# Global smooth solutions to 3D MHD with mixed partial dissipation and magnetic diffusion

## Abstract

In this paper, we prove the existence of global smooth solutions to the Cauchy problem of 3D incompressible magnetohydrodynamics (MHD) flows with mixed partial dissipation and magnetic diffusion if the initial condition is suitably small.

## 1 Introduction

In this paper, we consider the following 3D incompressible MHD equations with mixed partial dissipation and magnetic diffusion (see [1]), i.e.,

${u}_{t}+u\cdot \mathrm{\nabla }u=-\mathrm{\nabla }p+\mu {u}_{xx}+\mu {u}_{yy}+b\cdot \mathrm{\nabla }b,$
(1)
${b}_{t}+u\cdot \mathrm{\nabla }b=\eta {b}_{xx}+\eta {b}_{yy}+b\cdot \mathrm{\nabla }u,$
(2)
$divu=0,\phantom{\rule{2em}{0ex}}divb=0,$
(3)

associated with the initial data

$u\left(0,x\right)={u}_{0},\phantom{\rule{2em}{0ex}}b\left(0,x\right)={b}_{0}.$
(4)

Here $u=\left({u}_{1}\left(t,x\right),{u}_{2}\left(t,x\right),{u}_{3}\left(t,x\right)\right)$ is the velocity field, $b=\left({b}_{1}\left(t,x\right),{b}_{2}\left(t,x\right),{b}_{3}\left(t,x\right)\right)$ is the magnetic field, $p=p\left(t,x\right)$ is scalar pressure, $\mu >0$ is the kinematic viscosity, $\eta >0$ is the magnetic diffusion. For more background, we refer the reader to [2] for MHD and [1, 3] for MHD with mixed partial dissipation and magnetic diffusion. Without loss of generality, we assume that $\mu =\eta =1$ in the remainder of the paper.

To state the main results, we first introduce the following conventions and notations which will be used throughout this paper. Set

$\int f\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\triangleq {\int }_{{\mathbb{R}}^{3}}f\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\phantom{\rule{0.2em}{0ex}}\mathrm{d}y\phantom{\rule{0.2em}{0ex}}\mathrm{d}z,$

and that $\parallel \cdot \parallel$ is the ${L}^{2}$ norm, i.e.,

Our main result of this paper can be stated as follows.

Theorem 1.1 Assume that ${u}_{0}\in {H}^{2}$ and ${b}_{0}\in {H}^{2}$ with $div{u}_{0}=div{b}_{0}=0$ and ${\parallel {u}_{0}\parallel }_{{H}^{1}}+{\parallel {b}_{0}\parallel }_{{H}^{1}}\le \epsilon$, where ε is a sufficiently small positive number. Then (1)-(4) admit global smooth solutions.

Remark 1.1 Theorem 1.1 is Theorem 1.2 in [1], which has not been proved in their paper. We would also emphasize that our proof of Theorem 1.1 is clearer for deducing the desired a priori estimates in Lemma 2.3 (see the next section) than that of Proposition 3.1 in [1].

The rest of the paper is organized as follows. In Section 2, we deduce the desired a priori estimates to complete the proof of Theorem 1.1. We finish the proof of Theorem 1.1 in Section 3 by the method of vanishing viscosities.

## 2 A priori estimates

In this section, we deduce the desired a priori estimates in order to finish the main result. Before we begin to prove the main theorem of this paper, we first state the following useful lemma that was deduced in [1].

Lemma 2.1 Assume that f, g, h, ${f}_{x}$, ${g}_{y}$, ${h}_{z}$ are all in ${L}^{2}\left({\mathbb{R}}^{3}\right)$. Then we have

$\int |fgh|\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le C{\parallel f\parallel }^{\frac{1}{2}}{\parallel g\parallel }^{\frac{1}{2}}{\parallel h\parallel }^{\frac{1}{2}}{\parallel {f}_{x}\parallel }^{\frac{1}{2}}{\parallel {g}_{y}\parallel }^{\frac{1}{2}}{\parallel {h}_{z}\parallel }^{\frac{1}{2}}.$

Clearly, the standard energy estimate shows that

$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left({\parallel u\parallel }^{2}+{\parallel b\parallel }^{2}\right)+{\parallel {u}_{x}\parallel }^{2}+{\parallel {u}_{y}\parallel }^{2}+{\parallel {b}_{x}\parallel }^{2}+{\parallel {b}_{y}\parallel }^{2}=0.$
(5)

We denote that $\omega =\mathrm{\nabla }×u$ and $j=\mathrm{\nabla }×b$. Thus, applying the operator ‘×’ to (1) and (2), together with (3), we deduce that

${\omega }_{t}+u\cdot \mathrm{\nabla }\omega -\omega \cdot \mathrm{\nabla }u={\omega }_{xx}+{\omega }_{yy}+b\cdot \mathrm{\nabla }j-j\cdot \mathrm{\nabla }b,$
(6)
${j}_{t}+u\cdot \mathrm{\nabla }j={j}_{xx}+{j}_{yy}+b\cdot \mathrm{\nabla }\omega +{\epsilon }_{ijk}\left({\partial }_{j}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}-{\partial }_{j}{u}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{b}_{k}\right),$
(7)

where ${\epsilon }_{ijk}$ is defined as follows:

The first key lemma is the following.

