In this section, we shall give the proof of Theorem 1.1. We will assume that \(\mu =\chi =\gamma =\kappa =1\) throughout this paper.
Let \([0,T^{*})\) be the maximal time interval for the existence of the local smooth solution. If \(T^{*}\geq T\), the conclusion is obviously valid, but for \(T^{*}< T\), we would show that
$$\begin{aligned} {\lim \sup }_{t\rightarrow T^{*}}\bigl( \bigl\Vert \nabla u(\cdot,t) \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \nabla \omega (\cdot,t) \bigr\Vert ^{2}_{L^{2}}+ \bigl\Vert \nabla b(\cdot,t) \bigr\Vert ^{2}_{L^{2}}\bigr) \leq C \end{aligned}$$
under the assumption of (1.3) and (1.4). Hence, according to the definition of \(T^{*}\), this leads to a contradiction.
Step 1: \(L^{2}\)-energy estimate A standard energy method says
$$\begin{aligned} \begin{aligned} &\bigl\Vert \bigl(u(t),\omega (t),b(t)\bigr) \bigr\Vert ^{2}_{L^{2}}+2 \int _{0}^{t} \Vert \nabla u \Vert _{L^{2}}^{2}\,d \tau +2 \int _{0}^{t} \Vert \nabla \omega \Vert _{L^{2}}^{2}\,d\tau \\ &\quad{}+2 \int _{0}^{t} \Vert \nabla \cdot \omega \Vert _{L^{2}}^{2}\,d\tau +2 \int _{0}^{t} \Vert \omega \Vert _{L^{2}}^{2}\,d\tau \leq \bigl\Vert (u_{0},\omega _{0},b_{0}) \bigr\Vert _{L^{2}}^{2}. \end{aligned} \end{aligned}$$
(4.1)
Step 2: \(H^{1}\)-Horizontal energy estimate
We first establish the horizontal gradient of the velocity u and magnetic field b. Taking \(\nabla _{h}\) on both sides of Eqs. (1.1)1 and (1.1)3, multiplying by \(\nabla _{h} u\) and \(\nabla _{h} b\), respectively, and integrating over \({\mathbf{R}}^{3}\), we get
$$\begin{aligned} \begin{aligned}& \frac{1}{2}\frac{d}{dt} \bigl( \Vert \nabla _{h} u \Vert _{L^{2}}^{2}+ \Vert \nabla _{h} b \Vert _{L^{2}}^{2}\bigr)+2 \Vert \nabla _{h} \nabla u \Vert _{L^{2}}^{2}\\ &\quad =- \int _{R^{3}}\nabla _{h}(u \cdot \nabla u)\cdot \nabla _{h} u\,dx \\ &\qquad{}+ \int _{R^{3}} \bigl(\nabla _{h}(b\cdot \nabla b)\cdot \nabla _{h} u+ \nabla _{h}(b\cdot \nabla u)\cdot \nabla _{h} b-\nabla _{h}(u\cdot \nabla b)\cdot \nabla _{h} b \bigr) \,dx \\ &\qquad{}+ \int _{R^{3}}\nabla _{h}(\nabla \times \omega )\cdot \nabla _{h} u \,dx \\ &\quad:=A_{1}+A_{2}+A_{3}. \end{aligned} \end{aligned}$$
(4.2)
We start to estimate \(A_{2}\). From (1.1)4 and Lemma 2.1 and the fact \(\nabla \cdot u=\nabla \cdot b=0\), we know that \((p\geq 3)\)
$$\begin{aligned} \begin{aligned} A_{2}={}& \int _{R^{3}}\sum_{k=1}^{2} \partial _{k}(b\cdot \nabla b) \partial _{k} u \,dx+ \int _{R^{3}}\sum_{k=1}^{2} \partial _{k}(b\cdot \nabla u)\partial _{k} b \,dx\\ &{}- \int _{R^{3}}\sum_{k=1}^{2} \partial _{k}(u \cdot \nabla b)\partial _{k} b \,dx \\ ={}& \int _{R^{3}}\sum_{k=1}^{2}( \partial _{k} b\cdot \nabla b\partial _{k} u+\partial _{k} b\cdot \nabla u\partial _{k} b)\,dx- \int _{{\mathbf{R}}^{3}} \sum_{k=1}^{2} \partial _{k} u\cdot \nabla b\partial _{k} b \,dx \\ \leq {}&C \int _{R^{3}} \vert \nabla b \vert \vert \nabla u \vert \vert \nabla b \vert \,dx\leq C \Vert \nabla b \Vert _{\dot{M}_{p,q}} \Vert \nabla b \Vert _{L^{2}} \Vert \nabla u \Vert _{\dot{H}^{ \frac{3}{p}}} \\ \leq {}&C \Vert \nabla b \Vert _{\dot{M}_{p,q}} \Vert \nabla b \Vert _{L^{2}} \Vert \nabla u \Vert ^{1- \frac{3}{p}}_{L^{2}} \Vert \Delta u \Vert ^{\frac{3}{p}}_{L^{2}}. \end{aligned} \end{aligned}$$
(4.3)
Thanks to the Hölder and Young inequalities, one deduces
$$\begin{aligned} \begin{aligned} A_{3}={}& \int _{R^{3}}\sum_{k=1}^{2} \partial _{k}(\nabla \times \omega )\cdot \partial _{k} u \,dx - \int _{R^{3}}\sum_{k=1}^{2} \nabla \times \omega \cdot \partial _{k}\partial _{k} u \,dx \\ \leq{} &C \Vert \nabla \times \omega \Vert _{L^{2}} \Vert \nabla _{h}\nabla u \Vert _{L^{2}} \\ \leq {}&\frac{1}{2} \Vert \nabla _{h}\nabla u \Vert ^{2}_{L^{2}}+C \Vert \nabla \omega \Vert ^{2}_{L^{2}}. \end{aligned} \end{aligned}$$
