On the Global Well-Posedness of Rotating Magnetohydrodynamics Equations with Fractional Dissipation

Abstract

This work considers the three-dimensional incompressible rotating magnetohydrodynamics equation spaces with fractional dissipation ${(-\Delta )}^{\wp }$ for $\frac{1}{2}<\wp \le 1$. Furthermore, we use the Littlewood–Paley decomposition and frequency localization techniques to establish the global well-posedness of fractional rotating magnetohydrodynamics equations in a more generalized Besov spaces characterized by the time evolution semigroup related to the generalized linear Stokes–Coriolis operator.

1. Introduction

In this paper, we study the following three-dimensional fractional rotating magnetohydrodynamics (MHD) equations:

$$\left\{\begin{array}{cc}{u}_t+(u·\nabla )u+\mu {(-\Delta )}^{\wp }u+\Im e_3\times u+\nabla P=(B·\nabla )B\hfill & \mathrm{in}{\Re }^3\times {\Re }_{+},\hfill \\ B_t+(u·\nabla )B+\gamma {(-\Delta )}^{\wp }B=(B·\nabla )u\hfill & \mathrm{in}{\Re }^3\times {\Re }_{+},\hfill \\ divu=0,divB=0\hfill & \mathrm{in}{\Re }^3\times {\Re }_{+},\hfill \\ {\left.u\right|}_{t=0}=u_0,{\left.B\right|}_{t=0}=B_0\hfill & \mathrm{in}{\Re }^3,\hfill \end{array}\right.$$

(1)

where u is the incompressible velocity field, ℑ denotes the speed of rotation around a vertical unit vector $e_3=(0,0,1)$, $P=p+\frac{1}{2}{\left|B\right|}^2$ in which B is the magnetic field and p is pressure, and $\mu$ and $\gamma$ represent the viscosity coefficient and diffusion of the magnetic field, respectively. For convenience, we use $\mu =\gamma =1.$

For $\wp =1$, Equation (1) explains why the Earth has a nonzero large-scale magnetic field, the polarity of which appears to change over several hundred centuries.

When $\Im =0$, Equation (1) becomes the following fractional MHD equations:

$$\left\{\begin{array}{cc}{u}_t+(u·\nabla )u+{(-\Delta )}^{\wp }u+\nabla P=(B·\nabla )B\hfill & \mathrm{in}{\Re }^3\times {\Re }_{+},\hfill \\ B_t+(u·\nabla )B+{(-\Delta )}^{\wp }B=(B·\nabla )u\hfill & \mathrm{in}{\Re }^3\times {\Re }_{+},\hfill \\ divu=0,divB=0\hfill & \mathrm{in}{\Re }^3\times {\Re }_{+},\hfill \\ {\left.u\right|}_{t=0}=u_0,{\left.B\right|}_{t=0}=B_0\hfill & \mathrm{in}{\Re }^3.\hfill \end{array}\right.$$

(2)

Equation (2) studies the magnetic characteristics of electrically conducted fluids. Since Duvaut and Lions [1] developed a globally Leray–Hopf weak solution and a locally strong solution to the 3D inviscid MHD equations, the MHD equations remain a challenging open problem in terms of determining whether there is always a globally smooth solution for smooth initial data. Sermange and Temam further investigated the properties and conditions of these solutions [2]. Melo and Santos [3] proved the existence and decay rates of a unique global asymptotic solution for Equation (2) in Sobolev–Gevrey spaces, where Zhao and Li [4] obtained the time decay rate of weak solutions to (2) in ${\mathbb{R}}^2$. Liu et al. [5] obtained the existence of attractors for (2) with damping and overcame the main difficulty in dealing with the nonlinear term. For the detailed study related to the existence of a solution for fractional MHD equations, we refer the readers to [6,7,8,9,10].

When $B=0,\Im \ne 0$ and $\wp =1$, Equation (1) corresponds to the following Navier–Stokes equations with Coriolis force:

$$\left\{\begin{array}{cc}{u}_t+(u·\nabla )u-\Delta u+\Im e_3\times u+\nabla P=0\hfill & \mathrm{in}{\Re }^3\times {\Re }_{+},\hfill \\ divu=0,divB=0\hfill & \mathrm{in}{\Re }^3\times {\Re }_{+},\hfill \\ {\left.u\right|}_{t=0}=u_0,\hfill & \mathrm{in}{\Re }^3.\hfill \end{array}\right.$$

(3)

Equation (3) has received considerable attention due to its importance in geophysical flows. The existence of a solution for Equation (3) starts with the work of Babin et al. [11,12,13], who established the global existence and regularity of solutions for Equation (3) with periodic initial velocity, when the rotation speed ℑ is sufficiently large. Under the smallness condition of initial data $u_0$, Giga et al. [14] proved the uniform mild solution in $F{M}_0^{-1}\left({\Re }^3\right)$. The uniform global well-posedness was established by Hieber and Shibata [15] in $H^{\frac{1}{2}}\left({\Re }^3\right)$. More recently, for the fractional case $\frac{1}{2}<\wp \le 1$, Wang and Wu [16] obtained the global well-posedness for small initial data belonging to Lei–Lin spaces ${\mathcal{X}}^s$.

When $B=0,\Im =0$ and $\wp =1$, Equation (1) becomes the following classical Navier–Stokes equations:

$$\left\{\begin{array}{cc}{u}_t+(u·\nabla )u-\Delta u+\nabla P=0\hfill & \mathrm{in}{\Re }^3\times {\Re }_{+},\hfill \\ divu=0,divB=0\hfill & \mathrm{in}{\Re }^3\times {\Re }_{+},\hfill \\ {\left.u\right|}_{t=0}=u_0,\hfill & \mathrm{in}{\Re }^3.\hfill \end{array}\right.$$

(4)

Equation (4) is one of the most fundamental mathematical model in fluid dynamics. Numerous publications have addressed the global well-posedness of Equation (4) in critical spaces. This work originates with the work of Fujita and Kato [17], who established the existence of a strong solution to the Cauchy problem of Equation (4) under the smallness condition of $u_0$ in $H^{-1+\frac{n}{2}}\left({\Re }^n\right)$. In [18], they rewrote Equation (4) to an integral form and obtained the local well-posedness in some Sobolev and Lebesgue spaces. These results led to an intensive study of well-posedness for Navier–Stokes equations in various function spaces in recent years, such as [19,20,21,22]. As far as the fractional case is concerned, many authors have studied the existence of solutions. For reference we refer the readers to [23,24,25].

This paper considers the global well-posedness of Equation (1) in some function space characterized by the time evolution semigroup $T_{\Im ,\wp }\left(t\right)$. We obtain the global well-posedness of Equation (1) in scaling subcritical spaces $X_{\Im }^{s,p,\pi }\left({\Re }^3\right)$ for $\frac{1}{2}<\wp \le 1$, $\frac{2\wp }{\pi }+\frac{3}{p}<s+(2\wp -1)$ and the critical spaces $X_{\Im }^p\left({\Re }^3\right)$ for $\frac{7}{8}<\wp \le \frac{5}{4},\frac{3}{2\wp -1}<p\le 4$. The following are the main results of the paper.

Theorem 1.

Let $\wp ,s,p$, and π satisfy

$$\begin{array}{c}\frac{1}{2}<\wp \le 1,3-3\wp <s<\frac{3(15-4\wp )}{2(9+8\wp )},\\ \frac{1}{3}+\frac{s}{9}\le \frac{1}{p}<min\left\{\frac{1}{4}+\frac{5}{16\wp }-\frac{s}{8\wp },\frac{2\wp -1}{3}+\frac{s}{3}\right\},\\ max\left\{0,\frac{2}{p}-\frac{1}{2\wp }\left(1+\frac{3}{p}-s\right)\right\}<\frac{1}{\pi }<min\left\{\frac{1}{2},1-\frac{1}{2\wp }\left(1+\frac{3}{p}-s\right),\frac{1}{2}+\frac{1}{8\wp }-\frac{3}{2\wp p}+\frac{s}{4\wp }\right\}.\end{array}$$

Then, for $u_0,B_0\in X_{\Im }^{s,p,\pi }{\left({\Re }^3\right)}^3\cap {\dot{H}}^s{\left({\Re }^3\right)}^3$ with $div{u}_0=0,div{B}_0=0$ and

$${∥u_0∥}_{X_{\Im }^{s,p,\pi }}+{∥B_0∥}_{X_{\Im }^{s,p,\pi }}\lesssim {|\Im |}^{1-\frac{1}{2\wp }\left(1+\frac{3}{p}-s\right)-\frac{1}{\pi }},$$

Equation (1) has a unique global solution

$$u,B\in L^{\pi }\left(0,\infty ;{\dot{W}}^{s,p}{\left({\Re }^3\right)}^3\right)\cap C\left([0,\infty );{\dot{H}}^s{\left({\Re }^3\right)}^3\right)$$

such that $divu=0$ and $divB=0$.

