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Regularity criterion for 3D nematic liquid crystal flows in terms of finite frequency parts in \(\dot{B}_{\infty,\infty }^{-1}\)

Abstract

In this paper, we establish the regularity criterion for the weak solution of nematic liquid crystal flows in three dimensions when the \(L^{\infty }(0,T;\dot{B}_{\infty,\infty }^{-1})\)-norm of a suitable low frequency part of \((u,\nabla d)\) is bounded by a scaling invariant constant and the initial data \((u_{0},\nabla d_{0})\). Our result refines the corresponding one in (Liu and Zhao in J. Math. Anal. Appl. 407:557-566, 2013) and that in (Ri in Nonlinear Anal. TMA 190:111619, 2020).

Introduction

This note focuses on the regularity criteria for the following 3D nematic liquid crystal fluid flow:

$$ \textstyle\begin{cases} \partial _{t} u-\nu \Delta u+(u\cdot \nabla )u+\nabla \pi =-\lambda \nabla \cdot (\nabla d\odot \nabla d), &(x,t)\in \mathbf{R}^{3}\times (0, +\infty ), \\ \partial _{t} d+(u\cdot \nabla )d=\gamma (\Delta d+ \vert \nabla d \vert ^{2} d ), &(x,t)\in \mathbf{R}^{3}\times (0,+ \infty ), \\ \operatorname{div} u=0, & (x,t)\in \mathbf{R}^{3}\times (0,+\infty ), \\ (u,d)|_{t=0}=( u_{0},d_{0}), &x\in \mathbf{R}^{3}, \end{cases} $$
(1.1)

where \(u(x,t)\) is the unknown velocity field, \(d(x,t):\mathbf{R}^{3}\times (0,+\infty )\rightarrow \mathbb{S}^{2} \), the unit sphere in \(\mathbf{R}^{3}\), is the unknown (averaged) macroscopic/continuum molecule orientation of the nematic liquid crystal flow and π is the scalar pressure. ν, λ, γ are positive constants that represent viscosity, the competition between kinetic energy and potential energy, and the microscopic elastic relaxation time for the molecular orientation field. The notation \(\nabla d\odot \nabla d\) denotes the \(3\times 3\) matrix whose \((i,j)\) entry is given by \(\partial _{i} d\cdot \partial _{j} d\) (\(1\leq i,j\leq 3\)).

It is well-known that Ericksen and Leslie ([35, 8] established the hydrodynamic theory of liquid crystals in 1960s. Lin [9] first introduced the above liquid crystal flow (1.1). Later Lin and Liu [11] obtained the global existence theorem for a weak solution and the local existence for the strong solution to the system (1.1).

We first introduce the definition of Morrey spaces.

Definition 1.1

For \(1\le p\le q\le \infty \), we call \(\dot{M}_{p,q}(\mathbf{R}^{3})\) a Morrey space, if and only if

$$\begin{aligned} \Vert f \Vert _{\dot{M}_{p,q}(\mathbf{R}^{3})}=\sup_{x\in \mathbf{R}^{3}, 0< R< \infty }R^{\frac{3}{q}-\frac{3}{p}} \biggl( \int _{B(x,R)} \bigl\vert f(y) \bigr\vert ^{p}\,dy \biggr)^{ \frac{1}{p}}< +\infty, \end{aligned}$$

here \(B(x,R)\) denotes the ball in \(\mathbf{R}^{3}\) with center x and radius R.

In 2008, Fan and Guo [4] showed that, if u satisfies one of the following conditions:

$$\begin{aligned} &u\in L^{s} \bigl(0, T; \dot{M}_{p,q} \bigl( \mathbf{R}^{3} \bigr) \bigr)\quad \text{with }\frac{2}{s}+\frac{3}{p}=1, p \ge 3, p\ge q\ge 1, \\ &\nabla u\in L^{s} \bigl(0, T; \dot{M}_{p,q} \bigl( \mathbf{R}^{3} \bigr) \bigr)\quad \text{with }\frac{2}{s}+\frac{3}{p}=2, p \ge \frac{3}{2}, p\ge q\ge 1, \end{aligned}$$

then \((u,d)\) is extended beyond \(t=T\). Later Liu, Zhao and Cui [12] obtained the regularity criterion to the system (1.1) under the assumption that \(\partial _{3}u\in L^{\beta }(0,T; L^{\alpha })\) with \(\frac{2}{\beta }+\frac{3}{\alpha }\le 1, \alpha >3\). Recently, Wei, Li and Yao [16] proved that, if the weak solution \((u,d)\) satisfies

$$\begin{aligned} u_{3}, \nabla d\in L^{\beta } \bigl(0,T; L^{\alpha } \bigl( \mathbf{R}^{3} \bigr) \bigr), \quad\text{with }\frac{2}{\beta }+\frac{3}{\alpha } \le \frac{3}{4}+ \frac{1}{\alpha }, \alpha >\frac{10}{3}, \end{aligned}$$

then \((u,b)\) can be extended beyond \(t=T\). Liu and Zhao [13] proved that the solution \((u, d)\) to (1.1) is smooth up to time T provided that

$$\begin{aligned} \bigl\Vert (u,\nabla d) \bigr\Vert _{L^{\infty }(0,T; B^{-1}_{\infty,\infty }(\mathbf{R}^{3}))} \le \varepsilon _{0}. \end{aligned}$$

When \(d=0\), the system (1.1) becomes an incompressible Navier–Stokes equation. There is a large literature on the regularity criteria on the Navier–Stokes equation; see [1, 6, 7, 15].

By traditional turbulence theory, viscous incompressible flows develop in such a way that energy is transferred from large scales to neighboring smaller scales. Hence, it is important to study regularity for the Navier–Stokes equation based on various wave-number band parts of weak solutions is important since it reveals in a way the relationship between regularity of weak solutions and turbulent flows. Cheskidov and Shvydkoy [2] proved that a Leray–Hopf weak solution u to the Navier–Stokes equation is regular in \((0, T]\) if

$$\begin{aligned} \bigl\Vert u^{k} \bigr\Vert _{B_{\infty,\infty }^{-1}(\mathbf{R}^{3})}< C\nu, \end{aligned}$$

where \(u^{k}\) is high frequency part of u with Fourier models \(|\xi |\ge k\). Kim, Kwak and Yoo [5] proved that, if sufficiently high frequency parts of a weak solution to the Navier–Stokes equation on a torus belong to Serrin’s class, then the weak solution is regular. Very recently, Ri [14] proved that a Leray–Hopf weak solution u to 3D Navier–Stokes equations is regular if the \(L^{\infty }(0, T; B_{\infty,\infty }^{-1}(\mathbf{R}^{3}))\)-norm of a suitable low frequency part of u is bounded by a scaling invariant constant depending on the kinematic viscosity ν and initial value \(u_{0}\). Motivated by [2, 5, 13] and [14], we will investigate the regularity criteria for the weak solution \((u,d)\) to the liquid crystal fluid flows (1.1) in the critical function space \(L^{\infty }(0,T;\dot{B}_{\infty,\infty }^{-1}(\mathbf{R}^{3}))\) based on low and medium frequency parts, respectively. Before stating our result, we shall present some symbols and notations.

Let

$$ u_{k}:= \int _{0}^{k} u_{[s]}\,ds,\qquad u^{k}:= \int _{k}^{\infty }u_{[s]}\,ds,\quad u_{h,k}:=u_{k}-u_{h}, 0< h< k< \infty.$$
(1.2)

Here

$$ u_{[k]}(t,x)=\frac{1}{(2\pi )^{\frac{3}{2}}} \int _{ \vert \xi \vert =k} \hat{u}(t,\xi )e^{ix\cdot \xi }\,d\sigma _{\xi }, $$

and û denotes Fourier transform of u. Our result is stated as follows.

