# The estimate of lifespan and local well-posedness for the non-resistive MHD
equations in homogeneous Besov spaces

Weikui $\mbox{Ye}^{1}$111email<EMAIL_ADDRESS>Wei $\mbox{Luo}^{1}$222email:
<EMAIL_ADDRESS>and Zhaoyang $\mbox{Yin}^{1,2}$333email:
<EMAIL_ADDRESS>
${}^{1}\mbox{Department}$ of Mathematics, Sun Yat-sen University,
Guangzhou, 510275, China
${}^{2}\mbox{Faculty}$ of Information Technology,
Macau University of Science and Technology, Macau, China

###### Abstract

In this paper, we mainly investigate the Cauchy problem of the non-resistive
MHD equation. We first establish the local existence in the homogeneous Besov
space $\dot{B}^{\frac{d}{p}-1}_{p,1}\times\dot{B}^{\frac{d}{p}}_{p,1}$ with
$p<\infty$, and give a lifespan $T$ of the solution which depends on the norm
of the Littlewood-Paley decomposition of the initial data. Then, we prove that
if the initial data $(u^{n}_{0},b^{n}_{0})\rightarrow(u_{0},b_{0})$ in
$\dot{B}^{\frac{d}{p}-1}_{p,1}\times\dot{B}^{\frac{d}{p}}_{p,1}$, then the
corresponding existence times $T_{n}\rightarrow T$, which implies that they
have a common lower bound of the lifespan. Finally, we prove that the data-to-
solutions map depends continuously on the initial data when $p\leq 2d$.
Therefore the non-resistive MHD equation is local well-posedness in the
homogeneous Besov space in the Hadamard sense. Our obtained result improves
considerably the recent results in [13, 5, 11].

Keywords: The non-resistive MHD equation, Lifespan, Continuous dependence,
Local well-posedness.
Mathematics Subject Classification: 35Q53, 35B30, 35B44, 35D10, 76W05.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Local existence and uniqueness
  4. 4 Continuous dependence

##  1 Introduction

In this paper, we mainly investigate the Cauchy problem of the incompressible
non-resistive magnetohydrodynamic (MHD) equations:

$\left\\{\begin{array}[]{lll}u_{t}-\Delta u+\nabla P=b\nabla b-u\nabla u,\\\
b_{t}+u\nabla b=b\nabla u,\\\ div~{}u=div~{}b=0,\\\
(u,b)|_{t=0}:=(u_{0},b_{0}),\end{array}\right.$ (1.1)

where the unknowns are the vector fields $u=(u_{1},u_{2},...,u_{d}),\quad
b=(b_{1},b_{2},...,b_{d})$ and the scalar function $P$. Here, $u$ and $b$ are
the velocity and magnetic, respectively, while $P$ denotes the pressure. The
magnetohydrodynamic equation is a coupled system of the Navier-Stokes equation
and Maxwell’s equation. This model describes the interactions between the
magnetic field and the fluid of moving electrically charged particles such as
plasmas, liquid metals, and electrolytes. For more physical background, we
refer to [4, 8].

The MHD equations are of great interest in mathematics and physics. Let’s
review some well-posedness results about the MHD equations. In the case when
there is full magnetic diffusion in system (1.1), G. Duvaut and J.-L. Lions,
[9] firstly established the local existence and uniqueness result in the
Sobolev spaces. They also prove the global existence of strong solutions with
small initial data. M. Sermange, and R. Temam[18] proved the global well-
posedness in the Sobolev spaces with $d=2$. For the system (1.1) with magnetic
diffusion, one may refer to the survey paper[14] written by F-H. Lin and the
references therein for recent progress in this direction.