Theorem 2.

Let $\wp ,s,p$ satisfy

$$\frac{7}{8}<\wp \le \frac{5}{4},0\le s<2\wp -1,max\left\{\frac{1}{4},\frac{s}{3}\right\}\le \frac{1}{p}<\frac{2\wp -1}{3}$$

Then, for $u_0,B_0\in X_{\Im }^p{\left({\Re }^3\right)}^3\cap {\dot{H}}^s{\left({\Re }^3\right)}^3$ with $div{u}_0=0$, $B_0=0$ and

$${∥u_0∥}_{X_{\Im }^p}+{∥B_0∥}_{X_{\Im }^p}\le \delta ,$$

Equation (1) has a unique global solution

$$u,B\in C\left([0,\infty );{\dot{H}}^s{\left({\Re }^3\right)}^3\right)$$

such that

$$\underset{t>0}{sup}{t}^{\frac{1}{2\wp }\left(1-\frac{3}{p}\right)}{\parallel u\left(t\right)\parallel }_{L^p}+\underset{t>0}{sup}{t}^{\frac{1}{2\wp }\left(1-\frac{3}{p}\right)}{\parallel B\left(t\right)\parallel }_{L^p}\le 2\left({∥u_0∥}_{X_{\Im }^p}+{∥B_0∥}_{X_{\Im }^p}\right),$$

$divu=0$ and $divB=0$.

Throughout this paper we write $f\lesssim g$ to denote $f\le Kg$, where K is positive constant.

2. Preliminaries

This section gives a brief review of important definitions and useful lemmas. By substituting $B=0$ into Equation (1), we get the fractional Navier–Stokes equations with Coriolis force. We need to define an equivalent fractional Stokes–Coriolis semigroup $T_{\Im ,\wp }$. Therefore, we consider the following fractional linear problem:

$$\left\{\begin{array}{cc}{u}_t+{(-\Delta )}^{\wp }u+\Im e_3\times u+\nabla p=0\hfill & \mathrm{in}{\Re }^3\times {\Re }_{+}\hfill \\ divu=0\hfill & \mathrm{in}{\Re }^3\times {\Re }_{+}\hfill \\ {\left.u\right|}_{t=0}=u_0\hfill & \mathrm{in}{\Re }^3\hfill \end{array}\right.$$

(5)

The solution of Equation (5) can be obtained using the fractional Stokes–Coriolis semigroup $T_{\Im ,\wp }$, which has an explicit representation of the following form [15,16]:

$$\begin{array}{cc}\hfill T_{\Im ,\wp }\left(t\right)u& ={\mathcal{F}}^{-1}\left[cos\left(\Im \frac{{\eta }_3}{\left|\eta \right|}t\right)I+sin\left(\Im \frac{{\eta }_3}{\left|\eta \right|}t\right)R\left(\eta \right)\right]*\left({\mathrm{e}}^{-{(-\Delta )}^{\wp }t}u\right),\hfill \end{array}$$

(6)

where I denotes the unit matrix in $M_{3\times 3}(\Re )$ and $N\left(\eta \right)$ denotes the skew-symmetric matrix given by

$$R\left(\eta \right):=\frac{1}{\left|\eta \right|}\left(\begin{array}{ccc}0& {\eta }_3& -{\eta }_2\\ -{\eta }_3& 0& {\eta }_1\\ {\eta }_2& -{\eta }_1& 0\end{array}\right),\eta \in {\Re }^3\setminus \left\{0\right\}.$$

Therefore, we can write the semigroup as

$${\mathcal{A}}_{\Im ,\wp }\left(t\right)=\left(\begin{array}{cc}{T}_{\Im ,\wp }\left(t\right)& 0\\ 0& S_{\wp }\left(t\right)\end{array}\right),$$

where $S_{\wp }\left(t\right):={\mathrm{e}}^{-{(-\Delta )}^{\wp }t}={\mathcal{F}}^{-1}\left({\mathrm{e}}^{-{\left|\eta \right|}^{2\wp }t}\right)$. Using the semigroup ${\mathcal{A}}_{\Im ,\wp }\left(t\right)$, we can transform Equation (1) into the following integral form:

$$\begin{array}{ccc}\hfill \left(\begin{array}{c}u\\ B\end{array}\right)=& {\mathcal{A}}_{\Im ,\wp }\left(t\right)\left(\begin{array}{c}{u}_0\\ B_0\end{array}\right)\hfill & \hfill -{\int }_0^t{\mathcal{A}}_{\Im ,\wp }(t-y)\aleph \left(\begin{array}{c}div(u\otimes u-B\otimes B)\\ div(u\otimes B-B\otimes u)\end{array}\right)\left(y\right)dy,\end{array}$$

(7)

where $\aleph =I-\nabla {(-\Delta )}^{-1}\mathrm{div}$ is the Leray–Hopf projection.

If $(u,B)$ satisfies Equation (7) in a suitable function space, then $(u,B)$ is a solution to Equation (1).

Now, we give the definitions of the function spaces $X_{\Im }^{s,p,\pi }\left({\Re }^3\right)$ and $X_{\Im }^p\left({\Re }^3\right)$ of Besov type, which are generated by the linear semigroup $T_{\Im ,\wp }$. The set of all tempered distributions is denoted by ${\mathcal{S}}^{\prime }\left({\Re }^3\right)$. Let $-\infty <s,\Im <\infty$ and the function spaces $X_{\Im }^{s,p,\pi }\left({\Re }^3\right)$ and $X_{\Im }^p\left({\Re }^3\right)$ are defined as

$$X_{\Im }^{s,p,\pi }\left({\Re }^3\right)=\{u\in {\mathcal{S}}^{\prime }{|\parallel u\parallel }_{X_{\Im }^{s,p,\pi }}<\infty \},$$

$${\parallel u\parallel }_{X_{\Im }^{s,p,\pi }}={\parallel T_{\Im }\left(t\right)u\parallel }_{L_t^{\pi }(0,\infty ;{\dot{W}}_x^{s,p}\left({\Re }^3\right))}$$

and

$$X_{\Im }^p\left({\Re }^3\right)=\{u\in {\mathcal{S}}^{\prime }{|\parallel u\parallel }_{X_{\Im }^p}<\infty \},$$

$${\parallel u\parallel }_{X_{\Im }^p}=\underset{t>0}{sup}{t}^{\frac{1}{2\wp }\left(1-\frac{3}{p}\right)}{∥T_{\Im }\left(t\right)u∥}_{L^p}.$$

When $\Im =0$, $T_{\Im ,\wp }\left(t\right)$ becomes $T_{0,\wp }\left(t\right)u=e^{-t{(-\Delta )}^{\wp }}\left(t\right)u$. By [26], for $1\le \pi <\infty$, we have

$${\parallel u\parallel }_{X_0^{s,p,\pi }}=\parallel t^{\frac{1}{\pi }}{\parallel (-\Delta )}^{\frac{s}{2}}{e}^{-t{(-\Delta )}^{\wp }}{u\parallel }_{L^p}{\parallel }_{L^{\pi }(0,\infty ;\frac{dt}{t})}{\simeq \parallel (-\Delta )}^{\frac{s}{2}}{u\parallel }_{{\dot{B}}_{p,\pi }^{-\frac{2\wp }{\pi }}}={\parallel u\parallel }_{{\dot{B}}_{p,\pi }^{s-\frac{2\wp }{\pi }}}.$$

(8)

For $3<p\le \infty$, we see that

$${\parallel u\parallel }_{X_0^p}=\parallel t^{\frac{1}{2\wp }(1-\frac{3}{p})}\parallel e^{-t{(-\Delta )}^{\wp }}{u\parallel }_{L^p}{\parallel }_{L^{\pi }(0,\infty ;\frac{dt}{t})}\simeq {\parallel u\parallel }_{{\dot{B}}_{p,\pi }^{-1+\frac{3}{p}}}.$$

(9)

Hence, the function spaces and $X_{\Im }^{s,p,\pi }\left({\Re }^3\right)$ and $X_{\Im }^p\left({\Re }^3\right)$ can be considered as a generalization of the Besov spaces ${\dot{B}}_{p,\pi }^{s-\frac{2\wp }{\pi }}$ and ${\dot{B}}_{p,\pi }^{-1+\frac{3}{p}}$, respectively.