Lemma 2.2 If $\left(u,b\right)$ solves (1)-(4) and the initial data satisfies

${\parallel {u}_{0}\parallel }^{2}+{\parallel {b}_{0}\parallel }^{2}\le \frac{1}{4C}\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}{\parallel {\omega }_{0}\parallel }^{2}+{\parallel {j}_{0}\parallel }^{2}\le \frac{1}{4},$
(8)

where C is a suitable large number, then the vorticity ω and the current density j satisfy

${\parallel \omega \parallel }^{2}+{\parallel j\parallel }^{2}\le 1,\phantom{\rule{2em}{0ex}}{\int }_{0}^{t}\left({\parallel {\omega }_{x}\parallel }^{2}+{\parallel {\omega }_{y}\parallel }^{2}+{\parallel {j}_{x}\parallel }^{2}+{\parallel {j}_{y}\parallel }^{2}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\le 1\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\phantom{\rule{0.25em}{0ex}}t\ge 0.$

Proof Multiplying (6) and (7) by ω and j, respectively, then integrating the resulting equations by parts, after adding the two equalities together, we finally deduce

$\begin{array}{r}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left({\parallel \omega \parallel }^{2}+{\parallel j\parallel }^{2}\right)+{\parallel {\omega }_{x}\parallel }^{2}+{\parallel {\omega }_{y}\parallel }^{2}+{\parallel {j}_{x}\parallel }^{2}+{\parallel {j}_{y}\parallel }^{2}\\ \phantom{\rule{1em}{0ex}}=\int \left(\omega \cdot \mathrm{\nabla }\omega \cdot \omega -j\cdot \mathrm{\nabla }b\cdot \omega +{\epsilon }_{ijk}\left({\partial }_{j}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}-{\partial }_{j}{u}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{b}_{k}\right){j}_{i}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x=I+J+K+L.\end{array}$
(9)

We have to estimate each term on the right-hand side of (9). Some of the terms are the same as in [1] and are proved here for completeness. First, I can be written as

$\begin{array}{rcl}I& =& \int \omega \cdot \mathrm{\nabla }u\cdot \omega \phantom{\rule{0.2em}{0ex}}\mathrm{d}x=\int {\omega }_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{u}_{j}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =& \int {\omega }_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{u}_{j}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\omega }_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{u}_{j}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\omega }_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{j}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x={I}_{1}+{I}_{2}+{I}_{3}.\end{array}$

With the help of Lemma 2.1, we deduce that

$\begin{array}{rcl}{I}_{1}& =& \int {\omega }_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{u}_{j}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le C{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel {u}_{x}\parallel }^{\frac{1}{2}}{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}\\ \le & \frac{1}{24}{\parallel {\omega }_{x}\parallel }^{2}+\frac{1}{16}{\parallel {\omega }_{y}\parallel }^{2}+C{\parallel {u}_{x}\parallel }^{2}{\parallel \omega \parallel }^{4}.\end{array}$

Similarly, we obtain that

${I}_{2}=\int {\omega }_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{u}_{j}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le \frac{1}{24}{\parallel {\omega }_{x}\parallel }^{2}+\frac{1}{16}{\parallel {\omega }_{y}\parallel }^{2}+C{\parallel {u}_{y}\parallel }^{2}{\parallel \omega \parallel }^{4},$

and

$\begin{array}{rl}{I}_{3}& =\int {\omega }_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{j}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x=\int \left({\partial }_{x}{u}_{2}-{\partial }_{y}{u}_{1}\right)\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{j}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le \frac{1}{24}{\parallel {\omega }_{x}\parallel }^{2}+\frac{1}{16}{\parallel {\omega }_{y}\parallel }^{2}+C\left({\parallel {u}_{x}\parallel }^{2}+{\parallel {u}_{y}\parallel }^{2}\right){\parallel \omega \parallel }^{4}.\end{array}$

In order to bound J, we rewrite the integrand explicitly as follows:

$\begin{array}{rcl}J& =& -\int j\cdot \mathrm{\nabla }b\cdot \omega \phantom{\rule{0.2em}{0ex}}\mathrm{d}x=-\int {j}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{b}_{j}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =& -\int {j}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{b}_{j}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {j}_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{b}_{j}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {j}_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{j}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x={J}_{1}+{J}_{2}+{J}_{3}.\end{array}$

Due to Lemma 2.1, we see that

$\begin{array}{rl}{J}_{1}& =-\int {j}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{b}_{j}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le C{\parallel j\parallel }^{\frac{1}{2}}{\parallel {b}_{x}\parallel }^{\frac{1}{2}}{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}{\parallel {\partial }_{x}{b}_{z}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{22}{\parallel {j}_{x}\parallel }^{2}+\frac{1}{16}{\parallel {\omega }_{y}\parallel }^{2}+C{\parallel {b}_{x}\parallel }^{2}\left({\parallel \omega \parallel }^{4}+{\parallel j\parallel }^{4}\right).\end{array}$

Similarly,

$\begin{array}{c}\begin{array}{r}{J}_{2}=-\int {j}_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{b}_{j}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le \frac{1}{16}{\parallel {\omega }_{y}\parallel }^{2}+\frac{1}{22}{\parallel {j}_{x}\parallel }^{2}+\frac{1}{26}{\parallel {j}_{y}\parallel }^{2}+C{\parallel {b}_{y}\parallel }^{2}\left({\parallel \omega \parallel }^{4}+{\parallel j\parallel }^{4}\right),\end{array}\hfill \\ \begin{array}{rl}{J}_{3}& =-\int {j}_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{j}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x=-\int \left({\partial }_{x}{b}_{2}-{\partial }_{y}{b}_{1}\right)\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{j}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le \frac{1}{24}{\parallel {\omega }_{x}\parallel }^{2}+\frac{1}{22}{\parallel {j}_{x}\parallel }^{2}+\frac{1}{26}{\parallel {j}_{y}\parallel }^{2}+C\left({\parallel {b}_{x}\parallel }^{2}+{\parallel {b}_{y}\parallel }^{2}\right)\left({\parallel \omega \parallel }^{4}+{\parallel j\parallel }^{4}\right).\end{array}\hfill \end{array}$

Now, let us turn to bound L,

$\begin{array}{rl}L& =-\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{j}{u}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =-\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{u}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{u}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ ={L}_{1}+{L}_{2}+{L}_{3}.\end{array}$