In the physics of plasmas the magnetic diffusion is very small such that it
can be neglected. In this case, the study of well-posedness will become more
difficult. We refer to [1, 12, 15, 17, 16] about the global existence results
with initial data sufficiently close to the equilibrium. The $L^{2}$ decay
rate was studied by R. Agapito and M. Schonbek[2]. C. Fefferman, D.
McCormicket, J. Robinson and J. Rodrigo consider the critical Sobolev space
about the well-posedness of the system (1.1). They obtained a local existence
result in $\mathbb{R}^{d},d=2,3$ with the initial data $(u_{0},B_{0})\in
H^{s}(\mathbb{R}^{d})\times H^{s}(\mathbb{R}^{d}),s>d/2$ in [10] and
$(u_{0},B_{0})\in H^{s-1-\epsilon}(\mathbb{R}^{d})\times
H^{s}(\mathbb{R}^{d}),s>d/2,0<\epsilon<1$ in [11]. J. Chemin, D. McCormicket,
J. Robinson and J. Rodrigo [5] improved Fefferman et al.’s results to the
inhomogenous Besov space with the initial data $(u_{0},B_{0})\in
B^{\frac{d}{2}-1}_{2,1}(\mathbb{R}^{d})\times
B^{\frac{d}{2}}_{2,1}(\mathbb{R}^{d}),(d=2,3)$ and also proved the uniqueness
with $d=3$. R. Wan in [19] obtained the uniqueness with $d=2$. Recently, J.
Li, W, Tan and Z. Yin in [13] obtained the existence and uniqueness of
solutions to (1.1) with the initial data
$(u_{0},B_{0})\in\dot{B}^{\frac{d}{p}-1}_{p,1}(\mathbb{R}^{d})\times\dot{B}^{\frac{d}{p}}_{p,1}(\mathbb{R}^{d})$
$(1\leq p\leq 2d)$.

However, whether or not the solution for the non-resistive MHD equations is
local well-posedness (local existence, uniqueness and continuous dependence of
the solution) in homogeneous Besov spaces is an open problem which was
proposed by Chemin et al. in [5]. In [13], J. Li, W, Tan and Z. Yin proved the
local existence and uniqueness of solutions to (1.1) in
$(u_{0},B_{0})\in\dot{B}^{\frac{d}{p}-1}_{p,1}(\mathbb{R}^{d})\times\dot{B}^{\frac{d}{p}}_{p,1}(\mathbb{R}^{d})$
$(1\leq p\leq 2d)$. But the continuous dependence of the solution for the
Cauchy problem of the non-resistive MHD equations in homogeneous Besov spaces
has not been proved yet. In the paper, our aim is to solve this open problem
by establishing the local well-posedness for the Cauchy problem (1.1) in
homogeneous Besov spaces. Meanwhile, we generalized the local existence’s
index from $1\leq p\leq 2d$[13] to $1\leq p<\infty$.

For convenience, we transform the system (1.1) into an equivalent form of
compressible type. By using $divu=divB=0$, we have

$u\nabla u=div(u\otimes u),\quad B\nabla B=div(B\otimes B),\quad B\nabla
u=div(u\otimes B).$

Therefore, the system (1.2) is formally equivalent to the following equations

$\left\\{\begin{array}[]{lll}u_{t}-\Delta u=\mathbb{P}(b\nabla b-u\nabla
u),\\\ b_{t}+u\nabla b=b\nabla u,\\\
(u,b)|_{t=0}:=(u_{0},b_{0}),\end{array}\right.$ (1.2)

where $\mathbb{P}=I+\nabla(-\Delta)^{-1}div$ is the Leray project operator,
and the initial data is divergence free $div~{}u_{0}=div~{}b_{0}=0$.

To solve (1.2), the main difficulty is that the system is only partially
parabolic, owing to the magnetic equation which is of hyperbolic type. This
means it’s hard to get the concrete expression for the lifespan $T$
(especially the lower bound of $T$), which creates the main difficulty for
proving the continuous dependence. Therefore, we would like to present a
general functional framework to deal with the local existence of the solution
of (1.2) in the homogeneous Besov spaces. By obtaining the expression of the
lifespan, we get the uniformly lower bound of the lifespan $T$ by a
constructive way (see the key Lemma 4.1 below). Finally, we use the frequency
decomposition (see Theorem 4.4 below) to get the continuous dependence.
Our main theorem can be stated as follows.

######  Theorem 1.1.

Let
$(u_{0},b_{0})\in\dot{B}^{\frac{d}{p}-1}_{p,1}(\mathbb{R}^{d})\times\dot{B}^{\frac{d}{p}}_{p,1}(\mathbb{R}^{d})$
with $d\geq 2$. Then there exists a positive time $T$ such that
(1) Local existence: if $p\in[1,\infty)$, then the system (1.2) has a local
solution $(u,b)$ in $E^{p}_{T}$ with

$E^{p}_{T}:=C([0,T];\dot{B}^{\frac{d}{p}-1}_{p,1}(\mathbb{R}^{d}))\cap
L^{1}([0,T];\dot{B}^{\frac{d}{p}+1}_{p,1}(\mathbb{R}^{d}))\times
C([0,T];\dot{B}^{\frac{d}{p}}_{p,1}(\mathbb{R}^{d})).$

(2) Uniqueness: if $p\in[1,2d]$, then the solution of (1.2) is unique.
(3) Continuous dependence: if $p\in[1,2d]$, then the solution depends
continuously on the initial data in $E^{p}_{T}$.

