\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{amssymb}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 65, pp. 1--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/65\hfil Magneto-hydrodynamic equations]
{Global well-posedness and decay results for 3D generalized
magneto-hydrodynamic equations in critical Fourier-Besov-Morrey spaces}

\author[A. El Baraka, M. Toumlilin \hfil EJDE-2017/65\hfilneg]
{Azzeddine El Baraka, Mohamed Toumlilin}

\address{Azzeddine El Baraka \newline
University Mohamed Ben Abdellah,
FST Fes-Saiss, Laboratory AAFA,
Department of Mathematics, B.P. 2202 Route Immouzer,
Fes 30000, Morocco}
\email{azzeddine.elbaraka@usmba.ac.ma, az.elbaraka@gmail.com}

\address{Mohamed Toumlilin \newline
University Mohamed Ben Abdellah,
FST Fes-Saiss, Laboratory AAFA,
Department of Mathematics, B.P. 2202 Route Immouzer,
Fes 30000, Morocco}
\email{mohamed.toumlilin@usmba.ac.ma}

\thanks{Submitted December 28, 2016. Published March 4, 2017.}
\subjclass[2010]{35Q30, 76D05, 76D03}
\keywords{Magneto-hydrodynamic equations;  global well-posedness; 
\hfill\break\indent Fourier-Besov-Morrey space}

\begin{abstract}
 This article concerns the Cauchy problem of the 3D generalized incompressible
 magneto-hydrodynamic (GMHD) equations. By using the Fourier localization
 argument and the Littlewood-Paley theory as in \cite{cw,xcfz}, we
 obtain global well-posedness results of the GMHD equations with small initial
 data  belonging to the critical Fourier-Besov-Morrey spaces.
 Moreover, we prove that the corresponding global solution decays to zero
 as time approaches infinity.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

We investigate the generalized magneto-hydrodynamic equations in the whole
space $\mathbb{R}^3$,
\begin{equation}
\label{GMHD}
\begin{gathered}
u_{t}+u\cdot\nabla u+\mu(-\Delta)^{\alpha} u-b\cdot\nabla b+\nabla\pi
=0\quad\text{in } (0, +\infty)\times\mathbb{R}^3,\\
\nabla\cdot u=0,\quad \nabla\cdot b=0,\\
b_{t}+u\cdot\nabla b+\nu(-\Delta)^{\alpha} b- b\cdot\nabla u= 0 \quad
\text{in } (0, +\infty)\times\mathbb{R}^3,\\
(u,b)|_{t=0}=(u_0,b_0),
\end{gathered}
\end{equation}
where $u=u(t,x)\in\mathbb{R}^3$ denotes the velocity field of the flow,
$b(t,x)$ denotes the magnetic field, $\pi(t,x): \mathbb{R}^3\to \mathbb{R}$
represents the pressure function, $\mu\geq0$ and $\nu\geq0$ are
real positive parameters, $\nabla\cdot u = 0$ and $\nabla\cdot b = 0 $
represent the incompressible conditions, and $u_0$ and $b_0$ are for given
initial velocity and initial magnetic field with $\nabla\cdot u_0 = 0 $ and
$\nabla\cdot b_0 = 0 $, respectively. The operator $(-\Delta)^{\alpha}$ is
the Fourier multiplier with symbol $|\xi|^{2\alpha}$.

 The GMHD system plays a fundamental role in applied sciences such as astrophysics,
geophysics, and plasma physics. The first equation of system \eqref{GMHD} reflects the
conservation of momentum, the third equation of system \eqref{GMHD} is the
magnetic induction equation, and the second equation of system \eqref{GMHD}
specifies the conservation of mass.

 When $\alpha= 1$, the GMHD system
\eqref{GMHD} becomes the usual MHD equations, which describes the macroscopic
behavior of the electrically conducting incompressible fluids in a magnetic
field; when $\alpha= 1$ and $b=0$, the GMHD equations reduce to the
Navier-Stokes equations. The study of the generalized \eqref{GMHD} equations
will improve our understanding of the Navier-Stokes equations and the MHD
equations, which has drawn much attention during the past twenty more years.
Let's take this opportunity to briefly quote some works; Duvaut and Lions
\cite{21} constructed a global Leray-Hopf weak solution and a local strong
solution of the 3D incompressible MHD system, C. Cao, J. Wu \cite{34} proved
global regularity of classical solutions for the MHD equations with mixed
partial dissipation and magnetic diffusion, and they also give the global
existence, conditional regularity and uniqueness of a weak solution for 2D MHD
equations with only magnetic diffusion. For more results in this direction,
see \cite{35,36} and reference therein.

 On the other hand, there are
numerous important progresses on the fundamental issue of the blow-up criteria
or regularity criteria to the system \eqref{GMHD} (see
\cite{21,23,24,32,25,26,27,29,30,31} and the references cited therein for more
details).

 For the GMHD system \eqref{GMHD}, the global-in-time weak
solution for any given divergence free initial value $(u_0, b_0)\in
L^2(\mathbb{R}^n)$ was proved by Wu \cite{jwu}, the local-in-time
existence and uniqueness of smooth solution for any sufficient smooth initial
data $(u_0, b_0)$ was established by Yuan \cite{37}, and Liu, Zhao and Cui
\cite{38} obtained the global existence and stability of solutions for system
\eqref{GMHD} with small initial data $(u_0, b_0)$ belonging to the
pseudomeasure space $\mathcal{PM}^{\alpha}$, where $\mathcal{PM}^{\alpha}$ is
defined by
\[
\mathcal{PM}^{\alpha}:=\{f\in\mathcal{S'}:\hat{f}\in L_{loc}
^1(\mathbb{R}^3),\|f\|_{\mathcal{PM}^{\alpha}} :=\operatorname{ess\,sup}_{\xi\in
\mathbb{R}^3}|\xi|^{\alpha}|\hat{f}(\xi)|<\infty\}\,.
\]
Recently, Wang and Wang \cite{33} and Ye \cite{yz} obtained the global
existence results for classical 3-D MHD $(\alpha= 1)$ and GMHD $( \frac{1}
{2}\leq\alpha\leq1)$, respectively.

To give a clearer introduction to
our results in this paper, we first note that system \eqref{GMHD} enjoys
scaling properties. Clearly, if $(u(t, x), b(t, x))$ is a solution to system
\eqref{GMHD}, then $(u^{\lambda}(t,x), b^{\lambda}(t,x))$ is also a solution
of (1.1) corresponding to the initial data
$(u_0^{\lambda}, b_0^{\lambda})$, where
\begin{gather*}
u^{\lambda}(t,x):= \lambda^{2\alpha-1} u\bigl(\lambda^{2\alpha}t,\lambda x\bigr),
\quad  b^{\lambda}(t,x):=\lambda^{2\alpha-1} b\bigl( \lambda^{2\alpha}t,\lambda x\bigr), \\
u^{\lambda}_0(x):=\lambda^{2\alpha-1} u_0( \lambda x), \quad
b^{\lambda}_0(x):=\lambda^{2\alpha-1} b_0(\lambda x).
\end{gather*}
In this article, we use $\mathcal{F\dot{N}}_{p,\lambda,q}^{s}$ to denote
the homogenous Fourier Besov-Morrey spaces, $C$ will denote constants which
can be different at different places, ${\mathsf{U}}\lesssim{\mathsf{V}}$ means
that there exists a constant $C>0$ such that ${\mathsf{U}}\leq C{\mathsf{V}}$,
and $p'$ is the conjugate of $p$ satisfying 
$\frac{1}{p}+\frac{1}{p'}= 1$ for $1\leq p\leq\infty$.

 Motivated by the works
\cite{yz,xcfz,32,jbe}, the aim of this article is to prove the global existence
and the decay property of the system \eqref{GMHD} in the Fourier Besov-Morrey
spaces $\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}(\mathbb{R}^3)$.

\section{Preliminaries and statement of main result}

The proofs of the results presented in this paper are based on a dyadic
partition of unity in the Fourier variables, the so-called, homogeneous
Littlewood-Paley decomposition. We recall briefly this construction below. We
start with a dyadic decomposition of $\mathcal{\mathbb{R}}^n$.

Suppose $\chi\in C_0^{\infty}(\mathcal{\mathbb{R}}^n),\;\varphi\in
C_0^{\infty}(\mathcal{\mathbb{R}}^n\setminus\{0\})$ satisfying
\begin{gather*}
\operatorname{supp}\chi\subset\{\xi\in{\mathbb{R}}^n:|\xi|\leq\frac43\},\\
\operatorname{supp}\varphi\subset\{\xi\in{\mathbb{R}}^n:\frac34\leq|\xi
|\leq\frac83\},\\
\chi(\xi)+\sum_{j\geq0}\varphi(2^{-j}\xi)=1,\quad\xi\in\mathcal{\mathbb{R}}^n,\\
\sum_{j\in\mathbb{Z}}\varphi(2^{-j}\xi)=1,\quad\xi\in\mathcal{\mathbb{R}}^n\backslash\{0\},
\end{gather*}
and denote $\varphi_j(\xi)=\varphi(2^{-j}\xi)$ and $\mathcal{P}$ the set of
all polynomials. The space of tempered distributions is denoted by 
$S'$. The homogeneous dyadic blocks $\dot{\Delta}_j$ and the homogeneous
low-frequency cutoff operators $\dot{S}_j$ are defined for all
$j\in\mathbb{Z}$ by
%\label{e2.1}
\begin{gather*} 
\dot{\Delta}_ju=\varphi(2^{-j}D)u=2^{jn}\int h(2^jy)u(x-y)\,dy, \\ 
\dot{S}_ju=\sum_{k\leq j-1}\dot{\Delta}_ku=\chi(2^{-j}D)u=2^{jn}
\int \tilde{h}(2^jy)u(x-y)\,dy,
 \end{gather*}
where $h=\mathcal{F}^{-1}\varphi$ and $\tilde{h}=\mathcal{F}^{-1}\chi$.

 First, we recall the definition of Morrey spaces which are a
complement of $L^p$ spaces.

\begin{definition}[\cite{ae,cri}]\rm
 For $1\leq p<\infty$, $0\leq\lambda<n$, the
Morrey space $\mathrm{M}_p^{\lambda}=\mathrm{M}_p^{\lambda}(\mathbb{R}^n)$
is defined as the set of functions $f\in
L_{loc}^p(\mathbb{R}^n)$ such that
\begin{equation}\label{ms}
\|f\|_{\mathrm{M}_p^{\lambda}}=\sup_{x_0\in\mathbb{R}^n}
\sup_{r>0}r^{-\lambda/p} \|f\|_{L^p(B(x_0,r))}<\infty,
\end{equation}
where $B(x_0,r)$ denotes the ball in $\mathbb{R}^n$ with center
$x_0$ and radius $r$. the space $\mathrm{M}_p^{\lambda}$ endowed
with the norm $\|f\|_{\mathrm{M}_p^{\lambda}}$ is a Banach space.
In the case $p=1,\, \mathrm{M}_p^{\lambda}$ should be understood
as a space of Radon measures and
$\|f\|_{\mathrm{L}^1(B(x_0,r))}$ denoting the total variation of
f on $B(x_0,r)$. For various reasons we find it convenient to
include $L^{\infty}$ among the Morrey spaces, but the indices in the
notation $\mathrm{M}_p^{\lambda}$ will always be restricted to
$1\leq p < \infty $, $0 \leq \lambda < n$, notwithstanding that
\eqref{ms} makes sense for $\lambda=n$ and the resulting space is
equivalent to $L^{\infty}$(irrespective of the value of $p$).
It is not difficult to see  that the relation
$\mathrm{M}_{p_1}^{\lambda}\hookrightarrow
\mathrm{M}_{p_2}^{\mu}$ provided
$\frac{n-\mu}{p_2}\geq\frac{n-\lambda}{p_1}$ and $ p_2\leq p_1$, and 
$\mathrm{M}_p^{0}=L^p$.\\
If $1\leq p_1,p_2,p_3<\infty$ and
$0\leq\lambda_1, \lambda_2, \lambda_3<n$ with 
$\frac{1}{p_3}=\frac{1}{p_1}+\frac{1}{p_2}$ and 
$\frac{\lambda_3}{p_3}=\frac{\lambda_1}{p_1}+\frac{\lambda_2}{p_2}$,
then we have the H\"{o}lder type inequality
\begin{equation*}
\|fg\|_{\mathrm{M}_{p_3}^{\lambda_3}}\leq\|f\|_{\mathrm{M}_{p_1}^{\lambda_1}}
\|g\|_{\mathrm{M}_{p_2}^{\lambda_2}}\,.
\end{equation*}
Also, for $1\leq p<\infty$ and $0\leq\lambda<n$,
\begin{equation}\label{ym}
\|\varphi*g\|_{\mathrm{M}_p^{\lambda}}\leq\|\varphi\|_{L^1}
\|g\|_{\mathrm{M}_p^{\lambda}},
\end{equation}
for all $\varphi\in L^1$ and $g\in\mathrm{M}_p^{\lambda}$.
\end{definition}