By Lemma 2.1, we have that

$\begin{array}{rl}{L}_{1}& =-\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{u}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le C{\parallel {u}_{x}\parallel }^{\frac{1}{2}}{\parallel j\parallel }^{\frac{1}{2}}{\parallel j\parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{24}{\parallel {\omega }_{x}\parallel }^{2}+\frac{1}{22}{\parallel {j}_{x}\parallel }^{2}+\frac{1}{26}{\parallel {j}_{y}\parallel }^{2}+C{\parallel {u}_{x}\parallel }^{2}{\parallel j\parallel }^{4}.\end{array}$

Similarly,

${L}_{2}=-\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{u}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le \frac{1}{16}{\parallel {\omega }_{y}\parallel }^{2}+\frac{1}{22}{\parallel {j}_{x}\parallel }^{2}+\frac{1}{26}{\parallel {j}_{y}\parallel }^{2}+C{\parallel {u}_{y}\parallel }^{2}{\parallel j\parallel }^{4}.$

As for ${L}_{3}$, we should split it into three parts:

$\begin{array}{c}\begin{array}{rl}{L}_{3}& =-\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =-\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ ={L}_{31}+{L}_{32}+{L}_{33},\end{array}\hfill \\ \begin{array}{rl}{L}_{31}& =-\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel {b}_{x}\parallel }^{\frac{1}{2}}{\parallel j\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel {\partial }_{x}{b}_{z}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{24}{\parallel {\omega }_{x}\parallel }^{2}+\frac{1}{22}{\parallel {j}_{x}\parallel }^{2}+\frac{1}{26}{\parallel {j}_{y}\parallel }^{2}+C{\parallel {b}_{x}\parallel }^{2}\left({\parallel \omega \parallel }^{4}+{\parallel j\parallel }^{4}\right).\end{array}\hfill \end{array}$

Similarly,

${L}_{32}=-\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le \frac{1}{24}{\parallel {\omega }_{x}\parallel }^{2}+\frac{1}{26}{\parallel {j}_{y}\parallel }^{2}+C{\parallel {b}_{y}\parallel }^{2}\left({\parallel \omega \parallel }^{4}+{\parallel j\parallel }^{4}\right).$

To bound ${L}_{33}$, using the incompressibility condition $divu=0$, we deduce that

$\begin{array}{rl}{L}_{33}& =-\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =\int {\epsilon }_{ijk}\left({\partial }_{x}{u}_{1}+{\partial }_{y}{u}_{2}\right)\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x={L}_{33}^{1}+{L}_{33}^{2}.\end{array}$

We get the ${L}_{33}^{1}$ and ${L}_{33}^{2}$ as follows:

$\begin{array}{rl}{L}_{33}^{1}& =\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{u}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le C{\parallel {u}_{x}\parallel }^{\frac{1}{2}}{\parallel j\parallel }^{\frac{1}{2}}{\parallel j\parallel }^{\frac{1}{2}}{\parallel {\partial }_{x}{u}_{z}\parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{24}{\parallel {\omega }_{x}\parallel }^{2}+\frac{1}{22}{\parallel {j}_{x}\parallel }^{2}+\frac{1}{26}{\parallel {j}_{y}\parallel }^{2}+C{\parallel {u}_{x}\parallel }^{2}{\parallel j\parallel }^{4},\end{array}$

and

${L}_{33}^{2}\le \frac{1}{16}{\parallel {\omega }_{y}\parallel }^{2}+\frac{1}{22}{\parallel {j}_{x}\parallel }^{2}+\frac{1}{26}{\parallel {j}_{y}\parallel }^{2}+C{\parallel {u}_{y}\parallel }^{2}{\parallel j\parallel }^{4}.$

To bound K, we should divide it into three parts:

$\begin{array}{rl}K& =\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{j}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =\int {\epsilon }_{i1k}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\epsilon }_{i2k}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\epsilon }_{i3k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ ={K}_{1}+{K}_{2}+{K}_{3}.\end{array}$

Similarly, we deduce that

$\begin{array}{rl}{K}_{1}& =\int {\epsilon }_{i1k}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le C{\parallel {b}_{x}\parallel }^{\frac{1}{2}}{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel j\parallel }^{\frac{1}{2}}{\parallel {\partial }_{x}{b}_{z}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{24}{\parallel {\omega }_{x}\parallel }^{2}+\frac{1}{22}{\parallel {j}_{x}\parallel }^{2}+\frac{1}{26}{\parallel {j}_{y}\parallel }^{2}+C{\parallel {b}_{x}\parallel }^{2}\left({\parallel \omega \parallel }^{4}+{\parallel j\parallel }^{4}\right),\end{array}$

and

${K}_{2}=\int {\epsilon }_{i2k}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le \frac{1}{24}{\parallel {\omega }_{x}\parallel }^{2}+\frac{1}{26}{\parallel {j}_{y}\parallel }^{2}+C{\parallel {b}_{y}\parallel }^{2}\left({\parallel \omega \parallel }^{4}+{\parallel j\parallel }^{4}\right).$

For ${K}_{3}$, we have

$\begin{array}{rl}{K}_{3}& =\int {\epsilon }_{i3k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =\int {\epsilon }_{i3k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\epsilon }_{i3k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\epsilon }_{i3k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ ={K}_{31}+{K}_{32}+{K}_{33}.\end{array}$

Thus,

$\begin{array}{rl}{K}_{31}& =\int {\epsilon }_{i3k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le C{\parallel j\parallel }^{\frac{1}{2}}{\parallel {u}_{x}\parallel }^{\frac{1}{2}}{\parallel j\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{24}{\parallel {\omega }_{x}\parallel }^{2}+\frac{1}{22}{\parallel {j}_{x}\parallel }^{2}+\frac{1}{26}{\parallel {j}_{y}\parallel }^{2}+C{\parallel {u}_{x}\parallel }^{2}{\parallel j\parallel }^{4},\end{array}$

and

${K}_{32}=\int {\epsilon }_{i3k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le \frac{1}{16}{\parallel {\omega }_{y}\parallel }^{2}+\frac{1}{22}{\parallel {j}_{x}\parallel }^{2}+\frac{1}{26}{\parallel {j}_{y}\parallel }^{2}+C{\parallel {u}_{y}\parallel }^{2}{\parallel j\parallel }^{4}.$