(4.4)
Now it is time to deal with the first term \(A_{1}\). Integrating by parts, we get
$$\begin{aligned} \begin{aligned} A_{1}={}& \int _{R^{3}}u\cdot \nabla u\cdot \Delta _{h} u\,dx \\ ={}& \int _{R^{3}}\sum_{k=1}^{2} \sum_{i,j=1}^{2}u_{i}\partial _{i} u_{j} \partial _{k}\partial _{k} u_{j} \,dx+ \int _{{\mathbf{R}}^{3}}\sum_{k=1}^{2} \sum_{i=1}^{2}u_{i}\partial _{i} u_{3}\partial _{k}\partial _{k} u_{3} \,dx \\ &{}+ \int _{R^{3}}\sum_{k=1}^{2} \sum_{j=1}^{3}u_{3}\partial _{3} u_{j} \partial _{k}\partial _{k} u_{j} \,dx \\ :={}&A_{11}+A_{12}+A_{13}. \end{aligned} \end{aligned}$$
(4.5)
Using integration by parts again and applying the fact that \(\operatorname{div}u=0\), it yields
$$\begin{aligned} \begin{aligned} A_{11}={}&{-} \int _{R^{3}}\sum_{k=1}^{2} \sum_{i,j=1}^{2}(\partial _{k} u_{i} \partial _{i} u_{j}\partial _{k} u_{j}+u_{i}\partial _{i} \partial _{k} u_{j} \partial _{k} u_{j}) \,dx \\ ={}& \int _{R^{3}}\sum_{k=1}^{2} \sum_{i,j=1}^{2}\partial _{k} u_{i} \partial _{i} u_{j}\partial _{k} u_{j}\,dx+\frac{1}{2} \int _{R^{3}}\sum_{k=1}^{2} \sum_{i,j=1}^{2}\partial _{i} u_{i}\partial _{k} u_{j}\partial _{k} u_{j} \,dx \\ ={}& \int _{R^{3}}\sum_{k=1}^{2} \sum_{i,j=1}^{2}\partial _{k} u_{i} \partial _{i} u_{j}\partial _{k} u_{j}\,dx-\frac{1}{2} \int _{R^{3}}\sum_{k=1}^{2} \sum_{j=1}^{2}\partial _{3} u_{3}\partial _{k} u_{j}\partial _{k} u_{j} \,dx \\ ={}& \int _{R^{3}}\sum_{k=1}^{2} \sum_{i,j=1}^{2}\partial _{k} u_{i} \partial _{i} u_{j}\partial _{k} u_{j}\,dx-\frac{1}{2} \int _{R^{3}}\sum_{k=1}^{2} \sum_{j=1}^{2}\partial _{3} u_{3}\partial _{k} u_{j}\partial _{k} u_{j} \,dx \\ ={}&{-} \int _{R^{3}}\partial _{3}u_{3} \bigl((\partial _{1}u_{2})^{2}+ \partial _{2}u_{1} \partial _{1}u_{2}+(\partial _{2}u_{1})^{2}+( \partial _{1}u_{1})^{2}- \partial _{1}u_{1}\partial _{2}u_{2}+(\partial _{2}u_{2})^{2} \bigr)\,dx \\ &{}-\frac{1}{2} \int _{R^{3}}\sum_{k=1}^{2} \sum_{j=1}^{2}\partial _{3} u_{3} \partial _{k} u_{j}\partial _{k} u_{j} \end{aligned} \end{aligned}$$
(4.6)
and
$$\begin{aligned} A_{12}={}&{-} \int _{R^{3}}\sum_{k=1}^{2} \sum_{i=1}^{2}(\partial _{k} u_{i} \partial _{i} u_{3}\partial _{k} u_{3}+u_{i}\partial _{i}\partial _{k} u_{3} \partial _{k} u_{3}) \,dx \\ ={}& \int _{R^{3}}\sum_{k=1}^{2} \sum_{i=1}^{2}(\partial _{k} \partial _{i} u_{i} u_{3}\partial _{k} u_{3}+\partial _{k} u_{i} u_{3}\partial _{k} \partial _{i} u_{3}) \,dx \\ &{}+ \frac{1}{2} \int _{R^{3}}\sum_{k=1}^{2} \sum_{i=1}^{2} \partial _{i} u_{i}\partial _{k} u_{3}\partial _{k} u_{3} \,dx \\ ={}& \int _{R^{3}}\sum_{k=1}^{2} \sum_{i=1}^{2}(\partial _{k} \partial _{i} u_{i} u_{3}\partial _{k} u_{3}+\partial _{k} u_{i} u_{3}\partial _{k} \partial _{i} u_{3}) \,dx\\ &{}- \frac{1}{2} \int _{R^{3}}\sum_{k=1}^{2} \partial _{3} u_{3}\partial _{k} u_{3} \partial _{k} u_{3} \,dx \\ ={}& \int _{R^{3}}\sum_{k=1}^{2} \sum_{i=1}^{2}(\partial _{k} \partial _{i} u_{i} u_{3}\partial _{k} u_{3}+\partial _{k} u_{i} u_{3}\partial _{k} \partial _{i} u_{3}) \,dx \\ &{}+ \int _{R^{3}}\sum_{k=1}^{2} u_{3}\partial _{3} \partial _{k} u_{3} \partial _{k} u_{3} \,dx. \end{aligned}$$
(4.7)
Substituting (4.6) and (4.7) into (4.5) yields
$$\begin{aligned} A_{1}\leq C \int _{{\mathbf{R}}^{3}} \vert u_{3} \vert \vert \nabla u \vert \vert \nabla _{h} \nabla u \vert \,dx. \end{aligned}$$
(4.8)
