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)

$$divu=0,divb=0,$$

(3)

associated with the initial data

$$u(0,x)=u_0,b(0,x)=b_0.$$

(4)

Here $u=(u_1(t,x),u_2(t,x),u_3(t,x))$ is the velocity field, $b=(b_1(t,x),b_2(t,x),b_3(t,x))$ is the magnetic field, $p=p(t,x)$ 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

$$\int f\mathrm{d}x\triangleq {\int }_{{\mathbb{R}}^3}f\mathrm{d}x\mathrm{d}y\mathrm{d}z,$$

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

$$\begin{array}{c}\parallel f\parallel ={(\int f^2\mathrm{d}x)}^{\frac{1}{2}},\hfill \\ H^m\triangleq W^{m,2}\left({\mathbb{R}}^3\right)={\{\sum _{|\alpha |\le m}\int {\left|D^{\alpha }u\right|}^2\mathrm{d}x\}}^{1/2}\text{with the norm }{\parallel \cdot \parallel }_{H^m}.\hfill \end{array}$$

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 $L^2({\mathbb{R}}^3)$. Then we have

$$\int |fgh|\mathrm{d}x\le C{\parallel f\parallel }^{\frac{1}{2}}{\parallel g\parallel }^{\frac{1}{2}}{\parallel h\parallel }^{\frac{1}{2}}{\parallel f_x\parallel }^{\frac{1}{2}}{\parallel g_y\parallel }^{\frac{1}{2}}{\parallel h_z\parallel }^{\frac{1}{2}}.$$

Clearly, the standard energy estimate shows that

$$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}({\parallel u\parallel }^2+{\parallel b\parallel }^2)+{\parallel u_x\parallel }^2+{\parallel u_y\parallel }^2+{\parallel b_x\parallel }^2+{\parallel b_y\parallel }^2=0.$$

(5)

We denote that $\omega =\mathrm{\nabla }\times u$ and $j=\mathrm{\nabla }\times b$. 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)

$$j_t+u\cdot \mathrm{\nabla }j=j_{xx}+j_{yy}+b\cdot \mathrm{\nabla }\omega +{\epsilon }_{ijk}({\partial }_j{b}_l{\partial }_l{u}_k-{\partial }_j{u}_l{\partial }_l{b}_k),$$

(7)

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

$${\epsilon }_{ijk}=\{\begin{array}{cc}1\hfill & \text{if }(i,j,k)\text{ is an even permutation},\hfill \\ -1\hfill & \text{if }(i,j,k)\text{ is an odd permutation},\hfill \\ 0\hfill & \text{others}.\hfill \end{array}$$

The first key lemma is the following.

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

$${\parallel u_0\parallel }^2+{\parallel b_0\parallel }^2\le \frac{1}{4C}\mathit{\text{and}}{\parallel {\omega }_0\parallel }^2+{\parallel j_0\parallel }^2\le \frac{1}{4},$$

(8)

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

$${\parallel \omega \parallel }^2+{\parallel j\parallel }^2\le 1,{\int }_0^t({\parallel {\omega }_x\parallel }^2+{\parallel {\omega }_y\parallel }^2+{\parallel j_x\parallel }^2+{\parallel j_y\parallel }^2)\mathrm{d}s\le 1\mathit{\text{for all}}t\ge 0.$$

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

$$\begin{array}{r}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}({\parallel \omega \parallel }^2+{\parallel j\parallel }^2)+{\parallel {\omega }_x\parallel }^2+{\parallel {\omega }_y\parallel }^2+{\parallel j_x\parallel }^2+{\parallel j_y\parallel }^2\\ =\int (\omega \cdot \mathrm{\nabla }\omega \cdot \omega -j\cdot \mathrm{\nabla }b\cdot \omega +{\epsilon }_{ijk}({\partial }_j{b}_l{\partial }_l{u}_k-{\partial }_j{u}_l{\partial }_l{b}_k)j_i)\mathrm{d}x=I+J+K+L.\end{array}$$

(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

$$\begin{array}{rcl}I& =& \int \omega \cdot \mathrm{\nabla }u\cdot \omega \mathrm{d}x=\int {\omega }_i{\partial }_i{u}_j{\omega }_j\mathrm{d}x\\ =& \int {\omega }_1{\partial }_x{u}_j{\omega }_j\mathrm{d}x+\int {\omega }_2{\partial }_y{u}_j{\omega }_j\mathrm{d}x+\int {\omega }_3{\partial }_z{u}_j{\omega }_j\mathrm{d}x=I_1+I_2+I_3.\end{array}$$

With the help of Lemma 2.1, we deduce that

$$\begin{array}{rcl}{I}_1& =& \int {\omega }_1{\partial }_x{u}_j{\omega }_j\mathrm{d}x\le C{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel u_x\parallel }^{\frac{1}{2}}{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel {\omega }_x\parallel }^{\frac{1}{2}}{\parallel {\omega }_y\parallel }^{\frac{1}{2}}{\parallel {\omega }_x\parallel }^{\frac{1}{2}}\\ \le & \frac{1}{24}{\parallel {\omega }_x\parallel }^2+\frac{1}{16}{\parallel {\omega }_y\parallel }^2+C{\parallel u_x\parallel }^2{\parallel \omega \parallel }^4.\end{array}$$

Similarly, we obtain that

$$I_2=\int {\omega }_2{\partial }_y{u}_j{\omega }_j\mathrm{d}x\le \frac{1}{24}{\parallel {\omega }_x\parallel }^2+\frac{1}{16}{\parallel {\omega }_y\parallel }^2+C{\parallel u_y\parallel }^2{\parallel \omega \parallel }^4,$$

and

$$\begin{array}{rl}{I}_3& =\int {\omega }_3{\partial }_z{u}_j{\omega }_j\mathrm{d}x=\int ({\partial }_x{u}_2-{\partial }_y{u}_1){\partial }_z{u}_j{\omega }_j\mathrm{d}x\\ \le \frac{1}{24}{\parallel {\omega }_x\parallel }^2+\frac{1}{16}{\parallel {\omega }_y\parallel }^2+C({\parallel u_x\parallel }^2+{\parallel u_y\parallel }^2){\parallel \omega \parallel }^4.\end{array}$$

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

$$\begin{array}{rcl}J& =& -\int j\cdot \mathrm{\nabla }b\cdot \omega \mathrm{d}x=-\int j_i{\partial }_i{b}_j{\omega }_j\mathrm{d}x\\ =& -\int j_1{\partial }_x{b}_j{\omega }_j\mathrm{d}x-\int j_2{\partial }_y{b}_j{\omega }_j\mathrm{d}x-\int j_3{\partial }_z{b}_j{\omega }_j\mathrm{d}x=J_1+J_2+J_3.\end{array}$$

Due to Lemma 2.1, we see that

$$\begin{array}{rl}{J}_1& =-\int j_1{\partial }_x{b}_j{\omega }_j\mathrm{d}x\le C{\parallel j\parallel }^{\frac{1}{2}}{\parallel b_x\parallel }^{\frac{1}{2}}{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel j_x\parallel }^{\frac{1}{2}}{\parallel {\partial }_x{b}_z\parallel }^{\frac{1}{2}}{\parallel {\omega }_y\parallel }^{\frac{1}{2}}\\ \le \frac{1}{22}{\parallel j_x\parallel }^2+\frac{1}{16}{\parallel {\omega }_y\parallel }^2+C{\parallel b_x\parallel }^2({\parallel \omega \parallel }^4+{\parallel j\parallel }^4).\end{array}$$

Similarly,

$$\begin{array}{c}\begin{array}{r}{J}_2=-\int j_2{\partial }_y{b}_j{\omega }_j\mathrm{d}x\le \frac{1}{16}{\parallel {\omega }_y\parallel }^2+\frac{1}{22}{\parallel j_x\parallel }^2+\frac{1}{26}{\parallel j_y\parallel }^2+C{\parallel b_y\parallel }^2({\parallel \omega \parallel }^4+{\parallel j\parallel }^4),\end{array}\hfill \\ \begin{array}{rl}{J}_3& =-\int j_3{\partial }_z{b}_j{\omega }_j\mathrm{d}x=-\int ({\partial }_x{b}_2-{\partial }_y{b}_1){\partial }_z{b}_j{\omega }_j\mathrm{d}x\\ \le \frac{1}{24}{\parallel {\omega }_x\parallel }^2+\frac{1}{22}{\parallel j_x\parallel }^2+\frac{1}{26}{\parallel j_y\parallel }^2+C({\parallel b_x\parallel }^2+{\parallel b_y\parallel }^2)({\parallel \omega \parallel }^4+{\parallel j\parallel }^4).\end{array}\hfill \end{array}$$