As for ${K}_{33}$, using $divb=0$, we obtain

$\begin{array}{rl}{K}_{33}& =\int {\epsilon }_{i3k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x=-\int {\epsilon }_{i3k}\left({\partial }_{x}{b}_{1}+{\partial }_{y}{b}_{2}\right)\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le \frac{1}{24}{\parallel {\omega }_{x}\parallel }^{2}+\frac{1}{26}{\parallel {j}_{y}\parallel }^{2}+C\left({\parallel {b}_{x}\parallel }^{2}+{\parallel {b}_{y}\parallel }^{2}\right)\left({\parallel \omega \parallel }^{4}+{\parallel j\parallel }^{4}\right).\end{array}$

Substituting all the above estimates into (9), we conclude that

$\begin{array}{r}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left({\parallel \omega \parallel }^{2}+{\parallel j\parallel }^{2}\right)+\frac{1}{2}\left({\parallel {\omega }_{x}\parallel }^{2}+{\parallel {\omega }_{y}\parallel }^{2}+{\parallel {j}_{x}\parallel }^{2}+{\parallel {j}_{y}\parallel }^{2}\right)\\ \phantom{\rule{1em}{0ex}}\le C\left({\parallel {u}_{x}\parallel }^{2}+{\parallel {u}_{y}\parallel }^{2}+{\parallel {b}_{x}\parallel }^{2}+{\parallel {b}_{y}\parallel }^{2}\right)\left({\parallel \omega \parallel }^{4}+{\parallel j\parallel }^{4}\right).\end{array}$

Let ${\parallel \omega \parallel }^{2}+{\parallel j\parallel }^{2}\le 1$, we deduce, with the assumption (8) on the initial data, that

${\parallel \omega \parallel }^{2}+{\parallel j\parallel }^{2}+{\int }_{0}^{t}\left({\parallel {\omega }_{x}\parallel }^{2}+{\parallel {\omega }_{y}\parallel }^{2}+{\parallel {j}_{x}\parallel }^{2}+{\parallel {j}_{y}\parallel }^{2}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\le 1.$

Thus, the proof of Lemma 2.2 is completed. □

Now, we turn to deduce the higher order estimates about the solution.

Lemma 2.3 If $\left(u,b\right)$ is the solution of (1)-(4), then

${\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}+{\int }_{0}^{t}\left({\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{2}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\le C.$
(10)

Proof Multiplying (6) and (7) by Δω and Δj, respectively, then integrating the resultant equations by parts, after adding the two equalities together, we finally obtain

$\begin{array}{r}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right)+\left({\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{2}\right)\\ \phantom{\rule{1em}{0ex}}=\int \left[u\cdot \mathrm{\nabla }\omega \cdot \mathrm{\Delta }\omega +u\cdot \mathrm{\nabla }j\cdot \mathrm{\Delta }j-\omega \cdot \mathrm{\nabla }\omega \cdot \mathrm{\Delta }\omega +j\cdot \mathrm{\nabla }b\cdot \mathrm{\Delta }\omega -b\cdot \mathrm{\nabla }j\cdot \mathrm{\Delta }\omega \\ \phantom{\rule{2em}{0ex}}-b\cdot \mathrm{\nabla }\omega \cdot \mathrm{\Delta }j-{\epsilon }_{ijk}\left({\partial }_{j}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}-{\partial }_{j}{u}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{b}_{k}\right)\mathrm{\Delta }{j}_{i}\right]\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \phantom{\rule{1em}{0ex}}=M+N+P+Q+R+S.\end{array}$
(11)

Now, we turn to bound each term on the right-hand side of (11). Similar as the proof of Lemma 2.2, keeping in mind Lemma 2.1 and the divergence-free property of u and b, we deduce

$\begin{array}{rl}M& =\int u\cdot \mathrm{\nabla }\omega \cdot \mathrm{\Delta }\omega \phantom{\rule{0.2em}{0ex}}\mathrm{d}x=\int {u}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{kk}^{2}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x=-\int {\partial }_{k}{u}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =-\int {\partial }_{x}{u}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {\partial }_{y}{u}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {\partial }_{z}{u}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ ={M}_{1}+{M}_{2}+{M}_{3}.\end{array}$

We estimate each term as follows:

$\begin{array}{rl}{M}_{1}& =-\int {\partial }_{x}{u}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le C{\parallel {u}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel {\partial }_{x}{u}_{z}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+C\parallel {u}_{x}\parallel \parallel {\omega }_{x}\parallel {\parallel \mathrm{\nabla }\omega \parallel }^{2}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+C\left({\parallel {u}_{x}\parallel }^{2}+{\parallel {\omega }_{x}\parallel }^{2}\right){\parallel \mathrm{\nabla }\omega \parallel }^{2}.\end{array}$

Similarly,

$\begin{array}{rl}{M}_{2}& =-\int {\partial }_{y}{u}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel {u}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel {\partial }_{y}{u}_{z}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+C\left({\parallel {u}_{y}\parallel }^{2}+{\parallel {\omega }_{y}\parallel }^{2}\right){\parallel \mathrm{\nabla }\omega \parallel }^{2}.\end{array}$

Now, we turn to bound ${M}_{3}$,

$\begin{array}{rl}{M}_{3}& =-\int {\partial }_{z}{u}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =-\int {\partial }_{z}{u}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {\partial }_{z}{u}_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {\partial }_{z}{u}_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ ={M}_{31}+{M}_{32}+{M}_{33}.\end{array}$

