nirenberg inequalities (see [21]):

$$\begin{aligned} \Vert \Delta d\Vert _{L^{4}}\le C\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{\frac{9}{10}}\Vert \nabla \Delta ^{2} d\Vert _{L^{2}}^{\frac{1}{10}}\ \ \text {and}\ \ \Vert \Delta ^{2} d\Vert _{L^{4}}\le C\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{\frac{1}{10}}\Vert \nabla \Delta ^{2} d\Vert _{L^{2}}^{\frac{9}{10}}. \end{aligned}$$

For $J_{3}$, by using Lemmas 2.4 and 2.5 again, we obtain that

$$\begin{aligned} J_{3}&\le \Big |\int \limits _{\mathbb {R}^{3}}\big [\nabla \Delta (u\cdot \nabla d)-\nabla d\cdot \nabla \Delta u\big ]\cdot \nabla \Delta ^{2}d \hbox {d}x\Big |\nonumber \\&\le C\Big (\Vert \Delta d\Vert _{L^{4}}\Vert \Delta u\Vert _{L^{4}}+\Vert \Delta ^{2} d\Vert _{L^{3}}\Vert u\Vert _{L^{6}}\Big )\Vert \nabla \Delta ^{2}d\Vert _{L^{2}}\nonumber \\&\le C\Big (\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{\frac{1}{2}}\Vert \nabla \Delta u\Vert _{L^{2}}^{\frac{1}{2}} \Vert \Delta d\Vert _{L^{2}}^{\frac{5}{8}}\Vert \Delta ^{2}d\Vert _{L^{2}}^{\frac{3}{8}}+\Vert \nabla u\Vert _{L^{2}}\Vert \Delta d\Vert _{L^{2}}^{\frac{1}{6}}\Vert \nabla \Delta ^{2}d\Vert _{L^{2}}^{\frac{5}{6}}\Big ) \Vert \nabla \Delta ^{2}d\Vert _{L^{2}}\nonumber \\&\le \frac{1}{6}\Vert \nabla \Delta ^{2}d\Vert _{L^{2}}^{2}+C\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}\Vert \nabla \Delta u\Vert _{L^{2}} \Vert \Delta d\Vert _{L^{2}}^{\frac{5}{4}}\Vert \Delta ^{2}d\Vert _{L^{2}}^{\frac{3}{4}}+C\Vert \nabla u\Vert _{L^{2}}^{12}\Vert \Delta d\Vert _{L^{2}}^{2}\nonumber \\&\le \frac{1}{6}\Vert \nabla \Delta ^{2}d\Vert _{L^{2}}^{2}+C\Vert \nabla u\Vert _{L^{2}}^{12}\Vert \Delta d\Vert _{L^{2}}^{2}+C\Big (1\!+\!\frac{\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}}{\sqrt{1+\ln (e+\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }})}}\Big )\nonumber \\&\ \ \ \times \ln (e+\Vert \nabla \Delta u\Vert _{L^{2}}+\Vert \Delta ^{2} d\Vert _{L^{2}}) \Vert \nabla \Delta u\Vert _{L^{2}} \Vert \Delta d\Vert _{L^{2}}^{\frac{5}{4}}\Vert \Delta ^{2} d\Vert _{L^{2}}^{\frac{3}{4}}, \end{aligned}$$

(3.22)

where we have used the following Gagliardo–Nirenberg inequalities:

$$\begin{aligned}&\Vert \Delta u\Vert _{L^{4}}\le C\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{\frac{1}{2}}\Vert \nabla \Delta u\Vert _{L^{2}}^{\frac{1}{2}},\ \ \Vert \Delta d\Vert _{L^{4}}\le C\Vert \Delta d\Vert _{L^{2}}^{\frac{5}{8}}\Vert \Delta ^{2}d\Vert _{L^{2}}^{\frac{3}{8}},\nonumber \\&\Vert \Delta ^{2} d\Vert _{L^{3}}\le C\Vert \Delta d\Vert _{L^{2}}^{\frac{1}{6}}\Vert \nabla \Delta ^{2}d\Vert _{L^{2}}^{\frac{5}{6}}. \end{aligned}$$