Now, let us turn to bound L,

$$\begin{array}{rl}L& =-\int {\epsilon }_{ijk}{\partial }_j{u}_l{\partial }_l{u}_k{j}_i\mathrm{d}x\\ =-\int {\epsilon }_{ijk}{\partial }_x{u}_l{\partial }_l{u}_k{j}_i\mathrm{d}x-\int {\epsilon }_{ijk}{\partial }_y{u}_l{\partial }_l{u}_k{j}_i\mathrm{d}x-\int {\epsilon }_{ijk}{\partial }_z{u}_l{\partial }_l{u}_k{j}_i\mathrm{d}x\\ =L_1+L_2+L_3.\end{array}$$

By Lemma 2.1, we have that

$$\begin{array}{rl}{L}_1& =-\int {\epsilon }_{ijk}{\partial }_x{u}_l{\partial }_l{u}_k{j}_i\mathrm{d}x\le C{\parallel u_x\parallel }^{\frac{1}{2}}{\parallel j\parallel }^{\frac{1}{2}}{\parallel j\parallel }^{\frac{1}{2}}{\parallel j_x\parallel }^{\frac{1}{2}}{\parallel j_y\parallel }^{\frac{1}{2}}{\parallel {\omega }_x\parallel }^{\frac{1}{2}}\\ \le \frac{1}{24}{\parallel {\omega }_x\parallel }^2+\frac{1}{22}{\parallel j_x\parallel }^2+\frac{1}{26}{\parallel j_y\parallel }^2+C{\parallel u_x\parallel }^2{\parallel j\parallel }^4.\end{array}$$

Similarly,

$$L_2=-\int {\epsilon }_{ijk}{\partial }_y{u}_l{\partial }_l{u}_k{j}_i\mathrm{d}x\le \frac{1}{16}{\parallel {\omega }_y\parallel }^2+\frac{1}{22}{\parallel j_x\parallel }^2+\frac{1}{26}{\parallel j_y\parallel }^2+C{\parallel u_y\parallel }^2{\parallel j\parallel }^4.$$

As for $L_3$, we should split it into three parts:

$$\begin{array}{c}\begin{array}{rl}{L}_3& =-\int {\epsilon }_{ijk}{\partial }_z{u}_l{\partial }_l{u}_k{j}_i\mathrm{d}x\\ =-\int {\epsilon }_{ijk}{\partial }_z{u}_1{\partial }_x{u}_k{j}_i\mathrm{d}x-\int {\epsilon }_{ijk}{\partial }_z{u}_2{\partial }_y{u}_k{j}_i\mathrm{d}x-\int {\epsilon }_{ijk}{\partial }_z{u}_3{\partial }_z{u}_k{j}_i\mathrm{d}x\\ =L_{31}+L_{32}+L_{33},\end{array}\hfill \\ \begin{array}{rl}{L}_{31}& =-\int {\epsilon }_{ijk}{\partial }_z{u}_1{\partial }_x{u}_k{j}_i\mathrm{d}x\\ \le C{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel b_x\parallel }^{\frac{1}{2}}{\parallel j\parallel }^{\frac{1}{2}}{\parallel {\omega }_x\parallel }^{\frac{1}{2}}{\parallel {\partial }_x{b}_z\parallel }^{\frac{1}{2}}{\parallel j_y\parallel }^{\frac{1}{2}}\\ \le \frac{1}{24}{\parallel {\omega }_x\parallel }^2+\frac{1}{22}{\parallel j_x\parallel }^2+\frac{1}{26}{\parallel j_y\parallel }^2+C{\parallel b_x\parallel }^2({\parallel \omega \parallel }^4+{\parallel j\parallel }^4).\end{array}\hfill \end{array}$$

Similarly,

$$L_{32}=-\int {\epsilon }_{ijk}{\partial }_z{u}_2{\partial }_y{u}_k{j}_i\mathrm{d}x\le \frac{1}{24}{\parallel {\omega }_x\parallel }^2+\frac{1}{26}{\parallel j_y\parallel }^2+C{\parallel b_y\parallel }^2({\parallel \omega \parallel }^4+{\parallel j\parallel }^4).$$

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

$$\begin{array}{rl}{L}_{33}& =-\int {\epsilon }_{ijk}{\partial }_z{u}_3{\partial }_z{u}_k{j}_i\mathrm{d}x\\ =\int {\epsilon }_{ijk}({\partial }_x{u}_1+{\partial }_y{u}_2){\partial }_z{u}_k{j}_i\mathrm{d}x=L_{33}^1+L_{33}^2.\end{array}$$

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

$$\begin{array}{rl}{L}_{33}^1& =\int {\epsilon }_{ijk}{\partial }_x{u}_1{\partial }_z{u}_k{j}_i\mathrm{d}x\le C{\parallel u_x\parallel }^{\frac{1}{2}}{\parallel j\parallel }^{\frac{1}{2}}{\parallel j\parallel }^{\frac{1}{2}}{\parallel {\partial }_x{u}_z\parallel }^{\frac{1}{2}}{\parallel j_x\parallel }^{\frac{1}{2}}{\parallel j_y\parallel }^{\frac{1}{2}}\\ \le \frac{1}{24}{\parallel {\omega }_x\parallel }^2+\frac{1}{22}{\parallel j_x\parallel }^2+\frac{1}{26}{\parallel j_y\parallel }^2+C{\parallel u_x\parallel }^2{\parallel j\parallel }^4,\end{array}$$

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:

$$\begin{array}{rl}K& =\int {\epsilon }_{ijk}{\partial }_j{b}_l{\partial }_l{u}_k{j}_i\mathrm{d}x\\ =\int {\epsilon }_{i1k}{\partial }_x{b}_l{\partial }_l{u}_k{j}_i\mathrm{d}x+\int {\epsilon }_{i2k}{\partial }_y{b}_l{\partial }_l{u}_k{j}_i\mathrm{d}x+\int {\epsilon }_{i3k}{\partial }_z{b}_l{\partial }_l{u}_k{j}_i\mathrm{d}x\\ =K_1+K_2+K_3.\end{array}$$

Similarly, we deduce that

$$\begin{array}{rl}{K}_1& =\int {\epsilon }_{i1k}{\partial }_x{b}_l{\partial }_l{u}_k{j}_i\mathrm{d}x\le C{\parallel b_x\parallel }^{\frac{1}{2}}{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel j\parallel }^{\frac{1}{2}}{\parallel {\partial }_x{b}_z\parallel }^{\frac{1}{2}}{\parallel {\omega }_x\parallel }^{\frac{1}{2}}{\parallel j_y\parallel }^{\frac{1}{2}}\\ \le \frac{1}{24}{\parallel {\omega }_x\parallel }^2+\frac{1}{22}{\parallel j_x\parallel }^2+\frac{1}{26}{\parallel j_y\parallel }^2+C{\parallel b_x\parallel }^2({\parallel \omega \parallel }^4+{\parallel j\parallel }^4),\end{array}$$

and

$$K_2=\int {\epsilon }_{i2k}{\partial }_y{b}_l{\partial }_l{u}_k{j}_i\mathrm{d}x\le \frac{1}{24}{\parallel {\omega }_x\parallel }^2+\frac{1}{26}{\parallel j_y\parallel }^2+C{\parallel b_y\parallel }^2({\parallel \omega \parallel }^4+{\parallel j\parallel }^4).$$