Similarly, we can deduce that

$\begin{array}{c}\begin{array}{rl}{M}_{31}& =-\int {\partial }_{z}{u}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+C\left({\parallel \omega \parallel }^{2}+{\parallel {\omega }_{x}\parallel }^{2}\right){\parallel \mathrm{\nabla }\omega \parallel }^{2},\end{array}\hfill \\ \begin{array}{rl}{M}_{32}& =-\int {\partial }_{z}{u}_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel {\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+C\left({\parallel \omega \parallel }^{2}+{\parallel {\omega }_{x}\parallel }^{2}\right){\parallel \mathrm{\nabla }\omega \parallel }^{2},\end{array}\hfill \\ \begin{array}{rl}{M}_{33}=& -\int {\partial }_{z}{u}_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x=\int \left({\partial }_{x}{u}_{1}+{\partial }_{y}{u}_{2}\right)\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le & C{\parallel {u}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel {\partial }_{x}{u}_{z}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}\\ +C{\parallel {u}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel {\partial }_{y}{u}_{z}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}\\ \le & \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+C\left({\parallel {u}_{x}\parallel }^{2}+{\parallel {u}_{y}\parallel }^{2}+{\parallel {\omega }_{x}\parallel }^{2}+{\parallel {\omega }_{y}\parallel }^{2}\right){\parallel \mathrm{\nabla }\omega \parallel }^{2}.\end{array}\hfill \end{array}$

As for N, integrating by parts, we deduce that

$\begin{array}{rl}N& =\int u\cdot \mathrm{\nabla }j\cdot \mathrm{\Delta }j\phantom{\rule{0.2em}{0ex}}\mathrm{d}x=\int {u}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{kk}^{2}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x=-\int {\partial }_{k}{u}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =-\int {\partial }_{x}{u}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {\partial }_{y}{u}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {\partial }_{z}{u}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ ={N}_{1}+{N}_{2}+{N}_{3}.\end{array}$

Similarly, we have

$\begin{array}{c}\begin{array}{rl}{N}_{1}& =-\int {\partial }_{x}{u}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel {u}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{50}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{2}+C\left({\parallel {u}_{x}\parallel }^{2}+{\parallel {\omega }_{x}\parallel }^{2}\right){\parallel \mathrm{\nabla }j\parallel }^{2},\end{array}\hfill \\ \begin{array}{rl}{N}_{2}& =-\int {\partial }_{y}{u}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel {u}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{50}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{2}+C\left({\parallel {u}_{y}\parallel }^{2}+{\parallel {\omega }_{y}\parallel }^{2}\right){\parallel \mathrm{\nabla }j\parallel }^{2}.\end{array}\hfill \end{array}$

As for ${N}_{3}$, we obtain

$\begin{array}{rl}{N}_{3}& =-\int {\partial }_{z}{u}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =-\int {\partial }_{z}{u}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {\partial }_{z}{u}_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {\partial }_{z}{u}_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ ={N}_{31}+{N}_{32}+{N}_{33}.\end{array}$

Thus, we have

$\begin{array}{c}\begin{array}{rl}{N}_{31}& =-\int {\partial }_{z}{u}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{50}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{2}+C\left({\parallel \omega \parallel }^{2}+{\parallel {\omega }_{x}\parallel }^{2}\right){\parallel \mathrm{\nabla }j\parallel }^{2},\end{array}\hfill \\ \begin{array}{rl}{N}_{32}& =-\int {\partial }_{z}{u}_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{48}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{2}+C\left({\parallel \omega \parallel }^{2}+{\parallel {\omega }_{x}\parallel }^{2}\right){\parallel \mathrm{\nabla }j\parallel }^{2},\end{array}\hfill \\ \begin{array}{rl}{N}_{33}=& -\int {\partial }_{z}{u}_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le & C{\parallel {u}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}\\ +C{\parallel {u}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{y}\parallel }^{\frac{1}{2}}\\ \le & \frac{1}{50}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{2}+C\left({\parallel {u}_{x}\parallel }^{2}+{\parallel {u}_{y}\parallel }^{2}+{\parallel {\omega }_{x}\parallel }^{2}+{\parallel {\omega }_{y}\parallel }^{2}\right){\parallel \mathrm{\nabla }j\parallel }^{2}.\end{array}\hfill \end{array}$

We now turn to bound P,

$\begin{array}{rl}P& =-\int \omega \cdot \mathrm{\nabla }\omega \cdot \mathrm{\Delta }\omega \phantom{\rule{0.2em}{0ex}}\mathrm{d}x=-\int {\omega }_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{kk}^{2}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =\int {\partial }_{k}{\omega }_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\omega }_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x={P}_{1}+{P}_{2}.\end{array}$

For ${P}_{1}$, we have

$\begin{array}{c}\begin{array}{rl}{P}_{1}& =\int {\partial }_{k}{\omega }_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =\int {\partial }_{x}{\omega }_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\partial }_{y}{\omega }_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\partial }_{z}{\omega }_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ ={P}_{11}+{P}_{12}+{P}_{13},\end{array}\hfill \\ \begin{array}{rl}{P}_{11}& =\int {\partial }_{x}{\omega }_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+C\left({\parallel \omega \parallel }^{2}+{\parallel {\omega }_{x}\parallel }^{2}\right){\parallel \mathrm{\nabla }\omega \parallel }^{2},\end{array}\hfill \\ \begin{array}{rl}{P}_{12}& =\int {\partial }_{y}{\omega }_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel {\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel {\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+C\left({\parallel \omega \parallel }^{2}+{\parallel {\omega }_{x}\parallel }^{2}\right){\parallel \mathrm{\nabla }\omega \parallel }^{2}.\end{array}\hfill \end{array}$

For ${P}_{13}$, we see that

$\begin{array}{rl}{P}_{13}& =\int {\partial }_{z}{\omega }_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =\int {\partial }_{z}{\omega }_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\partial }_{z}{\omega }_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\partial }_{z}{\omega }_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ ={P}_{13}^{1}+{P}_{13}^{2}+{P}_{13}^{3}.\end{array}$