Theorem 1.2

Let \((u, d)\) be a weak solution to (1.1) with \((u_{0},d_{0})\in H^{3}(\mathbf{R}^{3})\times H^{4}(\mathbf{R}^{3}), \mathrm{div}u_{0}=0\). Assume that, for \(0< T<\infty \), there exists \(\delta \in (0,T)\) such that if \((u,d)\) is regular in \((0,T)\) the inequalities

$$\begin{aligned} \bigl\Vert (u_{\tilde{k}},\nabla d_{\tilde{k}}) \bigr\Vert _{L^{\infty }(T-\delta,T; \dot{B}_{\infty,\infty }^{-1})}< C_{1} \end{aligned}$$
(1.3)

and

$$\begin{aligned} &\bigl\Vert (u_{\tilde{k}/2,\tilde{k}},\nabla d_{\tilde{k}/2,\tilde{k}}) \bigr\Vert _{L^{\infty }(T-\delta,T;\dot{B}_{\infty,\infty }^{-1})} \\ &\quad < C_{2} \bigl( \Vert u_{0} \Vert _{L^{2}}+ \Vert \nabla d_{0} \Vert _{L^{2}} \bigr)^{-1} \bigl( \bigl\Vert \nabla u_{0}^{\tilde{k}} \bigr\Vert _{L^{2}}+ \bigl\Vert \Delta d_{0}^{\tilde{k}} \bigr\Vert _{L^{2}} \bigr)^{-1} \end{aligned}$$
(1.4)

hold. Then \((u,d)\) is regular on \((0,T]\), where \(\tilde{k}>0\) is defined by

$$\begin{aligned} \tilde{k}=C_{3} \bigl( \bigl\Vert \nabla u_{0}^{\tilde{k}} \bigr\Vert _{L^{2}}+ \bigl\Vert \Delta d_{0}^{ \tilde{k}} \bigr\Vert _{L^{2}} \bigr)^{2}, \end{aligned}$$

and the \(C_{i}, i=1, 2, 3\), are absolute constants.

Remark 1.1

Theorem 1.2 can be regarded as the generalization of Theorem 1.1 in [13] and Theorem 1.1 in [14].

The rest of this paper is organized as follows. Some useful facts are presented in Sect. 2. The proof of Theorem 1.2 is given in Sect. 3.

Preliminaries and some basic facts

In order to define Besov spaces, we first introduce the Littlewood–Paley decomposition theory. Let \(\mathcal{S}(\mathbf{R}^{n})\) be the Schwartz class of rapidly decreasing functions.

For given \(f \in \mathcal{S}(\mathbf{R}^{n})\), its Fourier transform \(\mathcal{F}(f)=\hat{f}\) and its inverse Fourier transform \(\mathcal{F}^{-1}(f)=\breve{f}\) are given by

$$\begin{aligned} \hat{f}(\xi )= \int _{\mathbf{R}^{n}}e^{-ix\cdot \xi }f(x)\,dx \end{aligned}$$

and

$$\begin{aligned} \check{f}(x)=\frac{1}{(2\pi )^{n}} \int _{\mathbf{R}^{n}}e^{ix\cdot \xi }f(x)\,d\xi, \end{aligned}$$

respectively. Let us choose two nonnegative radial functions \(\chi,\varphi \in \mathcal{S}(\mathbf{R}^{n})\) satisfying \(\operatorname{supp} \chi \subset B=\{\xi \in \mathbf{R}^{n}:|\xi |\le \frac{4}{3}\}\) and \(\operatorname{supp} \varphi \subset \mathcal{C}=\{\xi \in \mathbf{R}^{n}:\frac{3}{4}\le | \xi |\le \frac{8}{3}\}\) such that

$$\begin{aligned} \sum_{j\in \mathbf{Z}}\varphi \bigl(2^{-j}\xi \bigr)=1, \quad \text{for any }\xi \in \mathbf{R}^{n}\backslash \{0\} \end{aligned}$$

and

$$\begin{aligned} \chi (\xi )+\sum_{j\ge 0}\varphi \bigl(2^{-j} \xi \bigr)=1, \quad \text{for any }\xi \in \mathbf{R}^{n}. \end{aligned}$$

For \(j\in \mathbf{Z}\), the homogeneous Littlewood–Paley projection operators \({S}_{j}\) and \(\dot{\Delta }_{j}\) are defined by

$$\begin{aligned} \dot{S}_{j}f=\chi \bigl(2^{-j}D \bigr)f=2^{nj} \int _{\mathbf{R}^{n}}\tilde{h} \bigl(2^{j}y \bigr)f(x-y)\,dy, \quad \text{where }\tilde{h}=\mathcal{F}^{-1}\chi, \end{aligned}$$

and

$$\begin{aligned} \dot{\Delta }_{j}f=\varphi \bigl(2^{-j}D \bigr)f=2^{nj} \int _{\mathbf{R}^{n}}h \bigl(2^{j}y \bigr)f(x-y)\,dy,\quad \text{where }h= \mathcal{F}^{-1}\varphi. \end{aligned}$$

\(\dot{\Delta }_{j}\) is a frequency projection to the annulus \(\{|\xi |\sim 2^{j}\}\), and \(\dot{S}_{j}\) is a frequency projection to the ball \(\{|\xi |\le 2^{j}\}\). Let \(s\in \mathbf{R},p,q\in [1,\infty ]\). The homogeneous Besov space \(\dot{B}^{s}_{p,q}(\mathbf{R}^{n})\) is presented by the distributions \(f\in \mathcal{S}'_{h} \) such that

$$\begin{aligned} \biggl(\sum_{j\in \mathbf{Z}}2^{jsq} \Vert \dot{ \Delta }_{j} f \Vert _{L^{p}}^{q} \biggr)^{\frac{1}{q}}< \infty, \end{aligned}$$

with the norm

$$\begin{aligned} \Vert f \Vert _{\dot{B}_{p,q}^{s}(\mathbf{R}^{n})}= \textstyle\begin{cases} (\sum_{j\in \mathbf{Z}}2^{jsq} \Vert \dot{\Delta }_{j} f \Vert _{L^{p}}^{q} )^{\frac{1}{q}}, & 1\le q< \infty, \\ \sup_{j\in \mathbf{Z}} \{2^{js} \Vert \dot{\Delta }_{j} f \Vert _{L^{p}} \}, & q= \infty. \end{cases}\displaystyle \end{aligned}$$
(2.1)

On the other hand, we recall some facts that can be found in [14]. If \(u\in L^{2}(\mathbf{R}^{3})\), then it follows from the definition of \(u_{k}\) and \(u^{k}\) that

$$\begin{aligned} \bigl(u_{k},u^{k} \bigr)=0, \quad\forall k>0. \end{aligned}$$
(2.2)

Moreover, for \(0\leq r< s\), by Plancherel’s theorem,

$$ \begin{aligned} & \Vert u_{k} \Vert _{\dot{H}^{s}}= \bigl\Vert \vert \xi \vert ^{s} \hat{u}_{k} \bigr\Vert _{L^{2}}\leq k^{s-r} \bigl\Vert \vert \xi \vert ^{r} \hat{u}_{k} \bigr\Vert _{L^{2}}= k^{s-r} \Vert u_{k} \Vert _{\dot{H}^{r}}, \\ & \bigl\Vert u^{k} \bigr\Vert _{\dot{H}^{s}}= \bigl\Vert \vert \xi \vert ^{s}\hat{u}_{k} \bigr\Vert _{L^{2}} \geq k^{s-r} \bigl\Vert \vert \xi \vert ^{r}\hat{u}_{k} \bigr\Vert _{L^{2}}= k^{s-r} \Vert u_{k} \Vert _{\dot{H}^{r}}. \end{aligned} $$
(2.3)

Since \(\| \Delta u\|_{L^{2}}\sim \|\nabla ^{2} u\|_{L^{2}}, \forall u\in \dot{H}^{2}(\mathbb{R}^{3})\), we have

$$ k \bigl\Vert \nabla u^{k} \bigr\Vert _{L^{2}}\leq \bigl\Vert \nabla ^{2}u^{k} \bigr\Vert _{L^{2}}\leq c \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}},\quad \forall u\in H^{2}\bigl(\mathbb{R}^{3}\bigr), $$
(2.4)

with some \(c>0\). Moreover, it can be easily seen that

$$ (u_{k}v_{l})^{m}=0,\quad \forall u,v \in L^{2} \bigl(\mathbb{R}^{3} \bigr), \forall k,l>0, \forall m>k+l, $$
(2.5)

because the Fourier transform of \(u_{k}v_{l}\) is supported in \(\{\xi \in \mathbb{R}^{3}: \vert \xi \vert \leq k+l\}\).