For $K_3$, we have

$$\begin{array}{rl}{K}_3& =\int {\epsilon }_{i3k}{\partial }_z{b}_l{\partial }_l{u}_k{j}_i\mathrm{d}x\\ =\int {\epsilon }_{i3k}{\partial }_z{b}_1{\partial }_x{u}_k{j}_i\mathrm{d}x+\int {\epsilon }_{i3k}{\partial }_z{b}_2{\partial }_y{u}_k{j}_i\mathrm{d}x+\int {\epsilon }_{i3k}{\partial }_z{b}_3{\partial }_z{u}_k{j}_i\mathrm{d}x\\ =K_{31}+K_{32}+K_{33}.\end{array}$$

Thus,

$$\begin{array}{rl}{K}_{31}& =\int {\epsilon }_{i3k}{\partial }_z{b}_1{\partial }_x{u}_k{j}_i\mathrm{d}x\le C{\parallel j\parallel }^{\frac{1}{2}}{\parallel u_x\parallel }^{\frac{1}{2}}{\parallel j\parallel }^{\frac{1}{2}}{\parallel {\omega }_x\parallel }^{\frac{1}{2}}{\parallel j_x\parallel }^{\frac{1}{2}}{\parallel j_y\parallel }^{\frac{1}{2}}\\ \le \frac{1}{24}{\parallel {\omega }_x\parallel }^2+\frac{1}{22}{\parallel j_x\parallel }^2+\frac{1}{26}{\parallel j_y\parallel }^2+C{\parallel u_x\parallel }^2{\parallel j\parallel }^4,\end{array}$$

and

$$K_{32}=\int {\epsilon }_{i3k}{\partial }_z{b}_2{\partial }_y{u}_k{j}_i\mathrm{d}x\le \frac{1}{16}{\parallel {\omega }_y\parallel }^2+\frac{1}{22}{\parallel j_x\parallel }^2+\frac{1}{26}{\parallel j_y\parallel }^2+C{\parallel u_y\parallel }^2{\parallel j\parallel }^4.$$

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

$$\begin{array}{rl}{K}_{33}& =\int {\epsilon }_{i3k}{\partial }_z{b}_3{\partial }_z{u}_k{j}_i\mathrm{d}x=-\int {\epsilon }_{i3k}({\partial }_x{b}_1+{\partial }_y{b}_2){\partial }_z{u}_k{j}_i\mathrm{d}x\\ \le \frac{1}{24}{\parallel {\omega }_x\parallel }^2+\frac{1}{26}{\parallel j_y\parallel }^2+C({\parallel b_x\parallel }^2+{\parallel b_y\parallel }^2)({\parallel \omega \parallel }^4+{\parallel j\parallel }^4).\end{array}$$

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

$$\begin{array}{r}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}({\parallel \omega \parallel }^2+{\parallel j\parallel }^2)+\frac{1}{2}({\parallel {\omega }_x\parallel }^2+{\parallel {\omega }_y\parallel }^2+{\parallel j_x\parallel }^2+{\parallel j_y\parallel }^2)\\ \le C({\parallel u_x\parallel }^2+{\parallel u_y\parallel }^2+{\parallel b_x\parallel }^2+{\parallel b_y\parallel }^2)({\parallel \omega \parallel }^4+{\parallel j\parallel }^4).\end{array}$$

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

$${\parallel \omega \parallel }^2+{\parallel j\parallel }^2+{\int }_0^t({\parallel {\omega }_x\parallel }^2+{\parallel {\omega }_y\parallel }^2+{\parallel j_x\parallel }^2+{\parallel j_y\parallel }^2)\mathrm{d}s\le 1.$$

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 $(u,b)$ is the solution of (1)-(4), then

$${\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2+{\int }_0^t({\parallel \mathrm{\nabla }{\omega }_x\parallel }^2+{\parallel \mathrm{\nabla }{\omega }_y\parallel }^2+{\parallel \mathrm{\nabla }{j}_x\parallel }^2+{\parallel \mathrm{\nabla }{j}_y\parallel }^2)\mathrm{d}s\le C.$$

(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

$$\begin{array}{r}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2)+({\parallel \mathrm{\nabla }{\omega }_x\parallel }^2+{\parallel \mathrm{\nabla }{\omega }_y\parallel }^2+{\parallel \mathrm{\nabla }{j}_x\parallel }^2+{\parallel \mathrm{\nabla }{j}_y\parallel }^2)\\ =\int [u\cdot \mathrm{\nabla }\omega \cdot \mathrm{\Delta }\omega +u\cdot \mathrm{\nabla }j\cdot \mathrm{\Delta }j-\omega \cdot \mathrm{\nabla }\omega \cdot \mathrm{\Delta }\omega +j\cdot \mathrm{\nabla }b\cdot \mathrm{\Delta }\omega -b\cdot \mathrm{\nabla }j\cdot \mathrm{\Delta }\omega \\ -b\cdot \mathrm{\nabla }\omega \cdot \mathrm{\Delta }j-{\epsilon }_{ijk}({\partial }_j{b}_l{\partial }_l{u}_k-{\partial }_j{u}_l{\partial }_l{b}_k)\mathrm{\Delta }{j}_i]\mathrm{d}x\\ =M+N+P+Q+R+S.\end{array}$$

(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

$$\begin{array}{rl}M& =\int u\cdot \mathrm{\nabla }\omega \cdot \mathrm{\Delta }\omega \mathrm{d}x=\int u_i{\partial }_i{\omega }_j{\partial }_{kk}^2{\omega }_j\mathrm{d}x=-\int {\partial }_k{u}_i{\partial }_i{\omega }_j{\partial }_k{\omega }_j\mathrm{d}x\\ =-\int {\partial }_x{u}_i{\partial }_i{\omega }_j{\partial }_x{\omega }_j\mathrm{d}x-\int {\partial }_y{u}_i{\partial }_i{\omega }_j{\partial }_y{\omega }_j\mathrm{d}x-\int {\partial }_z{u}_i{\partial }_i{\omega }_j{\partial }_z{\omega }_j\mathrm{d}x\\ =M_1+M_2+M_3.\end{array}$$

We estimate each term as follows:

$$\begin{array}{rl}{M}_1& =-\int {\partial }_x{u}_i{\partial }_i{\omega }_j{\partial }_x{\omega }_j\mathrm{d}x\le C{\parallel u_x\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_x\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_y\parallel }^{\frac{1}{2}}{\parallel {\partial }_x{u}_z\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_x\parallel }^2+\frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_y\parallel }^2+C\parallel u_x\parallel \parallel {\omega }_x\parallel {\parallel \mathrm{\nabla }\omega \parallel }^2\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_x\parallel }^2+\frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_y\parallel }^2+C({\parallel u_x\parallel }^2+{\parallel {\omega }_x\parallel }^2){\parallel \mathrm{\nabla }\omega \parallel }^2.\end{array}$$

Similarly,

$$\begin{array}{rl}{M}_2& =-\int {\partial }_y{u}_i{\partial }_i{\omega }_j{\partial }_y{\omega }_j\mathrm{d}x\\ \le C{\parallel u_y\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_x\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_y\parallel }^{\frac{1}{2}}{\parallel {\partial }_y{u}_z\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_x\parallel }^2+\frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_y\parallel }^2+C({\parallel u_y\parallel }^2+{\parallel {\omega }_y\parallel }^2){\parallel \mathrm{\nabla }\omega \parallel }^2.\end{array}$$

Now, we turn to bound $M_3$,

$$\begin{array}{rl}{M}_3& =-\int {\partial }_z{u}_i{\partial }_i{\omega }_j{\partial }_z{\omega }_j\mathrm{d}x\\ =-\int {\partial }_z{u}_1{\partial }_x{\omega }_j{\partial }_z{\omega }_j\mathrm{d}x-\int {\partial }_z{u}_2{\partial }_y{\omega }_j{\partial }_z{\omega }_j\mathrm{d}x-\int {\partial }_z{u}_3{\partial }_z{\omega }_j{\partial }_z{\omega }_j\mathrm{d}x\\ =M_{31}+M_{32}+M_{33}.\end{array}$$