\begin{definition}[homogeneous Besov-Morrey spaces] \label{def2} \rm
Let $s\in\mathbb{R}$, $1\leq p<+\infty$, $1\leq q\leq+\infty$, and
$0\leq\lambda<n$, the space
$\mathcal{\dot{N}}_{p,\lambda,q}^{s}(\mathbb{R}^n)$ is defined by
\begin{equation*}
\mathcal{\dot{N}}_{p,\lambda,q}^{s}(\mathbb{R}^n)=\Big\{u\in
\mathcal{Z}'(\mathbb{R}^n);\;\;\;\|  u\|
_{\mathcal{\dot{N}}_{p,\lambda,q}^{s}(\mathbb{R}^n)}<\infty\Big\}\,.
\end{equation*}
Here
\[ %\label{nc}
\|u\|_{\mathcal{\dot{N}}_{p,\lambda,q}^{s}(\mathbb{R}^n)}
=\begin{cases}
\big\{\sum_{j\in\mathbb{Z}} 2^{jqs}\|\dot{\Delta}_ju\|_{\mathrm{M}_p^{\lambda}} ^q
\big\}^{1/q} & \text{for }q<\infty,\\
\sup_{j\in\mathbb{Z}}{\sup}2^{js}\|\dot{\Delta}_ju\|
_{\mathrm{M}_p^{\lambda}}& \text{for } q=\infty\,.
\end{cases}
\]
The space $\mathcal{Z}'(\mathbb{R}^n)$ denotes the topological
dual of the space
\[
\mathcal{Z}(\mathbb{R}^n)
=\big\{f\in\mathcal{S}(\mathbb{R}^n);\partial^{\alpha
}\widehat{f}(0)=0\text{ for every multi-index }\alpha\big\},
\]
and it can be identified to the quotient space
$\mathcal{S'}(\mathbb{R}^n)/\mathcal{P}$, where $\mathcal{P}$
denotes the set of all polynomials on $\mathbb{R}^n$.
We refer to \cite[chap.~8]{ws} for more details.
\end{definition}

\begin{definition}[homogeneous Fourier-Besov-Morrey spaces] \label{def3}\rm
Let $s\in\mathbb{R}$, $0\leq\lambda<n$, $1\leq p<+\infty$ and 
$1\leq q\leq+\infty$. The space
$\mathcal{F\dot{N}}_{p,\lambda,q}^{s}(\mathbb{R}^n)$ denotes the
set of all $u\in \mathcal{Z'}(\mathbb{R}^n) $ such that
\begin{equation}
\|u\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{s} (\mathbb{R}^n)}
=\Big\{\sum_{j\in\mathbb{Z}}2^{jqs}\| \widehat{\dot{\Delta} _ju}
\| _{\mathrm{M}_p^{\lambda}}^q\Big\}^{1/q} <+\infty,  \label{fbts}
\end{equation}
with suitable modification made when $q = \infty$.
\end{definition}

Note that the space
$\mathcal{F\dot{N}}_{p,\lambda,q}^{s}(\mathbb{R}^n)$ equipped with
the norm \eqref{fbts} is a Banach space.
We also notice that the  Fourier-Besov-Morrey spaces are independent
of the choice of $\varphi_j$, and the advantage of working in
these spaces lies in they are more adapted than the classical
Besov-Morrey-spaces for estimating the bilinear paraproduct using
H\"{o}lder's inequality directly, instead of Bernstein's inequality.
Now, we recall the definition of the mixed space-time spaces used in
\cite{cw,xcfz}.

\begin{definition} \rm
Let $s\in\mathbb{ R}$, $1\leq p<\infty$, $1\leq q,\rho\leq\infty$,
$0\leq\lambda<n$, and $I=[0,T)$, $T\in(0,\infty]$.
The space-time norm is defined on $u(t,x)$ by
\begin{align*}
&\|u(t,x)\|_{\pounds^{\rho}(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{s})} \\
&= \Big\{\sum_{j\in \mathbb{Z}}2^{jqs}\| \widehat{\dot{\Delta}_ju}\|
_{L^{\rho}(I,\mathrm{M}_p^{\lambda})} ^q \Big\}^{1/q},
\end{align*}
and denote by
$\pounds^{\rho}(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{s})$ the set of
distributions in $S'(\mathbb{R}\times\mathbb{R}^n)/\mathcal{P}$
with finite
$\|.\|_{\pounds^{\rho}(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{s})}$
norm.
\end{definition}

Our first main result is the following theorem.

\begin{theorem}\label{thm1}
Let $1\leq p<\infty$, $1\leq q\leq 2$, $0\leq\lambda<3$, and
$\frac{1}{2}<\alpha<1+\frac{3}{2p'}+\frac{\lambda}{2p}$. Then there
exists a constant $C_0(\alpha,p,q)$ such that, for any
$(u_0,b_0)\in \mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}$ satisfying $\nabla\cdot u_0=\nabla\cdot b_0=0$ and
\[
\|(u_0,b_0)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}
\leq C_0\min\{\mu,\nu\},
\]
the Cauchy problem \eqref{GMHD} admits a unique global solution 
\[
(u,b)\in
\mathcal{C}\big([0,\infty);\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha
+\frac{3}{p'}+\frac{\lambda}{p}}\big)\cap
\pounds^1\big([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}
+\frac{\lambda}{p}}\big),
\]
and it satisfies
\begin{align*}
&\|(u,b)\|_{\pounds^{\infty}\big([0,\infty);
 \mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\big)
\cap \pounds^1\big([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}
 +\frac{\lambda}{p}}\big)}\\
&\leq2\Big(1+(\frac{16}{9})^{\alpha}\Big)
\|(u_0,b_0)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+
\frac{3}{p'}+\frac{\lambda}{p}}}\,.
\end{align*}
\end{theorem}

\begin{remark} \rm
Theorem \ref{thm1} extends  the result of \cite{1} from Fourier-Herz 
spaces to Fourier-Besov-Morrey spaces, in fact, for
$\lambda=0$, $\mathcal{F\dot{N}}_{1,0,q}^{s}
=\mathrm{F\dot{B}}_{1,q}^{s}=\mathcal{\dot{B}}_{q}^{s}$ where 
$\mathcal{\dot{B}}_{q}^{ s}$ is the homogeneous
Fourier-Herz spaces (see  definition \ref{defFB}).

In addition, this result also holds in the Fourier-Besov spaces, in
fact,  for $\lambda=0$, $\mathcal{F\dot{N}}_{p,0,q}^{s}=\mathrm{F\dot{B}}_{p,q}^{s}$
where $\mathrm{F\dot{B}}_{p,q}^{s}$ is the homogeneous Fourier-Besov
spaces.

We also remark that for general $\alpha$ and $b=0$, the equation of
system \eqref{GMHD} becomes the Fractional Navier-Stokes equations.
\end{remark}

Our second purpose of this paper is to prove the non-blowup at large time and
the norm of global solution in 
$\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}$ 
goes to zero at infinity.

\begin{theorem}\label{thm2}
Let $1\leq p,q\leq2$, $0\leq\lambda<\frac{p}{2}$, and 
$\frac{5}{6}+\frac{\lambda}{3p}<\alpha\leq1$. Assume that 
$(u,b)\in \mathcal{C}\big([0,\infty);
\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\big)$
is a global solution of system \eqref{GMHD} given by Theorem
\ref{thm1}, then
\begin{align*}
\lim_{t \to \infty}\sup\Big(\|u(t)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}
^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}
+\|b(t)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}\Big)=0\,.
\end{align*}
\end{theorem}

Recently, Zhuan \cite{yz} obtained the same property in the space
$\chi^{s}=\mathrm{F\dot{B}}_{1,1}^{s}=\mathcal{F\dot{N}}_{1,0,1}^{s}$.
Therefore, Theorem \ref{thm2} improves and extends his result.

\begin{remark}\label{critical space} \rm
The Fourier-Besov-Morrey space
$\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}(\mathbb{R}^3)$
is critical for \eqref{GMHD}. For this, set $u_{0,\gamma}(\xi)=
\gamma^{2\alpha-1}u_0(\gamma \xi)$, then
$\widehat{u_{0,\gamma}}(\xi)=
\gamma^{2\alpha-4}\widehat{u}_0(\gamma^{-1}\xi)$. Next, setting
\[
f_j(\xi)=\varphi(2^{-j+[\log_2\gamma]-\log_2\gamma}\xi)\widehat{u_{0,\gamma}}(\xi),
\]
we can obtain
\begin{align*}
&2^{j(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p})}\|f_j\|_{\mathrm{M}_p^{\lambda}}\\
&=2^{j(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p})}
\sup_{x_0\in\mathbb{R}^3}\sup_{r>0}r^{-\lambda/p}
\|\varphi(2^{-j+[\log_2\gamma]-\log_2\gamma}\xi)
\widehat{u_{0,\gamma}}(\xi)\|_{L^p(B(x_0,r))}\\
&= 2^{([\log_2\gamma]-\log_2\gamma)(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p})}
2^{(j-[\log_2\gamma])(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p})}
\|\varphi(2^{-j+[\log_2\lambda]}\eta)\widehat{u_0}(\eta)
 \|_{\mathrm{M}_p^{\lambda}}\\
&\approx 2^{(j-[\log_2\gamma])(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p})}
 \|\varphi(2^{-j+[\log_2\lambda]}\eta)\widehat{u_0}(\eta)
 \|_{\mathrm{M}_p^{\lambda}}\,.
\end{align*}
This implies
\begin{align*}
\Big\{\sum_{j\in\mathbb{Z}}
2^{j(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p})q}\|f_j\|_{\mathrm{M}_p^{\lambda}}^q
\Big\}^{1/q}
\approx\|u_0\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}\,,
\end{align*}
and since
\begin{align*}
\varphi_j(\xi)\widehat{u_{0,\gamma}}(\xi)={\sum_{|k-j|\leq2}}
\varphi_j(\xi)f_{k}(\xi)
\end{align*}
we easily get
\begin{align*}
\|u_{0,\gamma}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}
\approx\|u_0\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}\,.
\end{align*}
Similarly,
\[
\|b_{0,\gamma}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}
\approx\|b_0\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}},
\]
thus $\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}$
is a critical space for the GMHD equations \eqref{GMHD}
\end{remark} 

Now we introduce the Fourier-Besov spaces which contain some
known spaces applied in studying Navier-Stokes equations. The norm of
Fourier-Besov spaces $F\dot{B}_{p,q}^{s}$\cite{xcfz} is defined as follows.