Proof of Theorem 1.2

For convenience, we assume \(\mu =\lambda =1\) throughout the proof of Theorem 1.2.

Proof

Assume that a weak solution \((u,d)\) of (1.1) is regular in \((0,T)\), but not in \((0,T]\). Then \(\lim_{t\rightarrow T-0}\|\nabla u(t)\|_{L^{2}}+\| \Delta d(t)\|_{L^{2}}=\infty \). Notice that, for all smooth solutions to system (1.1), one has the following basic energy law (see [10]):

$$ \begin{aligned} & \bigl\Vert u(\cdot,t) \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla d(\cdot,t) \bigr\Vert _{L^{2}}^{2}+ \int _{0}^{t} \bigl( \bigl\Vert \nabla u(\cdot, \tau ) \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \bigl(\Delta d+ \vert \nabla d \vert ^{2} \bigr) (\cdot,\tau ) \bigr\Vert _{L^{2}}^{2} \bigr)\,d\tau \\ &\quad\leq \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2}, \end{aligned} $$
(3.1)

for all \(0< t<\infty \). By (1.2), one has

$$ \bigl\Vert \nabla u(t) \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Delta d(t) \bigr\Vert _{L^{2}}^{2}\leq k^{2} \Vert u_{0} \Vert _{L^{2}}^{2}+k^{2} \Vert \nabla d_{0} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla u^{k}(t) \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Delta d^{k}(t) \bigr\Vert _{L^{2}}^{2}. $$

Thus,

$$ \lim_{t\rightarrow T-0} \bigl\Vert \nabla u^{k} (t) \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \Delta d^{k} (t) \bigr\Vert ^{2}_{L^{2}}=\infty. $$
(3.2)

We can see from [13] that, if there exists a positive constant \(\varepsilon _{0}>0\) such that

$$\begin{aligned} \bigl\Vert (u,\nabla d) \bigr\Vert _{L^{\infty }(0,T;\dot{B}_{\infty,\infty }^{-1})}\leq \varepsilon _{0}, \end{aligned}$$

then the solution \((u,d)\) is smooth up to time T.

Now we multiply the first equation of (1.1) with \(-\Delta u^{k}\) and integrate over \(\mathbf{R}^{3}\) to get by (2.2)

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \bigl\Vert \nabla u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2}= \bigl(u \cdot \nabla u,\Delta u^{k} \bigr)+ \biggl(\Delta d\cdot \nabla d+ \frac{1}{2} \nabla \vert \nabla d \vert ^{2},\Delta u^{k} \biggr). \end{aligned}$$
(3.3)

Applying to the second equation of (1.1) and making an \(L^{2}\) inner product with respect to \(\nabla \Delta d^{k}\), we can verify

$$ \begin{aligned} \frac{1}{2}\frac{d}{dt} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2}=& \bigl(\nabla (u\cdot \nabla d),\nabla \Delta d^{k} \bigr)+ \bigl( \nabla \bigl( \vert \nabla d \vert ^{2} d \bigr),\nabla \Delta d^{k} \bigr). \end{aligned} $$
(3.4)

Adding (3.3) and (3.4) gives rise to

$$ \begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl( \bigl\Vert \nabla u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \bigr)+ \bigl( \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \bigr) \\ &\quad= \bigl(u\cdot \nabla u,\Delta u^{k} \bigr)+ \bigl(\Delta d\cdot \nabla d,\Delta u^{k} \bigr)+ \frac{1}{2} \bigl(\nabla \vert \nabla d \vert ^{2},\Delta u^{k} \bigr) \\ & \qquad{}+ \bigl(\nabla (u\cdot \nabla d),\nabla \Delta d^{k} \bigr)+ \bigl( \nabla \bigl( \vert \nabla d \vert ^{2} d \bigr),\nabla \Delta d^{k} \bigr) \\ &\quad:=I_{1}+I_{2}+I_{3}+I_{4}+I_{5}. \end{aligned} $$
(3.5)

Next we estimate \(I_{1}\)\(I_{5}\), respectively. From [14], we have

$$ \begin{aligned} \vert I_{1} \vert ={}& \bigl\vert \bigl(u\cdot \nabla u,\Delta u^{k} \bigr) \bigr\vert \\ \leq {}& C k^{2} \Vert u_{0} \Vert _{L^{2}}^{2} \Vert u_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} +C \Vert u_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2} \\ &{}+C k^{-\frac{1}{2}} \bigl\Vert \nabla u^{k} \bigr\Vert _{L^{2}} \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2}+ \frac{1}{4} \Vert \Delta u_{k,2k} \Vert _{L^{2}}^{2}. \end{aligned} $$
(3.6)

Since \(d=d_{k}+d^{k}\), we write

$$\begin{aligned} (\Delta d\cdot \nabla )d=(\Delta d_{k}\cdot \nabla )d_{k}+ \bigl( \Delta d^{k}\cdot \nabla \bigr)d_{k}+(\Delta d_{k}\cdot \nabla )d^{k}+ \bigl( \Delta d^{k} \cdot \nabla \bigr)d^{k}. \end{aligned}$$

Then

$$ \begin{aligned}I_{2}={}&\bigl(\Delta d\cdot \nabla d,\Delta u^{k} \bigr) \\ = {}&\bigl((\Delta d_{k}\cdot \nabla )d_{k},\Delta u^{k} \bigr)+ \bigl( \bigl(\Delta d^{k} \cdot \nabla \bigr)d^{k},\Delta u^{k} \bigr)+ \bigl((\Delta d_{k}\cdot \nabla )d^{k}, \Delta u^{k} \bigr)\\ &{}+ \bigl( \bigl(\Delta d^{k}\cdot \nabla \bigr)d_{k},\Delta u^{k} \bigr) \\ :={}&I_{21}+I_{22}+I_{23}+I_{24}. \end{aligned} $$
(3.7)

Note that \(d_{k}=d_{\frac{k}{2}}+d_{\frac{k}{2},k}\) and the Fourier transform of \((\Delta d_{k}\cdot \nabla )d_{k}\) is supported in \(\{ \vert \xi \vert \leq 2k\}\), thus we deduce

$$ \begin{aligned} I_{21}&= \bigl((\Delta d_{k}\cdot \nabla )d_{k},\Delta u^{k} \bigr) \\ &= \bigl( \bigl[(\Delta d_{k}\cdot \nabla )d_{k} \bigr]_{k,2k},\Delta u_{k,2k} \bigr) \\ &= \bigl( \bigl[(\Delta d_{k}\cdot \nabla )d_{\frac{k}{2}}+(\Delta d_{k} \cdot \nabla )d_{\frac{k}{2},k} \bigr]_{k,2k},\Delta u_{k,2k} \bigr) \\ &= \bigl( \bigl[(\Delta d_{\frac{k}{2}}\cdot \nabla )d_{\frac{k}{2}}+(\Delta d_{ \frac{k}{2},k}\cdot \nabla )d_{\frac{k}{2}}+(\Delta d_{k}\cdot \nabla )d_{\frac{k}{2},k} \bigr]_{k,2k},\Delta u_{k,2k} \bigr) \\ &= \bigl((\Delta d_{k}\cdot \nabla )d_{\frac{k}{2},k},\Delta u_{k,2k} \bigr)+ \bigl(( \Delta d_{\frac{k}{2},k}\cdot \nabla )d_{\frac{k}{2}},\Delta u_{k,2k} \bigr) \\ &:=I_{211}+I_{212}, \end{aligned} $$
(3.8)

where we used the fact \([(\Delta d_{\frac{k}{2}}\cdot \nabla )d_{\frac{k}{2}}]_{k,2k}=0\). Thanks to the Hölder inequality, the Young inequality and (3.1), we get

$$ \begin{aligned} \vert I_{211} \vert &\leq \Vert \Delta d_{k} \Vert _{L^{2}} \Vert \nabla d_{ \frac{k}{2},k} \Vert _{L^{\infty }} \Vert \Delta u_{k,2k} \Vert _{L^{2}} \\ &\leq Ck \Vert \nabla d_{k} \Vert _{L^{2}} \Vert \nabla d_{k/2,k} \Vert _{L^{\infty }} \Vert \Delta u_{k,2k} \Vert _{L^{2}} \\ &\leq Ck^{2} \Vert \nabla d_{k} \Vert _{L^{2}}^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2}+\frac{1}{8} \Vert \Delta u_{k,2k} \Vert _{L^{2}}^{2} \\ &\leq Ck^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+\frac{1}{8} \Vert \Delta u_{k,2k} \Vert _{L^{2}}^{2}. \end{aligned} $$
(3.9)