Similarly, we can deduce that

$$\begin{array}{c}\begin{array}{rl}{M}_{31}& =-\int {\partial }_z{u}_1{\partial }_x{\omega }_j{\partial }_z{\omega }_j\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({\parallel \omega \parallel }^2+{\parallel {\omega }_x\parallel }^2){\parallel \mathrm{\nabla }\omega \parallel }^2,\end{array}\hfill \\ \begin{array}{rl}{M}_{32}& =-\int {\partial }_z{u}_2{\partial }_y{\omega }_j{\partial }_z{\omega }_j\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({\parallel \omega \parallel }^2+{\parallel {\omega }_x\parallel }^2){\parallel \mathrm{\nabla }\omega \parallel }^2,\end{array}\hfill \\ \begin{array}{rl}{M}_{33}=& -\int {\partial }_z{u}_3{\partial }_z{\omega }_j{\partial }_z{\omega }_j\mathrm{d}x=\int ({\partial }_x{u}_1+{\partial }_y{u}_2){\partial }_z{\omega }_j{\partial }_z{\omega }_j\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({\parallel u_x\parallel }^2+{\parallel u_y\parallel }^2+{\parallel {\omega }_x\parallel }^2+{\parallel {\omega }_y\parallel }^2){\parallel \mathrm{\nabla }\omega \parallel }^2.\end{array}\hfill \end{array}$$

As for N, integrating by parts, we deduce that

$$\begin{array}{rl}N& =\int u\cdot \mathrm{\nabla }j\cdot \mathrm{\Delta }j\mathrm{d}x=\int u_i{\partial }_i{j}_j{\partial }_{kk}^2{j}_j\mathrm{d}x=-\int {\partial }_k{u}_i{\partial }_i{j}_j{\partial }_k{j}_j\mathrm{d}x\\ =-\int {\partial }_x{u}_i{\partial }_i{j}_j{\partial }_x{j}_j\mathrm{d}x-\int {\partial }_y{u}_i{\partial }_i{j}_j{\partial }_y{j}_j\mathrm{d}x-\int {\partial }_z{u}_i{\partial }_i{j}_j{\partial }_z{j}_j\mathrm{d}x\\ =N_1+N_2+N_3.\end{array}$$

Similarly, we have

$$\begin{array}{c}\begin{array}{rl}{N}_1& =-\int {\partial }_x{u}_i{\partial }_i{j}_j{\partial }_x{j}_j\mathrm{d}x\\ \le C{\parallel u_x\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel j_x\parallel }^{\frac{1}{2}}{\parallel {\omega }_x\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_x\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_y\parallel }^{\frac{1}{2}}\\ \le \frac{1}{50}{\parallel \mathrm{\nabla }{j}_x\parallel }^2+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_y\parallel }^2+C({\parallel u_x\parallel }^2+{\parallel {\omega }_x\parallel }^2){\parallel \mathrm{\nabla }j\parallel }^2,\end{array}\hfill \\ \begin{array}{rl}{N}_2& =-\int {\partial }_y{u}_i{\partial }_i{j}_j{\partial }_y{j}_j\mathrm{d}x\\ \le C{\parallel u_y\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel j_y\parallel }^{\frac{1}{2}}{\parallel {\omega }_y\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_x\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_y\parallel }^{\frac{1}{2}}\\ \le \frac{1}{50}{\parallel \mathrm{\nabla }{j}_x\parallel }^2+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_y\parallel }^2+C({\parallel u_y\parallel }^2+{\parallel {\omega }_y\parallel }^2){\parallel \mathrm{\nabla }j\parallel }^2.\end{array}\hfill \end{array}$$

As for $N_3$, we obtain

$$\begin{array}{rl}{N}_3& =-\int {\partial }_z{u}_i{\partial }_i{j}_j{\partial }_z{j}_j\mathrm{d}x\\ =-\int {\partial }_z{u}_1{\partial }_x{j}_j{\partial }_z{j}_j\mathrm{d}x-\int {\partial }_z{u}_2{\partial }_y{j}_j{\partial }_z{j}_j\mathrm{d}x-\int {\partial }_z{u}_3{\partial }_z{j}_j{\partial }_z{j}_j\mathrm{d}x\\ =N_{31}+N_{32}+N_{33}.\end{array}$$

Thus, we have

$$\begin{array}{c}\begin{array}{rl}{N}_{31}& =-\int {\partial }_z{u}_1{\partial }_x{j}_j{\partial }_z{j}_j\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({\parallel \omega \parallel }^2+{\parallel {\omega }_x\parallel }^2){\parallel \mathrm{\nabla }j\parallel }^2,\end{array}\hfill \\ \begin{array}{rl}{N}_{32}& =-\int {\partial }_z{u}_2{\partial }_y{j}_j{\partial }_z{j}_j\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({\parallel \omega \parallel }^2+{\parallel {\omega }_x\parallel }^2){\parallel \mathrm{\nabla }j\parallel }^2,\end{array}\hfill \\ \begin{array}{rl}{N}_{33}=& -\int {\partial }_z{u}_3{\partial }_z{j}_j{\partial }_z{j}_j\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({\parallel u_x\parallel }^2+{\parallel u_y\parallel }^2+{\parallel {\omega }_x\parallel }^2+{\parallel {\omega }_y\parallel }^2){\parallel \mathrm{\nabla }j\parallel }^2.\end{array}\hfill \end{array}$$

We now turn to bound P,

$$\begin{array}{rl}P& =-\int \omega \cdot \mathrm{\nabla }\omega \cdot \mathrm{\Delta }\omega \mathrm{d}x=-\int {\omega }_i{\partial }_i{u}_j{\partial }_{kk}^2{\omega }_j\mathrm{d}x\\ =\int {\partial }_k{\omega }_i{\partial }_i{u}_j{\partial }_k{\omega }_j\mathrm{d}x+\int {\omega }_i{\partial }_k{\partial }_i{u}_j{\partial }_k{\omega }_j\mathrm{d}x=P_1+P_2.\end{array}$$

For $P_1$, we have

$$\begin{array}{c}\begin{array}{rl}{P}_1& =\int {\partial }_k{\omega }_i{\partial }_i{u}_j{\partial }_k{\omega }_j\mathrm{d}x\\ =\int {\partial }_x{\omega }_i{\partial }_i{u}_j{\partial }_x{\omega }_j\mathrm{d}x+\int {\partial }_y{\omega }_i{\partial }_i{u}_j{\partial }_y{\omega }_j\mathrm{d}x+\int {\partial }_z{\omega }_i{\partial }_i{u}_j{\partial }_z{\omega }_j\mathrm{d}x\\ =P_{11}+P_{12}+P_{13},\end{array}\hfill \\ \begin{array}{rl}{P}_{11}& =\int {\partial }_x{\omega }_i{\partial }_i{u}_j{\partial }_x{\omega }_j\mathrm{d}x\\ \le C{\parallel {\omega }_x\parallel }^{\frac{1}{2}}{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel {\omega }_x\parallel }^{\frac{1}{2}}{\parallel {\omega }_x\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_x\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_x\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_x\parallel }^2+C({\parallel \omega \parallel }^2+{\parallel {\omega }_x\parallel }^2){\parallel \mathrm{\nabla }\omega \parallel }^2,\end{array}\hfill \\ \begin{array}{rl}{P}_{12}& =\int {\partial }_y{\omega }_i{\partial }_i{u}_j{\partial }_y{\omega }_j\mathrm{d}x\\ \le C{\parallel {\omega }_y\parallel }^{\frac{1}{2}}{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel {\omega }_y\parallel }^{\frac{1}{2}}{\parallel {\omega }_x\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_y\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_y\parallel }^{\frac{1}{2}}\\ \le \frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_y\parallel }^2+C({\parallel \omega \parallel }^2+{\parallel {\omega }_x\parallel }^2){\parallel \mathrm{\nabla }\omega \parallel }^2.\end{array}\hfill \end{array}$$