\begin{definition}\label{defFB} \rm
For $s\in\mathbb{ R}$, $1\leq p,q\leq \infty$, set
\[
\|u\|_{F\dot{B}_{p,q}^{s}} 
= \begin{cases}
\Big(\sum_{j\in \mathbb{Z}}2^{jqs}\|\widehat{\dot{\Delta} _ju}\|
_{L^p}^q \Big)^{1/q} & \quad \text{for }q<\infty,\\
\sup_{j\in \mathbb{Z}}2^{js}\|\widehat{\dot{\Delta} _ju}\|
_{L^p}  & \quad\text{for } q=\infty\,.
\end{cases} 
\]
One defines the homogenous Fourier-Besov spaces $F\dot{B}_{p,q}^{s}$ by
\[
F\dot{B}_{p,q}^{s}=\{u\in \mathcal{S'}(\mathbb{R}^3)/\mathcal{P}:
\|u\|_{F\dot{B}_{p,q}^{s}}<\infty\}\,.
\]
\end{definition}

Particularly, for $p=1$ Cannone and Wu introduced the Fourier-Herz spaces
$\mathcal{\dot{B}}_{q}^{ s}$ \cite{cw} with the norm associated
\[
\|u\|_{\mathcal{\dot{B}}_{q}^{ s}} 
= \begin{cases}
\big(\sum_{j\in\mathbb{Z}}2^{jqs}\|\widehat{\dot{\Delta} _ju}\| _{L^1}^q
\big)^{1/q} & \text{for } q<\infty,\\
\sup_{j\in\mathbb{Z}}2^{js}\|\widehat{\dot{\Delta} _ju}\| _{L^1} & \text{for }
q=\infty\,.
\end{cases}
\]
Clearly, we have $\mathcal{\dot{B}}_{q}^{ s}=F\dot{B}_{1,q}^{s}$.
The space $\chi^{-1}$ introduced by Lei and Lin \cite{ll} is
\begin{align*}
\chi^{-1}=\{u\in\mathcal{S'}(\mathbb{R}^3);\int_{\mathbb{R}^3}
|\xi|^{-1}|\widehat{u}|d\xi<\infty\}.
\end{align*}


We have $\chi^{-1}=F\dot{B}_{1,1}^{-1}=\mathcal{\dot{B}}_1^{ -1}$.
We finish this section with a Bernstein type lemma in Fourier variables in
Morrey spaces. 

\begin{lemma}[\cite{e6}] \label{bm}
Let $1\leq q\leq p<\infty$,
$0\leq\lambda_1, \lambda_2<n$, $\frac{n-\lambda_1}{p}\leq\frac{n-\lambda_2}{q}$,
and let $\gamma$ be a multiindex. If
$\operatorname{supp}(\widehat{f})\subset\{|\xi|\leq A2^{j}\}$ then there is a
constant $C>0$ independent of $f$ and $j$ such that
\begin{equation}\label{b}
\|(i\xi)^{\gamma}\widehat{f}\|_{\mathrm{M}_{q}^{\lambda_2}}
\leq C2^{j|\gamma|+j(\frac{n-\lambda_2}{q}
 -\frac{n-\lambda_1}{p})}\|\widehat{f}\|_{\mathrm{M}_p^{\lambda_1}}.
\end{equation}
\end{lemma}

Note that
 \[
\|(i\xi)^{\gamma}\widehat{f}\|_{\mathrm{M}_{q}^{\lambda_2}} 
\leq C2^{j|\gamma|}\|1.\widehat{f}\|_{\mathrm{M}_p^{\lambda_2}} 
\leq 2^{j|\gamma|} C2^{j(\frac{n-\lambda_2}{q}-\frac{n-\lambda_1}{p}
)}\|\widehat{f}\|_{\mathrm{M}_p^{\lambda_1}},
\]
which gives \eqref{b}.

\section{Well-posedness}

First, we consider the linear nonhomogeneous dissipative equation
\begin{equation}
\label{HE}
\begin{gathered} 
u_{t}+\mu(-\Delta)^{\alpha}u=f(t,x)\quad (t,x)\in \mathbb{R}^{+}\times \mathbb{R}^3\\ 
u(0,x) = u_0(x)\quad x\in \mathbb{R}^3,
 \end{gathered} 
\end{equation}
for which we recall the following result.

\begin{lemma}[\cite{toum}] \label{le}
Let $I=[0,T)$, $0< T\leq \infty$, $s\in\mathbb{R}$, $0\leq\lambda<n$,
$1\leq p<\infty$, and $1\leq q\leq \infty$. Assume that 
$u_0\in \mathcal{F\dot{N}}_{p,\lambda,q}^{s}$ and
$f\in \pounds^1(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{s})$.
Then the solution $u(t,x)$ to the Cauchy problem \eqref{HE} satisfies
\begin{equation}\label{32}
\begin{aligned}
&\|u\|_{\pounds^{\infty}(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{s})}
+\mu\|u\|_{\pounds^1(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{s+2\alpha})}\\
&\leq (1+(\frac{16}{9})^{\alpha})(\|u_0\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{s
}}+\|f\|_{\pounds^1(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{s })}).
\end{aligned}
\end{equation}
If in addition $q$ is finite, then $u$ belongs to
$\mathcal{C}(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{s})$.
\end{lemma}


\begin{proof}

Inequality \eqref{32} was proved in \cite{toum}.
 Now, we shall briefly
present the proof of the continuity of $u(t,x)$ in time $t$ when $1\leq
q<\infty$. By using the definition of the Fourier-Besov-Morrey spaces, we
have
\begin{equation} \label{c1}
\begin{aligned}
&\|u(t_1)-u(t_2)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{s}}^q \\
&\leq\sum_{j\leq N}(2^{js}\|\hat{u}_j(t_1)-\hat{u}_j(t_2)\|_{\mathrm{M}
_p^{\lambda}})^q+2\sum_{j> N}(2^{js}\|\hat{u}_j(t)\|_{L^{\infty
}(I,\mathrm{M}_p^{\lambda})} )^q\,\,
\end{aligned}
\end{equation}
where $\hat{u}_j=\varphi_j\hat{u}$. For any small constant
$\varepsilon >0$, let $N$ be large enough such that
\begin{equation} \label{c2}
\sum_{j> N}2^{jsq}\|\hat{u}_j(t)\|_{L^{\infty}(I,\mathrm{M}_p^{\lambda})} ^q
\leq\frac{\varepsilon}{4}\,.
\end{equation}
By using Taylor's formula, we obtain
\begin{equation} \label{c3}
\begin{aligned}
&\sum_{j\leq N}(2^{js}\|\hat{u}_j(t_1)-\hat{u}_j
(t_2)\|_{\mathrm{M}_p^{\lambda}})^q \\
& \leq |t_1-t_2|^q\sum_{j\leq N}2^{jsq}\|\partial_{t}\hat{u}_j(t)\|_{
L^1(I,\mathrm{M}_p^{\lambda})}^q\\
& \lesssim|t_1-t_2|^q\|\partial_{t}u(t)\|_{\pounds ^1(I,\mathcal{F\dot
{N}}_{p,\lambda,q}^{s})}^q\\
& \lesssim|t_1-t_2|^q\Big(\mu^q\|(-\Delta)^{\alpha}u\|_{\pounds ^1
(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{s})}^q+ \|f\|_{\pounds ^1
(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{s})}^q\Big)\\
& \lesssim|t_1-t_2|^q\Big(\mu^q\|u\|_{\pounds ^1(I,\mathcal{F\dot
{N}}_{p,\lambda,q}^{s+2\alpha})}^q+ \|f\|_{\pounds ^1(I,\mathcal{F\dot{N}
}_{p,\lambda,q}^{s})}^q\Big)\\
& \lesssim|t_1-t_2|^q\Big(2+\Big(\frac{16}{9}\Big)^{\alpha}
\Big)^q\Big(\|u_0\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{s }}^q
+\|f\|_{\pounds ^1(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{s})}^q\Big)\,.
\end{aligned}
\end{equation}
Combining \eqref{c1}, \eqref{c2}, and \eqref{c3}, we obtain the continuity of
$u$ in time $t$. The proof  is complete.
\end{proof}

\begin{lemma}[\cite{toum}] \label{BE}
Let $1\leq p<\infty$, $1\leq \rho\leq \infty$, $1\leq q \leq 2$,
\[
\frac{1}{2}<\alpha<\frac{2+\frac{3}{p'}+\frac{\lambda}{p}}{4-\frac{2}{\rho}}\,,
\]
$0\leq\lambda<3$, and set
\begin{equation*}
X=\pounds^{\infty}(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}})\cap \pounds^{\rho}
(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{2\alpha}{\rho}+\frac{\lambda}{p}}),
\end{equation*}
with the norm
\begin{equation*}
\|u\|_X=\|u\|_{\pounds^{\infty}(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha
+\frac{3}{p'}+\frac{\lambda}{p}})}
+\min\{\mu,\nu\}\|u\|_{\pounds^{\rho}(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha
+\frac{3}{p'}+\frac{2\alpha}{\rho}+\frac{\lambda}{p}})}\,.
\end{equation*}
There exists a constant $C=C(\alpha,p,q)>0$ depending on
$\alpha,p,q$ such that
\begin{equation*}\label{es}
\| \nabla.(u\otimes
v)\|_{\pounds^{\rho}(I,\mathcal{F\dot{N}}_{p,\lambda,q}^{1-4\alpha+\frac{3}{p'}+\frac{2\alpha}{\rho}+\frac{\lambda}{p}}
)}\leq C (\min\{\mu,\nu\})^{-1}\|u\|_X\|v\|_X\,.
\end{equation*}
\end{lemma}


\begin{proof}[Proof of Theorem \ref{thm1}]