Similarly,

$$ \begin{aligned} \vert I_{212} \vert &\leq \Vert \Delta d_{\frac{k}{2},k} \Vert _{L^{\infty }} \Vert \nabla d_{\frac{k}{2}} \Vert _{L^{2}} \Vert \Delta u_{k,2k} \Vert _{L^{2}} \\ &\leq Ck \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }} \Vert \nabla d_{ \frac{k}{2}} \Vert _{L^{2}} \Vert \Delta u_{k,2k} \Vert _{L^{2}} \\ &\leq Ck^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \Vert \nabla d_{ \frac{k}{2}} \Vert _{L^{2}}^{2}+\frac{1}{8} \Vert \Delta u_{k,2k} \Vert _{L^{2}}^{2} \\ &\leq Ck^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+\frac{1}{8} \Vert \Delta u_{k,2k} \Vert _{L^{2}}^{2}, \end{aligned} $$
(3.10)

which along with (3.9) implies

$$ \vert I_{21} \vert \leq Ck^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+\frac{1}{4} \Vert \Delta u_{k,2k} \Vert _{L^{2}}^{2}. $$
(3.11)

With the help of Hölder’s inequality, (2.4), Gagaliardo–Nirenberg’s inequality, Sobolev’s embedding and Young’s inequality, one has

$$ \begin{aligned} \vert I_{22} \vert &\leq \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{3}} \bigl\Vert \nabla d^{k} \bigr\Vert _{L^{6}} \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}} \\ &\leq C \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}}^{\frac{1}{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{ \frac{1}{2}} \bigl\Vert \nabla ^{2} d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}} \\ &\leq Ck^{-\frac{1}{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}} \\ &\leq Ck^{-\frac{1}{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigl( \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \bigr). \end{aligned} $$
(3.12)

By the definition of the \(\dot{B}_{\infty,\infty }^{-1}\)-norm, we have

$$ \Vert u_{k} \Vert _{L^{\infty }}\leq Ck \Vert u_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}},\quad \forall k>0. $$
(3.13)

From the Hölder inequality, (3.13) and the Young inequality, we can conclude that

$$ \begin{aligned} \vert I_{23} \vert \leq {}& \Vert \Delta d_{k} \Vert _{L^{\infty }} \bigl\Vert \nabla d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}} \\ \leq {}&Ck \Vert \nabla d_{k} \Vert _{L^{\infty }} \bigl\Vert \nabla d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}} \\ \leq {}&Ck^{2} \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl\Vert \nabla d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}} \\ \leq {}&C \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}} \\ \leq {}&C \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl( \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \bigr). \end{aligned} $$
(3.14)

Similarly,

$$ \begin{aligned} \vert I_{24} \vert \leq {}& \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \Vert \nabla d_{k} \Vert _{L^{\infty }} \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}} \\ \leq {}&Ck \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}} \\ \leq {}&C \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}} \\ \leq {}&C \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl( \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \bigr). \end{aligned} $$
(3.15)

Combining (3.7), (3.11), (3.12), (3.14) and (3.15), one arrives at

$$ \begin{aligned} \vert I_{2} \vert \leq {}& C k^{-\frac{1}{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigl( \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \bigr) \\ &{}+C \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl( \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \bigr) \\ &{}+Ck^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+ \frac{1}{4} \Vert \Delta u_{k,2k} \Vert _{L^{2}}^{2}. \end{aligned} $$
(3.16)

To estimate \(I_{3}\), we make the following decomposition:

$$\begin{aligned} \frac{1}{2}\nabla \vert \nabla d \vert ^{2}&= \frac{1}{2}\nabla \bigl\vert \nabla d^{k}+\nabla d_{k} \bigr\vert ^{2} \leq \nabla \bigl\vert \nabla d^{k} \bigr\vert ^{2}+ \nabla \vert \nabla d_{k} \vert ^{2} \\ &=2\nabla d^{k}\cdot \nabla ^{2}d^{k}+2 \nabla d_{k}\cdot \nabla ^{2}d_{k}. \end{aligned}$$

Then

$$\begin{aligned} \vert I_{3} \vert \leq 2 \bigl\vert \bigl(\nabla d^{k}\cdot \nabla ^{2}d^{k},\Delta u^{k} \bigr) \bigr\vert +2 \bigl\vert \bigl( \nabla d_{k}\cdot \nabla ^{2}d_{k},\Delta u^{k} \bigr) \bigr\vert :=I_{31}+I_{32}. \end{aligned}$$
(3.17)

Applying the same method to the bound (3.12) gives rise to

$$ \begin{aligned} I_{31}\leq {}& \bigl\Vert \nabla d^{k} \bigr\Vert _{L^{3}} \bigl\Vert \nabla ^{2} d^{k} \bigr\Vert _{L^{6}} \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}} \\ \leq {}& C \bigl\Vert \nabla d^{k} \bigr\Vert _{\dot{H}^{\frac{1}{2}}} \bigl\Vert \Delta d^{k} \bigr\Vert _{ \dot{H}^{1}} \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}} \\ \leq {}& C \bigl\Vert \nabla d^{k} \bigr\Vert _{L^{2}}^{\frac{1}{2}} \bigl\Vert \nabla d^{k} \bigr\Vert _{ \dot{H}^{1}}^{\frac{1}{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{\dot{H}^{1}} \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}} \\ \leq {}& C \bigl\Vert \nabla d^{k} \bigr\Vert _{L^{2}}^{\frac{1}{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}}^{ \frac{1}{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}} \\ \leq {}& Ck^{-\frac{1}{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}} \\ \leq {}& Ck^{-\frac{1}{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigl( \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2} \bigr). \end{aligned} $$
(3.18)

Similarly to (3.8), we have

$$ \begin{aligned} I_{32}&=2 \bigl\vert \bigl( \nabla d_{k}\cdot \nabla ^{2}d_{k},\Delta u^{k} \bigr) \bigr\vert \\ &=2 \bigl\vert \bigl( \nabla d_{k}\cdot \nabla ^{2}d_{\frac{k}{2},k},\Delta u_{k,2k} \bigr)+ \bigl( \nabla d_{\frac{k}{2},k}\cdot \nabla ^{2}d_{\frac{k}{2}},\Delta u_{k,2k} \bigr) \bigr\vert \\ &\leq 2 \bigl\vert \bigl(\nabla d_{k}\cdot \nabla ^{2}d_{\frac{k}{2},k},\Delta u_{k,2k} \bigr) \bigr\vert +2 \bigl\vert \bigl( \nabla d_{\frac{k}{2},k}\cdot \nabla ^{2}d_{\frac{k}{2}}, \Delta u_{k,2k} \bigr) \bigr\vert \\ &:=I_{321}+I_{322}. \end{aligned} $$
(3.19)

Using Hölder’s inequality, (2.4), Young’s inequality and (3.1), one can verify

$$ \begin{aligned} I_{321}\leq {}&2 \Vert \nabla d_{k} \Vert _{L^{2}} \Vert \Delta d_{ \frac{k}{2},k} \Vert _{L^{\infty }} \Vert \Delta u_{k,2k} \Vert _{L^{2}} \\ \leq {}& Ck \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }} \Vert \nabla d_{k} \Vert _{L^{2}} \Vert \Delta u_{k,2k} \Vert _{L^{2}} \\ \leq {}& Ck^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+\frac{1}{8} \Vert \Delta u_{k,2k} \Vert _{L^{2}}^{2}. \end{aligned} $$
(3.20)