For $P_{13}$, we see that

$$\begin{array}{rl}{P}_{13}& =\int {\partial }_z{\omega }_i{\partial }_i{u}_j{\partial }_z{\omega }_j\mathrm{d}x\\ =\int {\partial }_z{\omega }_1{\partial }_x{u}_j{\partial }_z{\omega }_j\mathrm{d}x+\int {\partial }_z{\omega }_2{\partial }_y{u}_j{\partial }_z{\omega }_j\mathrm{d}x+\int {\partial }_z{\omega }_3{\partial }_z{u}_j{\partial }_z{\omega }_j\mathrm{d}x\\ =P_{13}^1+P_{13}^2+P_{13}^3.\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}{P}_{13}^1& =\int {\partial }_z{\omega }_1{\partial }_x{u}_j{\partial }_z{\omega }_j\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({\parallel u_x\parallel }^2+{\parallel {\omega }_x\parallel }^2){\parallel \mathrm{\nabla }\omega \parallel }^2,\end{array}\hfill \\ \begin{array}{rl}{P}_{13}^2& =\int {\partial }_z{\omega }_2{\partial }_y{u}_j{\partial }_z{\omega }_j\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({\parallel u_y\parallel }^2+{\parallel {\omega }_y\parallel }^2){\parallel \mathrm{\nabla }\omega \parallel }^2,\end{array}\hfill \\ \begin{array}{rl}{P}_{13}^2=& \int {\partial }_z{\omega }_3{\partial }_z{u}_j{\partial }_z{\omega }_j\mathrm{d}x\\ =& \int {\partial }_z({\partial }_x{u}_2-{\partial }_y{u}_1){\partial }_z{u}_j{\partial }_z{\omega }_j\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({\parallel \omega \parallel }^2+{\parallel {\omega }_x\parallel }^2+{\parallel {\omega }_y\parallel }^2){\parallel \mathrm{\nabla }\omega \parallel }^2.\end{array}\hfill \end{array}$$

Now, we turn to $P_2$,

$$\begin{array}{rl}{P}_2& =\int {\omega }_i{\partial }_k{\partial }_i{u}_j{\partial }_k{\omega }_j\mathrm{d}x\\ =\int {\omega }_1{\partial }_k{\partial }_x{u}_j{\partial }_k{\omega }_j\mathrm{d}x+\int {\omega }_2{\partial }_k{\partial }_y{u}_j{\partial }_k{\omega }_j\mathrm{d}x+\int {\omega }_3{\partial }_k{\partial }_z{u}_j{\partial }_k{\omega }_j\mathrm{d}x\\ =P_{21}+P_{22}+P_{23}.\end{array}$$

Similarly,

$$\begin{array}{c}\begin{array}{rl}{P}_{21}& =\int {\omega }_1{\partial }_k{\partial }_x{u}_j{\partial }_k{\omega }_j\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({\parallel \omega \parallel }^2+{\parallel {\omega }_x\parallel }^2){\parallel \mathrm{\nabla }\omega \parallel }^2,\end{array}\hfill \\ \begin{array}{rl}{P}_{22}& =\int {\omega }_2{\partial }_k{\partial }_y{u}_j{\partial }_k{\omega }_j\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({\parallel \omega \parallel }^2+{\parallel {\omega }_x\parallel }^2){\parallel \mathrm{\nabla }\omega \parallel }^2,\end{array}\hfill \\ \begin{array}{rl}{P}_{23}=& \int {\omega }_3{\partial }_k{\partial }_z{u}_j{\partial }_k{\omega }_j\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({\parallel u_x\parallel }^2+{\parallel u_y\parallel }^2+{\parallel {\omega }_x\parallel }^2+{\parallel {\omega }_y\parallel }^2){\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 \mathrm{d}x=\int j_i{\partial }_i{b}_j{\partial }_{kk}^2{\omega }_j\mathrm{d}x\\ =-\int {\partial }_k{j}_i{\partial }_i{b}_j{\partial }_k{\omega }_j\mathrm{d}x-\int j_i{\partial }_k{\partial }_i{b}_j{\partial }_k{\omega }_j\mathrm{d}x=Q_1+Q_2,\end{array}\hfill \\ \begin{array}{rl}{Q}_1& =-\int {\partial }_x{j}_i{\partial }_i{b}_j{\partial }_x{\omega }_j\mathrm{d}x-\int {\partial }_y{j}_i{\partial }_i{b}_j{\partial }_y{\omega }_j\mathrm{d}x-\int {\partial }_z{j}_i{\partial }_i{b}_j{\partial }_z{\omega }_j\mathrm{d}x\\ =Q_{11}+Q_{12}+Q_{13},\end{array}\hfill \\ \begin{array}{rl}{Q}_{11}& =-\int {\partial }_x{j}_i{\partial }_i{b}_j{\partial }_x{\omega }_j\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({\parallel j\parallel }^2+{\parallel j_x\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2),\end{array}\hfill \\ \begin{array}{rl}{Q}_{12}& =-\int {\partial }_y{j}_i{\partial }_i{b}_j{\partial }_y{\omega }_j\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({\parallel j\parallel }^2+{\parallel j_y\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2).\end{array}\hfill \end{array}$$

As for $Q_{13}$, we see that

$$\begin{array}{rl}{Q}_{13}& =-\int {\partial }_z{j}_i{\partial }_i{b}_j{\partial }_z{\omega }_j\mathrm{d}x\\ =-\int {\partial }_z{j}_1{\partial }_x{b}_j{\partial }_z{\omega }_j\mathrm{d}x-\int {\partial }_z{j}_2{\partial }_y{b}_j{\partial }_z{\omega }_j\mathrm{d}x-\int {\partial }_z{j}_3{\partial }_z{b}_j{\partial }_z{\omega }_j\mathrm{d}x\\ =Q_{13}^1+Q_{13}^2+Q_{13}^3.\end{array}$$

Thus, we can deduce that

$$\begin{array}{c}\begin{array}{rl}{Q}_{13}^1& =-\int {\partial }_z{j}_1{\partial }_x{b}_j{\partial }_z{\omega }_j\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({\parallel b_x\parallel }^2+{\parallel j_x\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2),\end{array}\hfill \\ \begin{array}{rl}{Q}_{13}^2& =-\int {\partial }_z{j}_2{\partial }_y{b}_j{\partial }_z{\omega }_j\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({\parallel b_y\parallel }^2+{\parallel j_y\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2),\end{array}\hfill \\ \begin{array}{rl}{Q}_{13}^3=& -\int {\partial }_z{j}_3{\partial }_z{b}_j{\partial }_z{\omega }_j\mathrm{d}x=-\int {\partial }_z({\partial }_x{b}_2-{\partial }_y{b}_1){\partial }_z{b}_j{\partial }_z{\omega }_j\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({\parallel j\parallel }^2+{\parallel j_x\parallel }^2+{\parallel j_y\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2).\end{array}\hfill \end{array}$$

Now, we turn to $Q_2$,

$$\begin{array}{rl}{Q}_2& =-\int b\cdot \mathrm{\nabla }j\cdot \mathrm{\Delta }\omega \mathrm{d}x=-\int j_i{\partial }_k{\partial }_i{b}_j{\partial }_k{\omega }_j\mathrm{d}x\\ =-\int j_1{\partial }_k{\partial }_x{b}_j{\partial }_k{\omega }_j\mathrm{d}x-\int j_2{\partial }_k{\partial }_y{b}_j{\partial }_k{\omega }_j\mathrm{d}x-\int j_3{\partial }_k{\partial }_z{b}_j{\partial }_k{\omega }_j\mathrm{d}x\\ =Q_{21}+Q_{22}+Q_{23}.\end{array}$$

We deduce that

$$\begin{array}{c}\begin{array}{rl}{Q}_{21}& =-\int j_1{\partial }_k{\partial }_x{b}_j{\partial }_k{\omega }_j\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({\parallel j\parallel }^2+{\parallel j_x\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2),\end{array}\hfill \\ \begin{array}{rl}{Q}_{22}& =-\int j_2{\partial }_k{\partial }_y{b}_j{\partial }_k{\omega }_j\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({\parallel j\parallel }^2+{\parallel j_y\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2),\end{array}\hfill \\ \begin{array}{rl}{Q}_{23}=& -\int j_3{\partial }_k{\partial }_z{b}_j{\partial }_k{\omega }_j\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({\parallel b_x\parallel }^2+{\parallel j_x\parallel }^2+{\parallel b_y\parallel }^2+{\parallel j_y\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2).\end{array}\hfill \end{array}$$