We  use the Banach fixed point theorem to ensure the existence of global
mild solutions with small initial data. Note that the functions here are
vector fields, whose norm is the sum of the norms of the three components.
According to Duhamel's principle, the mild solution $(u, b)$ for system
\eqref{GMHD} can be represented as
\begin{equation}\label{df}
\begin{gathered}
u=e^{-t\mu(-\Delta)^{\alpha}}u_0-\int_0^{t}e^{-\mu(t-\tau)(-\Delta
)^{\alpha}}\mathbb{P}\nabla\cdot(u\otimes u- b\otimes b)(\cdot, \tau
)\,d\tau:=\psi_1(u,b),\\
b=e^{-t\nu(-\Delta)^{\alpha}}b_0-\int_0^{t}e^{-\nu(t-\tau)(-\Delta
)^{\alpha}}\mathbb{P}\nabla\cdot(u\otimes b- b\otimes u)(\cdot,\tau
)\,d\tau:=\psi_2(u,b),
\end{gathered}
\end{equation}
where $\mathbb{P}=Id-\nabla\Delta^{-1}\nabla$ is the Leray-Hopf projector,
which is a pseudo differential operator of order $0$.
 Let
\[
X=\pounds ^{\infty}\Big([0,\infty);\mathcal{F\dot{N}}_{p,\lambda
,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)\cap\pounds ^1
\Big([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}
+\frac{\lambda}{p}}\Big)\,.
\]
 For $u,b\in X$, we define the norm of vector $(u,b)$ as
\[
\|(u,b)\|_X=\|u\|_X+\|b\|_X.
\]
Let 
\[
B_{\mu}(u,v):=\int_0^{t}e^{-\mu(t-\tau)(-\Delta)^{\alpha}}
\mathbb{P}\nabla.(u\otimes v)(\tau,x)d\tau.
\]
 It is clear that the system
\eqref{df} can be rewritten as
\[
(u,b)=(\psi_1(u,b),\psi_2(u,b)):=\Phi(u,b).
\]
We note that $B_{\mu}(u,b)$ can be thought as the solution to the heat
equation \eqref{HE}  with $u_0=0$ and force
$f=\mathbb{P}\nabla.(u\otimes v)$. According to Lemma \ref{le} with 
$s=1-2\alpha+\frac{3}{p'} +\frac{\lambda}{p}$ and Lemma \ref{BE} with $\rho=1$, 
and the fact that $\mathbb{P}$ is an homogeneous Fourier multiplier of degree $0$, 
we obtain
\begin{equation} \label{bc}
\begin{aligned}
\|B_{\mu}(u,b)\|_X
&\leq\big(1+\big(\frac{16}{9}\big)^{\alpha
}\big) \|\mathbb{P}\nabla.(u\otimes b)\|_{\pounds ^1(I,\mathcal{F\dot{N}
}_{p,\lambda,q}^{1-2\alpha+ \frac{3}{p'}+\frac{\lambda}{p}})}\\
&\leq\big(1+\big(\frac{16}{9}\big)^{\alpha}\big)C(\min\{\mu,\nu\})^{-1}
\|u\|_X\|b\|_X\,.
\end{aligned}
\end{equation}
We also notice that $e^{-\mu t(-\Delta)^{\alpha}}u_0$ is the solution to the
dissipative equation with $u_0=u_0$ and $f=0$. So, Lemma \ref{le} yields
\begin{equation}
\label{ci}
\|e^{-\mu t(-\Delta)^{\alpha}}u_0\|_X
\leq\big(1+\big(\frac{16}{9}\big)^{\alpha}\big)
\|u_0\|_{\mathcal{F\dot{N}}_{p,\lambda ,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}\,.
\end{equation}
By using the estimates \eqref{bc} and \eqref{ci}, we obtain
\begin{equation} \label{12}
\begin{aligned}
\|\psi_1(u,b)\|_X
&  \leq\big(1+\big(\frac{16}{9}\big)^{\alpha}\big) \|u_0\|_{\mathcal{F\dot
{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}} \\
&\quad +C\big(1+\big(\frac{16}{9}\big)^{\alpha}\big)(\min\{\mu,\nu\})^{-1}
\big(\|u\|_X^2+\|b\|_X^2\big) \,.
\end{aligned}
\end{equation}
Similarly, letting
\[
B_{\nu}(u,v):=\int_0^{t}e^{-\nu(t-\tau)(-\Delta)^{\alpha}
}\mathbb{P}\nabla.(u\otimes v)(\tau,x)d\tau,
\]
 we obtain
\begin{equation} \label{13}
\begin{aligned}
\|\psi_2(u,b)\|_X
& \leq\big(1+\big(\frac{16}{9}\big)^{\alpha}\big) \|b_0\|_{\mathcal{F\dot
{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}} \\
&\quad +2C\big(1+\big(\frac{16}{9}\big)^{\alpha}\big)(\min\{\mu,\nu\})^{-1}
\|u\|_X\|b\|_X\,.
\end{aligned}
\end{equation}
Since $\|(u_0,b_0)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac
{3}{p'}+\frac{\lambda}{p}}}\leq C_0\min\{\mu,\nu\}$, define
\begin{align*}
E=\Big\{(u,b)|(u,b)\in X,\|(u,b)\|_X\leq2\big(1+\big(\frac{16}
{9}\big)^{\alpha}\big) C_0\min\{\mu,\nu\}\Big\},
\end{align*}
where $C_0$ is a constant which can be chosen later. Combining
\eqref{ci}, \eqref{12}, and \eqref{13}, it follows that for
$(u,b)\in E$ we have
\begin{align*}
&\|\Phi(u,b)\|_X \\
& \leq \big(1+\big(\frac{16}{9}\big)^{\alpha}\big) \|(u_0,b_0
)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}} +C\big(1+\big(\frac{16}{9}\big)^{\alpha}\big)(\min
\{\mu,\nu\})^{-1}\|(u,b)\|_X^2\\
& \leq \big(1+\big(\frac{16}{9}\big)^{\alpha}\big)C_0\min\{\mu,\nu\}
+4C\big(1+\big(\frac{16}{9}\big)^{\alpha}\big)^3C_0^2\min\{\mu,\nu\},
\end{align*}
which implies that $\Phi(u,b)\in E$ when we choose $C_0$ small enough such
that $C_0<\frac{1}{16C\big(1+\big(\frac{16}{9}\big)^{\alpha}\big)^2}$.

On the other hand, for any $(u_1,b_1),\,(u_2,b_2)\in E$, we have
\begin{align*}
&\|\psi_1(u_1,b_1)-\psi_1(u_2,b_2)\|_X\\
& \leq \|B_{\mu}(u_1,u_1)-B_{\mu}(u_2,u_2)\|_X
+\|B_{\mu}(b_1 ,b_1)-B_{\mu}(b_2,b_2)\|_X\\
& \leq\|B_{\mu}(u_1,u_1-u_2)+B_{\mu}(u_1-u_2,u_2)\|_X +\|B_{\mu
}(b_1-b_2,b_2)+B_{\mu}(b_1,b_1-b_2)\|_X\\
& \leq C\big(1+\big(\frac{16}{9}\big)^{\alpha}\big) (\min\{\mu,\nu
\})^{-1}((\|u_1\|_X+\|u_2\|_X)\|u_1-u_2\|_X\\
&\quad +(\|b_1\|_X+\|b_2\|_X)\|b_1-b_2\|_X)\\
& \leq 4C\big(1+\big(\frac{16}{9}\big)^{\alpha}\big)^2 C_0(\|u_1
-u_2\|_X+\|b_1-b_2\|_X)\\
& \leq \frac{1}{4}(\|u_1-u_2\|_X+\|b_1-b_2\|_X).
\end{align*}
Similarly,
\begin{align*}
\lefteqn{\|\psi_2(u_1,b_1)-\psi_2(u_2,b_2)\|_X}\\
& \leq \|B_{\nu}(u_2,b_2)-B_{\nu}(u_1,b_1)\|_X+\|B_{\nu}(b_2
,u_2)-B_{\nu}(b_1,u_1)\|_X\\
& \leq 4C\big(1+\big(\frac{16}{9}\big)^{\alpha}\big)^2 C_0(\|u_1
-u_2\|_X+\|b_1-b_2\|_X)\\
& \leq \frac{1}{4}(\|u_1-u_2\|_X+\|b_1-b_2\|_X).
\end{align*}
Consequently,
\begin{align*}
\|\Phi(u_1,b_1)-\Phi(u_2,b_2)\|_X \leq\frac{1}{2}(\|u_1
-u_2\|_X+\|b_1-b_2\|_X).
\end{align*}
From the above estimate, we obtain that $\Phi$ is a contraction mapping from $E$
to $E$. By the Banach fixed point theorem, we conclude that $\Phi$ has a
unique fixed point $(u,b)\in E$ which is the solution of system \eqref{GMHD}.
The proof  is complete. 
\end{proof}

\section{Decay property}

In this section, we first introduce the following interpolation inequality
which have their own interest in the sequel. 

\begin{lemma}\label{16}
Let $\alpha<\frac{5}{4}+\frac{\lambda}{2p}$,
$s>\frac{5}{2}-2\alpha+\frac{\lambda}{p}$,
and $1\leq p,q\leq2$. Then we have
\[
\|(u,v)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}
\lesssim\|(u,v)\|_{L^2}^{1-\frac{5/2-2\alpha+\lambda/p}{s}}
\|(u,v)\|_{\dot{H}^{s}}^{\frac{5/2-2\alpha+\lambda/p}{s}}\,.
\]
\end{lemma}


\begin{proof}
By definition of Fourier Besov-Morrey spaces and H\"{o}lder's inequality we
have
\begin{align*}
&\|u\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}\\
& =\Big\{\sum_{j\in\mathbb{Z}} 2^{j(1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p})q}\|\varphi_j\hat{u}\|_{\mathrm{M}_p^{\lambda}}^q
\Big\}^{1/q}\\
& \lesssim \Big\{\sum_{j\in\mathbb{Z}} 2^{j(1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p})q}\Big(\sup_{x_0\in\mathbb{R}^3} \sup_{r>0}
r^{-\lambda/p} \|\varphi_j\hat{u}\|_{L^2(B(x_0,r))}2^{(\frac
{3}{p}-\frac{3}{2})j}\Big)^q \Big\}^{1/q}\\
& \lesssim\Big\{\sum_{j\leq M} 2^{j(\frac{5}{2}-2\alpha+\frac{\lambda}{p})q}
\|\varphi_j\hat{u}\|_{L^2(\mathbb{R}^3)}^q \Big\}^{1/q}+\Big\{\sum_{j>
M} 2^{j(\frac{5}{2}-2\alpha+\frac{\lambda}{p}-s)q} 2^{jsq}\|\varphi_j\hat
{u}\|_{L^2(\mathbb{R}^3)}^q \Big\}^{1/q}\\
& \lesssim2^{(\frac{5}{2}-2\alpha+\frac{\lambda}{p})M}\Big\{\sum
_{j\in\mathbb{Z}} \|\varphi_j\hat{u}\|_{L^2(\mathbb{R}^3)}^2
\Big\}^{1/2} +2^{(\frac{5}{2}-2\alpha+\frac{\lambda}{p}-s)M}\Big\{\sum
_{j\in\mathbb{Z}} 2^{jsq}\|\varphi_j\hat{u}\|_{L^2(\mathbb{R}^3)}^2
\Big\}^{1/2}\,.
\end{align*}
Taking $M$ such that $2^{M}=\big(\|u\|_{\dot{H}^{s}}/\|u\|_{L^2}\big)^{1/s}$,
using $F\dot{B}_{2,2}^{s}=\dot{B}_{2,2}^{s}=\dot{H}^{s}$ and 
$\dot{B}_{2,2}^{0}=L^2$, we obtain
\begin{align*}
\|u\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}
& \lesssim\Big(\frac{\|u\|_{\dot{H}^{s}}}{\|u\|_{L^2}}\Big)^{\frac{\frac
{5}{2}-2\alpha+\frac{\lambda}{p}}{s}}\|u\|_{F\dot{B}_{2,2}^{0}} +\Big(\frac
{\|u\|_{\dot{H}^{s}}}{\|u\|_{L^2}}\Big)^{\frac{\frac{5}{2}-2\alpha
+\frac{\lambda}{p}-s}{s}}\|u\|_{F\dot{B}_{2,2}^{s}}\\
& \lesssim \|u\|_{L^2}^{1-\frac{\frac{5}{2}-2\alpha+\frac{\lambda}{p}}{s}
}\|u\|_{\dot{H}^{s}}^{\frac{\frac{5}{2}-2\alpha+\frac{\lambda}{p}}{s}}\,.
\end{align*}
Similarly,
\[
\|v\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}\lesssim\|v\|_{L^2}^{1-\frac{\frac{5}{2}-2\alpha
+\frac{\lambda}{p}}{s}}\|v\|_{\dot{H}^{s}}^{\frac{\frac{5}{2}-2\alpha
+\frac{\lambda}{p}}{s}}\,.
\]
Finally,
\begin{align*}
&\|u\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}} +\|v\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha
+\frac{3}{p'}+\frac{\lambda}{p}}} \\
& \lesssim \|u\|_{L^2}^{1-\frac{\frac{5}{2}-2\alpha+\frac{\lambda}{p}}{s}
}\|u\|_{\dot{H}^{s}}^{\frac{\frac{5}{2}-2\alpha+\frac{\lambda}{p}}{s}}
+\|v\|_{L^2}^{1-\frac{\frac{5}{2}-2\alpha+\frac{\lambda}{p}}{s}}
\|v\|_{\dot{H}^{s}}^{\frac{\frac{5}{2}-2\alpha+\frac{\lambda}{p}}{s}}\,.
\end{align*}
This completes the proof. 
\end{proof}