Similarly,

$$ \begin{aligned} I_{322}\leq {}&2 \Vert \Delta d_{\frac{k}{2}} \Vert _{L^{2}} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }} \Vert \Delta u_{k,2k} \Vert _{L^{2}} \\ \leq {}& Ck \Vert \nabla d_{\frac{k}{2}} \Vert _{L^{2}} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }} \Vert \Delta u_{k,2k} \Vert _{L^{2}} \\ \leq {}& Ck^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+\frac{1}{8} \Vert \Delta u_{k,2k} \Vert _{L^{2}}^{2}, \end{aligned} $$
(3.21)

which along with (3.20) implies

$$ I_{32}\leq Ck^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+\frac{1}{4} \Vert \Delta u_{k,2k} \Vert _{L^{2}}^{2}. $$
(3.22)

From (3.17), (3.18) and (3.22), we can deduce

$$ \begin{aligned} \vert I_{3} \vert \leq{} &Ck^{-\frac{1}{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigl( \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \bigr) \\ &{}+Ck^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+ \frac{1}{4} \Vert \Delta u_{k,2k} \Vert _{L^{2}}^{2}. \end{aligned} $$
(3.23)

We now address the term \(I_{4}\). We decompose \(I_{4}\) into the following form:

$$ \begin{aligned} I_{4}= \bigl((\nabla u\cdot \nabla )d,\nabla \Delta d^{k} \bigr)+ \bigl((u \cdot \nabla )\nabla d, \nabla \Delta d^{k} \bigr):=I_{41}+I_{42}. \end{aligned} $$
(3.24)

Since

$$ (\nabla u\cdot \nabla )d= \bigl(\nabla u^{k}\cdot \nabla \bigr)d^{k}+ \bigl(\nabla u^{k} \cdot \nabla \bigr)d_{k}+(\nabla u_{k}\cdot \nabla )d^{k}+( \nabla u_{k} \cdot \nabla )d_{k}, $$

we can get

$$ \begin{aligned}I_{41}={}& \bigl( \bigl(\nabla u^{k}\cdot \nabla \bigr)d^{k},\nabla \Delta d^{k} \bigr)+ \bigl( \bigl( \nabla u^{k}\cdot \nabla \bigr)d_{k},\nabla \Delta d^{k} \bigr)+ \bigl((\nabla u_{k} \cdot \nabla )d^{k},\nabla \Delta d^{k} \bigr) \\ &{}+ \bigl((\nabla u_{k}\cdot \nabla )d_{k},\nabla \Delta d^{k} \bigr):=I_{411}+I_{412}+I_{413}+I_{414}. \end{aligned} $$
(3.25)

Similar to the estimate (3.12), one has

$$ \begin{aligned} \vert I_{411} \vert \leq {}& \bigl\Vert \nabla u^{k} \bigr\Vert _{L^{6}} \bigl\Vert \nabla d^{k} \bigr\Vert _{L^{3}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}& C \bigl\Vert \nabla u^{k} \bigr\Vert _{\dot{H}^{1}} \bigl\Vert \nabla d^{k} \bigr\Vert _{L^{2}}^{ \frac{1}{2}} \bigl\Vert \nabla d^{k} \bigr\Vert _{\dot{H}^{1}}^{\frac{1}{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}& Ck^{-\frac{1}{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}& Ck^{-\frac{1}{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigl( \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \bigr). \end{aligned} $$
(3.26)

The Hölder inequality, the Young inequality and (3.13) imply

$$ \begin{aligned} I_{412}\leq {}& \bigl\Vert \nabla u^{k} \bigr\Vert _{L^{2}} \Vert \nabla d_{k} \Vert _{L^{\infty }} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}& Ck \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl\Vert \nabla u^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}& C \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}& C \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl( \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \bigr). \end{aligned} $$
(3.27)

Similarly,

$$ \begin{aligned} I_{413}\leq {}& \Vert \nabla u_{k} \Vert _{L^{\infty }} \bigl\Vert \nabla d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}& Ck \Vert u_{k} \Vert _{L^{\infty }} \bigl\Vert \nabla d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}& Ck^{2} \Vert u_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl\Vert \nabla d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}& C \Vert u_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert ^{2}_{L^{2}}. \end{aligned} $$
(3.28)

Arguing as (3.8), we have

$$\begin{aligned} I_{414}= \bigl((\nabla u_{k}\cdot \nabla )d_{\frac{k}{2},k}+( \nabla u_{ \frac{k}{2},k}\cdot \nabla )d_{\frac{k}{2}},\nabla \Delta d_{k,2k} \bigr):=I_{4141}+I_{4142}. \end{aligned}$$

By Hölder’s inequality, (2.4) and Young’s inequality, we get

$$ \begin{aligned} \vert I_{4141} \vert \leq{} & \Vert \nabla u_{k} \Vert _{L^{2}} \Vert \nabla d_{ \frac{k}{2},k} \Vert _{L^{\infty }} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}} \\ \leq {}& Ck \Vert u_{k} \Vert _{L^{2}} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}} \\ \leq {}& Ck^{2} \Vert u_{k} \Vert _{L^{2}}^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2}+ \frac{1}{16} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}}^{2} \\ \leq {}& Ck^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+\frac{1}{16} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}}^{2}. \end{aligned} $$
(3.29)

Similarly,

$$ \begin{aligned} \vert I_{4142} \vert \leq {}& \Vert \nabla u_{\frac{k}{2},k} \Vert _{L^{\infty }} \Vert \nabla d_{\frac{k}{2}} \Vert _{L^{2}} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}} \\ \leq {}& Ck \Vert u_{\frac{k}{2},k} \Vert _{L^{\infty }} \Vert \nabla d_{\frac{k}{2}} \Vert _{L^{2}} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}} \\ \leq {}& Ck^{2} \Vert u_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \Vert \nabla d_{ \frac{k}{2}} \Vert _{L^{2}}^{2}+ \frac{1}{16} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}}^{2} \\ \leq {}& Ck^{2} \Vert u_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+ \frac{1}{16} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}}^{2}, \end{aligned} $$
(3.30)

which together with (3.29) reads

$$ \vert I_{414} \vert \leq Ck^{2} \bigl( \Vert u_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2}+ \Vert \nabla d_{ \frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigr) \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+\frac{1}{8} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}}^{2}. $$
(3.31)

Combining (3.26)–(3.28) and (3.31) yields

$$ \begin{aligned} \vert I_{41} \vert \leq {}& \frac{1}{8} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}}^{2}+Ck^{- \frac{1}{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigl( \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \bigr) \\ &{}+C \bigl( \Vert u_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}}+ \Vert \nabla d_{k} \Vert _{ \dot{B}_{\infty,\infty }^{-1}} \bigr) \bigl( \bigl\Vert \Delta u^{k} \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \nabla \Delta d^{k} \bigr\Vert ^{2}_{L^{2}} \bigr) \\ &{}+Ck^{2} \bigl( \Vert u_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2}+ \Vert \nabla d_{ \frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigr) \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr). \end{aligned} $$
(3.32)

To handle \(I_{42}\), we split \(I_{42}\) into

$$ \begin{aligned}I_{42}={}& \bigl((u_{k} \cdot \nabla )\nabla d^{k},\nabla \Delta d^{k} \bigr)+ \bigl( \bigl(u^{k} \cdot \nabla \bigr)\nabla d^{k},\nabla \Delta d^{k} \bigr)+ \bigl( \bigl(u^{k}\cdot \nabla \bigr) \nabla d_{k},\nabla \Delta d^{k} \bigr) \\ &{}+ \bigl((u_{k}\cdot \nabla )\nabla d_{k},\nabla \Delta d^{k} \bigr) \\ :={}&I_{421}+I_{422}+I_{423}+I_{424}. \end{aligned} $$
(3.33)

By Hölder’s inequality and (2.4), we get

$$ \begin{aligned} \vert I_{421} \vert \leq {}& \Vert u_{k} \Vert _{L^{\infty }} \bigl\Vert \nabla ^{2} d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}& \Vert u_{k} \Vert _{L^{\infty }} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}& c \Vert u_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2}. \end{aligned} $$
(3.34)