Here we start to estimate R as follows:

$$\begin{array}{rl}R& =-\int b\cdot \mathrm{\nabla }j\cdot \mathrm{\Delta }\omega \mathrm{d}x-\int b\cdot \mathrm{\nabla }\omega \cdot \mathrm{\Delta }j\mathrm{d}x\\ =-\int b_i{\partial }_i{j}_j{\partial }_{kk}^2{\omega }_j\mathrm{d}x-\int b_i{\partial }_i{\omega }_j{\partial }_{kk}^2{j}_j\mathrm{d}x\\ =\int {\partial }_k{b}_i{\partial }_i{j}_j{\partial }_k{\omega }_j\mathrm{d}x+\int {\partial }_k{b}_i{\partial }_i{\omega }_j{\partial }_k{j}_j\mathrm{d}x=R_1+R_2.\end{array}$$

For $R_1$, we have

$$\begin{array}{rl}{R}_1& =\int {\partial }_k{b}_i{\partial }_i{j}_j{\partial }_k{\omega }_j\mathrm{d}x\\ =\int {\partial }_x{b}_i{\partial }_i{j}_j{\partial }_x{\omega }_j\mathrm{d}x+\int {\partial }_y{b}_i{\partial }_i{j}_j{\partial }_y{\omega }_j\mathrm{d}x+\int {\partial }_z{b}_i{\partial }_i{j}_j{\partial }_z{\omega }_j\mathrm{d}x\\ =R_{11}+R_{12}+R_{13}.\end{array}$$

We deduce that

$$\begin{array}{c}\begin{array}{rl}{R}_{11}& =\int {\partial }_x{b}_i{\partial }_i{j}_j{\partial }_x{\omega }_j\mathrm{d}x\\ \le C{\parallel b_x\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel {\omega }_x\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_x\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_x\parallel }^{\frac{1}{2}}{\parallel j_x\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_x\parallel }^2+\frac{1}{50}{\parallel \mathrm{\nabla }{j}_x\parallel }^2+C({\parallel b_x\parallel }^2+{\parallel {\omega }_x\parallel }^2){\parallel \mathrm{\nabla }j\parallel }^2,\end{array}\hfill \\ \begin{array}{rl}{R}_{12}& =\int {\partial }_y{b}_i{\partial }_i{j}_j{\partial }_y{\omega }_j\mathrm{d}x\\ \le C{\parallel b_y\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel {\omega }_y\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_x\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_y\parallel }^{\frac{1}{2}}{\parallel j_y\parallel }^{\frac{1}{2}}\\ \le \frac{1}{50}{\parallel \mathrm{\nabla }{j}_x\parallel }^2+\frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_y\parallel }^2+C({\parallel b_y\parallel }^2+{\parallel {\omega }_y\parallel }^2){\parallel \mathrm{\nabla }j\parallel }^2.\end{array}\hfill \end{array}$$

For $R_{13}$, we have

$$\begin{array}{rl}{R}_{13}& =\int {\partial }_z{b}_i{\partial }_i{j}_j{\partial }_z{\omega }_j\mathrm{d}x\\ =\int {\partial }_z{b}_1{\partial }_x{j}_j{\partial }_z{\omega }_j\mathrm{d}x+\int {\partial }_z{b}_2{\partial }_y{j}_j{\partial }_z{\omega }_j\mathrm{d}x+\int {\partial }_z{b}_3{\partial }_z{j}_j{\partial }_z{\omega }_j\mathrm{d}x=R_{13}^1+R_{13}^2+R_{13}^3.\end{array}$$

Similarly,

$$\begin{array}{c}\begin{array}{rl}{R}_{13}^1& =\int {\partial }_z{b}_1{\partial }_x{j}_j{\partial }_z{\omega }_j\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({\parallel j\parallel }^2+{\parallel j_x\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2),\end{array}\hfill \\ \begin{array}{rl}{R}_{13}^2& =\int {\partial }_z{b}_2{\partial }_y{j}_j{\partial }_z{\omega }_j\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({\parallel j\parallel }^2+{\parallel j_y\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2),\end{array}\hfill \\ \begin{array}{rl}{R}_{13}^3=& \int {\partial }_z{b}_3{\partial }_z{j}_j{\partial }_z{\omega }_j\mathrm{d}x\\ =& -\int ({\partial }_x{b}_1+{\partial }_y{b}_2){\partial }_z{j}_j{\partial }_z{\omega }_j\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({\parallel j\parallel }^2+{\parallel j_y\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2).\end{array}\hfill \end{array}$$

Now, we turn to $R_2$,

$$\begin{array}{rl}{R}_2& =\int {\partial }_k{b}_i{\partial }_i{\omega }_j{\partial }_k{j}_j\mathrm{d}x\\ =\int {\partial }_x{b}_i{\partial }_i{\omega }_j{\partial }_x{j}_j\mathrm{d}x+\int {\partial }_y{b}_i{\partial }_i{\omega }_j{\partial }_y{j}_j\mathrm{d}x+\int {\partial }_z{b}_i{\partial }_i{\omega }_j{\partial }_z{j}_j\mathrm{d}x=R_{21}+R_{22}+R_{23}.\end{array}$$

Thus, similarly, we deduce that

$$\begin{array}{c}\begin{array}{rl}{R}_{21}& =\int {\partial }_x{b}_i{\partial }_i{\omega }_j{\partial }_x{j}_j\mathrm{d}x\le C{\parallel b_x\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel j_x\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_x\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_x\parallel }^{\frac{1}{2}}{\parallel j_x\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_x\parallel }^2+\frac{1}{50}{\parallel \mathrm{\nabla }{j}_x\parallel }^2+C({\parallel b_x\parallel }^2+{\parallel j_x\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2),\end{array}\hfill \\ \begin{array}{rl}{R}_{22}& =\int {\partial }_y{b}_i{\partial }_i{\omega }_j{\partial }_y{j}_j\mathrm{d}x\le C{\parallel b_y\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \parallel }^{\frac{1}{2}}{\parallel j_y\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{\omega }_x\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_y\parallel }^{\frac{1}{2}}{\parallel j_y\parallel }^{\frac{1}{2}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_x\parallel }^2+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_y\parallel }^2+C({\parallel b_y\parallel }^2+{\parallel j_y\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2).\end{array}\hfill \end{array}$$

For $R_{23}$, we have

$$\begin{array}{rl}{R}_{23}& =\int {\partial }_z{b}_i{\partial }_i{\omega }_j{\partial }_z{j}_j\mathrm{d}x\\ =\int {\partial }_z{b}_1{\partial }_x{\omega }_j{\partial }_z{j}_j\mathrm{d}x+\int {\partial }_z{b}_2{\partial }_y{\omega }_j{\partial }_z{j}_j\mathrm{d}x+\int {\partial }_z{b}_3{\partial }_z{\omega }_j{\partial }_z{j}_j\mathrm{d}x\\ =R_{23}^1+R_{23}^2+R_{23}^3.\end{array}$$

Similarly,

$$\begin{array}{c}\begin{array}{rl}{R}_{23}^1& =\int {\partial }_z{b}_1{\partial }_x{\omega }_j{\partial }_z{j}_j\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({\parallel j\parallel }^2+{\parallel {\omega }_x\parallel }^2){\parallel \mathrm{\nabla }j\parallel }^2,\end{array}\hfill \\ \begin{array}{rl}{R}_{23}^2& =\int {\partial }_z{b}_2{\partial }_y{\omega }_j{\partial }_z{j}_j\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({\parallel j\parallel }^2+{\parallel j_y\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2),\end{array}\hfill \\ \begin{array}{rl}{R}_{23}^3=& \int {\partial }_z{b}_3{\partial }_z{\omega }_j{\partial }_z{j}_j\mathrm{d}x\\ =& -\int ({\partial }_x{b}_1+{\partial }_y{b}_2){\partial }_z{\omega }_j{\partial }_z{j}_j\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({\parallel b_x\parallel }^2+{\parallel b_y\parallel }^2+{\parallel j_x\parallel }^2+{\parallel j_y\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2).\end{array}\hfill \end{array}$$

Finally, for the last term S, we have

$$\begin{array}{rl}S& =-\int {\epsilon }_{ijk}({\partial }_j{b}_l{\partial }_l{u}_k-{\partial }_j{u}_l{\partial }_l{b}_k)\mathrm{\Delta }{j}_i\mathrm{d}x\\ =\int {\epsilon }_{ijk}{\partial }_m({\partial }_j{b}_l{\partial }_l{u}_k-{\partial }_j{u}_l{\partial }_l{b}_k){\partial }_m{j}_i\mathrm{d}x\\ =S_1+S_2.\end{array}$$