\begin{lemma}\label{pg}
Let $\frac{1}{2}<\alpha\leq1$ and $1\leq p,q\leq2$. Then we have
\begin{equation}\label{epg}
\|uv\|_{\dot{H}^{1-\alpha}}\leq
C\|u\|_{L^2}\|v\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}
+C\|u\|_{\dot{H}^{\alpha}}\|v\|_{\mathcal{F\dot{N}}_{p,\lambda,q}
^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\,.
\end{equation}
\end{lemma}


\begin{proof}
The argument of this lemma is similar to the proof of Lemma \ref{BE}. In fact,
let us introduce some notations about the standard localization operators. 
We set
\[
u_j=\dot{\Delta}_ju,\quad \dot{S}_ju=\sum_{k\leq j-1}\dot{\Delta}
_{k}u,\quad \widetilde{\dot{\Delta}}_ju=\sum_{|k-j|\leq1}\dot{\Delta}
_{k}u,\quad \forall j\in\mathbb{Z}\,.
\]
Using Bony's paraproduct decomposition and the quasi-orthogonality property
for the Littlewood-Paley decomposition, for fixed $j$, we have
\begin{align*}
\dot{\Delta}_j(uv) 
&=\sum_{|k-j|\leq4}\dot{\Delta}_j(\dot{S}_{k-1}u
\dot{\Delta}_{k}v)+ \sum_{|k-j|\leq4}\dot{\Delta}_j(\dot{S}_{k-1}v
\dot{\Delta}_{k}u)+\sum_{k\geq j-3}\dot{\Delta}_j(\dot{\Delta}_{k}u
\widetilde{\dot{\Delta}}_{k}v)\\
&=I_j+II_j+III_j\,.
\end{align*}
For the proof of this lemma, we can write
\begin{equation} \label{IIIj}
\begin{aligned}
\|uv\|_{F\dot{B}_{2,2}^{1-\alpha}} & \leq\Big\{ \sum
_{j\in\mathbb{Z}}2^{j(1-\alpha)2} \|\widehat{I_j}\|_{L^2}^2\Big\}^{1/2}
 + \Big\{ \sum_{j\in\mathbb{Z}}2^{j(1-\alpha)2} \|\widehat{II _j}\|_{L^2
}^2 \Big\}^{1/2}\\
&\quad + \Big\{ \sum_{j\in\mathbb{Z}}2^{j(1-\alpha)2} \|\widehat{III_j}\|_{L^2
}^2\Big\}^{1/2}\,.
\end{aligned}
\end{equation}
The terms $I_j$ and $II_j$ are symmetrical. Using Young's inequality,
and Lemma \ref{bm} with $|\gamma|=0$, we obtain
\begin{align*}
\|\widehat{I_j}\|_{L^2}
& \leq\sum_{|k-j|\leq4}\|\widehat{\dot{S}_{k-1}u
\dot{\Delta}_{k}v}\|_{L^2}\\
& \leq\sum_{|k-j|\leq4}\|\widehat{v}_{k}\|_{L^2}\sum_{l\leq k-2}\|
\widehat{u}_{l}\|_{L^1}\\
& \lesssim\sum_{|k-j|\leq4}\|\widehat{v}_{k}\|_{L^2}\sum_{l\leq
k-2}2^{l(\frac{3}{p'}+\frac{\lambda}{p})}\| \widehat{u}_{l}
\|_{\mathrm{M}_p^{\lambda}}\\
& \lesssim\sum_{|k-j|\leq4}\|\widehat{v}_{k}\|_{L^2}\sum_{l\leq
k-2}2^{l(\frac{3}{p'}+\frac{\lambda}{p})}2^{-l(2\alpha-1)}
2^{l(2\alpha-1)}\| \widehat{u}_{l}\|_{\mathrm{M}_p^{\lambda}}\\
& \lesssim\sum_{|k-j|\leq4}2^{k(2\alpha-1)}\|\widehat{v}_{k}\|_{L^2}
\|u\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}\,.
\end{align*}
Consequently
\begin{equation} \label{Ij}
\Big\{ \sum_{j\in\mathbb{Z}}2^{j(1-\alpha)2} \|\widehat{I_j
}\|_{L^2}^2 \Big\}^{1/2}\lesssim\|v\|_{\dot{H}^{\alpha}}
\|u\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}\,.
\end{equation}
To estimate the term $II_j$, we make a minor modification to get
\begin{align*}
\|\widehat{II_j}\|_{L^2} & \leq\sum_{|k-j|\leq4}\|\widehat{\dot{S}_{k-1}v
\dot{\Delta}_{k}u}\|_{L^2}\\
& \leq\sum_{|k-j|\leq4}\|\widehat{u}_{k}\|_{L^1}\sum_{l\leq k-2}\|
\widehat{v}_{l}\|_{L^2}\\
& \lesssim\sum_{|k-j|\leq4}2^{k(\frac{3}{p'}+\frac{\lambda}{p}
)}\|\widehat{u}_{k}\|_{\mathrm{M}_p^{\lambda}}\sum_{l\leq k-2}\| \widehat
{u}_{l}\|_{L^2}\,.
\end{align*}
So we have
\[
\|\widehat{II_j}\|_{L^2}^2\lesssim\sum_{|k-j|\leq4}2^{2k(\frac
{3}{p'}+\frac{\lambda}{p})}\|\widehat{u}_{k}\|_{\mathrm{M}
_p^{\lambda}}^2\|v\|_{F\dot{B}_{2,2}^{0}}^2\,.
\]
This leads to
\begin{equation} \label{IIj}
\Big\{ \sum_{j\in\mathbb{Z}}2^{j(1-\alpha)2} \|\widehat{II _j
}\|_{L^2}^2 \Big\}^{1/2}\lesssim\|u\|_{\mathcal{F\dot{N}}_{p,\lambda
,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\|v\|_{L^2}\,.
\end{equation}
Now, we estimate the last term, let
\[
III_{jk}:=\dot{\Delta}_j\Big(\sum_{|i-k|\leq1}\dot{\Delta}_{i}v\dot{\Delta
}_{k}u\Big)=\sum_{i=-1}^1\dot{\Delta}_j(\dot{\Delta}_{k}u\dot{\Delta
}_{i+k}v).
\]
The estimate of the so-called ``remainder term'' requires a different
approach.
 First we use the Young inequality \eqref{ym} in  Morrey
spaces, and Lemma \ref{bm} with $|\gamma|=0$, we obtain
\begin{align*}
&2^{j(1-\alpha)}\|\widehat{III_{jk}}\|_{L^2} \\
& \leq \sum_{i=-1}^12^{j\alpha}2^{j(1-2\alpha)} \|\hat{u}_{k}
\|_{L^2}\|\hat{v}_{i+k}\|_{L^1}\\
& \lesssim \sum_{i=-1}^12^{j\alpha}2^{(1-2\alpha)(j-k)}2^{(2\alpha-1)i}2^{(
1-2\alpha)i} 2^{(1-2\alpha)k}\|\hat{u}_{k}\|_{L^2} \\
&\quad\times 2^{(\frac{3}{p'}+\frac{\lambda}{p})(i+k)}
 \|\hat{v}_{i+k}\|_{\mathrm{M}_p^{\lambda}}\\
&:=\sum_{i=-1}^{1}2^{j\alpha}2^{(1-2\alpha)(j-k)}
2^{(2\alpha-1)i}2^{(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p})(i+k)}
\|\hat{u}_{k}\|_{L^2}
\|\hat{v}_{i+k}\|_{\mathrm{M}_p^{\lambda}}\,.
\end{align*}


Taking the $l^2-$norm on both sides in the above estimate, and using the
H\"{o}lder's inequalities for series, we obtain
\begin{equation} \label{IIId}
\begin{aligned}
&\Big\{ \sum_{j\in\mathbb{Z}}2^{j(1-\alpha)2}
\|\widehat{III _j}\|_{L^2}^2 \Big\}^{1/2} \\
& \lesssim\Big\{\sum_{j\in\mathbb{Z}}\Big(\sum_{l\leq3}2^{j\alpha}
2^{(1-2\alpha)l}2^{(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p})(j-l)}
\|\hat{u}_{j-l}\|_{L^2} \|\hat{v}_{j-l}\|_{\mathrm{M}
_p^{\lambda}}\Big)^2\Big\}^{1/2}\\
& \lesssim\sum_{j\in\mathbb{Z}}\sum_{l\leq3}2^{j\alpha}2^{(1-2\alpha
)l}2^{(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p})(j-l)} \|\hat
{u}_{j-l}\|_{L^2}\|\hat{v}_{j-l}\|_{\mathrm{M}_p^{\lambda}
}\\
& \lesssim\sum_{l\leq3}2^{(1-\alpha)l}\Big\{\sum_{j\in\mathbb{Z}}
2^{2\alpha(j-l)}\|\hat{u}_{j-l}\|_{L^2}^2\Big\}^{1/2}
\|v\|_{\mathcal{F\dot{N}}_{p,\lambda,2}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}\\
& \lesssim\|u\|_{\dot{H}^{\alpha}} \|v\|_{\mathcal{F\dot{N}}_{p,\lambda
,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\,,
\end{aligned}
\end{equation}
where we have used the fact that $1\leq q\leq2$ implies
\[
\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}
\hookrightarrow\mathcal{F\dot{N}}_{p,\lambda,2}^{1-2\alpha
+\frac{3}{p'}+\frac{\lambda}{p}}
\]
and $F\dot{B}_{2,2}^{\alpha}=\dot{H}^{\alpha}$.
For the case $\alpha=1$, we have
\begin{equation} \label{IIIId}
\begin{aligned}
&\Big(\sum_{j\in\mathbb{Z}} \|\widehat{III_j}\|_{L^2 }^2\Big)^{1/2}\\
& \leq\sum_{j\in\mathbb{Z}}\Big(\sum_{k\geq j-3} \|\varphi_j(\xi)\times
\sum_{i=-1}^1 \hat{u}_{k}*\hat{v}_{k+i}\|_{L^2}\Big)\\
& \leq\sup_{\xi}\Big(\sum_{j\in\mathbb{Z}}\varphi_j(\xi)\Big)\sum
_{k\in\mathbb{Z}} \|\sum_{i=-1}^1 \hat{u}_{k}*\hat{v}_{k+i}\|_{L^2}\\
& \leq\sum_{i=-1}^1\sum_{k\in\mathbb{Z}} \| \hat{u}_{k}\|_{L^2}\|\hat
{v}_{k+i}\|_{L^1}\\
& \leq\sum_{i=-1}^1\sum_{k\in\mathbb{Z}} 2^{(\frac{3}{p'}
+\frac{\lambda}{p})(k+i)} \|\hat{u}_{k}\|_{L^2} \|\hat{v}_{k+i}
\|_{\mathrm{M}_p^{\lambda}}\\
& \leq\sum_{i=-1}^1\sum_{k\in\mathbb{Z}}2^{-(1-2\alpha)(k+i)} \|\hat{u}
_{k}\|_{L^2} 2^{(k+i)(1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}
)}\|\hat{v}_{k+i}\|_{\mathrm{M}_p^{\lambda}}\\
& \leq C\sum_{k\in\mathbb{Z}}2^{k} \|\hat{u}_{k}\|_{L^2} 2^{k(-1+\frac
{3}{p'}+\frac{\lambda}{p})}\|\hat{v}_{k}\|_{\mathrm{M}_p^{\lambda}}\\
& \leq C \|u\|_{\dot{H}^1} \|v\|_{\mathcal{F\dot{N}}_{p,\lambda,2}
^{-1+\frac{3}{p'}+\frac{\lambda}{p}}}\\
& \leq C \|u\|_{\dot{H}^1} \|v\|_{\mathcal{F\dot{N}}_{p,\lambda,q}
^{-1+\frac{3}{p'}+\frac{\lambda}{p}}}\,.
\end{aligned}
\end{equation}
Estimates \eqref{IIIj}, \eqref{Ij}, \eqref{IIj},  \eqref{IIId} and \ref{IIIId}
yield \eqref{epg}\,.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2}]
The proof is largely based on the idea from the work of
Gallagher-Iftimie-Planchon \cite{32} and (see also \cite{yz,jbe,xcfz} and
\cite[chap.~11]{LR}).
 Let $\varepsilon>0$ be any constant small enough
such that $\varepsilon\leq C_0\min\{\mu,\nu\}$, where $C_0$ is the
constant given in Theorem \ref{thm1} and $\mu,\nu$ are the viscosity
coefficient in \eqref{GMHD}. For $k\in\mathbb{N}$, define
\begin{align*}
\mathcal{A}_{k}=\{\xi\in\mathbb{R}^3;|\xi|\leq k\text{ and } |\hat{u}_0
|+|\hat{b}_0|\leq k\}\,.
\end{align*}
Clearly $(\mathcal{F}^{-1}(\chi_{\mathcal{A}_{k}}\hat{u}_0),\mathcal{F}
^{-1}(\chi_{\mathcal{A}_{k}}\hat{b}_0))$ converge to $(u_0,b_0)$ in
$\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}$.