Thanks to the Hölder inequality, (2.1) and the Young inequality, we obtain for \(\alpha >3\)
$$\begin{aligned} \begin{aligned} A_{1}&\leq C \Vert u_{3} \Vert _{L^{\alpha }} \Vert \nabla _{h}\nabla u \Vert _{L^{2}} \Vert \nabla u \Vert _{L^{\frac{2\alpha }{\alpha -2}}} \\ &\leq C \Vert u_{3} \Vert _{L^{\alpha }} \Vert \nabla _{h}\nabla u \Vert _{L^{2}} \Vert \nabla u \Vert _{L^{2}}^{1-\frac{3}{\alpha }} \Vert \nabla u \Vert _{L^{6}}^{\frac{3}{\alpha }} \\ &\leq C \Vert u_{3} \Vert _{L^{\alpha }} \Vert \nabla _{h}\nabla u \Vert _{L^{2}} \Vert \nabla u \Vert _{L^{2}}^{1-\frac{3}{\alpha }} \Vert \nabla _{h}\nabla u \Vert _{L^{2}}^{ \frac{2}{\alpha }} \Vert \partial _{3}\nabla u \Vert _{L^{2}}^{\frac{1}{\alpha }} \\ &\leq \frac{1}{2} \Vert \nabla _{h}\nabla u \Vert _{L^{2}}^{2}+C \Vert u_{3} \Vert _{L^{\alpha }}^{ \frac{2\alpha }{\alpha -2}} \Vert \nabla u \Vert _{L^{2}}^{ \frac{2(\alpha -3)}{\alpha -2}} \Vert \Delta u \Vert _{L^{2}}^{ \frac{2}{\alpha -2}}, \end{aligned} \end{aligned}$$
(4.9)
which, along with (4.3), (4.4) and (4.2), gives
$$\begin{aligned} \begin{aligned} &\frac{d}{dt}\bigl( \Vert \nabla _{h} u \Vert _{L^{2}}^{2}+ \Vert \nabla _{h} b \Vert _{L^{2}}^{2}\bigr)+2 \Vert \nabla _{h} \nabla u \Vert _{L^{2}}^{2} \\ &\quad\leq C \Vert u_{3} \Vert _{L^{\alpha }}^{\frac{2\alpha }{\alpha -2}} \Vert \nabla u \Vert _{L^{2}}^{ \frac{2(\alpha -3)}{\alpha -2}} \Vert \Delta u \Vert _{L^{2}}^{ \frac{2}{\alpha -2}}+C \Vert \nabla \omega \Vert ^{2}_{L^{2}} \\ &\qquad{}+C \Vert \nabla b \Vert _{\dot{M}_{p,q}} \Vert \nabla b \Vert _{L^{2}} \Vert \nabla u \Vert ^{1- \frac{3}{p}}_{L^{2}} \Vert \Delta u \Vert ^{\frac{3}{p}}_{L^{2}}. \end{aligned} \end{aligned}$$
(4.10)
Integrating over \([0, t)\) and using (4.1), one can verify
$$\begin{aligned} \begin{aligned} &\sup_{0\leq \tau \leq t}\bigl( \Vert \nabla _{h} u \Vert _{L^{2}}^{2}+ \Vert \nabla _{h} b \Vert _{L^{2}}^{2}\bigr)+2 \int _{0}^{t} \Vert \nabla _{h} \nabla u \Vert _{L^{2}}^{2} \,d \tau \\ &\quad\leq C\bigl( \Vert \nabla _{h} u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla _{h} b_{0} \Vert _{L^{2}}^{2}+1\bigr)+C \int _{0}^{t} \Vert u_{3} \Vert _{L^{\alpha }}^{\frac{2\alpha }{\alpha -2}} \Vert \nabla u \Vert _{L^{2}}^{\frac{2(\alpha -3)}{\alpha -2}} \Vert \Delta u \Vert _{L^{2}}^{ \frac{2}{\alpha -2}}\,d\tau \\ &\qquad{}+C \int _{0}^{t} \Vert \nabla b \Vert _{\dot{M}_{p,q}} \Vert \nabla b \Vert _{L^{2}} \Vert \nabla u \Vert ^{1-\frac{3}{p}}_{L^{2}} \Vert \Delta u \Vert ^{\frac{3}{p}}_{L^{2}}\,d \tau. \end{aligned} \end{aligned}$$
(4.11)
Step 3: \(H^{1}\)-full energy estimate
Multiplying \({\text{(1.1)}}_{1}, {\text{(1.1)}}_{2}\) and (1.1)3 by \(-\Delta u, -\Delta \omega \) and \(-\Delta b\), respectively, and integrating over \({\mathbf{R}}^{3}\), then adding them we obtain
$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl( \Vert \nabla u \Vert _{L^{2}}^{2}+ \Vert \nabla \omega \Vert _{L^{2}}^{2}+ \Vert \nabla b \Vert _{L^{2}}^{2}\bigr)+2 \Vert \Delta u \Vert _{L^{2}}^{2}+ \Vert \Delta \omega \Vert _{L^{2}}^{2}+ \Vert \nabla \nabla \cdot \omega \Vert _{L^{2}}^{2} \\ &\quad= \int _{R^{3}}u\cdot \nabla u\cdot \Delta u \,dx- \int _{{\mathbf{R}}^{3}}b \cdot \nabla b\cdot \Delta u \,dx+ \int _{R^{3}}u\cdot \nabla \omega \cdot \Delta \omega \,dx \\ &\qquad{}-2 \int _{R^{3}}\nabla \times u\cdot \Delta \omega\,dx+ \int _{R^{3}}u \cdot \nabla b\cdot \Delta b\,dx- \int _{R^{3}}b\cdot \nabla u\cdot \Delta b\,dx \\ &\quad :=B_{1}+B_{2}+B_{3}+B_{4}+B_{5}+B_{6}, \end{aligned} \end{aligned}$$
(4.12)
where we have used the inequalities