We first consider $S_1$,

$$\begin{array}{rl}{S}_1& =\int {\epsilon }_{ijk}{\partial }_m({\partial }_j{b}_l{\partial }_l{u}_k){\partial }_m{j}_i\mathrm{d}x\\ =\int {\epsilon }_{ijk}{\partial }_x({\partial }_j{b}_l{\partial }_l{u}_k){\partial }_x{j}_i\mathrm{d}x+\int {\epsilon }_{ijk}{\partial }_y({\partial }_j{b}_l{\partial }_l{u}_k){\partial }_y{j}_i\mathrm{d}x+\int {\epsilon }_{ijk}{\partial }_z({\partial }_j{b}_l{\partial }_l{u}_k){\partial }_z{j}_i\mathrm{d}x\\ =S_{11}+S_{12}+S_{13}.\end{array}$$

We deduce that

$$\begin{array}{c}\begin{array}{rl}{S}_{11}=& \int {\epsilon }_{ijk}{\partial }_x({\partial }_j{b}_l{\partial }_l{u}_k){\partial }_x{j}_i\mathrm{d}x=\int {\epsilon }_{ijk}{\partial }_x{\partial }_j{b}_l{\partial }_l{u}_k{\partial }_x{j}_i\mathrm{d}x+\int {\epsilon }_{ijk}{\partial }_j{b}_l{\partial }_x{\partial }_l{u}_k{\partial }_x{j}_i\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({\parallel j_x\parallel }^2+{\parallel \omega \parallel }^2+{\parallel j\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2),\end{array}\hfill \\ \begin{array}{rl}{S}_{12}=& \int {\epsilon }_{ijk}{\partial }_y({\partial }_j{b}_l{\partial }_l{u}_k){\partial }_y{j}_i\mathrm{d}x=\int {\epsilon }_{ijk}{\partial }_y{\partial }_j{b}_l{\partial }_l{u}_k{\partial }_y{j}_i\mathrm{d}x+\int {\epsilon }_{ijk}{\partial }_j{b}_l{\partial }_y{\partial }_l{u}_k{\partial }_y{j}_i\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({\parallel \omega \parallel }^2+{\parallel j_y\parallel }^2+{\parallel j\parallel }^2+{\parallel j_x\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2).\end{array}\hfill \end{array}$$

As for $S_{13}$, we see that

$$\begin{array}{rl}{S}_{13}=& \int {\epsilon }_{ijk}{\partial }_z({\partial }_j{b}_l{\partial }_l{u}_k){\partial }_z{j}_i\mathrm{d}x\\ =& \int {\epsilon }_{ijk}{\partial }_z{\partial }_j{b}_l{\partial }_l{u}_k{\partial }_z{j}_i\mathrm{d}x+\int {\epsilon }_{ijk}{\partial }_j{b}_l{\partial }_z{\partial }_l{u}_k{\partial }_z{j}_i\mathrm{d}x\\ =& \int {\epsilon }_{ijk}{\partial }_z{\partial }_x{b}_l{\partial }_l{u}_k{\partial }_z{j}_i\mathrm{d}x+\int {\epsilon }_{ijk}{\partial }_z{\partial }_y{b}_l{\partial }_l{u}_k{\partial }_z{j}_i\mathrm{d}x+\int {\epsilon }_{ijk}{\partial }_z{\partial }_z{b}_l{\partial }_l{u}_k{\partial }_z{j}_i\mathrm{d}x\\ +\int {\epsilon }_{ijk}{\partial }_x{b}_l{\partial }_z{\partial }_l{u}_k{\partial }_z{j}_i\mathrm{d}x+\int {\epsilon }_{ijk}{\partial }_y{b}_l{\partial }_z{\partial }_l{u}_k{\partial }_z{j}_i\mathrm{d}x+\int {\epsilon }_{ijk}{\partial }_z{b}_l{\partial }_z{\partial }_l{u}_k{\partial }_z{j}_i\mathrm{d}x\\ =& S_{13}^1+S_{13}^2+S_{13}^3+S_{13}^4+S_{13}^5+S_{13}^6.\end{array}$$

We deduce each term step by step as follows:

$$\begin{array}{c}\begin{array}{rl}{S}_{13}^1& =\int {\epsilon }_{ijk}{\partial }_z{\partial }_x{b}_l{\partial }_l{u}_k{\partial }_z{j}_i\mathrm{d}x\\ \le C{\parallel j_x\parallel }^{\frac{1}{2}}{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_y\parallel }^{\frac{1}{2}}{\parallel {\omega }_x\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_x\parallel }^{\frac{1}{2}}\\ \le \frac{1}{50}{\parallel \mathrm{\nabla }{j}_x\parallel }^2+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_y\parallel }^2+C({\parallel \omega \parallel }^2+{\parallel j_x\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2),\end{array}\hfill \\ \begin{array}{rl}{S}_{13}^2& =\int {\epsilon }_{ijk}{\partial }_z{\partial }_y{b}_l{\partial }_l{u}_k{\partial }_z{j}_i\mathrm{d}x\\ \le C{\parallel j_y\parallel }^{\frac{1}{2}}{\parallel \omega \parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_x\parallel }^{\frac{1}{2}}{\parallel {\omega }_y\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }{j}_y\parallel }^{\frac{1}{2}}\\ \le \frac{1}{50}{\parallel \mathrm{\nabla }{j}_x\parallel }^2+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_y\parallel }^2+C({\parallel \omega \parallel }^2+{\parallel j_y\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2).\end{array}\hfill \end{array}$$

For $S_{13}^3$, we have

$$\begin{array}{rl}{S}_{13}^3& =\int {\epsilon }_{ijk}{\partial }_z{\partial }_z{b}_l{\partial }_l{u}_k{\partial }_z{j}_i\mathrm{d}x\\ =\int {\epsilon }_{ijk}{\partial }_z{\partial }_z{b}_1{\partial }_x{u}_k{\partial }_z{j}_i\mathrm{d}x+\int {\epsilon }_{ijk}{\partial }_z{\partial }_z{b}_2{\partial }_y{u}_k{\partial }_z{j}_i\mathrm{d}x+\int {\epsilon }_{ijk}{\partial }_z{\partial }_z{b}_3{\partial }_z{u}_k{\partial }_z{j}_i\mathrm{d}x\\ =S_{13}^{31}+S_{13}^{32}+S_{13}^{33}.\end{array}$$

Thus, we deduce that

$$\begin{array}{c}\begin{array}{rl}{S}_{13}^{31}& =\int {\epsilon }_{ijk}{\partial }_z{\partial }_z{b}_1{\partial }_x{u}_k{\partial }_z{j}_i\mathrm{d}x\\ \le C{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel u_x\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}}\\ \le \frac{1}{50}{\parallel \mathrm{\nabla }{j}_x\parallel }^2+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_y\parallel }^2+C({\parallel u_x\parallel }^2+{\parallel {\omega }_x\parallel }^2){\parallel \mathrm{\nabla }j\parallel }^2,\end{array}\hfill \\ \begin{array}{rl}{S}_{13}^{32}& =\int {\epsilon }_{ijk}{\partial }_z{\partial }_z{b}_2{\partial }_y{u}_k{\partial }_z{j}_i\mathrm{d}x\\ \le C{\parallel \mathrm{\nabla }j\parallel }^{\frac{1}{2}}{\parallel u_y\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}}\\ \le \frac{1}{50}{\parallel \mathrm{\nabla }{j}_x\parallel }^2+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_y\parallel }^2+C({\parallel u_y\parallel }^2+{\parallel {\omega }_y\parallel }^2){\parallel \mathrm{\nabla }j\parallel }^2,\end{array}\hfill \\ \begin{array}{rl}{S}_{13}^{33}=& \int {\epsilon }_{ijk}{\partial }_z{\partial }_z{b}_3{\partial }_z{u}_k{\partial }_z{j}_i\mathrm{d}x\\ =& -\int {\epsilon }_{ijk}{\partial }_z({\partial }_x{b}_1+{\partial }_y{b}_2){\partial }_z{u}_k{\partial }_z{j}_i\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}_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_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 \mathrm{\nabla }{j}_y\parallel }^{\frac{1}{2}}{\parallel \mathrm{\nabla }\omega \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({\parallel \omega \parallel }^2+{\parallel j_x\parallel }^2+{\parallel j_y\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2),\end{array}\hfill \\ \begin{array}{rl}{S}_{13}^4& =\int {\epsilon }_{ijk}{\partial }_x{b}_l{\partial }_z{\partial }_l{u}_k{\partial }_z{j}_i\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}}\\ \le \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_x\parallel }^2+\frac{1}{48}{\parallel \mathrm{\nabla }{j}_y\parallel }^2+C({\parallel b_x\parallel }^2+{\parallel j_x\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2),\end{array}\hfill \\ \begin{array}{rl}{S}_{13}^5& =\int {\epsilon }_{ijk}{\partial }_y{b}_l{\partial }_z{\partial }_l{u}_k{\partial }_z{j}_i\mathrm{d}x\\ \le 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({\parallel b_y\parallel }^2+{\parallel j_y\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2).\end{array}\hfill \end{array}$$