######  Remark 1.2.

Comparing to [13], we generalized the local existence’s index from
$p\in[1,2d]$ to $p\in[1,\infty)$ and prove the continuous dependence with
$p\in[1,2d]$ in homogeneous Besov spaces.

The remainder of the paper is organized as follows. In Section 2 we introduce
some useful preliminaries. In Section 3, we prove the local existence and the
uniqueness of the solution to (1.2) with the expression of local time being
given. In Section 4, we firstly prove that if the initial data
$(u^{n}_{0},B^{n}_{0})$ tends to $(u_{0},B_{0})$ in
$B^{\frac{d}{p}-1}_{p,1}\times B^{\frac{d}{p}}_{p,1}$, then their local
existence times $T_{n}\rightarrow T$, which implies that they have the common
existence time $T-\delta$ ($\delta$ is small enough). Then we use the method
of frequency decomposition to obtain the continuous dependence.

Notations:  Throughout, we donate
$\dot{B}^{s}_{p,r}(\mathbb{R}^{d}))=\dot{B}^{s}_{p,r}$,
$\|u\|_{\dot{B}^{s}_{p,r}(\mathbb{R}^{d})}+\|v\|_{\dot{B}^{s}_{p,r}(\mathbb{R}^{d})}=\|u,v\|_{\dot{B}^{s}_{p,r}}$
and $C([0,T];\dot{B}^{s}_{p,r}(\mathbb{R}^{d}))=C_{T}(\dot{B}^{s}_{p,r})$,
$L^{p}([0,T];\dot{B}^{s}_{p,r}(\mathbb{R}^{d}))=L^{p}_{T}(\dot{B}^{s}_{p,r})$.
For convenience, we donate $C_{E_{0}}\approx C(1+E_{0}+e^{E_{0}})$ for $C$
large enough.

##  2 Preliminaries

In this section, we will recall some propositons and lemmas on the Littlewood-
Paley decomposition and Besov spaces.

######  Proposition 2.1.

[3] Let $\mathcal{C}$ be the annulus
$\\{\xi\in\mathbb{R}^{d}:\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\\}$. There exist
radial functions $\chi$ and $\varphi$, valued in the interval $[0,1]$,
belonging respectively to $\mathcal{D}(B(0,\frac{4}{3}))$ and
$\mathcal{D}(\mathcal{C})$, and such that

$\forall\xi\in\mathbb{R}^{d},\ \chi(\xi)+\sum_{j\geq 0}\varphi(2^{-j}\xi)=1,$
$\forall\xi\in\mathbb{R}^{d}\backslash\\{0\\},\
\sum_{j\in\mathbb{Z}}\varphi(2^{-j}\xi)=1,$ $|j-j^{\prime}|\geq
2\Rightarrow\mathrm{Supp}\ \varphi(2^{-j}\cdot)\cap\mathrm{Supp}\
\varphi(2^{-j^{\prime}}\cdot)=\emptyset,$ $j\geq 1\Rightarrow\mathrm{Supp}\
\chi(\cdot)\cap\mathrm{Supp}\ \varphi(2^{-j}\cdot)=\emptyset.$

The set $\widetilde{\mathcal{C}}=B(0,\frac{2}{3})+\mathcal{C}$ is an annulus,
and we have

$|j-j^{\prime}|\geq 5\Rightarrow 2^{j}\mathcal{C}\cap
2^{j^{\prime}}\widetilde{\mathcal{C}}=\emptyset.$

Further, we have

$\forall\xi\in\mathbb{R}^{d},\ \frac{1}{2}\leq\chi^{2}(\xi)+\sum_{j\geq
0}\varphi^{2}(2^{-j}\xi)\leq 1,$
$\forall\xi\in\mathbb{R}^{d}\backslash\\{0\\},\
\frac{1}{2}\leq\sum_{j\in\mathbb{Z}}\varphi^{2}(2^{-j}\xi)\leq 1.$