Similarly to (3.12), one has

$$ \begin{aligned} \vert I_{422} \vert \leq {}& \bigl\Vert u^{k} \bigr\Vert _{L^{6}} \bigl\Vert \nabla ^{2} d^{k} \bigr\Vert _{L^{3}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}&C \bigl\Vert u^{k} \bigr\Vert _{L^{6}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{3}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}& C \bigl\Vert \nabla u^{k} \bigr\Vert _{L^{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}}^{1/2} \bigl\Vert \Delta d^{k} \bigr\Vert _{\dot{H}^{1}}^{1/2} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}& C \bigl\Vert \nabla u^{k} \bigr\Vert _{L^{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}}^{1/2} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{1/2} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}& Ck^{-1/2} \bigl\Vert \nabla u^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2}. \end{aligned} $$
(3.35)

Hölder’s inequality, (2.4), and Young’s inequality guarantee

$$ \begin{aligned} \vert I_{423} \vert \leq{} & \bigl\Vert u^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla ^{2} d_{k} \bigr\Vert _{L^{\infty }} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq{} & \bigl\Vert u^{k} \bigr\Vert _{L^{2}} \Vert \Delta d_{k} \Vert _{L^{\infty }} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq{} & ck \bigl\Vert u^{k} \bigr\Vert _{L^{2}} \Vert \nabla d_{k} \Vert _{L^{\infty }} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}& c \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}} \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty, \infty }^{-1}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq{} & c \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl( \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \bigr). \end{aligned} $$
(3.36)

Similarly to (3.8), we write

$$\begin{aligned} I_{424}= \bigl((u_{k}\cdot \nabla )\nabla d_{\frac{k}{2},k}, \nabla \Delta d_{k,2k} \bigr)+ \bigl((u_{ \frac{k}{2},k}\cdot \nabla ) \nabla d_{\frac{k}{2}},\nabla \Delta d_{k,2k} \bigr):=I_{4241}+I_{4242}. \end{aligned}$$

From the Hölder inequality and the Young inequality, we conclude

$$ \begin{aligned} \vert I_{4241} \vert \leq{} & \Vert u_{k} \Vert _{L^{2}} \bigl\Vert \nabla ^{2} d_{ \frac{k}{2},k} \bigr\Vert _{L^{\infty }} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}} \\ \leq {}&C \Vert u_{k} \Vert _{L^{2}} \Vert \Delta d_{\frac{k}{2},k} \Vert _{L^{\infty }} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}} \\ \leq {}& Ck \Vert u_{k} \Vert _{L^{2}} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}& Ck^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+\frac{1}{16} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \end{aligned} $$
(3.37)

and

$$ \begin{aligned} \vert I_{4242} \vert \leq {}& \Vert u_{\frac{k}{2},k} \Vert _{L^{\infty }} \bigl\Vert \nabla ^{2} d_{\frac{k}{2}} \bigr\Vert _{L^{2}} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}} \\ \leq {}&C \Vert u_{\frac{k}{2},k} \Vert _{L^{\infty }} \Vert \Delta d_{\frac{k}{2}} \Vert _{L^{2}} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}} \\ \leq {}& ck \Vert u_{\frac{k}{2},k} \Vert _{L^{\infty }} \Vert \nabla d_{\frac{k}{2}} \Vert _{L^{2}} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}} \\ \leq {}& Ck^{2} \Vert u_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+ \frac{1}{16} \bigl\Vert \nabla \triangle d^{k} \bigr\Vert _{L^{2}}^{2}. \end{aligned} $$
(3.38)

Therefore, by (3.24)–(3.38), we have

$$ \begin{aligned} \vert I_{4} \vert \leq {}& Ck^{-\frac{1}{2}} \bigl( \bigl\Vert \nabla u^{k} \bigr\Vert _{L^{2}}+ \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigr) \bigl( \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \bigr) \\ &{}+C \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl( \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \bigr) \\ &{}+C \Vert u_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2}+C k^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)) \\ &{}\times \bigl( \Vert u_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2}+ \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigr)+ \frac{1}{4} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}}^{2}. \end{aligned} $$
(3.39)

It is left to deal with the last term, \(I_{5}\). Using the fact that

$$\begin{aligned} \nabla \bigl( \vert \nabla d \vert ^{2}d \bigr)=2\nabla ^{2}d \nabla d d+ \vert \nabla d \vert ^{2} \nabla d, \end{aligned}$$

we can rewrite \(I_{5}\) as follows:

$$ I_{5}=2 \bigl(\nabla ^{2}d\nabla d d,\nabla \Delta d^{k} \bigr)+ \bigl( \vert \nabla d \vert ^{2} \nabla d, \nabla \Delta d^{k} \bigr):=I_{51}+I_{52}. $$
(3.40)

Since

$$\begin{aligned} 2\nabla ^{2}d\nabla d d= \bigl(2\nabla ^{2}d_{k} \nabla d_{k}+2\nabla ^{2}d_{k} \nabla d^{k}+2\nabla ^{2}d^{k}\nabla d_{k}+2 \nabla ^{2}d^{k}\nabla d^{k} \bigr)d, \end{aligned}$$

we have

$$ \begin{aligned} I_{51}={}&2 \bigl(\nabla ^{2}d_{k}\nabla d_{k} d,\nabla \Delta d^{k} \bigr)+2 \bigl( \nabla ^{2}d_{k}\nabla d^{k} d,\nabla \Delta d^{k} \bigr)+2 \bigl(\nabla ^{2}d^{k} \nabla d_{k} d,\nabla \Delta d^{k} \bigr) \\ &{}+2 \bigl(\nabla ^{2}d^{k}\nabla d^{k}d,\nabla \Delta d^{k} \bigr) \\ :={}&I_{511}+I_{512}+I_{513}+I_{514}. \end{aligned} $$
(3.41)

Reasoning as (3.8), one has

$$ \begin{aligned} I_{511}&= \bigl(\nabla ^{2}d_{k}\nabla d_{\frac{k}{2},k} d, \nabla \Delta d_{k,2k} \bigr)+ \bigl(\nabla ^{2}d_{\frac{k}{2},k}\nabla d_{ \frac{k}{2}} d,\nabla \Delta d_{k,2k} \bigr) \\ &:=I_{5111}+I_{5112}. \end{aligned} $$
(3.42)

Using \(|d|=1\), Hölder’s inequality, inequality (2.4) and Young’s inequality, we have

$$ \begin{aligned} \vert I_{5111} \vert \leq {}&2 \bigl\Vert \nabla ^{2} d_{k} \bigr\Vert _{L^{2}} \Vert \nabla d_{ \frac{k}{2},k} \Vert _{L^{\infty }} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}} \\ \leq {}&C \Vert \Delta d_{k} \Vert _{L^{2}} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}} \\ \leq {}&Ck \Vert \nabla d_{k} \Vert _{L^{2}} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}} \\ \leq {}&Ck^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+\frac{1}{8} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}}^{2}. \end{aligned} $$
(3.43)

Similarly,

$$ \begin{aligned} \vert I_{5112} \vert \leq {}& \bigl\Vert \nabla ^{2}d_{\frac{k}{2},k} \bigr\Vert _{L^{\infty }} \Vert \nabla d_{\frac{k}{2}} \Vert _{L^{2}} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}} \\ \leq {}& Ck \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }} \Vert \nabla d_{ \frac{k}{2}} \Vert _{L^{2}} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}} \\ \leq{} & Ck^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+\frac{1}{8} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}}^{2}, \end{aligned} $$
(3.44)

which all taken together implies

$$ \vert I_{511} \vert \leq Ck^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+\frac{1}{4} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}}^{2}. $$
(3.45)

By the fact \(|d|=1\), the Hölder inequality, (2.4) and (3.13), we can get

$$ \begin{aligned} \vert I_{512} \vert \leq{} &2 \bigl\Vert \nabla ^{2}d_{k} \bigr\Vert _{L^{\infty }} \bigl\Vert \nabla d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}&C \Vert \Delta d_{k} \Vert _{L^{\infty }} \bigl\Vert \nabla d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq{} & Ck \Vert \nabla d_{k} \Vert _{L^{\infty }} \bigl\Vert \nabla d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}&Ck^{2} \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl\Vert \nabla d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}&C \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2}. \end{aligned} $$
(3.46)