Thus, we can bound ${P}_{13}$ as follows:

$\begin{array}{c}\begin{array}{rl}{P}_{13}^{1}& =\int {\partial }_{z}{\omega }_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le C{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel {u}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+C\left({\parallel {u}_{x}\parallel }^{2}+{\parallel {\omega }_{x}\parallel }^{2}\right){\parallel \mathrm{\nabla }\omega \parallel }^{2},\end{array}\hfill \\ \begin{array}{rl}{P}_{13}^{2}& =\int {\partial }_{z}{\omega }_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le C{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel {u}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+C\left({\parallel {u}_{y}\parallel }^{2}+{\parallel {\omega }_{y}\parallel }^{2}\right){\parallel \mathrm{\nabla }\omega \parallel }^{2},\end{array}\hfill \\ \begin{array}{rl}{P}_{13}^{2}=& \int {\partial }_{z}{\omega }_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =& \int {\partial }_{z}\left({\partial }_{x}{u}_{2}-{\partial }_{y}{u}_{1}\right)\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le & C{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}\\ +C{\parallel {\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}\\ \le & \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+C\left({\parallel \omega \parallel }^{2}+{\parallel {\omega }_{x}\parallel }^{2}+{\parallel {\omega }_{y}\parallel }^{2}\right){\parallel \mathrm{\nabla }\omega \parallel }^{2}.\end{array}\hfill \end{array}$

Now, we turn to ${P}_{2}$,

$\begin{array}{rl}{P}_{2}& =\int {\omega }_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =\int {\omega }_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\omega }_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\omega }_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ ={P}_{21}+{P}_{22}+{P}_{23}.\end{array}$

Similarly,

$\begin{array}{c}\begin{array}{rl}{P}_{21}& =\int {\omega }_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+C\left({\parallel \omega \parallel }^{2}+{\parallel {\omega }_{x}\parallel }^{2}\right){\parallel \mathrm{\nabla }\omega \parallel }^{2},\end{array}\hfill \\ \begin{array}{rl}{P}_{22}& =\int {\omega }_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel {\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+C\left({\parallel \omega \parallel }^{2}+{\parallel {\omega }_{x}\parallel }^{2}\right){\parallel \mathrm{\nabla }\omega \parallel }^{2},\end{array}\hfill \\ \begin{array}{rl}{P}_{23}=& \int {\omega }_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le & C{\parallel {u}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}\\ +C{\parallel {u}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{y}\parallel }^{\frac{1}{2}}\\ \le & \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+C\left({\parallel {u}_{x}\parallel }^{2}+{\parallel {u}_{y}\parallel }^{2}+{\parallel {\omega }_{x}\parallel }^{2}+{\parallel {\omega }_{y}\parallel }^{2}\right){\parallel \mathrm{\nabla }\omega \parallel }^{2}.\end{array}\hfill \end{array}$

To bound Q, we see that

$\begin{array}{c}\begin{array}{rl}Q& =\int j\cdot \mathrm{\nabla }b\cdot \mathrm{\Delta }\omega \phantom{\rule{0.2em}{0ex}}\mathrm{d}x=\int {j}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{kk}^{2}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =-\int {\partial }_{k}{j}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {j}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x={Q}_{1}+{Q}_{2},\end{array}\hfill \\ \begin{array}{rl}{Q}_{1}& =-\int {\partial }_{x}{j}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {\partial }_{y}{j}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {\partial }_{z}{j}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ ={Q}_{11}+{Q}_{12}+{Q}_{13},\end{array}\hfill \\ \begin{array}{rl}{Q}_{11}& =-\int {\partial }_{x}{j}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le C{\parallel {j}_{x}\parallel }^{\frac{1}{2}}{\parallel j\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{50}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+C\left({\parallel j\parallel }^{2}+{\parallel {j}_{x}\parallel }^{2}\right)\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right),\end{array}\hfill \\ \begin{array}{rl}{Q}_{12}& =-\int {\partial }_{y}{j}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le C{\parallel {j}_{y}\parallel }^{\frac{1}{2}}{\parallel j\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{2}+C\left({\parallel j\parallel }^{2}+{\parallel {j}_{y}\parallel }^{2}\right)\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right).\end{array}\hfill \end{array}$

As for ${Q}_{13}$, we see that

$\begin{array}{rl}{Q}_{13}& =-\int {\partial }_{z}{j}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =-\int {\partial }_{z}{j}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {\partial }_{z}{j}_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {\partial }_{z}{j}_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ ={Q}_{13}^{1}+{Q}_{13}^{2}+{Q}_{13}^{3}.\end{array}$

Thus, we can deduce that

$\begin{array}{c}\begin{array}{rl}{Q}_{13}^{1}& =-\int {\partial }_{z}{j}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel {b}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+\frac{1}{50}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+C\left({\parallel {b}_{x}\parallel }^{2}+{\parallel {j}_{x}\parallel }^{2}\right)\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right),\end{array}\hfill \\ \begin{array}{rl}{Q}_{13}^{2}& =-\int {\partial }_{z}{j}_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel {b}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+\frac{1}{50}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+C\left({\parallel {b}_{y}\parallel }^{2}+{\parallel {j}_{y}\parallel }^{2}\right)\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right),\end{array}\hfill \\ \begin{array}{rl}{Q}_{13}^{3}=& -\int {\partial }_{z}{j}_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x=-\int {\partial }_{z}\left({\partial }_{x}{b}_{2}-{\partial }_{y}{b}_{1}\right)\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le & C{\parallel {j}_{x}\parallel }^{\frac{1}{2}}{\parallel j\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}\\ +C{\parallel {j}_{y}\parallel }^{\frac{1}{2}}{\parallel j\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}\\ \le & \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{50}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{2}+C\left({\parallel j\parallel }^{2}+{\parallel {j}_{x}\parallel }^{2}+{\parallel {j}_{y}\parallel }^{2}\right)\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right).\end{array}\hfill \end{array}$