Then, there is $k\in\mathbb{N}$ such that
\[
\|u_0-\mathcal{F}^{-1}(\chi_{\mathcal{A}_{k}}\hat{u}_0)\|_{\mathcal{F\dot
{N}}_{p, \lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}
+\|b_0-\mathcal{F}^{-1}(\chi_{\mathcal{A}_{k}}\hat{b}_0)\|_{\mathcal{F\dot
{N}}_{p, \lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}
\leq\frac{\varepsilon}{2}\,.
\]
Put
\begin{gather*}
u_{0,k}=\mathcal{F}^{-1}(\chi_{\mathcal{A}_{k}}\hat{u}_0),\quad
b_{0,k} =\mathcal{F}^{-1}(\chi_{\mathcal{A}_{k}}\hat{b}_0),\\
w_{0,k}=u_0-\mathcal{F}^{-1}(\chi_{\mathcal{A}_{k}}\hat{u}_0),\quad
d_{0,k} =b_0-\mathcal{F}^{-1}(\chi_{\mathcal{A}_{k}}\hat{b}_0)\,.
\end{gather*}
Then $u_{0,k},b_{0,k}\in\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha
+\frac{3}{p'}+\frac{\lambda}{p}}\cap\mathrm{L}^2$, and we have shown
that
\begin{equation} \label{elb}
\|w_{0,k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac
{3}{p'}+\frac{\lambda}{p}}} +\|d_{0,k}\|_{\mathcal{F\dot{N}
}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}
\leq \frac{\varepsilon}{2}\,.
\end{equation}
Now, we consider the  system
\begin{equation} \label{GMHD2}
\begin{gathered}
w_{t}+w\cdot\nabla w+ \mu(-\Delta)^{\alpha} w-d\cdot\nabla d+\nabla\pi
_{k}=0\quad\text{in } (0, +\infty)\times\mathbb{R}^3,\\
\nabla\cdot w=0,\quad \nabla\cdot d=0,\\
d_{t}+w\cdot\nabla d+\nu(-\Delta)^{\alpha} d- d\cdot\nabla u= 0 \quad
\text{in } (0, +\infty)\times\mathbb{R}^3,\\
(w,d)|_{t=0}=(w_{0,k},d_{0,k}).
\end{gathered}
\end{equation}
Since $\frac{\varepsilon}{2}\leq C_0\frac{\min\{\mu,\nu\}}{2}\leq C_0
\min\{\mu,\nu\}$, we deduce from Theorem \ref{thm1} that the system
\eqref{GMHD2} has a unique global solution
\[
(w_{k},d_{k})\in\mathcal{C}\big([0,\infty);\mathcal{F\dot{N}}_{p,\lambda
,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\big)\cap\pounds ^1
\big([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}
+\frac{\lambda}{p}}\big)\,,
\]
such that
\begin{align*}
&\Big\| (w_{k},d_{k})\Big\|_{\pounds ^{\infty}\big([0,\infty);\mathcal{F\dot
{N}}_{p,\lambda,q}^{1-2\alpha+ \frac{3}{p'}+\frac{\lambda}{p}
}\big)\cap\pounds ^1\big([0,\infty),\mathcal{F\dot{N}}_{p,\lambda
,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}\big)}\\
&\leq C \Big\|(w_{0,k},d_{0,k})\Big\|_{\mathcal{F\dot{N}}_{p,\lambda
,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\,.
\end{align*}
Moreover, for any $t\geq0$ we have
\begin{equation} \label{75}
\begin{aligned}
&\|w_{k}(t)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}
^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}} +\|d_{k}(t)\|_{\mathcal{F\dot{N}
}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\\
&+ \mu\|w_{k}\|_{\pounds ^1\big([0,t),\mathcal{F\dot{N}}_{p,\lambda
,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}}\big)} +\nu\|d_{k}
\|_{\pounds ^1\big([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac
{3}{p'}+\frac{\lambda}{p}}\big)}\\
& \leq C\|w_{0,k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac
{3}{p'}+\frac{\lambda}{p}}} +C\|d_{0,k}\|_{\mathcal{F\dot{N}
}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\,.
\end{aligned}
\end{equation}
Next, we take into consideration the difference $u_{k}=u-w_{k}$,
$b_{k}=b-d_{k}$, which satisfies
\begin{gather*}
(u_{k},b_{k})\in\mathcal{C}\Big([0,\infty);\mathcal{F\dot{N}}_{p,\lambda
,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}\Big)\cap\pounds ^1
\Big([0,\infty),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}
+\frac{\lambda}{p}}\Big),\\
\partial_{t}u_{k}+ \mu(-\Delta)^{\alpha} u_{k}+(u\cdot\nabla) u_{k}
+(u_{k}\cdot\nabla) w_{k} +\nabla\pi-\nabla\pi_{k}=(b\cdot\nabla) b_{k}
+(b_{k}\cdot\nabla) d_{k},\\
\partial_{t}b_{k}+\nu(-\Delta)^{\alpha} b_{k}+(u\cdot\nabla) b_{k}+(u_{k}
\cdot\nabla) d_{k} = (b\cdot\nabla) u_{k}+(b_{k}\cdot\nabla) w_{k},\\
\nabla\cdot u_{k}=0,\quad \nabla\cdot b_{k}=0,
\end{gather*}
where $\pi$ and $\pi_{k}$ are the correspond pressures to the solutions $u$
and $w_{k}$, respectively. Taking the inner products of the first equation
with $u_{k}$ and of the second equation with $b_{k}$ and integrating by parts,
we can show that
\begin{equation}\label{eh1}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}(\| u_{k}\|_{L^2}^2 +\|b_{k}\|_{L^2}^2)
 +\mu\|(-\Delta)^{\frac{\alpha}{2}} u_{k}\|_{L^2}^2
 +\nu\|(-\Delta)^{\frac{\alpha}{2}}b_{k}\|_{L^2}^2\\
&\leq \Big|\int_{\mathbb{R}^3}(u_{k}\cdot\nabla) w_{k}\cdot u_{k}\,dx\Big|
 +\Big|\int_{\mathbb{R}^3}(b_{k}\cdot\nabla) d_{k}\cdot u_{k}\,dx\Big|\\
&\quad +\Big|\int_{\mathbb{R}^3}(u_{k}\cdot\nabla )d_{k}\cdot b_{k}\,dx\Big|
  +\Big|\int_{\mathbb{R}^3}(b_{k}\cdot\nabla) w_{k}\cdot b_{k}\,dx\Big|\\
&:=I_1+I_2+I_3+I_{4}\,,
\end{aligned}
\end{equation}
where we have used the cancelation property
\[
\int_{\mathbb{R}^3}(b\cdot\nabla)b_{k}\cdot u_{k}\,dx +\int_{\mathbb{R}^3
}(b\cdot\nabla)u_{k}\cdot b_{k}\,dx=0\,.
\]
Integrating by parts, H\"{o}lder's inequality, and Lemma \ref{pg} yield
\begin{align*}
I_1
& := \big|\big<\nabla.(u_{k}\otimes w_{k}),u_{k}\big>\big|\\
& \le\|(-\Delta)^{\frac{1}{2}-\frac{\alpha}{2}}(u_{k}\otimes
w_{k})\|_{L^2} \|(-\Delta)^{\frac{\alpha}{2}}u_{k}\|_{L^2}\\
& \le C\|u_{k}\otimes w_{k}\|_{\dot{H}^{1-\alpha}} \|u_{k}
\|_{\dot{H}^{\alpha}}\\
& \le C\|u_{k}\|_{L^2}\|w_{k}\Big\|_{\mathcal{F\dot{N}
}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}
\|u_{k}\|_{\dot{H}^{\alpha}} +C\|u_{k}\|_{\dot{H}^{\alpha}
}^2 \|w_{k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha
 +\frac{3}{p'}+\frac{\lambda}{p}}}\\
& \le\frac{6C^2}{\mu}\|u_{k}\|_{L^2}^2\|w_{k}
\Big\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}^2 +\frac{\mu}{24}\|u_{k}\|_{\dot{H}^{\alpha}
}^2 +C\|u_{k}\|_{\dot{H}^{\alpha}}^2 \|w_{k}
\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}\,.
\end{align*}
By \eqref{elb} and \eqref{75} we have $\Big\|w_{k}\Big\|_{\mathcal{F\dot{N}
}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\leq
C\frac{\varepsilon}{2}$. We further assume $\varepsilon$ small enough such
that $C^2\varepsilon\leq\frac{\mu}{12}$, thus
\begin{equation} \label{eh2}
I_1\leq\frac{6C^2}{\mu}\|u_{k}\|_{L^2}^2
\|w_{k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac
{3}{p'}+\frac{\lambda}{p}}}^2 +\frac{\mu}{12}\|u_{k}
\|_{\dot{H}^{\alpha}}^2\,.
\end{equation}
To estimate $I_2$, we have
\begin{align*}
I_2
& = \big|\big<\nabla.(b_{k}\otimes d_{k}),u_{k}\big>\big|\\
& \le\|(-\Delta)^{\frac{1}{2}-\frac{\alpha}{2}}(b_{k}\otimes
d_{k})\|_{L^2} \|(-\Delta)^{\frac{\alpha}{2}}u_{k}\|_{L^2}\\
& \le C\|b_{k}\|_{L^2}\|d_{k}\Big\|_{\mathcal{F\dot{N}
}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}
\|u_{k}\|_{\dot{H}^{\alpha}} +C\|b_{k}\|_{\dot{H}^{\alpha}}
\|d_{k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac
{3}{p'}+\frac{\lambda}{p}}}\|u_{k}\|_{\dot{H}^{\alpha}}\\
& \le\frac{6C^2}{\mu}\|b_{k}\|_{L^2}^2\|d_{k}
\Big\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}^2 +\frac{\mu}{24}\|u_{k}\|_{\dot{H}^{\alpha}
}^2+\frac{6C^2}{\mu}\|b_{k}\|_{\dot{H}^{\alpha}}^2
\|d_{k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac
{3}{p'}+\frac{\lambda}{p}}}^2\\
& \quad+\frac{\mu}{24}\|u_{k}\|_{\dot{H}^{\alpha}}^2\,.
\end{align*}
By \eqref{75}, we have $\|d_{k}\|_{\mathcal{F\dot{N}}_{p,\lambda
,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\leq C\frac
{\varepsilon}{2}$. We take sufficiently small $\varepsilon$ such that
$C^2\varepsilon\leq\frac{2\nu^{1/2}\mu^{1/2}}{\sqrt{6}
\sqrt{12}}$, we obtain
\begin{equation}  \label{eh3}
I_2\leq\frac{6C^2}{\mu}\|b_{k}\|_{L^2}^2
\|d_{k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac
{3}{p'}+\frac{\lambda}{p}}}^2 +\frac{\nu}{12}\|b_{k}
\|_{\dot{H}^{\alpha}}^2+\frac{\mu}{12}\|u_{k}\|_{\dot{H}
^{\alpha}}^2\,.
\end{equation}
Similarly,
\begin{gather}\label{eh4}
I_3\leq\frac{6C^2}{\nu}\|u_{k}\|_{L^2}^2
\|d_{k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac
{3}{p'}+\frac{\lambda}{p}}}^2 +\frac{\nu}{12}\|b_{k}
\|_{\dot{H}^{\alpha}}^2+\frac{\mu}{12}\|u_{k}\|_{\dot{H}
^{\alpha}}^2\,, \\
\label{eh5}
I_{4}\leq\frac{6C^2}{\nu}\|b_{k}\|_{L^2}^2
\|w_{k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac
{3}{p'}+\frac{\lambda}{p}}}^2 +\frac{\nu}{12}\|b_{k}
\|_{\dot{H}^{\alpha}}^2\,.
\end{gather}
Combining \eqref{eh1}, \eqref{eh2}, \eqref{eh3}, \eqref{eh4}, and \eqref{eh5},
we obtain
\[ %\label{eh6}
\begin{aligned}
&\frac{d}{dt}(\| u_{k}\|_{L^2}^2 +\|b_{k}\|_{L^2}^2)
 +\mu\| u_{k}\|_{\dot{H}^{\alpha}}^2 +\nu\|b_{k}\|_{\dot{H}^{\alpha}}^2\\
&\leq \max\Big(\frac{12C^2}{\mu},\frac{12C^2}{\nu}\Big)
\Big(\|u_{k}\|_{L^2}^2+\|b_{k}\|_{L^2}^2\Big)
\Big(\|w_{k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}
 +\frac{\lambda}{p}}}^2 \\
&\quad + \|d_{k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}
 ^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}^2\Big)\,. \end{aligned}
\]
Integrating in time and using Gronwall's lemma we obtain
\begin{equation}\label{eh7}
\begin{aligned}
&\| u_{k}\|_{L^2}^2 +\|b_{k}\|_{L^2}^2+\mu\int_0^{t}\| u_{k}\|_{\dot{H}^{\alpha}}^2
+\nu\int_0^{t}\|b_{k}\|_{\dot{H}^{\alpha}}^2\\
&\leq \big(\|u_{0,k}\|_{L^2}^2+\|b_{0,k}\|_{L^2}^2\big)
\exp\Big\{\max\Big(\frac{12C^2}{\mu},\frac{12C^2}{\nu}\Big)\\