######  Definition 2.2.

[3] Denote $\mathcal{F}$ by the Fourier transform and $\mathcal{F}^{-1}$ by
its inverse. Let $u$ be a tempered distribution in
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$. For all $j\in\mathbb{Z}$, define

$\Delta_{j}u=0\,\ \text{if}\,\
j\leq-2,\quad\Delta_{-1}u=\mathcal{F}^{-1}(\chi\mathcal{F}u),\quad\Delta_{j}u=\mathcal{F}^{-1}(\varphi(2^{-j}\cdot)\mathcal{F}u)\,\
\text{if}\,\ j\geq 0,\quad S_{j}u=\sum_{j^{\prime}<j}\Delta_{j^{\prime}}u.$

Then the Littlewood-Paley decomposition is given as follows:

$u=\sum_{j\in\mathbb{Z}}\Delta_{j}u\quad\text{in}\
\mathcal{S}^{\prime}(\mathbb{R}^{d}).$

Let $s\in\mathbb{R},\ 1\leq p,r\leq\infty.$ The nonhomogeneous Besov space
$B^{s}_{p,r}(\mathbb{R}^{d})$ is defined by

$B^{s}_{p,r}=B^{s}_{p,r}(\mathbb{R}^{d})=\\{u\in
S^{\prime}(\mathbb{R}^{d}):\|u\|_{B^{s}_{p,r}(\mathbb{R}^{d})}=\Big{\|}(2^{js}\|\Delta_{j}u\|_{L^{p}})_{j}\Big{\|}_{l^{r}(\mathbb{Z})}<\infty\\}.$

Similarly, we can define the homogeneous Besov space.

$\dot{B}^{s}_{p,r}=\dot{B}^{s}_{p,r}(\mathbb{R}^{d}):=\\{u\in
S^{\prime}_{h}(\mathbb{R}^{d})|\|u\|_{\dot{B}^{s}_{p,r}}:=\|2^{sj}\|\dot{\Delta}_{j}u\|_{L^{p}(\mathbb{S}^{d})}\|_{l^{r}}\leq\infty\\},$

where the Littlewood-Paley operator $\dot{\Delta}_{j}$ is defined by

$\dot{\Delta}_{j}u=\mathcal{F}^{-1}(\varphi(2^{-j}\cdot)\mathcal{F}u)\,\
\text{if}\,\ j\in\mathbb{Z}.$

######  Lemma 2.3.

Let $s\in(-\frac{d}{p^{\prime}},\frac{d}{p}]$ ($s=\frac{d}{p},r=1$). Assume
$f^{n}$ is uniformly bounded in
$\dot{B}^{s}_{p,r}\cap\dot{B}^{-\delta}_{\infty,\infty}(\forall\delta>0)$ or
$\dot{B}^{s}_{p,r}\cap L^{\infty}$ . Then $\varphi f^{n}$ is bound in
$\dot{B}^{s}_{p,r}\cap\dot{B}^{s-\epsilon_{1}}_{p,r}$
$(0<\epsilon_{1}<s+\frac{d}{p^{\prime}})$, and the map $f^{n}\mapsto\varphi
f^{n}$ is compact in $\dot{B}^{s-\epsilon}_{p,r}$ $(0<\epsilon<\epsilon_{1})$,
where $\varphi\in S(\mathbb{R}^{d})$.

###### Proof.

The proof is based on Theorems 2.93-2.94 in [3], we omit it here. ∎

######  Lemma 2.4.

[6] Let $s_{1},s_{2}\leq\frac{d}{p}$ and
$s_{1}+s_{2}>d\max\\{0,\frac{2}{p}-1\\}$. Assume $f\in\dot{B}^{s_{1}}_{p,1}$
and $g\in\dot{B}^{s_{2}}_{p,1}$. Then there holds

$\|fg\|_{\dot{B}^{s_{1}+s_{2}-\frac{d}{p}}_{p,\infty}}\leq
C\|f\|_{\dot{B}^{s_{1}}_{p,\infty}}\|g\|_{\dot{B}^{s_{2}}_{p,1}}.$