Now, we turn to ${Q}_{2}$,

$\begin{array}{rl}{Q}_{2}& =-\int b\cdot \mathrm{\nabla }j\cdot \mathrm{\Delta }\omega \phantom{\rule{0.2em}{0ex}}\mathrm{d}x=-\int {j}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =-\int {j}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {j}_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {j}_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ ={Q}_{21}+{Q}_{22}+{Q}_{23}.\end{array}$

We deduce that

$\begin{array}{c}\begin{array}{rl}{Q}_{21}& =-\int {j}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel j\parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{50}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+C\left({\parallel j\parallel }^{2}+{\parallel {j}_{x}\parallel }^{2}\right)\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right),\end{array}\hfill \\ \begin{array}{rl}{Q}_{22}& =-\int {j}_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel j\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{2}+C\left({\parallel j\parallel }^{2}+{\parallel {j}_{y}\parallel }^{2}\right)\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right),\end{array}\hfill \\ \begin{array}{rl}{Q}_{23}=& -\int {j}_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le & C{\parallel {b}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}\\ +C{\parallel {b}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}\\ \le & \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{2}+C\left({\parallel {b}_{x}\parallel }^{2}+{\parallel {j}_{x}\parallel }^{2}+{\parallel {b}_{y}\parallel }^{2}+{\parallel {j}_{y}\parallel }^{2}\right)\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right).\end{array}\hfill \end{array}$

Here we start to estimate R as follows:

$\begin{array}{rl}R& =-\int b\cdot \mathrm{\nabla }j\cdot \mathrm{\Delta }\omega \phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int b\cdot \mathrm{\nabla }\omega \cdot \mathrm{\Delta }j\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =-\int {b}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{kk}^{2}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\int {b}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{kk}^{2}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =\int {\partial }_{k}{b}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\partial }_{k}{b}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x={R}_{1}+{R}_{2}.\end{array}$

For ${R}_{1}$, we have

$\begin{array}{rl}{R}_{1}& =\int {\partial }_{k}{b}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =\int {\partial }_{x}{b}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\partial }_{y}{b}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\partial }_{z}{b}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ ={R}_{11}+{R}_{12}+{R}_{13}.\end{array}$

We deduce that

$\begin{array}{c}\begin{array}{rl}{R}_{11}& =\int {\partial }_{x}{b}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel {b}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{50}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+C\left({\parallel {b}_{x}\parallel }^{2}+{\parallel {\omega }_{x}\parallel }^{2}\right){\parallel \mathrm{\nabla }j\parallel }^{2},\end{array}\hfill \\ \begin{array}{rl}{R}_{12}& =\int {\partial }_{y}{b}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel {b}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel {\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{50}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+\frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+C\left({\parallel {b}_{y}\parallel }^{2}+{\parallel {\omega }_{y}\parallel }^{2}\right){\parallel \mathrm{\nabla }j\parallel }^{2}.\end{array}\hfill \end{array}$

For ${R}_{13}$, we have

$\begin{array}{rl}{R}_{13}& =\int {\partial }_{z}{b}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =\int {\partial }_{z}{b}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\partial }_{z}{b}_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\partial }_{z}{b}_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x={R}_{13}^{1}+{R}_{13}^{2}+{R}_{13}^{3}.\end{array}$

Similarly,

$\begin{array}{c}\begin{array}{rl}{R}_{13}^{1}& =\int {\partial }_{z}{b}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel j\parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{50}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+C\left({\parallel j\parallel }^{2}+{\parallel {j}_{x}\parallel }^{2}\right)\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right),\end{array}\hfill \\ \begin{array}{rl}{R}_{13}^{2}& =\int {\partial }_{z}{b}_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel j\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{2}+C\left({\parallel j\parallel }^{2}+{\parallel {j}_{y}\parallel }^{2}\right)\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right),\end{array}\hfill \\ \begin{array}{rl}{R}_{13}^{3}=& \int {\partial }_{z}{b}_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =& -\int \left({\partial }_{x}{b}_{1}+{\partial }_{y}{b}_{2}\right)\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le & C{\parallel {b}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}\\ +C{\parallel {b}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}\\ \le & \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{2}+C\left({\parallel j\parallel }^{2}+{\parallel {j}_{y}\parallel }^{2}\right)\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right).\end{array}\hfill \end{array}$

Now, we turn to ${R}_{2}$,

$\begin{array}{rl}{R}_{2}& =\int {\partial }_{k}{b}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{k}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =\int {\partial }_{x}{b}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\partial }_{y}{b}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\partial }_{z}{b}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x={R}_{21}+{R}_{22}+{R}_{23}.\end{array}$

Thus, similarly, we deduce that

$\begin{array}{c}\begin{array}{rl}{R}_{21}& =\int {\partial }_{x}{b}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le C{\parallel {b}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{50}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+C\left({\parallel {b}_{x}\parallel }^{2}+{\parallel {j}_{x}\parallel }^{2}\right)\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right),\end{array}\hfill \\ \begin{array}{rl}{R}_{22}& =\int {\partial }_{y}{b}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le C{\parallel {b}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{2}+C\left({\parallel {b}_{y}\parallel }^{2}+{\parallel {j}_{y}\parallel }^{2}\right)\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right).\end{array}\hfill \end{array}$

For ${R}_{23}$, we have

$\begin{array}{rl}{R}_{23}& =\int {\partial }_{z}{b}_{i}\phantom{\rule{0.2em}{0ex}}{\partial }_{i}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =\int {\partial }_{z}{b}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\partial }_{z}{b}_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\partial }_{z}{b}_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ ={R}_{23}^{1}+{R}_{23}^{2}+{R}_{23}^{3}.\end{array}$