&\quad\times\Big(\int_0^{t}\|w_{k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}
^{1-\alpha+\frac{3}{p'} +\frac{\lambda}{p}}}^2
+ \int_0^{t}\|d_{k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}^2\Big)\Big\}\,.
\end{aligned}
\end{equation}
Since $q\leq2$, by H\"{o}lder's inequality, we obtain
\begin{align*}
&\int_0^{t}\|w_{k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}^2\\
 &\leq \Big\{ \sum_{j\in\mathbb{Z}}2^{j(1-\alpha+\frac{3}{p'}
+\frac{\lambda}{p})q} \Big(\int_0^{t}\|\varphi_j\hat{w}_{k}
 \|_{\mathrm{M}_p^{\lambda} }^2\Big)^{\frac{q}{2}}\Big\}^{2/q}\\
&\leq\Big\{ \sum_{j\in\mathbb{Z}}2^{j(1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p})\frac{q}{2}} 2^{j(1+\frac{3}{p'}
+\frac{\lambda}{p})\frac{q}{2}} \|\varphi_j\hat{w}_{k}\|_{L^{\infty}([0,t),
\mathrm{M}_p^{\lambda}) }^{\frac{q}{2}}\|\varphi_j\hat{w}_{k}\|_{L^1([0,t),
\mathrm{M}_p^{\lambda}) }^{\frac{q}{2}}\Big\}^{2/q}\\
&\leq\|w_{k}\|_{\pounds^{\infty}([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha
+\frac{3}{p'}+\frac{\lambda}{p}})} \|w_{k}\|_{\pounds^1([0,t),
\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}})}\,.
\end{align*}
Similarly,
\[
\int_0^{t}\|d_{k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha
+\frac{3}{p'}+\frac{\lambda}{p}}}^2 \leq\|d_{k}
\|_{\pounds ^{\infty}([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}
^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}})} \|d_{k}
\|_{\pounds ^1([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac
{3}{p'}+\frac{\lambda}{p}})}\,.
\]
With the aid of \eqref{75}, we obtain
\begin{align*}
&\int_0^{t}\|w_{k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}^2+ \int_0^{t}\|d_{k}\|
_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}^2\\
&\leq \|w_{k}\|_{\pounds^{\infty}([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha
+\frac{3}{p'}+\frac{\lambda}{p}})} \|w_{k}\|_{\pounds^1([0,t),
\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}})}\\
&\quad+\|d_{k}\|_{\pounds^{\infty}([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha
+\frac{3}{p'}+\frac{\lambda}{p}})} \|d_{k}\|_{\pounds^1([0,t),
\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}})}\\
&\leq\frac{1}{2\mu}\Big(\|w_{k}\|_{\pounds^{\infty}([0,t),\mathcal{F\dot{N}}_{p,
\lambda,q}^{1-2\alpha+ \frac{3}{p'}+\frac{\lambda}{p}})}^2
+\mu^2\|w_{k}\|_{\pounds^1([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}
+ \frac{\lambda}{p}})}^2\Big)\\
&\quad+\frac{1}{2\nu}\Big(\|d_{k}\|_{\pounds^{\infty}([0,t),
\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+ \frac{3}{p'}
+\frac{\lambda}{p}})}^2 +\nu^2\|d_{k}\|_{\pounds^1([0,t),
\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}
+ \frac{\lambda}{p}})}^2\Big)\\
 &\leq\max\big(\frac{1}{2\mu},\frac{1}{2\nu}\big)\Big(\|w_{k}\|_{\pounds^{\infty}
([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+ \frac{\lambda}{p}})} +\|d_{k}\|_{\pounds^{\infty}([0,t),
\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+ \frac{\lambda}{p}})}\\ 
&\quad+\mu\|w_{k}\|_{\pounds^1([0,t),
\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}})}
+\nu\|d_{k}\|_{\pounds^1([0,t),\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}
+ \frac{\lambda}{p}})}\Big)^2\\ &\leq \max\big(\frac{C^2}{2\mu},
\frac{C^2}{2\nu}\big) \|(w_{0,k},d_{0,k})\|_{\mathcal{F\dot{N}}_{p,\lambda,q}
^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}^2\,.
\end{align*}
Consequently,
\begin{equation} \label{eh9}
\begin{aligned}
 &\| u_{k}\|_{L^2}^2 +\|b_{k}\|_{L^2}^2+\mu\int_0^{t}\| u_{k}\|_{\dot{H}^{\alpha}}^2
+\nu\int_0^{t}\|b_{k}\|_{\dot{H}^{\alpha}}^2 \\
&\leq \big(\|u_{0,k}\|_{L^2}^2+\|b_{0,k}\|_{L^2}^2\big)
\exp\big\{\max\Big(\frac{6C^{4}}{\nu\mu},\frac{6C^{4}}{\nu^2},
\frac{6C^{4}}{\mu^2}\Big) \\
&\quad\times  \|(w_{0,k},d_{0,k})\|_{\mathcal{F\dot{N}}
 _{p,\lambda,q}^{1-2\alpha+ \frac{3}{p'}+\frac{\lambda}{p}}}^2\big\}\,.
 \end{aligned}
\end{equation}
Now, noting $\sigma=\frac{5-4\alpha+2\lambda/p}{2\alpha}$ and using Lemma
\ref{16} we  obtain
%\label{eh10}
\begin{align*}
&\int_0^{t}\big(\|u_{k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}^{\frac{2}{\sigma}}
+\|b_{k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}}^{\frac{2}{\sigma}}\big)\\
&\lesssim C^{\frac{2}{\sigma}}\int_0^{t}
\big(\|u_{k}\|_{L^2}^2 +\|b_{k}\|_{L^2}^2\big)^{(\frac{1-\sigma}{\sigma})}
\big(\|u_{k}\|_{\dot{H}^{\alpha}}+\|b_{k}\|_{\dot{H}^{\alpha}}\big)^2\,,
\end{align*}
Now \eqref{eh9} yields
%\label{eh11}
\begin{align*}
&(\| u_{k}\|_{L^2}^2 +\|b_{k}\|_{L^2}^2)^{(\frac{1-\sigma}{\sigma})} \\
&\lesssim \big(\|u_{0,k}\|_{L^2}^2+\|b_{0,k}\|_{L^2}^2\big)
 ^{(\frac{1-\sigma}{\sigma})}
 \exp\Big\{\max\Big(\frac{6C^{4}}{\nu\mu},
 \frac{6C^{4}}{\nu^2},\frac{6C^{4}}{\mu^2}\Big)\big(\frac{1-\sigma}{\sigma}\big) \\
&\quad\times \|(w_{0,k},d_{0,k})\|_{\mathcal{F\dot{N}}_{p,\lambda,q}
^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}^2\Big\}\,.
 \end{align*}
Thus
%\label{eh12}
\begin{align*}
&\int_0^{t}\big(\|u_{k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+ \frac{\lambda}{p}}}^{\frac{2}{\sigma}}
+\|b_{k}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
 + \frac{\lambda}{p}}}^{\frac{2}{\sigma}}\big) \\
&\lesssim C^{\frac{2}{\sigma}}\big(\|u_{0,k}\|_{L^2}^2+\|b_{0,k}\|_{L^2}^2\big)
 ^{(\frac{1- \sigma}{\sigma})}
\exp\Big\{\max\Big(\frac{6C^{4}}{\nu\mu},\frac{6C^{4}}{\nu^2},
 \frac{6C^{4}}{\mu^2}\Big)\big(\frac{1-\sigma}{\sigma}\big) \\
&\quad\times \|(w_{0,k},d_{0,k})\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha
 +\frac{3}{p'} +\frac{\lambda}{p}}}^2\Big\}
\int_0^{t} \big(\|u_{k}\|_{\dot{H}^{\alpha}}+\|b_{k}\|_{\dot{H}^{\alpha}}\big)^2\,.
\end{align*}
Using again \eqref{eh9},
%\label{eh13}
\begin{align*}
&\int_0^{t}\big(\|u_{k}\|_{\dot{H}^{\alpha}}+\|b_{k}\|_{\dot{H}^{\alpha}}\big)^2 \\
&\lesssim (\min\{\mu,\nu\})^{-1} \big(\|u_{0,k}\|_{L^2}^2+\|b_{0,k}\|_{L^2}^2\big)
\exp\Big\{\max\Big(\frac{6C^{4}}{\nu\mu},\frac{6C^{4}}{\nu^2},
 \frac{6C^{4}}{\mu^2}\Big) \\
&\quad\times \|(w_{0,k},d_{0,k})\|_{\mathcal{F\dot{N}}
 _{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}^2\Big\}\,.
\end{align*}
Finally
\begin{align*}%\label{eh14}
&\int_0^{\infty}\big(\|u_{k}\|_{\mathcal
{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+ \frac{\lambda}{p}}}
^{\frac{4\alpha}{5-4\alpha+2\lambda/p}} +\|b_{k}\|_{\mathcal{F\dot{N}
}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+ \frac{\lambda}{p}}}^{\frac{4\alpha
}{5-4\alpha+2\lambda/p}}\big)\\
& \lesssim C^{\frac{4\alpha}{5-4\alpha+2\lambda/p}}(\min\{\mu,\nu
\})^{-1}\big(\|u_{0,k}\|_{L^2} +\|b_{0,k}\|_{L^2
}\big)^{\frac{4\alpha}{5-4\alpha+2\lambda/p}}\\
& \quad\times \exp\Big\{\max\Big(\frac{6C^{4}}{\nu\mu},\frac{6C^{4}}{\nu^2
},\frac{6C^{4}}{\mu^2}\Big) \big(\frac{2\alpha}{5-4\alpha+2\lambda
/p}\big) \|(w_{0,k},d_{0,k})\|_{\mathcal{F\dot{N}}_{p,\lambda
,q}^{1-2\alpha+\frac{3}{p'} +\frac{\lambda}{p}}}^2\Big\}\,.
\end{align*}
So by continuity of $u_{k}$ and $b_{k}$ in $\mathcal{F\dot{N}}_{p,\lambda
,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}$, there exists a time
$t_0$ such that
\[
\|u_{k}(t_0)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac
{3}{p'}+\frac{\lambda}{p}}} +\|b_{k}(t_0)\|_{\mathcal{F\dot{N}
}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\leq
\frac{\varepsilon}{2}\,.
\]
Then we have
\begin{align*}
&\|u(t_0)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac
{3}{p'}+\frac{\lambda}{p}}} +\|b(t_0)\|_{\mathcal{F\dot{N}}_{p,\lambda
,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\\
& \leq \|u_{k}(t_0)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac
{3}{p'}+\frac{\lambda}{p}}} +\|w_{k}(t_0)\|_{\mathcal{F\dot{N}
}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\\
&  +\|b_{k}(t_0)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac
{3}{p'}+\frac{\lambda}{p}}} +\|d_{k}(t_0)\|_{\mathcal{F\dot{N}
}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\\
& \leq\frac{\varepsilon}{2}+\frac{\varepsilon}{2}\,.
\end{align*}
Now, we consider the generalized magneto-hydrodynamic equations starting
at  $t=t_0$,
\begin{gather*}
u_{t}+u\cdot\nabla u+\mu(-\Delta)^{\alpha} u-b\cdot\nabla b+\nabla\pi=0,\\
\nabla\cdot u=0,\quad \nabla\cdot b=0,\\
b_{t}+u\cdot\nabla b+\nu(-\Delta)^{\alpha} b- b\cdot\nabla u= 0 ,\\
u(t_0,x)=u(t_0),\quad b(t_0,x)=b(t_0).
\end{gather*}
By Theorem \ref{thm1} and using the method described in the proof of
\eqref{75}, we immediately obtain
\begin{align*}
&\|u(t)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}
{p'}+\frac{\lambda}{p}}} +\|b(t)\|_{\mathcal{F\dot{N}}_{p,\lambda
,q}^{1-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\\
 &+ \mu\|u\|_{\pounds ^1\Big([t_0,t),\mathcal{F\dot{N}}_{p,\lambda,q}
^{1+\frac{3}{p'}+\frac{\lambda}{p}}\Big)} 
+\nu\|b\|_{\pounds ^1\Big([t_{0},t),\mathcal{F\dot{N}}_{p,\lambda,q}
^{1+\frac{3}{p'}+\frac{\lambda}{p}}\Big)}\\
& \leq C\|u(t_0)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-2\alpha+\frac{3}{p'}
+\frac{\lambda}{p}}} +C\|b(t_0)\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1
-2\alpha+\frac{3}{p'}+\frac{\lambda}{p}}}\leq C\varepsilon\,,
\end{align*}
for all $t\geq t_0$. This completes the proof.
\end{proof}