######  Definition 2.5.

[3] Let $s\in\mathbb{R},1\leq p,q,r\leq\infty$ and $T\in(0,\infty].$ The
functional space $\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,r})$ is defined as the
set of all the distributions $f(t)$ satisfying
$\|f\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,r})}:=\|(2^{ks}\|\dot{\Delta}_{k}f(t)\|_{L^{q}_{T}L^{p}})_{k}\|_{l^{r}}<\infty.$

By Minkowski’s inequality, it is easy to find that

$\|f\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,r})}\leq\|f\|_{L^{q}_{T}(\dot{B}^{s}_{p,r})}\quad
q\leq
r,\quad\quad\quad\|f\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,r})}\geq\|f\|_{L^{q}_{T}(\dot{B}^{s}_{p,r})}\quad
q\geq r.$

Finally, we state some useful results about the heat equation and the
transport equation

$\left\\{\begin{array}[]{l}u_{t}+\Delta u=G,\ x\in\mathbb{R}^{d},\ t>0,\\\
u(0,x)=u_{0}(x),\end{array}\right.$ (2.1)
$\left\\{\begin{array}[]{l}f_{t}+v\cdot\nabla f=g,\ x\in\mathbb{R}^{d},\
t>0,\\\ f(0,x)=f_{0}(x),\end{array}\right.$ (2.2)

which are crucial to the proof of our main theorem later.

######  Lemma 2.6.

[7] Let $s\in\mathbb{R},1\leq q,q_{1},p,r\leq\infty$ with $q_{1}\leq q$.
Assume $u_{0}$ in $\dot{B}^{s}_{p,r}$, and $G$ in
$\widetilde{L}^{q_{1}}_{T}(\dot{B}^{s}_{p,r})$. Then (2.1) has a unique
solution $u$ in $\widetilde{L}^{q}_{T}(\dot{B}^{s+\frac{2}{q}}_{p,r})$
satisfying

$\|u\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s+\frac{2}{q}}_{p,r})}\leq
C_{1}\Big{(}\|u_{0}\|_{\dot{B}^{s}_{p,r}}+\|G\|_{\widetilde{L}^{q_{1}}_{T}\dot{B}^{s+\frac{2}{q_{1}}-2}_{p,r}}\Big{)}.$

In particular, if $q_{1}=r=1$, by Minkowski’s inequality we have

$\|u\|_{L^{\infty}_{T}(\dot{B}^{s}_{p,1})\cap
L^{2}_{T}(\dot{B}^{s+1}_{p,1})\cap L^{1}_{T}(\dot{B}^{s+2}_{p,1})}\leq
C_{1}\Big{(}\|u_{0}\|_{\dot{B}^{s}_{p,1}}+\|G\|_{L^{1}_{T}\dot{B}^{s}_{p,1}}\Big{)}.$

######  Lemma 2.7.

[3] Let $s\in[\max\\{-\frac{d}{p},-\frac{d}{p^{\prime}}\\},\frac{d}{p}+1]$
($s=1+\frac{1}{p},r=1$;
$s=\max\\{-\frac{d}{p},-\frac{d}{p^{\prime}}\\},r=\infty$). There exists a
constant $C$ such that for all solutions $f\in L^{\infty}([0,T];B^{s}_{p,r})$
of (2.2) with initial data $f_{0}$ in $\dot{B}^{s}_{p,r}$, and $g$ in
$L^{1}([0,T];\dot{B}^{s}_{p,r})$, we have, for a.e. $t\in[0,T]$,

$\|f(t)\|_{\dot{B}^{s}_{p,r}}\leq
e^{C_{2}V(t)}\Big{(}\|f_{0}\|_{\dot{B}^{s}_{p,r}}+\int_{0}^{t}e^{-C_{2}V(t^{\prime})}\|g(t^{\prime})\|_{\dot{B}^{s}_{p,r}}dt^{\prime}\Big{)},$

where $V^{\prime}(t)=\|\nabla v\|_{\dot{B}^{\frac{d}{p}}_{p,r}\cap
L^{\infty}}$(if $s=1+\frac{1}{p},r=1$, $V^{\prime}(t)=\|\nabla
v\|_{\dot{B}^{\frac{d}{p}}_{p,1}}$).