Similarly,

$$ \begin{aligned} \vert I_{513} \vert \leq{} &2 \bigl\Vert \nabla ^{2} d^{k} \bigr\Vert _{L^{2}} \Vert \nabla d_{k} \Vert _{L^{\infty }} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}& C \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2}. \end{aligned} $$
(3.47)

Reasoning as (3.12) again, one has

$$ \begin{aligned} \vert I_{514} \vert \leq {}&2 \bigl\Vert \nabla ^{2} d^{k} \bigr\Vert _{L^{3}} \bigl\Vert \nabla d^{k} \bigr\Vert _{L^{6}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}& C \bigl\Vert \nabla ^{2} d^{k} \bigr\Vert _{L^{2}}^{\frac{1}{2}} \bigl\Vert \nabla ^{2} d^{k} \bigr\Vert _{\dot{H}^{1}}^{\frac{1}{2}} \bigl\Vert \nabla d^{k} \bigr\Vert _{\dot{H}^{1}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}&C \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}}^{\frac{1}{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{ \frac{1}{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}& Ck^{-\frac{1}{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2}. \end{aligned} $$
(3.48)

Therefore, inequalities (3.45)–(3.48) yield

$$ \begin{aligned} \vert I_{51} \vert \leq {}& Ck^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+C \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \\ &{}+Ck^{-\frac{1}{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2}+ \frac{1}{4} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}}^{2}. \end{aligned} $$
(3.49)

It is easy to get \(\Delta d\cdot d=- \vert \nabla d \vert ^{2}\) due to \(\vert d \vert =1\). Then \(\vert \nabla d \vert ^{2}\nabla d=-\Delta d\cdot d\nabla d\). Hence we decompose \(I_{52}\) in the following way:

$$ \begin{aligned} I_{52}={}&{-} \bigl(\Delta d\cdot d\nabla d,\nabla \Delta d^{k} \bigr) \\ ={}&{-} \bigl(\Delta d^{k}\cdot d\nabla d^{k},\nabla \Delta d^{k} \bigr)- \bigl(\Delta d^{k} \cdot d\nabla d_{k},\nabla \Delta d^{k} \bigr)\\ &{}- \bigl(\Delta d_{k}\cdot d\nabla d^{k}, \nabla \Delta d^{k} \bigr)- \bigl(\Delta d_{k}\cdot d\nabla d_{k},\nabla \Delta d^{k} \bigr) \\ :={}&I_{521}+I_{522}+I_{523}+I_{524}. \end{aligned} $$
(3.50)

Repeating the methods to prove (3.12), we obtain

$$ \begin{aligned} \vert I_{521} \vert \leq {}& \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{3}} \bigl\Vert \nabla d^{k} \bigr\Vert _{L^{6}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}&C \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}}^{\frac{1}{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{ \dot{H}^{1}}^{\frac{1}{2}} \bigl\Vert \nabla d^{k} \bigr\Vert _{\dot{H}^{1}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}&C \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}}^{\frac{1}{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{ \frac{1}{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}} \\ \leq {}&Ck^{-\frac{1}{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2}. \end{aligned} $$
(3.51)

Similarly to (3.46), we have

$$\begin{aligned} \vert I_{522} \vert + \vert I_{523} \vert \leq C \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2}. \end{aligned}$$
(3.52)

Similarly to (3.45), one has

$$\begin{aligned} \vert I_{524} \vert \leq Ck^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+\frac{1}{4} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}}^{2}. \end{aligned}$$
(3.53)

Thus

$$ \begin{aligned} \vert I_{52} \vert \leq{} &Ck^{-\frac{1}{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2}+C \Vert \nabla d_{k} \Vert _{\dot{B}_{ \infty,\infty }^{-1}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \\ &{}+Ck^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2} + \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+ \frac{1}{4} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}}^{2}. \end{aligned} $$
(3.54)

From (3.49) and (3.54), we deduce

$$ \begin{aligned} \vert I_{5} \vert \leq {}& Ck^{2} \Vert \nabla d_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr)+C \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \\ &{}+Ck^{-\frac{1}{2}} \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2}+ \frac{1}{2} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}}^{2}. \end{aligned} $$
(3.55)

Combining (3.6), (3.16), (3.23), (3.39) and (3.55), we have

$$ \begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl( \bigl\Vert \nabla u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \bigr)+ \bigl( \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \bigr) \\ &\quad\leq C_{1}k^{2} \bigl( \Vert u_{\frac{k}{2},k} \Vert _{L^{\infty }}^{2}+ \Vert \nabla d_{ \frac{k}{2},k} \Vert _{L^{\infty }}^{2} \bigr) \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr) \\ &\qquad{}+C_{2} \bigl( \Vert u_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}}+ \Vert \nabla d_{k} \Vert _{ \dot{B}_{\infty,\infty }^{-1}} \bigr) \bigl( \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \bigr) \\ &\qquad{}+C_{3} k^{-\frac{1}{2}} \bigl( \bigl\Vert \nabla u^{k} \bigr\Vert _{L^{2}}+ \bigl\Vert \Delta d^{k} \bigr\Vert _{L^{2}} \bigr) \bigl( \bigl\Vert \Delta u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \Delta d^{k} \bigr\Vert _{L^{2}}^{2} \bigr)+ \frac{3}{4} \Vert \Delta u_{k,2k} \Vert _{L^{2}}^{2} \\ &\qquad{}+\frac{3}{4} \Vert \nabla \Delta d_{k,2k} \Vert _{L^{2}}^{2} \end{aligned} $$
(3.56)

and

$$ \begin{aligned} &\frac{d}{dt} \bigl( \bigl\Vert \nabla u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \triangle d^{k} \bigr\Vert _{L^{2}}^{2} \bigr)\\ &\quad\leq \biggl[c_{1} k^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr) \bigl( \Vert u_{k/2,k} \Vert _{L^{\infty }}^{2}+ \Vert \nabla d_{k/2,k} \Vert _{L^{\infty }}^{2} \bigr) \\ &\qquad{}-\frac{1}{8} \bigl( \bigl\Vert \triangle u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \triangle d^{k} \bigr\Vert _{L^{2}}^{2} \bigr) \biggr]+ \biggl(c_{2} \bigl( \bigl\Vert u_{k}(t) \bigr\Vert _{\dot{B}_{\infty,\infty }^{-1}}+ \Vert \nabla d_{k} \Vert _{\dot{B}_{\infty,\infty }^{-1}} \bigr)-\frac{1}{4} \biggr) \\ &\qquad{}\times \bigl( \bigl\Vert \triangle u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \triangle d^{k} \bigr\Vert _{L^{2}}^{2} \bigr) \\ &\qquad{}+ \biggl(c_{3} k^{-1/2} \bigl( \bigl\Vert \nabla u^{k} \bigr\Vert _{L^{2}}+ \bigl\Vert \triangle d^{k} \bigr\Vert _{L^{2}} \bigr)- \frac{1}{8} \biggr) \bigl( \bigl\Vert \triangle u^{k} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \triangle d^{k} \bigr\Vert _{L^{2}}^{2} \bigr). \end{aligned} $$
(3.57)

Let

$$\begin{aligned} \begin{aligned} \tilde{k}=128\times 4 c_{3}^{2} \bigl( \bigl\Vert \nabla u_{0}^{\tilde{k}} \bigr\Vert _{L^{2}}+ \bigl\Vert \triangle d_{0}^{\tilde{k}} \bigr\Vert _{L^{2}} \bigr)^{2}. \end{aligned} \end{aligned}$$
(3.58)

Then

$$\begin{aligned} \bigl\Vert \nabla u_{0}^{\tilde{k}} \bigr\Vert _{L^{2}}+ \bigl\Vert \Delta d_{0}^{\tilde{k}} \bigr\Vert _{L^{2}}< \frac{\tilde{k}^{\frac{1}{2}}}{16 c_{3}}. \end{aligned}$$