\subsection*{Acknowledgments}
The authors warmly thank the anonymous referee for his/her careful reading of
the manuscript and some pertinent remarks that lead to various improvements to
this paper.

\begin{thebibliography}{99}

\bibitem{hb} H. Bahouri, J. Y. Chemin, R. Danchin; 
\emph{Fourier Analysis and Nonlinear Partial Differential Equations}, 
Grundlehren der Mathematischen Wissenschaften, Vol. 343. New York: 
Springer-Verlag (2011).

\bibitem{jbe} J. Benameur; 
\emph{Long time decay to the Lei--Lin solution of
3D Navier--Stokes equations,} J. Math. Anal. Appl., 422.1 (2015), 424-434.

\bibitem{cd} Q. Bie, Q. Wang, Z. A. Yao; 
\emph{On the well-posedness of the inviscid Boussinesq equations in the 
Besov-Morrey spaces,} Kinetic Related Models 8.3 (2015).

\bibitem{18} M. Cannone, C. Miao, N. Prioux, B. Yuan; 
\emph{The Cauchy problem for the magneto-hydrodynamic system,} 
Banach Center Publ., 74 (2006), 59-93.

\bibitem{cw} M. Cannone, G. Wu; 
\emph{Global well-posedness for Navier-Stokes equations in critical Fourier-Herz 
spaces}, Nonlinear Anal., 75 (2012), 3754-3760.

\bibitem{35} C. Cao, D. Regmi, J. Wu; 
\emph{The 2D MHD equations with horizontal dissipation and horizontal 
magnetic diffusion,} J. Differ. Equ., 254 (2013), 2661-2681.

\bibitem{23} C. Cao, J. Wu; 
\emph{Two regularity criteria for the 3D MHD equations,} 
J. Differ. Equ., 248 (2010), 2263-2274.

\bibitem{34} C. Cao, J. Wu; 
\emph{Global regularity for the 2D MHD equations with mixed partial dissipation 
and magnetic diffusion}, Adv. Math., 226 (2011), 1803-1822.

\bibitem{36} C. Cao, J. Wu, B. Yuan; 
\emph{The 2D incompressible magnetohydrodynamics equations with only magnetic
 diffusion,} SIAM J. Math. Anal., 46.1 (2014), 588-602.

\bibitem{24} Q. Chen, C. Miao, Z. Zhang; 
\emph{On the regularity criterion of weak solution for the 3D viscous 
magnetohydrodynamics equations,} Commun. Math. Phys., 284 (2008), 919-930.

\bibitem{21} G. Duvaut, J.-L. Lions; 
\emph{In\'equations en thermo\'elasticit\'e et magn\'etohydrodynamique,} 
Arch. Ration. Mech. Anal., 46 (1972), 241-279.

\bibitem{toum} A. El Baraka, M. Toumlilin; 
\emph{Global Well-Posedness for Fractional Navier-Stokes Equations in critical 
Fourier-Besov-Morrey Spaces,} Moroccan J. Pure and Appl. Anal., 3.1 (2017), 1-14.

\bibitem{e6} L. C. Ferreira, L. S. Lima; 
\emph{Self-similar solutions for active scalar equations in Fourier-Besov-Morrey 
spaces,} Monatsh. Math., 175.4 (2014), 491-509.

\bibitem{32} I. Gallagher, D. Iftimie, F. Planchon; 
\emph{Non-blowup at large times and stability for global solutions to the 
Navier-Stokes equations}, CR Math. Acad. Sci. Paris, 334.4 (2002), 289-292.

\bibitem{26} C. He, Y. Wang; 
\emph{On the regularity criteria for weak solutions to the magnetohydrodynamic 
equations,} J. Differ. Equ., 238 (2007), 1-17.

\bibitem{25} C. He, Z. Xin; 
\emph{On the regularity of solutions to the magnetohydrodynamic equations,} 
J. Differ. Equ., 213 (2005), 235-254.

\bibitem{ae} T. Kato; 
\emph{Strong solutions of the Navier-Stokes equations in Morrey spaces}, 
Bol. Soc. Brasil Mat., 22.2 (1992), 127-155.

\bibitem{ll} Z. Lei, F. Lin; 
\emph{Global mild solutions of Navier-Stokes equations}, Comm. Pure Appl. Math., 
64:9 (2011).

\bibitem{LR} P. G. Lemari\'e-Rieusset; 
\emph{The Navier-Stokes Problem in the 21st Century}, CRC Press 2016.

\bibitem{1} Q. Liu, J. Zhao; 
\emph{Global well-posedness for the generalized magneto-hydrodynamic equations 
in the critical Fourier-Herz spaces,} J. Math. Anal. Appl., 420.2 (2014), 
1301-1315.

\bibitem{38} Q. Liu, J. Zhao, S. Cui; 
\emph{Existence and regularizing rate estimates of solutions to a generalized 
magneto-hydrodynamic system in pseudomeasure spaces,} Ann. Mat. Pura Appl., 
191.2 (2012), 293-309.

\bibitem{27} C. Miao, B. Yuan; 
\emph{On well-posedness of the Cauchy problem for MHD system in Besov spaces,} 
Math. Methods Appl. Sci. 32 (2009), 53-76.

\bibitem{22} M. Sermange, R. Temam; 
\emph{Some mathematical questions related to the MHD equations,} 
Commun. Pure Appl. Math., 36 (1983), 635-664.

\bibitem{s} W. Sickel; 
\emph{Smoothness spaces related to Morrey spaces -a survey},
 I, Eurasian Math. J., 3.3 (2012), 110-149.

\bibitem{cri}M. E. Taylor;
 \emph{ Analysis on Morrey spaces and applications
to Navier-Stokes and other evolution equations}, Commun. Partial Differ. Equ.,
17 (1992), 1407-1456.

\bibitem{17} C. Tran, X. Yu, Z. Zhai; 
\emph{ Note on solution regularity of the generalized nagnetohydrodynamic 
equations with partial dissipation,} Nonlinear Anal., 85 (2013), 43-51.

\bibitem{29} Y. Wang; 
\emph{BMO and the regularity criterion for weak solutions to the magnetohydrodynamic 
equations,} J. Math. Anal. Appl., 328 (2007), 1082-1086.

\bibitem{33} Y. Wang, K. Wang; 
\emph{Global well-posedness of the three dimensional magnetohydrodynamics equations,}
 Nonlinear Anal. RWA, 17 (2014), 245-251.

\bibitem{jwu} J. Wu; 
\emph{Generalized MHD equations,} J. Differ. Equ., 195 (2003), 284-312.

\bibitem{15} J. Wu; 
\emph{ Global regularity for a class of generalized
magnetohydrodynamic equations,} J. Math. Fluid Mech.,13 (2011), 295-305.

\bibitem{xcfz} W. Xiao, J. Chen, D. Fan, X. Zhou; 
\emph{Global Well-Posedness and Long Time Decay of Fractional Navier-Stokes 
Equations in Fourier Besov Spaces,} Abstract and Applied Analysis., 2014 (2014).

\bibitem{30} X. Xu, Z. Ye, Z. Zhang; 
\emph{Remark on an improved regularity criterion for the 3D MHD equations,} 
Appl. Math. Lett., 42 (2015), 41-46.

\bibitem{37} J. Yuan; 
\emph{Existence theorem and regularity criteria for the
generalized MHD equations,} Nonlinear Anal. Real World Appl., 
11.3 (2010), 1640-1649.

\bibitem{ws} W. Yuan, W. Sickel, D. Yang; 
\emph{Morrey and Campanato Meet Besov, Lizorkin and Triebel}, 
Lecture Notes in Math. 2005. Springer, 2010.

\bibitem{16} Y. Zhou;  
\emph{Reguality criteria for the generalized viscous MHD equations,} 
Ann. Inst. H. Poincar\'e, Anal. Non Lin\'eaire, 24 (2007), 491-505.

\bibitem{31} Z. Zhou, S. Gala; 
\emph{Regularity criteria for the solutions to the 3D MHD equations in the
 multiplier space,} Z. Angew. Math. Phys., 61 (2010), 193-199.

\bibitem{yz} Y. Zhuan; 
\emph{Global well-posedness and decay results to 3D
generalized viscous magnetohydrodynamic equations,} Ann. Mat. Pura Appl.,
 195.4 (2016), 1111-1121.

\end{thebibliography}


\end{document}