Next, we recall the Littlewood–Paley decomposition tool. Let $\mathcal{S}\left({\Re }^3\right)$ be a Schwartz space and let $\psi \in \mathcal{S}\left({\Re }^3\right)$ satisfy the following conditions:

$$0\le {\widehat{\psi }}_0\left(\eta \right)\le 1\mathrm{for}\mathrm{all}\eta \in {\Re }^3,$$

$$supp{\widehat{\psi }}_0\subset :=\left\{\eta \in {\Re }^3:\frac{1}{2}\le \left|\eta \right|\le 2\right\}$$

and

$$\sum _{j\in \mathbb{Z}}{\widehat{\psi }}_0\left(2^{-j}\eta \right)=1\mathrm{for}\mathrm{all}\eta \in {\Re }^3\setminus \left\{0\right\},$$

where ${\psi }_j\left(x\right):=2^{3j}{\psi }_0\left(2^{j}x\right)$ and ${\Delta }_{j}u:={\psi }_j*u$ for $u\in {\mathcal{S}}^{\prime }\left({\Re }^3\right)$ and $j\in \mathbb{Z}$.

Now, we define the homogeneous Besov space ${\dot{B}}_{p,q}^s\left({\Re }^3\right)$. For $s\in \Re$ and $p,q\in [1,\infty ]$

$$\begin{array}{c}{\dot{B}}_{p,q}^s\left({\Re }^3\right):=\left\{u\in {\mathcal{S}}^{\prime }\left({\Re }^3\right)\mid {\parallel u\parallel }_{{\dot{B}}_{p,q}^s}<+\infty \right\},\\ {\parallel u\parallel }_{{\dot{B}}_{p,q}^s}:={\left(\sum _{j\in \mathbb{Z}}{2}^{jsq}{∥{\Delta }_{j}u∥}_{L^p}^q\right)}^{\frac{1}{q}}.\end{array}$$

Further, we write the operator $\mathcal{G}$ as

$${\mathcal{G}}_{\pm }\left(t\right)u\left(x\right):=e^{\pm it\frac{D_3}{\left|D\right|}}u\left(x\right):={\int }_{{\Re }^3}{e}^{ix·{\eta }^2\pm it\frac{{\eta }_3}{\left|\eta \right|}}\widehat{u}\left(\eta \right)d\eta \mathrm{for}x\in {\Re }^3\mathrm{and}t\in \Re .$$

So the operator $T_{\Im ,\wp }$ becomes

$$T_{\Im ,\wp }\left(t\right)u=\frac{1}{2}\mathcal{G}+(\Im t)\left[e^{-t{(-\Delta )}^{\wp }}(I+J)u\right]+\frac{1}{2}\mathcal{G}-(\Im t)\left[e^{-t{(-\Delta )}^{\wp }}(I-J)u\right],$$

where J is the singular integral operator matrix defined as

$$J:=\left(\begin{array}{ccc}0& R_3& -R_2\\ -R_3& 0& R_1\\ R_2& -R_1& 0\end{array}\right),$$

where $R_k$ for $k=1,2,3$ is the Riesz transforms in ${\Re }^3$. Next, we give the following lemmas as important tools to prove our main results.

Lemma 1

([26]). Let $u\in L^r\left({\Re }^n\right)$ and $1\le r\le p\le \infty$. Then, for $\wp ,\beta >0$ we have

$${∥e^{-t{(-\Delta )}^{\wp }}u∥}_{L^p}\lesssim t^{-\frac{n}{2\wp }\left(\frac{1}{r}-\frac{1}{p}\right)}{\parallel u\parallel }_{L^r},$$

and

$${∥{(-\Delta )}^{\frac{\beta }{2}}{e}^{-t{(-\Delta )}^{\wp }}u∥}_{L^p}\lesssim t^{-\frac{\beta }{2\wp }-\frac{n}{2\wp }\left(\frac{1}{r}-\frac{1}{p}\right)}{\parallel u\parallel }_{L^r}.$$

Lemma 2

([27]). Let $1\le p,q\le \infty$ and $-\infty <s_0\le s_1<\infty$. Then,

$${∥e^{-t{(-\Delta )}^{\wp }}u∥}_{{\dot{B}}_{p,q}^{s_1}}\lesssim t^{-\frac{1}{2\wp }\left(s_1-s_0\right)}{\parallel u\parallel }_{{\dot{B}}_{p,q}^{s_0}}.$$

Lemma 3

([28]). Let $1\le p_0\le p_1\le \infty ,1\le q\le \infty$ and $-\infty <s_0\le s_1<\infty$. Then,

$${∥e^{-t{(-\Delta )}^{\wp }}u∥}_{{\dot{B}}_{p_1,q}^{s_1}}\lesssim t^{-\frac{1}{2\wp }\left(s_1-s_0\right)-\frac{3}{2\wp }\left(\frac{1}{p_0}-\frac{1}{p_1}\right)}{\parallel u\parallel }_{{\dot{B}}_{p_0,\mathcal{A}}^{s_0}}.$$

Lemma 4 is used to obtain the dispersive estimate for the operator ${\mathcal{G}}_{\pm }\left(y\right)$.

Lemma 4

([29]). Let $y,s\in \Re$, $2\le p\le \infty$ and $1\le q\le \infty$. Then,

$${∥{\mathcal{G}}_{\pm }\left(y\right)u∥}_{{\dot{B}}_{p,q}^s}\lesssim {(1+|y\left|\right)}^{-\left(1-\frac{2}{p}\right)}{\parallel u\parallel }_{{\dot{B}}_{p^{\prime },q}^{s+3(1-2)}}$$

where $\frac{1}{p}+\frac{1}{p^{\prime }}=1$.

Lemma 5.

Let $1\le p_0\le 2\le p_1\le \infty$, $1\le q\le \infty .$ and $-\infty <s_0\le s_1<\infty .$ Then,

$${∥T_{\Im ,\wp }\left(t\right)u∥}_{{\dot{B}}_{p_1,q}^{s_1}}\lesssim t^{-\frac{1}{2\wp }\left(s_1-s_0\right)-\frac{3}{2a}\left(\frac{1}{p_0}-\frac{1}{p_1}\right)}{\parallel u\parallel }_{{\dot{B}}_{p_0,q}^{s_0}}.$$

Proof. 

According to Plancherel’s theorem, $\mathcal{R}$ is bounded in $L^2\left({\Re }^3\right)$, and using the semigroup estimate $e^{-t{(-\Delta )}^{\wp }}$ in $L^{p_0}-L^2$ and Lemmas 3, 4 we have

$$\begin{array}{cc}\hfill {∥e^{-\frac{t}{2}{(-\Delta )}^{\wp }}{\mathcal{G}}_{\pm }(\Im t)\left[e^{-\frac{t}{2}{(-\Delta )}^{\wp }}(I\pm \mathcal{R})u\right]∥}_{{\dot{B}}_{p_1,q}^{s_1}}\lesssim & t^{-\frac{1}{2\wp }\left(s_1-s_0\right)-\frac{3}{2\wp }\left(\frac{1}{2}-\frac{1}{p_1}\right)}{∥{\mathcal{G}}_{\pm }(\Im t)\left[e^{-\frac{t}{2}{(-\Delta )}^{\wp }}u\right]∥}_{{\dot{B}}_{2,q}^{s_0}}\hfill \\ \hfill \lesssim & t^{-\frac{1}{2a}\left(s_1-s_0\right)-\frac{3}{2\wp }\left(\frac{1}{2}-\frac{1}{p_1}\right)}{∥e^{-\frac{t}{2}{(-\Delta )}^{\wp }}u∥}_{{\dot{B}}_{2,q}^{s_0}}\hfill \\ \hfill \lesssim & t^{-\frac{1}{2\wp }\left(s_1-s_0\right)-\frac{3}{2\wp }\left(\frac{1}{p_0}-\frac{1}{p_1}\right)}{\parallel u\parallel }_{{\dot{B}}_{p_0,q}^{s_0}}.\hfill \end{array}$$

□

Lemma 6.