Since \(\lim_{t\rightarrow T-0}\|\nabla u^{\tilde{k}}(t)\|_{L^{2}}+\| \Delta d^{\tilde{k}}(t)\|_{L^{2}}=\infty \), there is some \(\delta \in (0,T)\) such that

$$\begin{aligned} & \bigl\Vert \nabla u^{\tilde{k}}(T- \delta ) \bigr\Vert _{L^{2}}+ \bigl\Vert \Delta d^{\tilde{k}}(T- \delta ) \bigr\Vert _{L^{2}}=\frac{\tilde{k}^{\frac{1}{2}}}{16 c_{3}}, \end{aligned}$$
(3.59)
$$\begin{aligned} & \bigl\Vert \nabla u^{\tilde{k}}(t) \bigr\Vert _{L^{2}}+ \bigl\Vert \Delta d^{\tilde{k}}(t) \bigr\Vert _{L^{2}}> \frac{\tilde{k}^{\frac{1}{2}}}{16 c_{3}}. \end{aligned}$$
(3.60)

From (3.60), we get for any \(t\in (T-\delta,T)\)

$$\begin{aligned} &c_{1}{\tilde{k}}^{2} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr) \bigl( \Vert u_{\tilde{\frac{k}{2}},\tilde{k}} \Vert _{L^{\infty }}^{2}+ \Vert \nabla d_{ \tilde{\frac{k}{2}},\tilde{k}} \Vert _{L^{\infty }}^{2} \bigr)-\frac{1}{8} \bigl( \bigl\Vert \Delta u^{\tilde{k}} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \Delta d^{\tilde{k}} \bigr\Vert _{L^{2}}^{2} \bigr) \\ &\quad\leq {\tilde{k}}^{2} \biggl[c_{1} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr) \bigl( \Vert u_{\tilde{\frac{k}{2}},\tilde{k}} \Vert _{L^{\infty }}^{2}+ \Vert \nabla d_{ \tilde{\frac{k}{2}},\tilde{k}} \Vert _{L^{\infty }}^{2} \bigr)-\frac{1}{8} \bigl( \bigl\Vert \nabla u^{\tilde{k}} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Delta d^{\tilde{k}} \bigr\Vert _{L^{2}}^{2} \bigr) \biggr] \\ &\quad\leq {\tilde{k}}^{2} \biggl[c_{1} \bigl( \Vert u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla d_{0} \Vert _{L^{2}}^{2} \bigr) \bigl( \Vert u_{\tilde{\frac{k}{2}},\tilde{k}} \Vert _{L^{\infty }}^{2}+ \Vert \nabla d_{ \tilde{\frac{k}{2}},\tilde{k}} \Vert _{L^{\infty }}^{2} \bigr)-\frac{1}{16} \frac{\tilde{k}}{256 c_{3}^{2}} \biggr] \\ &\quad\leq 0, \end{aligned}$$

provided

$$\begin{aligned} \begin{aligned} \bigl\Vert {u_{\frac{\tilde{k}}{2},\tilde{k}}}(t) \bigr\Vert _{L^{\infty }}+ \bigl\Vert {\nabla d_{ \frac{\tilde{k}}{2},\tilde{k}}(t)} \bigr\Vert _{L^{\infty }}< \frac{\tilde{k}^{\frac{1}{2}}}{32c_{3}\sqrt{c_{1}}( \Vert u_{0} \Vert _{L^{2}}+ \Vert \nabla d_{0} \Vert _{L^{2}})}, \quad\forall t\in (T-\delta,T). \end{aligned} \end{aligned}$$
(3.61)

In view of (3.58), the inequality (3.61) is equivalent to

$$\begin{aligned} \begin{aligned} &\bigl\Vert {u_{\frac{\tilde{k}}{2},\tilde{k}}}(t) \bigr\Vert _{B_{\infty,\infty }^{-1}}+ \bigl\Vert {\nabla d_{\frac{\tilde{k}}{2},\tilde{k}}}(t) \bigr\Vert _{B_{\infty,\infty }^{-1}} \\ &\quad< c \frac{1}{\tilde{k}^{\frac{1}{2}} 32 c_{3}\sqrt{c_{1}}( \Vert u_{0} \Vert _{L^{2}}+ \Vert \nabla d_{0} \Vert _{L^{2}})} \\ &\quad< c \frac{1}{16\sqrt{2}c_{3}( \Vert \nabla u_{0} \Vert _{L^{2}}+ \Vert \Delta d_{0} \Vert _{L^{2}})\times 32c_{3}\sqrt{c_{1}}( \Vert u_{0} \Vert _{L^{2}}+ \Vert \nabla d_{0} \Vert _{L^{2}})} \\ &\quad< c \frac{1}{512\sqrt{2}c_{3}^{2}\sqrt{c_{1}}( \Vert u_{0} \Vert _{L^{2}}+ \Vert \nabla d_{0} \Vert _{L^{2}})( \Vert \nabla u_{0}^{\tilde{k}} \Vert _{L^{2}}+ \Vert \Delta d_{0}^{\tilde{k}} \Vert _{L^{2}})}. \end{aligned} \end{aligned}$$
(3.62)

Thus, if (3.62) and

$$\begin{aligned} c_{2} ( \Vert u_{\tilde{k}} \Vert _{{L^{\infty }}(T-\delta,T;\dot{B}_{\infty, \infty }^{-1})}+ \Vert \nabla d_{\tilde{k}} \Vert _{{L^{\infty }}(T-\delta,T; \dot{B}_{\infty,\infty }^{-1})}\leq \frac{1}{4} \end{aligned}$$
(3.63)

hold, we can infer from (3.57) that

$$\begin{aligned} \begin{aligned} &\frac{d}{dt} \bigl( \bigl\Vert \nabla u^{\tilde{k}} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \Delta d^{ \tilde{k}} \bigr\Vert _{L^{2}}^{2} \bigr) \\ &\quad\leq \biggl(c_{3}\tilde{k}^{-\frac{1}{2}} \bigl( \bigl\Vert \nabla u^{\tilde{k}} \bigr\Vert _{L^{2}}+ \bigl\Vert \Delta d^{\tilde{k}} \bigr\Vert _{L^{2}} \bigr)-\frac{1}{8} \biggr) \bigl( \bigl\Vert \Delta u^{\tilde{k}} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \Delta d^{\tilde{k}} \bigr\Vert _{L^{2}}^{2} \bigr). \end{aligned} \end{aligned}$$
(3.64)

Since \(c_{3}{\tilde{k}}^{-\frac{1}{2}}(\|\nabla u^{\tilde{k}}(T-\delta )\|_{L^{2}}+ \|\Delta d^{\tilde{k}}(T-\delta )\|_{L^{2}})-\frac{1}{8}=c_{3} \tilde{k}^{-\frac{1}{2}}\frac{\tilde{k}^{\frac{1}{2}}}{16 c_{3}}- \frac{1}{8}<0\), there is a right neighborhood I of \(t=T-\delta \) such that

$$\begin{aligned} c_{3} {\tilde{k}}^{-\frac{1}{2}} \bigl( \bigl\Vert \nabla u^{\tilde{k}}(t) \bigr\Vert _{L^{2}}+ \bigl\Vert \Delta d^{\tilde{k}}(t) \bigr\Vert _{L^{2}} \bigr)-\frac{1}{8}< 0,\quad \forall t\in \mathrm{I.} \end{aligned}$$

Hence, it follows by (3.64) that the function \(t\to \|\nabla u^{\tilde{k}}\|_{L^{2}}+\|\Delta d^{\tilde{k}}\|_{L^{2}}\) decreases in I, which contradicts (3.59) and (3.60). Thus, when (3.62) and (3.63) are satisfied, u and d cannot blow up at \(t=T\), and u and d are regular in \((0,T]\). The proof of the theorem is completed. □

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11961032 and No. 11971209), the Natural Science Foundation of Jiangxi Province, China (No. 20191BAB201003).

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Chen, X., Cheng, H. Regularity criterion for 3D nematic liquid crystal flows in terms of finite frequency parts in \(\dot{B}_{\infty,\infty }^{-1}\). Bound Value Probl 2021, 23 (2021). https://doi.org/10.1186/s13661-021-01500-1

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MSC

  • 35Q35
  • 76D03

Keywords

  • Liquid crystal flow
  • Regularity criterion
  • Weak solution
  • \(\dot{B}_{\infty,\infty }^{-1}\)