######  Remark 2.8.

[3] If $div~{}v=0$, we can get the same result with a better indicator:
$\max\\{-\frac{d}{p},-\frac{d}{p^{\prime}}\\}-1<s<\frac{d}{p}+1$(or
$s=\max\\{-\frac{d}{p},-\frac{d}{p^{\prime}}\\}-1,r=\infty$).

######  Lemma 2.9.

Let $s\in(\max\\{1-\frac{d}{p},1-\frac{d}{p^{\prime}}\\},\frac{d}{p}]$
($s=\frac{d}{p},r=1$), $f_{0}\in\dot{B}^{s}_{p,r}$, $g_{0}\in
L^{1}_{T}(\dot{B}^{s}_{p,r})$ and $\nabla v\in
L^{1}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})$. If $v(t,x)$ satisfies one of the
following conditions $(\rho>1)$:
1) when $s>1$, $v\in
L^{\rho}_{T}(L^{\infty}\cap\dot{B}^{\frac{d}{p}}_{p,\infty})$;
2) when $s=1$, $v\in
L^{\rho}_{T}(L^{\infty}\cap\dot{B}^{\frac{d}{p}}_{p,r^{\prime}})$;
3) when $s<1$ and $1\leq p\leq 2$, $v\in
L^{\rho}_{T}(L^{\infty}\cap\dot{B}^{\frac{d}{p}}_{p,\infty}\cap\dot{B}^{\frac{d}{p^{\prime}}}_{p^{\prime},r^{\prime}})$;
4) when $s<1$ and $p\geq 2$, $v\in
L^{\rho}_{T}(L^{\infty}\cap\dot{B}^{\frac{d}{p}}_{p,r^{\prime}})$.
Then (2.2) has a unique solution $f\in C_{T}(\dot{B}^{s}_{p,1})$ with
$r<\infty$ ($f\in C_{Tw}(\dot{B}^{s}_{p,\infty})$ with $r=\infty$).

###### Proof.

Without loss of generality, we only give the proof with $s=\frac{d}{p},r=1$,
other cases are similar.

Firstly, we smooth out the data:

$f^{n}_{0}:=S_{n}f_{0},\quad g^{n}:=\rho_{n}*_{t}S_{n}g,\quad
v^{n}:=\rho_{n}*_{t}S_{n}v.$

Hence, the function

$f^{n}(t,x)=f^{n}_{0}(\psi^{-1}_{t}(x))+\int_{0}^{t}g^{n}(s,\psi_{s}(\psi^{-1}_{t}(x)))ds$

is a solution to

$\frac{d}{dt}f^{n}(t,\psi_{t}(x))=g^{n}(t,\psi_{t}(x)).$

Further, by Theorem 3.14 in [3], we have

$\displaystyle\|f^{n}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\leq
Ce^{\int_{0}^{T}\|v\|_{\dot{B}^{1+\frac{d}{2}}_{p,1}}ds}(\|f^{n}_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\int_{0}^{T}\|g^{n}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}ds).$
(2.3)

Then, setting
$\bar{f}^{n}:=f^{n}-\int_{0}^{t}g^{n}(s)ds=-\int_{0}^{t}v^{n}\nabla f^{n}ds$,
by the Bony decomposition, we get

$\|v^{n}\nabla
f^{n}\|_{L^{\rho}\dot{B}^{\frac{d}{p}-1}_{p,\infty}}\leq\left\\{\begin{array}[]{lll}\|v^{n}\|_{L^{\rho}(L^{\infty}\cap\dot{B}^{\frac{d}{p}}_{p,\infty})}\|f^{n}\|_{L^{\infty}\dot{B}^{\frac{d}{p}}_{p,1}},\quad\quad\quad\quad\text{when
}s>1,\\\
\|v^{n}\|_{L^{\rho}(L^{\infty}\cap\dot{B}^{\frac{d}{p}}_{p,\infty})}\|f^{n}\|_{L^{\infty}\dot{B}^{\frac{d}{p}}_{p,1}},\quad\quad\quad\quad\text{when
}s=1,\\\
\|v^{n}\|_{L^{\rho}(L^{\infty}\cap\dot{B}^{\frac{d}{p}}_{p,\infty}\cap\dot{B}^{\frac{d}{p^{\prime}}}_{p^{\prime},\infty})}\|f^{n}\|_{L^{\infty}\dot{B}^{\frac{d}{p}}_{p,1}},\quad\text{when
}s<1,1\leq p\leq 2,\\\
\|v^{n}\|_{L^{\rho}(L^{\infty}\cap\dot{B}^{\frac{d}{p}}_{p,\infty})}\|f^{n}\|_{L^{\infty}\dot{B}^{\frac{d}{p}}_{p,1}},\quad\quad\quad\quad\text{when
}s<1,p\geq 2.\\\ \end{array}\right.$ (2.4)