Let $1<p_0\le 2\le p_1<\infty$ and $0\le s<\infty$. Then,

$${∥{\partial }_x^{\beta }{T}_{\Im ,\wp }\left(t\right)u∥}_{H^s}\lesssim t^{-\frac{1}{2\wp }(s+|\beta \left|\right)-\frac{3}{2a}\left(\frac{1}{p_0}-\frac{1}{2}\right)}{\parallel u\parallel }_{L^{p_0}},$$

and

$${∥{\partial }_x^{\beta }{T}_{\Im ,\wp }\left(t\right)u∥}_{L^{p_1}}\lesssim t^{-\frac{\beta }{2\wp }-\frac{3}{2\wp }\left(\frac{1}{p_0}-\frac{1}{p_1}\right)}{\parallel u\parallel }_{L^{p_0}}.$$

Proof. 

By using Lemma 4 and the embeddings

$$L^p\left({\Re }^3\right)\hookrightarrow {\dot{B}}_{p,2}^0\left({\Re }^3\right),$$

$${\dot{B}}_{p,2}^0\left({\Re }^3\right)\hookrightarrow L^p\left({\Re }^3\right),\mathrm{for}p\in (1,2]$$

and

$${\dot{H}}^s\left({\Re }^3\right)={\dot{B}}_{2,2}^s,\mathrm{for}p\in [2,\infty ),$$

we can easily obtain the result. □

Lemma 7

([27]). Let $t>0,\Im \in \Re$ and $s\in \Re$. Then,

$${∥T_{\Im ,\wp }\left(t\right)u∥}_{{\dot{H}}^s}\lesssim {\parallel u\parallel }_{{\dot{H}}^s}.$$

Lemma 8.

Let $\wp ,p,q$ and π satisfy $\frac{1}{2}<\wp <\frac{5}{4},2<p<\frac{3}{2-\wp }$ and $1-\frac{1}{p}\le \frac{1}{q}<\frac{2\wp -1}{3}+\frac{1}{p}$, $max\left\{0,1-\frac{1}{2\wp }-\frac{3}{2\wp }\left(\frac{1}{q}-\frac{1}{p}\right)-\left(1-\frac{2}{p}\right)\right\}<\frac{1}{\pi }\le min\left\{\frac{1}{2},1-\frac{1}{2\wp }-\frac{3}{2\wp }\left(\frac{1}{q}-\frac{1}{p}\right)\right\}$. Then,

$${∥{\int }_0^t{T}_{\Im }(t-y)\aleph \nabla u\left(y\right)dy∥}_{L^{\pi }\left(0,\infty ;L^p\right)}\lesssim {|\Im |}^{-\left\{1-\frac{1}{2a}-\frac{3}{2a}\left(\frac{1}{q}-\frac{1}{p}\right)-\frac{1}{\pi }\right\}}{\parallel u\parallel }_{L^{\frac{\pi }{2}}\left(0,\infty ;L^q\right)}$$

(10)

for $u,B\in L^{\frac{\pi }{2}}\left(0,\infty ;L^q\right)\left({\Re }^3\right).$

Proof. 

By considering the boundedness of ℵ in $L^q\left({\Re }^3\right)$, using the embedding ${\dot{B}}_{p,2}^0\left({\Re }^3\right)\hookrightarrow L^p\left({\Re }^3\right)$ for $p\in [2,\infty )$, applying Lemmas 1, 3, and 4, we get

$${∥{\int }_0^t{T}_{\Im }(t-y)\aleph \nabla u\left(y\right)dy∥}_{L^{\pi }\left(0,\infty ;L^p\right)}\lesssim {∥{\int }_0^t{k}_{\Im }(t-y){\parallel u\left(y\right)\parallel }_{L^q}dy∥}_{L_t^{\pi }{(0,\infty )}^{\prime }}$$

where

$$k_{\Im }\left(t\right):={\{1+|\Im \left|t\right\}}^{-\left(1-\frac{2}{p}\right)}{t}^{-\frac{3}{2a}\left(\frac{1}{q}-\frac{1}{p}\right)-\frac{1}{2a}}$$

If $\frac{1}{\pi }<1-\frac{1}{2\wp }-\frac{3}{2\wp }\left(\frac{1}{q}-\frac{1}{p}\right)$, we have ${∥k_{\Im }∥}_{L^{\prime }}\lesssim {|\Im |}^{-\left\{1-\frac{1}{2\wp }-\frac{3}{2\wp }\left(\frac{1}{q}-\frac{1}{p}\right)-\frac{1}{\pi }\right\}}$. By using Young’s inequality, $\frac{1}{\pi }=\frac{1}{{\pi }^{\prime }}+\frac{2}{\pi }-1$, we obtain inequality (10). □

In order to prove the bilinear term in Equation (7), the following lemma is significant.

Lemma 9

([29]). Let $p,q$, and s satisfy the conditions $\frac{s}{3}<\frac{1}{p}<\frac{1}{2}+\frac{s}{6},\frac{1}{q}=\frac{2}{p}-\frac{s}{3}$ and $0\le s<3.$ Then,

$${\parallel uv\parallel }_{{\dot{W}}^{s,q}}\lesssim {\parallel u\parallel }_{{\dot{W}}^{s,p}}{\parallel v\parallel }_{{\dot{W}}^{s,p}},$$

where $u,v\in {\dot{W}}^{s,p}\left({\Re }^3\right)$.

Proof of Theorem 1: 

Let us define ${\parallel ·\parallel }_{Z_1}:={\parallel ·\parallel }_{L^{\pi }\left(0,\infty ;W^{s,p}{\left({\Re }^3\right)}^3\right)}$. It is easy to get

$${∥{\mathcal{A}}_{\Im ,\wp }\left(t\right)\left(\begin{array}{c}{u}_0\\ B_0\end{array}\right)∥}_{Z_1}\le {∥T_{\Im ,\wp }\left(t\right)u_0∥}_{Z_1}+\parallel S_{\wp }\left(t\right)B_0{\parallel }_{Z_1}=\parallel u_0{\parallel }_{X_{\Im }^{s,p,\pi }}+{\parallel B_0\parallel }_{X_{\Im }^{s,p,\pi }}.$$

(11)

Next, we write a complete metric space $({\mathcal{X}}_1,d_1)$ with mapping $\mathcal{J}$ as:

$$\begin{array}{cc}\hfill & {\mathcal{X}}_1:=\left\{u,B\in L^{\pi }\left(0,\infty ;{\dot{W}}^{s,p}{\left({\Re }^3\right)}^3\right)\mid {\parallel u\parallel }_{Z_1}⩽2{∥u_0∥}_{X_{\Im }^{s,p,\pi }},{\parallel B\parallel }_{Z_1}⩽2{∥B_0∥}_{X_{\Im }^{s,p,\pi }}\right\},\hfill \\ \hfill & d_1\left(\left(\begin{array}{c}u\\ B\end{array}\right),\left(\begin{array}{c}v\\ b\end{array}\right)\right):={\parallel u-v\parallel }_{Z_1}+{\parallel B-b\parallel }_{Z_1},\hfill \\ \hfill & \mathcal{J}\left(\begin{array}{c}u\\ B\end{array}\right)\left(t\right):={\mathcal{A}}_{\Im ,\wp }\left(\begin{array}{c}{u}_0\\ B_0\end{array}\right)\left(t\right)-B\left(\left(\begin{array}{c}u\\ B\end{array}\right),\left(\begin{array}{c}u\\ B\end{array}\right)\right),\hfill \end{array}$$

where

$$B\left(\left(\begin{array}{c}u\\ B\end{array}\right),\left(\begin{array}{c}u\\ B\end{array}\right)\right)={\int }_0^t{\mathcal{A}}_{\Im ,\wp }(t-y)\aleph \left(\begin{array}{c}div(u\otimes u-B\otimes B)\\ div(u\otimes B-B\otimes u)\end{array}\right)ydy.$$

(12)

We can write

$$\begin{array}{cc}\hfill {∥\mathcal{J}\left(\begin{array}{c}u\\ B\end{array}\right)\left(t\right)∥}_{Z_1}& \le {∥{\mathcal{A}}_{\Im ,\wp }\left(\begin{array}{c}{u}_0\\ B_0\end{array}\right)\left(t\right)∥}_{Z_1}+{∥B\left(\left(\begin{array}{c}u\\ B\end{array}\right),\left(\begin{array}{c}u\\ B\end{array}\right)\right)∥}_{Z_1}=I_1+I_2.\hfill \end{array}$$