For $J_{4}$, applying the Leibniz’s rule, the fact $|d|=1$, the Hölder inequality and the Young inequality, one obtains

$$\begin{aligned} J_{4}&=\int \limits _{\mathbb {R}^{3}}\Delta ^{2}(|\nabla d|^{2}d)\cdot \Delta ^{2}d \hbox {d}x=-\int \limits _{\mathbb {R}^{3}}\nabla \Delta (|\nabla d|^{2}d)\cdot \nabla \Delta ^{2}d \hbox {d}x\nonumber \\&=-\int \limits _{\mathbb {R}^{3}}\big [\nabla \Delta (|\nabla d|^{2}) d+3\Delta (|\nabla d|^{2})\nabla d +3\nabla (|\nabla d|^{2})\Delta d +|\nabla d|^{2}\nabla \Delta d\big ]\cdot \nabla \Delta ^{2}d\hbox {d}x\nonumber \\&\le C\Big (\Vert \nabla d\Vert _{L^{6}}^{2}\Vert \nabla \Delta d\Vert _{L^{6}}+\Vert \nabla d\Vert _{L^{6}}\Vert \Delta ^{2} d\Vert _{L^{3}}+\Vert \Delta d\Vert _{L^{4}}\Vert \nabla \Delta d\Vert _{L^{4}}\nonumber \\&\quad +\Vert \nabla d\Vert _{L^{6}}\Vert \Delta ^{2} d\Vert _{L^{6}}^{2}\Big ) \Vert \nabla \Delta ^{2} d\Vert _{L^{2}}\nonumber \\&\le C\Big (\Vert \Delta d\Vert _{L^{2}}^{\frac{7}{6}}\Vert \nabla \Delta ^{2} d\Vert _{L^{2}}^{\frac{5}{6}} +\Vert \Delta d\Vert _{L^{2}}^{\frac{7}{3}}\Vert \nabla \Delta ^{2} d\Vert _{L^{2}}^{\frac{2}{3}}\Big )\Vert \nabla \Delta ^{2} d\Vert _{L^{2}}\nonumber \\&\le \frac{1}{6}\Vert \nabla \Delta ^{2} d\Vert _{L^{2}}^{2}+C\Vert \Delta d\Vert _{L^{2}}^{14}, \end{aligned}$$

(3.23)

where we have used the following Gagliardo–Nirenberg inequalities:

$$\begin{aligned}&\Vert \Delta ^{2} d\Vert _{L^{3}}\le C\Vert \Delta d\Vert _{L^{2}}^{\frac{1}{6}}\Vert \nabla \Delta ^{2} d\Vert _{L^{2}}^{\frac{5}{6}};\ \ \ \Vert \Delta d\Vert _{L^{4}}\le C\Vert \Delta d\Vert _{L^{2}}^{\frac{3}{4}}\Vert \nabla \Delta ^{2}d\Vert _{L^{2}}^{\frac{1}{4}};\\&\Vert \nabla \Delta d\Vert _{L^{6}}\le C\Vert \Delta d\Vert _{L^{2}}^{\frac{1}{3}}\Vert \nabla \Delta ^{2}d\Vert _{L^{2}}^{\frac{2}{3}}; \ \ \ \Vert \Delta d\Vert _{L^{6}}\le C \Vert \Delta d\Vert _{L^{2}}^{\frac{2}{3}}\Vert \nabla \Delta ^{2} d\Vert _{L^{2}}^{\frac{1}{3}}. \end{aligned}$$

Putting the above estimates (3.20)–(3.23) into (3.19), and using (3.16), we conclude that

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}W(t)&\le C\Big (1+\frac{\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}}{\sqrt{1+\ln (e+\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }})}}\Big )\nonumber \\&\quad \ln (e+W(t))W(t) +C(e+W(t))^{14C\varepsilon }\nonumber \\&\quad +\,C\Big (1+\frac{\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}}{\sqrt{1+\ln (e+\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }})}} \Big )\ln (e+W(t))W(t)^{\frac{7}{8}+\frac{5C\varepsilon }{4}}. \end{aligned}$$

(3.24)

By selecting $\varepsilon$ sufficiently small such that $14C\varepsilon \le 1$ and $\frac{7}{8}+\frac{5C\varepsilon }{4} \le 1$, it follows from (3.24) that

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}W(t)&\le C\Big (1+\frac{\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty , \infty }}+\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}}{\sqrt{1+\ln (e+\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }})}}\Big )\ln (e+W(t))W(t). \end{aligned}$$

Note that by using the condition (3.1), we know the terms in the parentheses on the right-hand side of the above inequality are integrable in time. Then, applying Gronwall’s inequality to the above inequality, we get the boundness of $W(t)$ on $[T,T_{*}]$, which implies that we can extend the solution $(u,d)$ beyond the time $T_{*}$, this contradicts the assumption that $0<T_{*}<\infty$ is the maximal existence time. We complete the proof of Theorem 1.1.