$$\begin{aligned} \int _{R^{3}}(-\nabla \nabla \cdot \omega ) (-\Delta \omega ) \,dx={}&\sum_{i,j,k=1}^{3} \int _{R^{3}} \partial _{i}\partial _{j} \omega _{j}\partial _{k}\partial _{k} \omega _{i}\,dx \\ ={}&\sum_{i,j,k=1}^{3} \int _{R^{3}}\partial _{k}\partial _{j} \omega _{j} \partial _{k}\partial _{i}\omega _{i}\,dx \\ ={}& \Vert \nabla \nabla \cdot \omega \Vert _{L^{2}}^{2} \end{aligned}$$
and
$$\begin{aligned} \int _{R^{3}}\nabla \times \omega \cdot \Delta u\,dx= \int _{{\mathbf{R}}^{3}} \Delta \omega \cdot \nabla \times u\,dx. \end{aligned}$$
Now, we estimate \(B_{2}, B_{5}\) and \(B_{6}\). After applying integration by parts, \(\operatorname{div}u=\operatorname{div}b=0\), the Hölder inequality, Lemma 2.1, the Sobolev interpolation inequality and the Young inequality, we have
$$\begin{aligned} \begin{aligned} &B_{2}+B_{5}+B_{6}\\ &\quad= \int _{R^{3}}\nabla b\cdot \nabla b\cdot \nabla u\,dx - \int _{R^{3}}\nabla u\cdot \nabla b\cdot \nabla b\,dx + \int _{R^{3}} \nabla b\cdot \nabla u\cdot \nabla b\,dx \\ &\quad\leq C \int _{R^{3}} \vert \nabla b \vert \vert \nabla u \vert \vert \nabla b \vert \,dx \\ &\quad\leq C \Vert \nabla b \Vert _{\dot{M}_{p,q}} \Vert \nabla b \Vert _{L^{2}} \Vert \nabla u \Vert _{ \dot{H}^{\frac{3}{p}}} \\ &\quad\leq C \Vert \nabla b \Vert _{\dot{M}_{p,q}} \Vert \nabla b \Vert _{L^{2}} \Vert \nabla u \Vert _{L^{2}}^{1- \frac{3}{p}} \Vert \Delta u \Vert _{L^{2}}^{\frac{3}{p}} \\ &\quad\leq \frac{1}{4} \Vert \Delta u \Vert _{L^{2}}^{2}+C \Vert \nabla b \Vert _{\dot{M}_{p,q}}^{ \frac{2p}{2p-3}} \Vert \nabla b \Vert _{L^{2}}^{\frac{2p}{2p-3}} \Vert \nabla u \Vert _{L^{2}}^{ \frac{2(p-3)}{2p-3}} \\ &\quad\leq \frac{1}{4} \Vert \Delta u \Vert _{L^{2}}^{2}+C \Vert \nabla b \Vert _{\dot{M}_{p,q}}^{ \frac{2p}{2p-3}}\bigl( \Vert \nabla b \Vert _{L^{2}}^{2}+ \Vert \nabla u \Vert _{L^{2}}^{2}\bigr). \end{aligned} \end{aligned}$$
(4.13)
We can infer from the Hölder and the Young inequalities that
$$\begin{aligned} B_{4}\leq 2 \int _{R^{3}}\nabla (\nabla \times u)\cdot \nabla \omega\,dx \leq \frac{1}{4} \Vert \Delta u \Vert _{L^{2}}^{2}+C \Vert \nabla \omega \Vert _{L^{2}}^{2}. \end{aligned}$$
(4.14)
Applying integration by parts we obtain
$$\begin{aligned} B_{3}={}&{-} \int _{R^{3}}\nabla u\cdot \nabla \omega \cdot \nabla \omega\,dx \\ ={}& \int _{R^{3}}\nabla u\cdot \nabla (\nabla \omega )\cdot \omega \,dx+ \int _{R^{3}}\nabla (\nabla u)\cdot \nabla \omega \cdot \omega\,dx \\ :={}&B_{31}+B_{32}. \end{aligned}$$
(4.15)
Thanks to the Hölder inequality, the interpolation inequality with \(3\leq \alpha \leq 9\) and Lemma 3.1, we arrive at
$$\begin{aligned} \begin{aligned} B_{31}\leq {}& \int _{R^{3}} \vert \nabla u \vert \vert \Delta \omega \vert \vert \omega \vert \,dx\leq \Vert \Delta \omega \Vert _{L^{2}} \Vert \omega \Vert _{L^{\alpha }} \Vert \nabla u \Vert _{L^{ \frac{2\alpha }{\alpha -2}}} \\ \leq {}& \Vert \Delta \omega \Vert _{L^{2}} \Vert \omega \Vert _{L^{3}}^{ \frac{9-\alpha }{2\alpha }} \Vert \omega \Vert _{L^{9}}^{ \frac{3\alpha -9}{2\alpha }} \Vert \nabla u \Vert _{L^{2}}^{1-\frac{3}{\alpha }} \Vert \nabla u \Vert _{L^{6}}^{\frac{3}{\alpha }} \\ \leq {}& \Vert \Delta \omega \Vert _{L^{2}}\||\omega |^{\frac{3}{2}} \|_{L^{6}}^{1- \frac{3}{\alpha }} \Vert \nabla u \Vert _{L^{2}}^{1-\frac{3}{\alpha }} \Vert \Delta u \Vert _{L^{2}}^{ \frac{3}{\alpha }} \\ \leq {}& \Vert \Delta \omega \Vert _{L^{2}} \bigl\Vert \nabla \vert \omega \vert ^{\frac{3}{2}} \bigr\Vert _{L^{2}}^{1- \frac{3}{\alpha }} \Vert \nabla u \Vert _{L^{2}}^{1-\frac{3}{\alpha }} \Vert \Delta u \Vert _{L^{2}}^{ \frac{3}{\alpha }} \\ \leq {}&\frac{1}{4} \Vert \Delta \omega \Vert _{L^{2}}^{2}+C \bigl\Vert \nabla \vert \omega \vert ^{ \frac{3}{2}} \bigr\Vert _{L^{2}}^{2(1-\frac{3}{\alpha })} \Vert \nabla u \Vert _{L^{2}}^{2(1- \frac{3}{\alpha })} \Vert \Delta u \Vert _{L^{2}}^{\frac{6}{\alpha }} \\ \leq {}&\frac{1}{4} \Vert \Delta \omega \Vert _{L^{2}}^{2}+ \frac{1}{4} \Vert \Delta u \Vert _{L^{2}}^{2}+C \bigl\Vert \nabla \vert \omega \vert ^{\frac{3}{2}} \bigr\Vert _{L^{2}}^{2} \Vert \nabla u \Vert _{L^{2}}^{2}. \end{aligned} \end{aligned}$$
(4.16)
Similarly, the term \(B_{32}\) can be bounded as follows:
$$\begin{aligned} \begin{aligned} B_{32}\leq {}& \int _{R^{3}} \vert \Delta u \vert \vert \nabla \omega \vert \vert \omega \vert \,dx\leq \Vert \Delta u \Vert _{L^{2}} \Vert \omega \Vert _{L^{\alpha }} \Vert \nabla \omega \Vert _{L^{ \frac{2\alpha }{\alpha -2}}} \\ \leq {}& \Vert \Delta u \Vert _{L^{2}} \Vert \omega \Vert _{L^{3}}^{ \frac{9-\alpha }{2\alpha }} \Vert \omega \Vert _{L^{9}}^{ \frac{3\alpha -9}{2\alpha }} \Vert \nabla \omega \Vert _{L^{2}}^{1-\frac{3}{\alpha }} \Vert \nabla \omega \Vert _{L^{6}}^{\frac{3}{\alpha }} \\ \leq {}& \Vert \Delta u \Vert _{L^{2}}\||\omega |^{\frac{3}{2}} \|_{L^{6}}^{1- \frac{3}{\alpha }} \Vert \nabla \omega \Vert _{L^{2}}^{1-\frac{3}{\alpha }} \Vert \Delta \omega \Vert _{L^{2}}^{\frac{3}{\alpha }} \\ \leq {}& \Vert \Delta u \Vert _{L^{2}} \bigl\Vert \nabla \vert \omega \vert ^{\frac{3}{2}} \bigr\Vert _{L^{2}}^{1- \frac{3}{\alpha }} \Vert \nabla \omega \Vert _{L^{2}}^{1-\frac{3}{\alpha }} \Vert \Delta \omega \Vert _{L^{2}}^{\frac{3}{\alpha }} \\ \leq {}&\frac{1}{4} \Vert \Delta u \Vert _{L^{2}}^{2}+C \bigl\Vert \nabla \vert \omega \vert ^{\frac{3}{2}} \bigr\Vert _{L^{2}}^{2(1- \frac{3}{\alpha })} \Vert \nabla \omega \Vert _{L^{2}}^{2(1-\frac{3}{\alpha })} \Vert \Delta \omega \Vert _{L^{2}}^{\frac{6}{\alpha }} \\ \leq {}&\frac{1}{4} \Vert \Delta u \Vert _{L^{2}}^{2}+ \frac{1}{4} \Vert \Delta \omega \Vert _{L^{2}}^{2}+C \bigl\Vert \nabla \vert \omega \vert ^{\frac{3}{2}} \bigr\Vert _{L^{2}}^{2} \Vert \nabla \omega \Vert _{L^{2}}^{2}. \end{aligned} \end{aligned}$$
(4.17)
For the term \(B_{1}\), similar to \(A_{1}\), we find that
$$\begin{aligned} B_{1}={}& \int _{R^{3}}u\cdot \nabla u\cdot \Delta _{h} u\,dx+ \int _{R^{3}}u \cdot \nabla u\cdot \partial _{3}\partial _{3} u\,dx \\ ={}&{-} \int _{R^{3}}\sum_{k=1}^{2} \sum_{i,j=1}^{3}\partial _{k} u_{i} \partial _{i} u_{j}\partial _{k} u_{j} \,dx- \int _{{\mathbf{R}}^{3}}\sum_{k=1}^{2} \sum_{i,j=1}^{3}u_{i}\partial _{i}\partial _{k} u_{j}\partial _{k} u_{j} \,dx \\ &{}+ \int _{R^{3}}\sum_{i,j=1}^{3}u_{i} \partial _{i} u_{j}\partial _{3} \partial _{3} u_{j} \,dx \\ ={}&{-} \int _{R^{3}}\sum_{k=1}^{2} \sum_{i,j=1}^{3}\partial _{k} u_{i} \partial _{i} u_{j}\partial _{k} u_{j} \,dx - \int _{R^{3}}\sum_{i,j=1}^{3} \partial _{3} u_{i}\partial _{i} u_{j} \partial _{3} u_{j} \,dx \\ &{}- \int _{{ \mathbf{R}}^{3}}\sum_{i,j=1}^{3}u_{i} \partial _{i}\partial _{3} u_{j} \partial _{3} u_{j} \,dx \\ ={}&{-} \int _{R^{3}}\sum_{k=1}^{2} \sum_{i,j=1}^{3}\partial _{k} u_{i} \partial _{i} u_{j}\partial _{k} u_{j} \,dx - \int _{R^{3}}\sum_{j=1}^{3} \sum_{i=1}^{2}\partial _{3} u_{i}\partial _{i} u_{j}\partial _{3} u_{j} \,dx\\ &{}- \int _{{\mathbf{R}}^{3}}\sum_{j=1}^{3} \partial _{3} u_{3}\partial _{3} u_{j} \partial _{3} u_{j} \,dx \\ \leq{}& C \int _{R^{3}} \vert \nabla _{h} u \vert \vert \nabla u \vert ^{2}\,dx\leq C \Vert \nabla _{h} u \Vert _{L^{2}} \Vert \nabla u \Vert _{L^{4}}^{2} \\ \leq{}& C \Vert \nabla _{h} u \Vert _{L^{2}} \Vert \nabla u \Vert _{L^{2}}^{\frac{1}{2}} \Vert \nabla u \Vert _{L^{6}}^{\frac{3}{2}} \\ \leq {}&C \Vert \nabla _{h} u \Vert _{L^{2}} \Vert \nabla u \Vert _{L^{2}}^{\frac{1}{2}} \Vert \partial _{1}\nabla u \Vert _{L^{2}}^{\frac{1}{2}} \Vert \partial _{2}\nabla u \Vert _{L^{2}}^{ \frac{1}{2}} \Vert \partial _{3}\nabla u \Vert _{L^{2}}^{\frac{1}{2}} \\ \leq{}& C \Vert \nabla _{h} u \Vert _{L^{2}} \Vert \nabla u \Vert _{L^{2}}^{\frac{1}{2}} \Vert \nabla _{h}\nabla u \Vert _{L^{2}} \Vert \Delta u \Vert _{L^{2}}^{\frac{1}{2}}. \end{aligned}$$
(4.18)
Combining (4.13), (4.14), (4.16), (4.17) and (4.18), then (4.12) becomes
$$\begin{aligned} \begin{aligned} &\frac{d}{dt}\bigl( \Vert \nabla u \Vert _{L^{2}}^{2}+ \Vert \nabla \omega \Vert _{L^{2}}^{2}+ \Vert \nabla b \Vert _{L^{2}}^{2} \bigr)+ \Vert \Delta u \Vert _{L^{2}}^{2}+ \Vert \Delta \omega \Vert _{L^{2}}^{2}+ \Vert \nabla \omega \Vert _{L^{2}}^{2} \\ &\quad\leq C\bigl( \Vert \nabla b \Vert _{\dot{M}_{p,q}}^{\frac{2p}{2p-3}}+ \bigl\Vert \nabla \vert \omega \vert ^{\frac{3}{2}} \bigr\Vert _{L^{2}}^{2}\bigr) \bigl( \Vert \nabla u \Vert _{L^{2}}^{2}+ \Vert \nabla \omega \Vert _{L^{2}}^{2}+ \Vert \nabla b \Vert _{L^{2}}^{2} \bigr) \\ &\qquad{}+C \Vert \nabla \omega \Vert _{L^{2}}^{2}+C \Vert \nabla _{h} u \Vert _{L^{2}} \Vert \nabla u \Vert _{L^{2}}^{\frac{1}{2}} \Vert \nabla _{h}\nabla u \Vert _{L^{2}} \Vert \Delta u \Vert _{L^{2}}^{ \frac{1}{2}}. \end{aligned} \end{aligned}$$
(4.19)
Integrating over \([0,t]\) yields
$$\begin{aligned} \begin{aligned} &\bigl( \Vert \nabla u \Vert _{L^{2}}^{2}+ \Vert \nabla \omega \Vert _{L^{2}}^{2}+ \Vert \nabla b \Vert _{L^{2}}^{2} \bigr)+ \int _{0}^{t} \Vert \Delta u \Vert _{L^{2}}^{2}+ \Vert \Delta \omega \Vert _{L^{2}}^{2}\,d\tau \\ &\quad\leq C \int _{0}^{t}\bigl( \Vert \nabla b \Vert _{\dot{M}_{p,q}}^{\frac{2p}{2p-3}}+ \bigl\Vert \nabla \vert \omega \vert ^{\frac{3}{2}} \bigr\Vert _{L^{2}}^{2}\bigr) \bigl( \Vert \nabla u \Vert _{L^{2}}^{2}+ \Vert \nabla \omega \Vert _{L^{2}}^{2}+ \Vert \nabla b \Vert _{L^{2}}^{2} \bigr)\,d\tau \\ &\qquad{}+C\bigl( \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla \omega _{0} \Vert _{L^{2}}^{2}+ \Vert \nabla b_{0} \Vert _{L^{2}}^{2}+1 \bigr)+CG(t), \end{aligned} \end{aligned}$$
(4.20)
where
$$\begin{aligned} G(t)= \int _{0}^{t} \Vert \nabla _{h} u \Vert _{L^{2}} \Vert \nabla _{h}\nabla u \Vert _{L^{2}} \Vert \nabla u \Vert _{L^{2}}^{\frac{1}{2}} \Vert \Delta u \Vert _{L^{2}}^{\frac{1}{2}}\,d\tau. \end{aligned}$$
We proceed to estimate \(G(t)\). From (4.1) and (4.11), we deduce that
$$\begin{aligned} G(t)\leq {}&\sup_{0\leq \tau \leq t} \Vert \nabla _{h} u \Vert _{L^{2}}\biggl( \int _{0}^{t} \Vert \nabla _{h}\nabla u \Vert _{L^{2}}^{2}\,d\tau \biggr)^{\frac{1}{2}}\biggl( \int _{0}^{t} \Vert \nabla u \Vert _{L^{2}}^{2}\biggr)^{\frac{1}{4}}\biggl( \int _{0}^{t} \Vert \Delta u \Vert _{L^{2}}^{2}\,d \tau \biggr)^{\frac{1}{4}} \\ \leq{} &C\biggl(\sup_{0\leq \tau \leq t} \Vert \nabla _{h} u \Vert _{L^{2}}^{2}+ \int _{0}^{t} \Vert \nabla _{h}\nabla u \Vert _{L^{2}}^{2}\,d\tau \biggr) \biggl( \int _{0}^{t} \Vert \Delta u \Vert _{L^{2}}^{2}\,d \tau \biggr)^{\frac{1}{4}} \\ \leq {}&C \biggl(\bigl( \Vert \nabla _{h} u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla _{h} b_{0} \Vert _{L^{2}}^{2}+1\bigr)+ \int _{0}^{t} \Vert u_{3} \Vert _{L^{\alpha }}^{\frac {2\alpha }{\alpha -2}} \Vert \nabla u \Vert _{L^{2}}^{\frac{2(\alpha -3)}{\alpha -2}} \Vert \Delta u \Vert _{L^{2}}^{ \frac{2}{\alpha -2}}\,d\tau \\ &{}+ \int _{0}^{t} \Vert \nabla b \Vert _{\dot{M}_{p,q}} \Vert \nabla b \Vert _{L^{2}} \Vert \nabla u \Vert ^{1-\frac{3}{p}}_{L^{2}} \Vert \Delta u \Vert ^{\frac{3}{p}}_{L^{2}}\,d \tau \biggr) \biggl( \int _{0}^{t} \Vert \Delta u \Vert _{L^{2}}^{2}\,d\tau \biggr)^{\frac{1}{4}} \\ \leq{} &C\bigl( \Vert \nabla _{h} u_{0} \Vert _{L^{2}}^{\frac{8}{3}}+ \Vert \nabla _{h} b_{0} \Vert _{L^{2}}^{ \frac{8}{3}}+1\bigr)+C\biggl( \int _{0}^{t} \Vert u_{3} \Vert _{L^{\alpha }}^{ \frac {2\alpha }{\alpha -2}} \Vert \nabla u \Vert _{L^{2}}^{ \frac{2(\alpha -3)}{\alpha -2}} \Vert \Delta u \Vert _{L^{2}}^{ \frac{2}{\alpha -2}}\,d\tau \biggr)^{\frac{4}{3}} \\ &{}+\biggl( \int _{0}^{t} \Vert \nabla b \Vert _{\dot{M}_{p,q}} \Vert \nabla b \Vert _{L^{2}} \Vert \nabla u \Vert ^{1-\frac{3}{p}}_{L^{2}} \Vert \Delta u \Vert ^{\frac{3}{p}}_{L^{2}}\,d \tau \biggr)^{\frac{4}{3}}+\frac{1}{4} \int _{0}^{t} \Vert \Delta u \Vert _{L^{2}}^{2}\,d\tau \\ \leq {}&C\bigl( \Vert \nabla _{h} u_{0} \Vert _{L^{2}}^{\frac{8}{3}}+ \Vert \nabla _{h} b_{0} \Vert _{L^{2}}^{ \frac{8}{3}}+1\bigr)+\frac{1}{4} \int _{0}^{t} \Vert \Delta u \Vert _{L^{2}}^{2}\,d\tau \\ &{}+\biggl( \int _{0}^{t} \Vert u_{3} \Vert _{L^{\alpha }}^{\frac{2\alpha }{\alpha -3}} \Vert \nabla u \Vert _{L^{2}}^{2} \,d\tau \biggr)^{\frac{4(\alpha -3)}{3(\alpha -2)}}\biggl( \int _{0}^{t} \Vert \Delta u \Vert _{L^{2}}^{2}\,d\tau \biggr)^{\frac{4}{3(\alpha -2)}} \\ &{}+C\biggl( \int _{0}^{t} \Vert \nabla b \Vert _{\dot{M}_{p,q}}^{\frac{2p}{2p-3}} \Vert \nabla b \Vert _{L^{2}}^{\frac{2p}{2p-3}} \Vert \nabla u \Vert _{L^{2}}^{ \frac{2(p-3)}{2p-3}}\,d\tau \biggr)^{\frac{2(2p-3)}{3p}}\biggl( \int _{0}^{t} \Vert \Delta u \Vert _{L^{2}}^{2}\,d\tau \biggr)^{\frac{2}{p}} \\ \leq {}&C\bigl( \Vert \nabla _{h} u_{0} \Vert _{L^{2}}^{\frac{8}{3}}+ \Vert \nabla _{h} b_{0} \Vert _{L^{2}}^{ \frac{8}{3}}+1\bigr)+\frac{3}{4} \int _{0}^{t} \Vert \Delta u \Vert _{L^{2}}^{2}\,d\tau \\ &{}+C\biggl( \int _{0}^{t} \Vert u_{3} \Vert _{L^{\alpha }}^{\frac{2\alpha }{\alpha -3}} \Vert \nabla u \Vert _{L^{2}}^{2} \,d\tau \biggr)^{\frac{4(\alpha -3)}{3\alpha -10}}\\ &{}+C\biggl( \int _{0}^{t} \Vert \nabla b \Vert _{\dot{M}_{p,q}}^{\frac{2p}{2p-3}} \Vert \nabla b \Vert _{L^{2}}^{\frac{2p}{2p-3}} \Vert \nabla u \Vert _{L^{2}}^{\frac{2(p-3)}{2p-3}}\,d \tau \biggr)^{\frac{2(2p-3)}{3(p-2)}} \\ \leq {}&C\bigl( \Vert \nabla _{h} u_{0} \Vert _{L^{2}}^{\frac{8}{3}}+ \Vert \nabla _{h} b_{0} \Vert _{L^{2}}^{ \frac{8}{3}}+1\bigr)+\frac{3}{4} \int _{0}^{t} \Vert \Delta u \Vert _{L^{2}}^{2}\,d\tau \\ &{}+C\biggl( \int _{0}^{t} \Vert u_{3} \Vert _{L^{\alpha }}^{\frac{2\alpha }{\alpha -3}} \Vert \nabla u \Vert _{L^{2}}^{2} \,d\tau \biggr)^{\frac{4(\alpha -3)}{3\alpha -10}}\\ &{}+C\biggl( \int _{0}^{t} \Vert \nabla b \Vert _{\dot{M}_{p,q}}^{\frac{2p}{2p-3}}\bigl( \Vert \nabla b \Vert _{L^{2}}^{2}+ \Vert \nabla u \Vert _{L^{2}}^{2} \bigr)\,d\tau \biggr)^{ \frac{2(2p-3)}{3(p-2)}} \\ \leq{} &C\bigl( \Vert \nabla _{h} u_{0} \Vert _{L^{2}}^{\frac{8}{3}}+ \Vert \nabla _{h} b_{0} \Vert _{L^{2}}^{ \frac{8}{3}}+1\bigr)+\frac{3}{4} \int _{0}^{t} \Vert \Delta u \Vert _{L^{2}}^{2}\,d\tau \\ &{}+C\biggl( \int _{0}^{t} \Vert u_{3} \Vert _{L^{\alpha }}^{\frac{2\alpha }{\alpha -3}} \Vert \nabla u \Vert _{L^{2}}^{\frac{3\alpha -10}{2(\alpha -3)}} \Vert \nabla u \Vert _{L^{2}}^{ \frac{\alpha -2}{2(\alpha -3)}}\,d\tau \biggr)^{ \frac{4(\alpha -3)}{3\alpha -10}} \\ &{}+C\biggl( \int _{0}^{t} \Vert \nabla b \Vert _{\dot{M}_{p,q}}^{\frac{2p}{2p-3}}\bigl( \Vert \nabla b \Vert _{L^{2}}^{\frac{3(p-2)}{2p-3}}+ \Vert \nabla u \Vert _{L^{2}}^{ \frac{3(p-2)}{2p-3}} \bigr) \bigl( \Vert \nabla b \Vert _{L^{2}}^{\frac{p}{2p-3}}+ \Vert \nabla u \Vert _{L^{2}}^{\frac{p}{2p-3}}\bigr)\,d\tau \biggr)^{\frac{2(2p-3)}{3(p-2)}} \\ \leq {}&C\bigl( \Vert \nabla _{h} u_{0} \Vert _{L^{2}}^{\frac{8}{3}}+ \Vert \nabla _{h} b_{0} \Vert _{L^{2}}^{ \frac{8}{3}}+1\bigr)+\frac{3}{4} \int _{0}^{t} \Vert \Delta u \Vert _{L^{2}}^{2}\,d\tau \\ &{}+C\biggl( \int _{0}^{t} \Vert u_{3} \Vert _{L^{\alpha }}^{\frac{8\alpha }{3\alpha -10}} \Vert \nabla u \Vert _{L^{2}}^{2} \,d\tau \biggr) \biggl( \int _{0}^{t} \Vert \nabla u \Vert _{L^{2}}^{2}\,d \tau \biggr)^{\frac{\alpha -2}{3\alpha -10}} \\ &{}+C\biggl( \int _{0}^{t} \Vert \nabla b \Vert _{\dot{M}_{p,q}}^{\frac{4p}{3(p-2)}}\bigl( \Vert \nabla b \Vert _{L^{2}}^{2}+ \Vert \nabla u \Vert _{L^{2}}^{2} \bigr)\,d\tau \biggr) \biggl( \int _{0}^{t} \Vert \nabla b \Vert _{L^{2}}^{2}+ \Vert \nabla u \Vert _{L^{2}}^{2} \,d\tau \biggr)^{ \frac{p}{3(p-2)}} \\ \leq {}&C\bigl( \Vert \nabla _{h} u_{0} \Vert _{L^{2}}^{\frac{8}{3}}+ \Vert \nabla _{h} b_{0} \Vert _{L^{2}}^{ \frac{8}{3}}+1\bigr)+\frac{3}{4} \int _{0}^{t} \Vert \Delta u \Vert _{L^{2}}^{2}\,d\tau \\ &{}+C \int _{0}^{t}\bigl( \Vert u_{3} \Vert _{L^{\alpha }}^{\frac{8\alpha }{3\alpha -10}}+ \Vert \nabla b \Vert _{\dot{M}_{p,q}}^{\frac{4p}{3(p-2)}} \bigr) \bigl( \Vert \nabla u \Vert _{L^{2}}^{2}+ \Vert \nabla \omega \Vert _{L^{2}}^{2}+ \Vert \nabla b \Vert _{L^{2}}^{2}\bigr)\,d\tau. \end{aligned}$$
Inserting the above inequality into (4.20), we have
$$\begin{aligned} &\bigl( \Vert \nabla u \Vert _{L^{2}}^{2}+ \Vert \nabla \omega \Vert _{L^{2}}^{2}+ \Vert \nabla b \Vert _{L^{2}}^{2}\bigr)+ \int _{0}^{t}\bigl( \Vert \Delta u \Vert _{L^{2}}^{2}+ \Vert \Delta \omega \Vert _{L^{2}}^{2}\bigr)\,d\tau \\ &\quad\leq C\bigl( \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla \omega _{0} \Vert _{L^{2}}^{2}+ \Vert \nabla b_{0} \Vert _{L^{2}}^{2}+1\bigr) \\ &\qquad{}+C \int _{0}^{t}\bigl( \Vert \nabla b \Vert _{\dot{M}_{p,q}}^{\frac{2p}{2p-3}}+ \bigl\Vert \nabla \vert \omega \vert ^{\frac{3}{2}} \bigr\Vert _{L^{2}}^{2}+ \Vert u_{3} \Vert _{L^{\alpha }}^{ \frac{8\alpha }{3\alpha -10}}+ \Vert \nabla b \Vert _{\dot{M}_{p,q}}^{ \frac{4p}{3(p-2)}}\bigr) \\ &\qquad{}\times \bigl( \Vert \nabla u \Vert _{L^{2}}^{2}+ \Vert \nabla \omega \Vert _{L^{2}}^{2}+ \Vert \nabla b \Vert _{L^{2}}^{2}\bigr)\,d\tau, \end{aligned}$$
Gronwall’s inequality and Lemma 3.1 help to obtain
$$\begin{aligned} &\bigl( \Vert \nabla u \Vert _{L^{2}}^{2}+ \Vert \nabla \omega \Vert _{L^{2}}^{2}+ \Vert \nabla b \Vert _{L^{2}}^{2}\bigr)+ \int _{0}^{t}\bigl( \Vert \Delta u \Vert _{L^{2}}^{2}+ \Vert \Delta \omega \Vert _{L^{2}}^{2}\bigr)\,d\tau \\ &\quad\le C\bigl( \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla \omega _{0} \Vert _{L^{2}}^{2}+ \Vert \nabla b_{0} \Vert _{L^{2}}^{2}+1\bigr) \\ &\qquad{}\times \exp \biggl\{ \int _{0}^{T}\bigl( \Vert \nabla b \Vert _{\dot{M}_{p,q}}^{ \frac{2p}{2p-3}}+ \bigl\Vert \nabla \vert \omega \vert ^{\frac{3}{2}} \bigr\Vert _{L^{2}}^{2}+ \Vert u_{3} \Vert _{L^{\alpha }}^{\frac{8\alpha }{3\alpha -10}}+ \Vert \nabla b \Vert _{\dot{M}_{p,q}}^{ \frac{4p}{3(p-2)}}\bigr)\,d\tau \biggr\} \\ &\quad\le C\bigl( \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla \omega _{0} \Vert _{L^{2}}^{2}+ \Vert \nabla b_{0} \Vert _{L^{2}}^{2}+1\bigr)\\ &\qquad{}\times\exp \biggl\{ C \int _{0}^{T}(1+ \Vert u_{3} \Vert _{L^{\alpha }}^{\frac{8\alpha }{3\alpha -10}}+ \Vert \nabla b \Vert _{\dot{M}_{p,q}}^{ \frac{4p}{3(p-2)}} \,d\tau \biggr\} , \end{aligned}$$
which completes the proof of Theorem 1.1.