(13)

$I_1$ is estimated in inequality (11). To estimate $I_2$, let $\frac{1}{q}=\frac{2}{p}-\frac{s}{3}$, using Lemmas 7, 8, and Holder’s inequality, it is easy to obtain that

$$\begin{array}{cc}\hfill {∥{\int }_0^t{T}_{\Im ,\wp }(t-y)\aleph div(u\otimes u)\left(y\right)dy∥}_{Z_1}& \lesssim {|\Im |}^{-\left\{1-\frac{1}{2\wp }\left(1+\frac{3}{p}-s\right)-\frac{1}{\pi }\right\}}{\parallel u\parallel }_{Z_1}^2\hfill \\ \hfill & \lesssim {4|\Im |}^{-\left\{1-\frac{1}{2\wp }\left(1+\frac{3}{p}-s\right)-\frac{1}{\pi }\right\}}{∥u_0∥}_{X_{\Im }^{s,p,\pi }}^2.\hfill \end{array}$$

(14)

Similarly, we have

$${∥{\int }_0^t{T}_{\Im ,\wp }(t-y)\aleph div(B\otimes B)\left(y\right)dy∥}_{Z_1}\lesssim 4{|\Im |}^{-\left\{1-\frac{1}{2\wp }\left(1+\frac{3}{p}-s\right)-\frac{1}{\pi }\right\}}{∥B_0∥}_{X_{\Im }^{s,p,\pi }}^2,$$

(15)

$${∥{\int }_0^t{S}_{\wp }(t-y)\aleph div(u\otimes B)\left(y\right)dy∥}_{Z_1}\lesssim 4{|\Im |}^{-\left\{1-\frac{1}{2\wp }\left(1+\frac{3}{p}-s\right)-\frac{1}{\pi }\right\}}{∥u_0∥}_{X_{\Im }^{s,p,\pi }}{∥B_0∥}_{X_{\Im }^{s,p,\pi }},$$

(16)

and

$${∥{\int }_0^t{S}_{\wp }(t-y)\aleph div(B\otimes u)\left(y\right)dy∥}_{Z_1}\lesssim 4{|\Im |}^{-\left\{1-\frac{1}{2\wp }\left(1+\frac{3}{p}-s\right)-\frac{1}{\pi }\right\}}{∥B_0∥}_{X_{\Im }^{s,p,\pi }}{∥u_0∥}_{X_{\Im }^{s,p,\pi }}.$$

(17)

So from Equation (13), we have

$$\begin{array}{cc}\hfill {∥\mathcal{J}\left(\begin{array}{c}u\\ B\end{array}\right)\left(t\right)∥}_{Z_1}& \lesssim \parallel u_0{\parallel }_{X_{\Im }^{s,p,\pi }}+\parallel B_0{\parallel }_{X_{\Im }^{s,p,\pi }}+4{|\Im |}^{-\left\{1-\frac{1}{2\wp }\left(1+\frac{3}{p}-s\right)-\frac{1}{\pi }\right\}}{∥u_0∥}_{X_{\Im }^{s,p,\pi }}^2\hfill \\ & +{4|\Im |}^{-\left\{1-\frac{1}{2\wp }\left(1+\frac{3}{p}-s\right)-\frac{1}{\pi }\right\}}{∥B_0∥}_{X_{\Im }^{s,p,\pi }}^2\hfill \\ & +{4|\Im |}^{-\left\{1-\frac{1}{2\wp }\left(1+\frac{3}{p}-s\right)-\frac{1}{\pi }\right\}}{∥u_0∥}_{X_{\Im }^{s,p,\pi }}{∥B_0∥}_{X_{\Im }^{s,p,\pi }}\hfill \\ & +{4|\Im |}^{-\left\{1-\frac{1}{2\wp }\left(1+\frac{3}{p}-s\right)-\frac{1}{\pi }\right\}}{∥B_0∥}_{X_{\Im }^{s,p,\pi }}{∥u_0∥}_{X_{\Im }^{s,p,\pi }}\hfill \\ & =\parallel u_0{\parallel }_{X_{\Im }^{s,p,\pi }}\left(1+{4|\Im |}^{-\left\{1-\frac{1}{2\wp }\left(1+\frac{3}{p}-s\right)-\frac{1}{\pi }\right\}}\left({∥u_0∥}_{X_{\Im }^{s,p,\pi }}+{∥B_0∥}_{X_{\Im }^{s,p,\pi }}\right)\right)\hfill \\ & +\parallel B_0{\parallel }_{X_{\Im }^{s,p,\pi }}\left(1+{4|\Im |}^{-\left\{1-\frac{1}{2\wp }\left(1+\frac{3}{p}-s\right)-\frac{1}{\pi }\right\}}\left({∥B_0∥}_{X_{\Im }^{s,p,\pi }}+{∥u_0∥}_{X_{\Im }^{s,p,\pi }}\right)\right).\hfill \end{array}$$

(18)

We can also write

$$\begin{array}{cc}\hfill & {∥\mathcal{J}\left(\begin{array}{c}u\\ B\end{array}\right)\left(t\right)-\mathcal{J}\left(\begin{array}{c}v\\ b\end{array}\right)\left(t\right)∥}_{Z_1}\hfill \\ & ={∥B\left(\left(\begin{array}{c}u\\ B\end{array}\right),\left(\begin{array}{c}u-v\\ B-b\end{array}\right)\right)+B\left(\left(\begin{array}{c}u-v\\ B-b\end{array}\right),\left(\begin{array}{c}u\\ B\end{array}\right)\right)∥}_{Z_1}\hfill \\ & \le {|\Im |}^{-\left\{1-\frac{1}{2\wp }\left(1+\frac{3}{p}-s\right)-\frac{1}{\pi }\right\}}\left({\parallel u\parallel }_{Z_1}+{\parallel v\parallel }_{Z_1}+{\parallel B\parallel }_{Z_1}+{\parallel b\parallel }_{Z_1}\right)\left({\parallel u-v\parallel }_{Z_1}+{\parallel B-b\parallel }_{Z_1}\right)\hfill \\ & \lesssim {4|\Im |}^{-\left\{1-\frac{1}{2\wp }\left(1+\frac{3}{p}-s\right)-\frac{1}{\pi }\right\}}\left(\parallel u_0{\parallel }_{X_{\Im }^{s,p,\pi }}+{\parallel B_0\parallel }_{X_{\Im }^{s,p,\pi }}\right)\left({\parallel u-v\parallel }_{Z_1}+{\parallel B-b\parallel }_{Z_1}\right).\hfill \end{array}$$

(19)

Let us consider that $u_0,B_0\in X_{\Im }^{s,p,\pi }{\left({\Re }^3\right)}^3\cap {\dot{H}}^s{\left({\Re }^3\right)}^3$ satisfies

$$\underset{\Im \in \Re \setminus \left\{0\right\}}{sup}{|\Im |}^{-\left\{1-\frac{1}{2\wp }\left(1+\frac{3}{p}-s\right)-\frac{1}{\pi }\right\}}\left({∥u_0∥}_{X_{\Im }^{s,p,\pi }}+{∥u_0∥}_{X_{\Im }^{s,p,\pi }}\right)\le min\left\{\frac{1}{8K_1},\frac{1}{4K_2}\right\}.$$

(20)

Then, in the view of Equations (18) and (19), for all $\left(\begin{array}{c}u\\ B\end{array}\right)$, $\left(\begin{array}{c}v\\ b\end{array}\right)\in X_1$ we can obtain that

$${∥\mathcal{J}\left(\begin{array}{c}u\\ B\end{array}\right)∥}_{Z_1}\le 2\left({∥u_0∥}_{X_{\Im }^{s,p,\pi }}+{∥B_0∥}_{X_{\Im }^{s,p,\pi }}\right),$$

$${∥\mathcal{J}\left(\begin{array}{c}u\\ B\end{array}\right)-\mathcal{J}\left(\begin{array}{c}v\\ b\end{array}\right)∥}_{Z_1}\le \frac{1}{2}\left({\parallel u-v\parallel }_{Z_1}+{\parallel B-b\parallel }_{Z_1}\right).$$

Therefore, by using Banach’s contraction principle, there exists a unique solution $u,B\in X_1$ satisfying (7). Next, we need to show that the solution $u,B\in X_1$ also belongs to $C([0,\infty );H^s{\left(R^3\right)}^3)$.