This implies that $\bar{f}^{n}$ is uniformly bounded in
$C^{\beta}_{T}(\dot{B}^{\frac{d}{p}-1}_{p,\infty})\cap
L^{\infty}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})$. Lemma 2.3 guarantees that the
map

$\bar{f}^{n}\rightarrow\varphi\bar{f}^{n},\quad\forall\varphi\in
C^{\infty}_{0}$

is compact in $\dot{B}^{\frac{d}{p}-1}_{p,\infty}$. Combining Ascoli’s theorem
and Cantor’s diagonal process thus ensures that

$\varphi\bar{f}^{n}\rightarrow\varphi\bar{f}\quad in\quad
C_{T}(\dot{B}^{\frac{d}{p}-1}_{p,\infty}).$

By the Fatou property, we have
$\varphi\bar{f}^{n}\rightharpoonup\varphi\bar{f}\in
L^{\infty}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})$. By interpolation, we get

$\varphi\bar{f}^{n}\rightarrow\varphi\bar{f}\quad in\quad
C_{T}(\dot{B}^{\frac{d}{p}-\epsilon}_{p,1}),\quad 0<\epsilon<1.$

Finally, applying the above results we can pass the limit in the weak sense:

$f:=\lim_{n\rightarrow\infty}f^{n}=\bar{f}+\int_{0}^{t}g(s)ds.$

It is easily to deduce that $f(t,x)$ is a solution of (2.2) and $f\in
C_{T}(\dot{B}^{\frac{d}{p}}_{p,1})$ (For more details see Theorem 3.19 in
[3]). This completes the proof. ∎

######  Remark 2.10.

If $div~{}v=0$, we can get the same result with a better indicator:
$\max\\{-\frac{d}{p},-\frac{d}{p^{\prime}}\\}<s<\frac{d}{p}$
($s=\frac{d}{p},r=1$). The proof is similar to Lemma 2.10, we omit the detail
here.

######  Definition 2.11.

[3] Let $a>0$, $\mu(r)$ be a continue non-zero and non-decreasing function
from $[0,a]$ to $\mathbb{R}^{+}$, $\mu(0)=0$. We say that $\mu$ is an Osgood
modulus of continuity if

$\int_{0}^{a}\frac{1}{\mu(r)}dr=+\infty.$

######  Lemma 2.12.

[3] Let $\rho$ be a measurable function from $[0,T]$ to $[0,a]$, $\gamma$ a
locally integrable function from $[0,T]$ to $\mathbb{R}^{+}$, and $\mu$ be an
Osgood modulus of continuity. If for some $\rho_{0}\geq 0$,

$\rho(t)\leq\rho_{0}+\int_{0}^{t}\gamma(s)\mu(\rho(s))ds\quad for\quad
a.e.\quad t\in[0,T],$

then we have

$-M(\rho(t))+M(\rho_{0})\leq\int_{0}^{t}\gamma(s)ds\quad with\quad
M(x)=\int_{x}^{a}\frac{dr}{\mu(r)}.$ (2.5)

For example, if $\mu(r)=r$, we obtain the Gronwall inequality:

$\rho(t)\leq\rho_{0}e^{\int_{0}^{t}\gamma(s)ds},\quad M(x)=lna-lnx.$

If $\mu(r)=rln(e+c/r)$, it’s easy to check that it is still an Osgood modulus
of continuity. Then we have

$\rho(t)\leq\rho_{0}\frac{ce^{e^{\int_{0}^{t}\gamma(s)ds}}}{c-\rho_{0}(e^{\int_{0}^{t}\gamma(s)ds}-e)},\quad-M(\rho(t))+M(\rho_{0})\geq
ln[\frac{ln(e+\frac{c}{\rho_{0}})}{ln(e+\frac{c}{\rho(t)})}].$

Since $\gamma$ is locally integrable, we deduce that if $\rho_{0}$ small
enough such that $\rho_{0}\leq\frac{c}{2(e^{\int_{0}^{t}\gamma(s)ds}-e)}$,
then

$\rho(t)\leq 2\rho_{0}e^{e^{\int_{0}^{t}\gamma(s)ds}}.$

##  3 Local existence and uniqueness

We divide the proof of local existence and uniqueness into 4 steps:

Step 1: An iterative scheme.