For the last term $S_{13}^6$, we see that

$$\begin{array}{rl}{S}_{13}^6& =\int {\epsilon }_{ijk}{\partial }_z{b}_l{\partial }_z{\partial }_l{u}_k{\partial }_z{j}_i\mathrm{d}x\\ =\int {\epsilon }_{ijk}{\partial }_z{b}_1{\partial }_z{\partial }_x{u}_k{\partial }_z{j}_i\mathrm{d}x+\int {\epsilon }_{ijk}{\partial }_z{b}_2{\partial }_z{\partial }_y{u}_k{\partial }_z{j}_i\mathrm{d}x+\int {\epsilon }_{ijk}{\partial }_z{b}_3{\partial }_z{\partial }_z{u}_k{\partial }_z{j}_i\mathrm{d}x\\ =S_{13}^{61}+S_{13}^{62}+S_{13}^{63}.\end{array}$$

Then,

$$\begin{array}{c}\begin{array}{rl}{S}_{13}^{61}& =\int {\epsilon }_{ijk}{\partial }_z{b}_1{\partial }_z{\partial }_x{u}_k{\partial }_z{j}_i\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 }_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({\parallel j\parallel }^2+{\parallel {\omega }_x\parallel }^2){\parallel \mathrm{\nabla }j\parallel }^2,\end{array}\hfill \\ \begin{array}{rl}{S}_{13}^{62}& =\int {\epsilon }_{ijk}{\partial }_z{b}_2{\partial }_z{\partial }_y{u}_k{\partial }_z{j}_i\mathrm{d}x\le C{\parallel j\parallel }^{\frac{1}{2}}{\parallel {\omega }_y\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}{40}{\parallel \mathrm{\nabla }{\omega }_y\parallel }^2+\frac{1}{50}{\parallel \mathrm{\nabla }{j}_x\parallel }^2+C({\parallel j\parallel }^2+{\parallel {\omega }_y\parallel }^2){\parallel \mathrm{\nabla }j\parallel }^2,\end{array}\hfill \\ \begin{array}{rl}{S}_{13}^{63}=& \int {\epsilon }_{ijk}{\partial }_z{b}_3{\partial }_z{\partial }_z{u}_k{\partial }_z{j}_i\mathrm{d}x\\ =& -\int {\epsilon }_{ijk}({\partial }_x{b}_1+{\partial }_y{b}_2){\partial }_z{\partial }_z{u}_k{\partial }_z{j}_i\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_y\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({\parallel b_x\parallel }^2+{\parallel b_y\parallel }^2+{\parallel j_y\parallel }^2){\parallel \mathrm{\nabla }j\parallel }^2.\end{array}\hfill \end{array}$$

As for $S_2$, deduced by similar methods, we can obtain the following inequality (for simplicity we omit the details here):

$$\begin{array}{rl}{S}_2\le & \frac{1}{62}{\parallel \mathrm{\nabla }{\omega }_x\parallel }^2+\frac{1}{40}{\parallel \mathrm{\nabla }{\omega }_y\parallel }^2+\frac{1}{50}{\parallel \mathrm{\nabla }{j}_x\parallel }^2\\ +\frac{1}{48}{\parallel \mathrm{\nabla }{j}_y\parallel }^2+({\parallel {\omega }_x\parallel }^2+{\parallel \omega \parallel }^2+{\parallel j\parallel }^2+{\parallel {\omega }_y\parallel }^2\\ +{\parallel b_x\parallel }^2+{\parallel j_x\parallel }^2+{\parallel b_y\parallel }^2+{\parallel j_y\parallel }^2+{\parallel u_x\parallel }^2+{\parallel u_y\parallel }^2+{\parallel {\omega }_y\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2).\end{array}$$

Then, substituting all the above estimates into (11), we finally deduce that

$$\begin{array}{r}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2)+\frac{1}{2}({\parallel \mathrm{\nabla }{\omega }_x\parallel }^2+{\parallel \mathrm{\nabla }{\omega }_y\parallel }^2+{\parallel \mathrm{\nabla }{j}_x\parallel }^2+{\parallel \mathrm{\nabla }{j}_y\parallel }^2)\\ \le C({\parallel {\omega }_x\parallel }^2+{\parallel \omega \parallel }^2+{\parallel j\parallel }^2+{\parallel {\omega }_y\parallel }^2+{\parallel b_x\parallel }^2+{\parallel j_x\parallel }^2+{\parallel b_y\parallel }^2+{\parallel j_y\parallel }^2\\ +{\parallel u_x\parallel }^2+{\parallel u_y\parallel }^2+{\parallel {\omega }_y\parallel }^2)({\parallel \mathrm{\nabla }\omega \parallel }^2+{\parallel \mathrm{\nabla }j\parallel }^2).\end{array}$$

Then we complete the proof of Lemma 2.3 by Gronwall’s lemma. □

3 Proof of Theorem 1.1

In this section, we prove Theorem 1.1 by using the method of vanishing viscosity. To this end, we consider the following regularized problem:

$$u_t^{\epsilon }+u^{\epsilon }\cdot \mathrm{\nabla }{u}^{\epsilon }=-\mathrm{\nabla }{p}^{\epsilon }+u_{xx}^{\epsilon }+u_{yy}^{\epsilon }+\epsilon u_{zz}^{\epsilon }+b^{\epsilon }\cdot \mathrm{\nabla }{b}^{\epsilon },$$

(12)

$$b_t^{\epsilon }+u^{\epsilon }\cdot \mathrm{\nabla }{b}^{\epsilon }=b_{xx}^{\epsilon }+b_{yy}^{\epsilon }+\epsilon b_{zz}^{\epsilon }+b^{\epsilon }\cdot \mathrm{\nabla }{u}^{\epsilon },$$

(13)

$$div{u}^{\epsilon }=0,div{b}^{\epsilon }=0,$$

(14)

with smooth initial data

$$u^{\epsilon }(0,x)={\psi }_{\epsilon }\ast u_0,b^{\epsilon }(0,x)={\psi }_{\epsilon }\ast b_0,$$

(15)

where ${\psi }_{\epsilon }(x,y)={\epsilon }^{-2}\psi (x/\epsilon ,y/\epsilon )$ is the standard mollifier satisfying

$$\psi \ge 0,\psi \in C_0^{\mathrm{\infty }}\left({\mathbb{R}}^3\right)\text{and}\int \psi \mathrm{d}x=1.$$

Now, an application of the classical result shows that for any $T>0$, there exists a unique global smooth solution $(u^{\epsilon },b^{\epsilon })$ of (12)-(15) on ${\mathbb{R}}^3\times (0,T)$ satisfying the global bounds stated in Lemma 2.2 and 2.3, which are uniform in ε. So, by standard compactness arguments, we can extract a subsequence $(u^{{\epsilon }_j},b^{{\epsilon }_j})$ and pass to the limit as $j\to \mathrm{\infty }$ to get that the limit function $(u,b)$ is indeed a global smooth solution of the problem (1)-(4). The proof of Theorem 1.1 is therefore completed.