$$\begin{array}{cc}\hfill {∥\left(\begin{array}{c}u\\ B\end{array}\right)\left(t\right)∥}_{{\dot{H}}^s}& \le {∥{\mathcal{A}}_{\Im ,\wp }\left(\begin{array}{c}{u}_0\\ B_0\end{array}\right)\left(t\right)∥}_{{\dot{H}}^s}+{∥B\left(\left(\begin{array}{c}u\\ B\end{array}\right),\left(\begin{array}{c}u\\ B\end{array}\right)\right)∥}_{{\dot{H}}^s}=I_3+I_4.\hfill \end{array}$$

To obtain $I_1$, we have $I_3\lesssim {∥u_0∥}_{{\dot{H}}^s}+{∥B_0∥}_{{\dot{H}}^s}.$ For $I_4$, let $\frac{1}{q}=\frac{2}{p}-\frac{3}{s}$ with $q\in (1,2]$, applying Lemmas 6, 9 and Holder’s inequality, we have the following

$$\begin{array}{cc}\hfill I_4& \lesssim \int \frac{1}{{(t-y)}^{\left(\frac{3}{p\wp }-\frac{s}{2\wp }-\frac{3}{4\wp }+\frac{1}{2\wp }\right)}{\parallel u\otimes u\parallel }_{L^p}dy}+\int \frac{1}{{(t-y)}^{\left(\frac{3}{p\wp }-\frac{s}{2\wp }-\frac{3}{4\wp }+\frac{1}{2\wp }\right)}{\parallel B\otimes B\parallel }_{L^p}dy}\hfill \\ & +\int \frac{1}{{(t-y)}^{\left(\frac{3}{p\wp }-\frac{s}{2\wp }-\frac{3}{4\wp }+\frac{1}{2\wp }\right)}{\parallel u\otimes B\parallel }_{L^p}dy}+\int \frac{1}{{(t-y)}^{\left(\frac{3}{p\wp }-\frac{s}{2\wp }-\frac{3}{4\wp }+\frac{1}{2\wp }\right)}{\parallel B\otimes u\parallel }_{L^p}dy}\hfill \\ & \lesssim {\left({\int }_0^t\frac{1}{{(t-y)}^{\left(\frac{3}{p\wp }-\frac{s}{2\wp }-\frac{3}{4\wp }+\frac{1}{2\wp }\right){\left(\frac{\pi }{2}\right)}^{\prime }}}dy\right)}^{\frac{1}{{\left(\frac{\pi }{2}\right)}^{\prime }}}{\parallel u\parallel }_{L^{\pi }\left(0,\infty ;{\dot{W}}^{s,p}\right)}^2\hfill \\ & +{\left({\int }_0^t\frac{1}{{(t-y)}^{\left(\frac{3}{p\wp }-\frac{s}{2\wp }-\frac{3}{4\wp }+\frac{1}{2\wp }\right){\left(\frac{\pi }{2}\right)}^{\prime }}}dy\right)}^{\frac{1}{{\left(\frac{\pi }{2}\right)}^{\prime }}}{\parallel u\parallel }_{L^{\pi }\left(0,\infty ;{\dot{W}}^{s,p}\right)}^2\hfill \\ & +{\left({\int }_0^t\frac{1}{{(t-y)}^{\left(\frac{3}{p\wp }-\frac{s}{2\wp }-\frac{3}{4\wp }+\frac{1}{2\wp }\right){\left(\frac{\pi }{2}\right)}^{\prime }}}dy\right)}^{\frac{1}{{\left(\frac{\pi }{2}\right)}^{\prime }}}{\parallel u\parallel }_{L^{\pi }\left(0,\infty ;{\dot{W}}^{s,p}\right)}^2{\parallel B\parallel }_{L^{\pi }\left(0,\infty ;{\dot{W}}^{s,p}\right)}^2\hfill \\ & +{\left({\int }_0^t\frac{1}{{(t-y)}^{\left(\frac{3}{p\wp }-\frac{s}{2\wp }-\frac{3}{4\wp }+\frac{1}{2\wp }\right){\left(\frac{\pi }{2}\right)}^{\prime }}}dy\right)}^{\frac{1}{{\left(\frac{\pi }{2}\right)}^{\prime }}}{\parallel B\parallel }_{L^{\pi }\left(0,\infty ;{\dot{W}}^{s,p}\right)}^2{\parallel u\parallel }_{L^{\pi }\left(0,\infty ;{\dot{W}}^{s,p}\right)}^2.\hfill \end{array}$$

(21)

Since $\frac{1}{\pi }<\frac{1}{2}+\frac{1}{8\wp }+\frac{s}{4\wp }-\frac{3}{2\wp p}$, the time integral on the right hand side in inequality (21) converges and

$${\left({\int }_0^t\frac{1}{{(t-y)}^{\left(\frac{3}{p\wp }-\frac{s}{2\wp }-\frac{3}{4\wp }+\frac{1}{2\wp }{\left(\frac{\pi }{2}\right)}^{\prime }\right.}}dy\right)}^{\frac{1}{{\left(\frac{\pi }{2}\right)}^{\prime }}}\lesssim t^{2\left(\frac{1}{2}+\frac{1}{8\wp }+\frac{s}{4\wp }-\frac{3}{2\wp p}-\frac{1}{\pi }\right)}.$$

This implies that for all $t\ge 0$, $u,B\in {\dot{H}}^s{\left({\Re }^3\right)}^3$. Similarly, it is easy to get that $u,B\in C([0,\infty );{\dot{H}}^s{\left({\Re }^3\right)}^3)$, which finishes the proof of Theorem 1. □

Remark 1.

By substituting $B=0$ and $\wp =1$ in Theorem 1, we get Theorem 1.3, [30].

Proof of Theorem 2: 

Let us define

$${\parallel u\parallel }_{Y_2}:=\underset{t>0}{sup}{\parallel u\left(t\right)\parallel }_{{\dot{H}}^s}{\mathrm{and}\parallel u\parallel }_{Z_2}:=\underset{t>0}{sup}{t}^{\frac{1}{2}\left(1-\frac{3}{p}\right)}{\parallel u\left(t\right)\parallel }_{L^p}.$$

(22)

It is easy to get

$${∥{\mathcal{A}}_{\Im ,\wp }\left(t\right)\left(\begin{array}{c}{u}_0\\ B_0\end{array}\right)∥}_{Y_1}\le {∥T_{\Im ,\wp }\left(t\right)u_0∥}_{Y_1}+\parallel S_{\wp }\left(t\right)B_0{\parallel }_{Y_1}\lesssim \parallel u_0{\parallel }_{{\dot{H}}^s}+{\parallel B_0\parallel }_{{\dot{H}}^s}$$

(23)

and

$${∥{\mathcal{A}}_{\Im ,\wp }\left(t\right)\left(\begin{array}{c}{u}_0\\ B_0\end{array}\right)∥}_{Z_2}\le {∥T_{\Im ,\wp }\left(t\right)u_0∥}_{Z_2}+\parallel S_{\wp }\left(t\right)B_0{\parallel }_{Z_2}=\parallel u_0{\parallel }_{X_{\Im }^p}+{\parallel B_0\parallel }_{X_{\Im }^p}.$$

(24)