Similarly,

$\begin{array}{c}\begin{array}{rl}{R}_{23}^{1}& =\int {\partial }_{z}{b}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le C{\parallel j\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{50}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+C\left({\parallel j\parallel }^{2}+{\parallel {\omega }_{x}\parallel }^{2}\right){\parallel \mathrm{\nabla }j\parallel }^{2},\end{array}\hfill \\ \begin{array}{rl}{R}_{23}^{2}& =\int {\partial }_{z}{b}_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le C{\parallel j\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{50}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+\frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+C\left({\parallel j\parallel }^{2}+{\parallel {j}_{y}\parallel }^{2}\right)\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right),\end{array}\hfill \\ \begin{array}{rl}{R}_{23}^{3}=& \int {\partial }_{z}{b}_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =& -\int \left({\partial }_{x}{b}_{1}+{\partial }_{y}{b}_{2}\right)\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{\omega }_{j}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le & C{\parallel {b}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}\\ +C{\parallel {b}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}\\ \le & \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{2}+C\left({\parallel {b}_{x}\parallel }^{2}+{\parallel {b}_{y}\parallel }^{2}+{\parallel {j}_{x}\parallel }^{2}+{\parallel {j}_{y}\parallel }^{2}\right)\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right).\end{array}\hfill \end{array}$

Finally, for the last term S, we have

$\begin{array}{rl}S& =-\int {\epsilon }_{ijk}\left({\partial }_{j}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}-{\partial }_{j}{u}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{b}_{k}\right)\mathrm{\Delta }{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{m}\left({\partial }_{j}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}-{\partial }_{j}{u}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{b}_{k}\right)\phantom{\rule{0.2em}{0ex}}{\partial }_{m}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ ={S}_{1}+{S}_{2}.\end{array}$

We first consider ${S}_{1}$,

$\begin{array}{rl}{S}_{1}& =\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{m}\left({\partial }_{j}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\right)\phantom{\rule{0.2em}{0ex}}{\partial }_{m}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}\left({\partial }_{j}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\right)\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}\left({\partial }_{j}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\right)\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}\left({\partial }_{j}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\right)\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ ={S}_{11}+{S}_{12}+{S}_{13}.\end{array}$

We deduce that

$\begin{array}{c}\begin{array}{rl}{S}_{11}=& \int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}\left({\partial }_{j}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\right)\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x=\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}\phantom{\rule{0.2em}{0ex}}{\partial }_{j}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{j}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le & C{\parallel {j}_{x}\parallel }^{\frac{1}{2}}{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}\\ +C{\parallel j\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}\\ \le & \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_{x}\parallel }^{2}+\frac{1}{50}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{2}+C\left({\parallel {j}_{x}\parallel }^{2}+{\parallel \omega \parallel }^{2}+{\parallel j\parallel }^{2}\right)\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right),\end{array}\hfill \\ \begin{array}{rl}{S}_{12}=& \int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}\left({\partial }_{j}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\right)\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x=\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}\phantom{\rule{0.2em}{0ex}}{\partial }_{j}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{j}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le & C{\parallel {j}_{y}\parallel }^{\frac{1}{2}}{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}\\ +C{\parallel j\parallel }^{\frac{1}{2}}{\parallel {\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel {j}_{y}\parallel }^{\frac{1}{2}}{\parallel {j}_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}\\ \le & \frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_{y}\parallel }^{2}+\frac{1}{50}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{2}\\ +C\left({\parallel \omega \parallel }^{2}+{\parallel {j}_{y}\parallel }^{2}+{\parallel j\parallel }^{2}+{\parallel {j}_{x}\parallel }^{2}\right)\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right).\end{array}\hfill \end{array}$

As for ${S}_{13}$, we see that

$\begin{array}{rl}{S}_{13}=& \int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}\left({\partial }_{j}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\right)\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =& \int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}\phantom{\rule{0.2em}{0ex}}{\partial }_{j}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{j}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =& \int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ +\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =& {S}_{13}^{1}+{S}_{13}^{2}+{S}_{13}^{3}+{S}_{13}^{4}+{S}_{13}^{5}+{S}_{13}^{6}.\end{array}$

We deduce each term step by step as follows:

$\begin{array}{c}\begin{array}{rl}{S}_{13}^{1}& =\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel {j}_{x}\parallel }^{\frac{1}{2}}{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{x}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{50}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{2}+C\left({\parallel \omega \parallel }^{2}+{\parallel {j}_{x}\parallel }^{2}\right)\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right),\end{array}\hfill \\ \begin{array}{rl}{S}_{13}^{2}& =\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ \le C{\parallel {j}_{y}\parallel }^{\frac{1}{2}}{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{\frac{1}{2}}{\parallel {\omega }_{y}\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{\frac{1}{2}}\\ \le \frac{1}{50}{\parallel \mathrm{\nabla }{j}_{x}\parallel }^{2}+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_{y}\parallel }^{2}+C\left({\parallel \omega \parallel }^{2}+{\parallel {j}_{y}\parallel }^{2}\right)\left({\parallel \mathrm{\nabla }\omega \parallel }^{2}+{\parallel \mathrm{\nabla }j\parallel }^{2}\right).\end{array}\hfill \end{array}$

For ${S}_{13}^{3}$, we have

$\begin{array}{rl}{S}_{13}^{3}& =\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{l}\phantom{\rule{0.2em}{0ex}}{\partial }_{l}{u}_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ =\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{1}\phantom{\rule{0.2em}{0ex}}{\partial }_{x}{u}_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{2}\phantom{\rule{0.2em}{0ex}}{\partial }_{y}{u}_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{b}_{3}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{u}_{k}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}{j}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\\ ={S}_{13}^{31}+{S}_{13}^{32}+{S}_{13}^{33}.\end{array}$

Thus, we deduce that

$\begin{array}{c}\begin{array}{rl}{S}_{13}^{31}& =\int {\epsilon }_{ijk}\phantom{\rule{0.2em}{0ex}}{\partial }_{z}\end{array}\hfill \end{array}$