Next, we write a complete metric space $({\mathcal{X}}_2,d_2)$ with mapping $\mathcal{J}$ as:

$${\mathcal{X}}_2:=\left\{u,B\in C\left([0,\infty );{\dot{H}}^s{\left({\Re }^3\right)}^3\right)\right\}$$

such that ${\parallel u\parallel }_{Y_1}\lesssim 2{∥u_0∥}_{{\dot{H}}^s}{,\parallel B\parallel }_{Y_1}\lesssim 2{∥B_0∥}_{{\dot{H}}^s}{,\parallel u\parallel }_{Z_2}\le 2{∥u_0∥}_{X_{\Im }^p},{\parallel B\parallel }_{Z_2}⩽2{∥B_0∥}_{X_{\Im }^p}$,

$$\begin{array}{cc}\hfill & d_1\left(\left(\begin{array}{c}u\\ B\end{array}\right),\left(\begin{array}{c}v\\ b\end{array}\right)\right):={\parallel u-v\parallel }_{Y_1}+{\parallel B-b\parallel }_{Y_1},\hfill \\ \hfill & \mathcal{J}\left(\begin{array}{c}u\\ B\end{array}\right)\left(t\right):={\mathcal{A}}_{\Im ,\wp }\left(\begin{array}{c}{u}_0\\ B_0\end{array}\right)\left(t\right)-B\left(\left(\begin{array}{c}u\\ B\end{array}\right),\left(\begin{array}{c}u\\ B\end{array}\right)\right),\hfill \end{array}$$

where the bilinear term $B\left(\left(\begin{array}{c}u\\ B\end{array}\right),\left(\begin{array}{c}u\\ B\end{array}\right)\right)$ is defined in Equation (12). We can write

$$\begin{array}{cc}\hfill {∥\mathcal{J}\left(\begin{array}{c}u\\ B\end{array}\right)\left(t\right)∥}_{Y_1}& \le {∥{\mathcal{A}}_{\Im ,\wp }\left(\begin{array}{c}{u}_0\\ B_0\end{array}\right)\left(t\right)∥}_{Y_1}+{∥B\left(\left(\begin{array}{c}u\\ B\end{array}\right),\left(\begin{array}{c}u\\ B\end{array}\right)\right)∥}_{Y_1}=I_5+I_6.\hfill \end{array}$$

(25)

$I_5$ is estimated in inequality (23). For $I_6$, let $1<r\le 2$ and set $\frac{1}{r}=\frac{1}{s^{\prime }}+\frac{1}{p}$, where $\frac{1}{s^{\prime }}=\frac{1}{2}-\frac{s}{3}$ with $2\le s^{\prime }<\frac{6}{5-4\wp }$. Then, by Lemma 6, the boundedness of ℵ in $L^2\left({\Re }^n\right)$, and Holder’s inequality, we have

$$\begin{array}{c}\\ & {∥B\left(\left(\begin{array}{c}u\\ B\end{array}\right),\left(\begin{array}{c}u\\ B\end{array}\right)\right)∥}_{{\dot{H}}^s}\hfill \\ & \le {\int }_0^t{∥T_{\Im }(t-y)\aleph \nabla [u\left(y\right)\otimes u\left(y\right)]∥}_{{\dot{H}}^s}dy+{\int }_0^t{∥T_{\Im }(t-y)\aleph \nabla [B\left(y\right)\otimes B\left(y\right)]∥}_{{\dot{H}}^s}dy\hfill \\ & +{\int }_0^t{∥T_{\Im }(t-y)\aleph \nabla [u\left(y\right)\otimes B\left(y\right)]∥}_{{\dot{H}}^s}dy+{\int }_0^t{∥T_{\Im }(t-y)\aleph \nabla [B\left(y\right)\otimes u\left(y\right)]∥}_{{\dot{H}}^s}dy\hfill \\ & \lesssim {\int }_0^t\frac{1}{{(t-y)}^{\frac{1}{2\wp }+\frac{3}{2\wp p}}}{\parallel u\left(y\right)\otimes u\left(y\right)\parallel }_{L^r}dy+{\int }_0^t\frac{1}{{(t-y)}^{\frac{1}{2\wp }+\frac{3}{2\wp p}}}{\parallel B\left(y\right)\otimes B\left(y\right)\parallel }_{L^r}dy\hfill \\ & +{\int }_0^t\frac{1}{{(t-y)}^{\frac{1}{2\wp }+\frac{3}{2\wp p}}}{\parallel u\left(y\right)\otimes B\left(y\right)\parallel }_{L^r}dy+{\int }_0^t\frac{1}{{(t-y)}^{\frac{1}{2\wp }+\frac{3}{2\wp p}}}{\parallel B\left(y\right)\otimes u\left(y\right)\parallel }_{L^r}dy\hfill \\ & \lesssim {\int }_0^t\frac{1}{{(t-y)}^{\frac{1}{2\wp }+\frac{3}{2\wp p}}}{\parallel u\left(y\right)\parallel }_{L^{s^{\prime }}}{\parallel u\left(y\right)\parallel }_{L^p}dy+{\int }_0^t\frac{1}{{(t-y)}^{\frac{1}{2\wp }+\frac{3}{2\wp p}}}{\parallel B\left(y\right)\parallel }_{L^{s^{\prime }}}{\parallel B\left(y\right)\parallel }_{L^p}dy\hfill \\ & +{\int }_0^t\frac{1}{{(t-y)}^{\frac{1}{2\wp }+\frac{3}{2\wp p}}}{\parallel u\left(y\right)\parallel }_{L^{s^{\prime }}}{\parallel B\left(y\right)\parallel }_{L^p}dy+{\int }_0^t\frac{1}{{(t-y)}^{\frac{1}{2\wp }+\frac{3}{2\wp p}}}{\parallel B\left(y\right)\parallel }_{L^{s^{\prime }}}{\parallel u\left(y\right)\parallel }_{L^p}dy.\hfill \end{array}$$

Using ${\parallel ·\parallel }_{Y_1}$, ${\parallel ·\parallel }_{Z_2}$ defined in Equation (22) with $\frac{1}{2\wp }\left(1-\frac{3}{p}\right)<1$ and ${\dot{H}}^s\left({\Re }^3\right)\hookrightarrow L^{s^{\prime }}\left({\Re }^3\right)$, we have

$$\begin{array}{c}\\ & {∥B\left(\left(\begin{array}{c}u\\ B\end{array}\right),\left(\begin{array}{c}u\\ B\end{array}\right)\right)∥}_{{\dot{H}}^s}\hfill \\ & \lesssim {\parallel u\parallel }_{Y_1}{\parallel u\parallel }_{Z_2}{\int }_0^t\frac{1}{{(t-y)}^{\frac{1}{2\wp }+\frac{3}{2\wp p}}{y}^{\frac{1}{2\wp }\left(1-\frac{3}{p}\right)}}dy+{\parallel B\parallel }_{Y_1}{\parallel B\parallel }_{Z_2}{\int }_0^t\frac{1}{{(t-y)}^{\frac{1}{2\wp }+\frac{3}{2\wp p}}{y}^{\frac{1}{2\wp }\left(1-\frac{3}{p}\right)}}dy\hfill \\ & +{\parallel u\parallel }_{Y_1}{\parallel B\parallel }_{Z_2}{\int }_0^t\frac{1}{{(t-y)}^{\frac{1}{2\wp }+\frac{3}{2\wp p}}{y}^{\frac{1}{2\wp }\left(1-\frac{3}{p}\right)}}dy+{C\parallel B\parallel }_{Y_1}{\parallel u\parallel }_{Z_2}{\int }_0^t\frac{1}{{(t-y)}^{\frac{1}{2\wp }+\frac{3}{2\wp p}}{y}^{\frac{1}{2\wp }\left(1-\frac{3}{p}\right)}}dy\hfill \\ & \lesssim {\parallel u\parallel }_{Y_1}{\parallel u\parallel }_{Z_2}+{\parallel B\parallel }_{Y_1}{\parallel B\parallel }_{Z_2}+{\parallel u\parallel }_{Y_1}{\parallel B\parallel }_{Z_2}+{\parallel B\parallel }_{Y_1}{\parallel u\parallel }_{Z_2}.\hfill \end{array}$$

Therefore, from Equation (25), we have

$$\begin{array}{cc}\hfill {∥\mathcal{J}\left(\begin{array}{c}u\\ B\end{array}\right)\left(t\right)∥}_{Y_1}& \lesssim \parallel u_0{\parallel }_{{\dot{H}}^s}+{\parallel B_0\parallel }_{{\dot{H}}^s}\hfill \\ & +4\left({∥u_0∥}_{{\dot{H}}^s}{∥u_0∥}_{X_{\Im }^p}+{∥B_0∥}_{{\dot{H}}^s}{∥B_0∥}_{X_{\Im }^p}+{∥u_0∥}_{{\dot{H}}^s}{∥B_0∥}_{X_{\Im }^p}+{∥B_0∥}_{{\dot{H}}^s}{∥u_0∥}_{X_{\Im }^p}\right)\hfill \\ & \lesssim \left(\parallel u_0{\parallel }_{{\dot{H}}^s}+{\parallel B_0\parallel }_{{\dot{H}}^s}\right)\left(1+4\left({∥u_0∥}_{X_{\Im }^p}+{∥B_0∥}_{X_{\Im }^p}\right)\right).\hfill \end{array}$$

(26)

Similarly, we estimate the following

$$\begin{array}{cc}\hfill {∥\mathcal{J}\left(\begin{array}{c}u\\ B\end{array}\right)\left(t\right)∥}_{Z_2}& \le {∥{\mathcal{A}}_{\Im ,\wp }\left(\begin{array}{c}{u}_0\\ B_0\end{array}\right)\left(t\right)∥}_{Z_2}+{∥B\left(\left(\begin{array}{c}u\\ B\end{array}\right),\left(\begin{array}{c}u\\ B\end{array}\right)\right)∥}_{Z_2}=I_7+I_8.\hfill \end{array}$$

(27)

$I_7$ is estimated in inequality (24) and we have to estimate $I_8$. By using Lemma 6, Holder’s inequality, and the definition of $Z_2$ defined in (22), we have

$$\begin{array}{c}\\ & t^{\frac{1}{2\wp }\left(1-\frac{3}{p}\right)}{∥B\left(\left(\begin{array}{c}u\\ B\end{array}\right),\left(\begin{array}{c}u\\ B\end{array}\right)\right)∥}_{L^p}\hfill \\ & \le t^{\frac{1}{2\wp }\left(1-\frac{3}{p}\right)}\left({\int }_0^t{∥T_{\Im }(t-y)\aleph \nabla [u\left(y\right)\otimes u\left(y\right)]∥}_{L^p}dy+{\int }_0^t{∥T_{\Im }(t-y)\aleph \nabla [B\left(y\right)\otimes B\left(y\right)]∥}_{L^p}dy\right)\hfill \\ & +t^{\frac{1}{2\wp }\left(1-\frac{3}{p}\right)}\left({\int }_0^t{∥T_{\Im }(t-y)\aleph \nabla [u\left(y\right)\otimes B\left(y\right)]∥}_{L^p}dy+{\int }_0^t{∥T_{\Im }(t-y)\aleph \nabla [B\left(y\right)\otimes u\left(y\right)]∥}_{L^p}dy\right)\hfill \\ \hfill & \lesssim t^{\frac{1}{2\wp }\left(1-\frac{3}{p}\right)}\left({\int }_0^t\frac{1}{{(t-y)}^{\frac{1}{2\wp }+\frac{3}{2\wp p}}{\parallel u\left(y\right)\otimes v\left(y\right)\parallel }_{L^{\frac{p}{2}}}dy}+{\int }_0^t\frac{1}{{(t-y)}^{\frac{1}{2\wp }+\frac{3}{2\wp p}}{\parallel u\left(y\right)\otimes v\left(y\right)\parallel }_{L^{\frac{p}{2}}}dy}\right)\hfill \\ \hfill & +t^{\frac{1}{2\wp }\left(1-\frac{3}{p}\right)}\left({\int }_0^t\frac{1}{{(t-y)}^{\frac{1}{2\wp }+\frac{3}{2\wp p}}{\parallel u\left(y\right)\otimes v\left(y\right)\parallel }_{L^{\frac{p}{2}}}dy}+{\int }_0^t\frac{1}{{(t-y)}^{\frac{1}{2\wp }+\frac{3}{2\wp p}}{\parallel u\left(y\right)\otimes v\left(y\right)\parallel }_{L^{\frac{p}{2}}}dy}\right)\hfill \\ \hfill & \lesssim t^{\frac{1}{2\wp }\left(1-\frac{3}{p}\right)}\left({\parallel u\parallel }_{Z_2}{\parallel u\parallel }_{Z_2}+{\parallel B\parallel }_{Z_2}{\parallel B\parallel }_{Z_2}+{2\parallel u\parallel }_{Z_2}{\parallel B\parallel }_{Z_2}\right){\int }_0^t\frac{1}{{(t-y)}^{\frac{1}{2\wp }+\frac{3}{2\wp p}}{y}^{\frac{1}{\wp }\left(1-\frac{3}{p}\right)}}dy\hfill \\ \hfill & \lesssim {\parallel u\parallel }_{Z_2}{\parallel u\parallel }_{Z_2}+{\parallel B\parallel }_{Z_2}{\parallel B\parallel }_{Z_2}+{2\parallel u\parallel }_{Z_2}{\parallel B\parallel }_{Z_2}.\hfill \end{array}$$

Here, we notice that $\frac{1}{2\wp }+\frac{3}{2\wp p}<1$ as $p>\frac{3}{2\wp -1}$. As a result, from Equation (27), we have

$$\begin{array}{cc}\hfill {∥\mathcal{J}\left(\begin{array}{c}u\\ B\end{array}\right)\left(t\right)∥}_{Z_2}& \lesssim \parallel u_0{\parallel }_{X_{\Im }^p}+{\parallel B_0\parallel }_{X_{\Im }^p}+4\left({∥u_0∥}_{X_{\Im }^p}^2+{∥B_0∥}_{X_{\Im }^p}^2+2{∥u_0∥}_{X_{\Im }^p}{∥B_0∥}_{X_{\Im }^p}\right)\hfill \\ & \lesssim \left(\parallel u_0{\parallel }_{X_{\Im }^p}+{\parallel B_0\parallel }_{X_{\Im }^p}\right)\left(1+4\left({∥u_0∥}_{X_{\Im }^p}+{∥B_0∥}_{X_{\Im }^p}\right)\right).\hfill \end{array}$$

(28)

Furthermore, we can write

$$\begin{array}{cc}\hfill & {∥\mathcal{J}\left(\begin{array}{c}u\\ B\end{array}\right)\left(t\right)-\mathcal{J}\left(\begin{array}{c}v\\ b\end{array}\right)\left(t\right)∥}_{Y_1}\hfill \\ & ={∥B\left(\left(\begin{array}{c}u\\ B\end{array}\right),\left(\begin{array}{c}u-v\\ B-b\end{array}\right)\right)+B\left(\left(\begin{array}{c}u-v\\ B-b\end{array}\right),\left(\begin{array}{c}u\\ B\end{array}\right)\right)∥}_{Y_1}\hfill \\ & \lesssim \left({\parallel u\parallel }_{Y_1}+{\parallel v\parallel }_{Y_1}+{\parallel B\parallel }_{Y_1}+{\parallel b\parallel }_{Y_1}\right)\left({\parallel u-v\parallel }_{Y_1}+{\parallel B-b\parallel }_{Y_1}\right)\hfill \\ & \lesssim 4\left(\parallel u_0{\parallel }_{X_{\Im }^p}+{\parallel B_0\parallel }_{X_{\Im }^p}\right)\left({\parallel u-v\parallel }_{Y_1}+{\parallel B-b\parallel }_{Y_1}\right).\hfill \end{array}$$

(29)

Let us consider that $u_0,B_0\in X_{\Im }^p{\left({\Re }^3\right)}^3\cap {\dot{H}}^s{\left({\Re }^3\right)}^3$ satisfies

$$\underset{\Im \in \Re }{sup}\left({∥u_0∥}_{X_{\Im }^p}+{∥u_0∥}_{X_{\Im }^p}\right)\le min\left\{\frac{1}{8K_1},\frac{1}{4K_2}\frac{1}{4K_3}\right\}.$$

(30)

Then, from Equations (19), (26) and (28), for all $\left(\begin{array}{c}u\\ B\end{array}\right)$, $\left(\begin{array}{c}v\\ b\end{array}\right)\in X_1$ we can easily obtain that

$${∥\mathcal{J}\left(\begin{array}{c}u\\ B\end{array}\right)∥}_{Y_1}\lesssim 2\left({∥u_0∥}_{{\dot{H}}^s}+{∥B_0∥}_{{\dot{H}}^s}\right),$$

$${∥\mathcal{J}\left(\begin{array}{c}u\\ B\end{array}\right)∥}_{Z_2}\le 2\left({∥u_0∥}_{X_{\Im }^p}+{∥B_0∥}_{X_{\Im }^p}\right),$$

$${∥\mathcal{J}\left(\begin{array}{c}u\\ B\end{array}\right)-\mathcal{J}\left(\begin{array}{c}v\\ b\end{array}\right)∥}_{Y_1}\le \frac{1}{2}\left({\parallel u-v\parallel }_{Y_1}+{\parallel B-b\parallel }_{Z_1}\right).$$

As a result, by using Banach’s contraction principle, there exists a unique solution $u,B\in X_2$ that satisfies (7), which finishes the proof of Theorem 2. □

Remark 2.

By substituting $B=0$ and $\wp =1$ in Theorem 1, we get Theorem 1.5, [30].