Set $(u^{n}_{0},b^{n}_{0}):=(\dot{S}_{n}u_{0},\dot{S}_{n}b_{0})$ and define
the first term $(u^{0},b^{0}):=(e^{t\Delta}u_{0},e^{t\Delta}b_{0})$. Then we
introduce a sequence $(u^{n},b^{n})$ with the initial data
$(u^{n}_{0},b^{n}_{0})$ by solving the following linear transport and heat
conductive equations:

$\left\\{\begin{array}[]{lll}u^{n+1}_{t}-\Delta u^{n+1}=\mathbb{P}(b^{n}\nabla
b^{n}-u^{n}\nabla u^{n}),\\\ b^{n+1}_{t}+u^{n}\nabla b^{n+1}=b^{n}\nabla
u^{n},\\\
(u^{n}_{0},b^{n}_{0}):=(\dot{S}_{n}u_{0},\dot{S}_{n}b_{0}),\end{array}\right.$
(3.1)

where $\dot{S}_{n}g:=\sum_{k<n}\dot{\Delta}_{k}g$, it makes sense in Besov
spaces when $s<\frac{d}{p}$ or $s=\frac{d}{p},r=1$.

Step 2: Uniform estimates.

Taking advantage of Lemmas 2.6-2.7, we shall bound the approximating sequences
in $E^{p}_{T}$. Now we claim that there exists some $T$ independent of $n$
such that the solutions $(u^{n},b^{n})$ satisfy the following inequalities :

$(H_{1}):\quad\|b^{n}\|_{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})}+\|u^{n}\|_{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}-1}_{p,1})}\leq
6E_{0},$ $(H_{2}):\quad\|u^{n}\|_{A_{T}}\leq 2a,\quad
A_{T}:={L^{2}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})\cap
L^{1}_{T}(\dot{B}^{\frac{d}{p}+1}_{p,1}}),$

where
$E_{0}:=\|b_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\|u_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}$.
Now we suppose that $a$ is small enough such that ($a$ will be determined
later):

$\displaystyle a\leq\min\\{\sqrt{\frac{E_{0}}{4C_{1}}},c\\},$ (3.2)

where $c$ is any positive real number satisfying
$c\leq\frac{1}{12},e^{C_{2}c}\leq\frac{3}{2},4cC_{1}\leq\frac{1}{2}$. Suppose
that $T$ satisfies that

$\displaystyle C_{1}E^{2}_{0}T\leq\frac{1}{72}a,\quad 36C_{1}E_{0}T\leq
1,\quad\|e^{t\Delta}u_{0}\|_{A_{T}}\leq a,$ (3.3)

where $C_{1}$ and $C_{2}$ are the constants in Lemmas 2.6-2.7. (Indeed, we
should take $C_{1}$ and $C_{2}$ more large as we need.)

It’s easy to check that $(H_{1})-(H_{2})$ hold true for $n=0$. Now we will
show that if $(H_{1})-(H_{2})$ hold true for $n$, then they hold true for
$n+1$. In fact, by (3.2)-(3.3) and Lemmas 2.6-2.7, we have

$\displaystyle\|u^{n+1}\|_{A_{T}}$
$\displaystyle\leq\|e^{t\Delta}u_{0}\|_{A_{T}}+\|\mathbb{P}div(-u^{n}\otimes
u^{n}+b^{n}\otimes b^{n})\|_{L^{1}_{T}(\dot{B}^{\frac{d}{p}-1}_{p,1})}$
$\displaystyle\leq a+C_{1}4a^{2}+36C_{1}E^{2}_{0}T\leq 2a,$ (3.4)
$\displaystyle\|u^{n+1}\|_{L^{\infty}_{T}\dot{B}^{\frac{d}{p}-1}_{p,1}}$
$\displaystyle\leq\|e^{t\Delta}u_{0}\|_{L^{\infty}_{T}\dot{B}^{\frac{d}{p}-1}_{p,1}}+\|\mathbb{P}div(-u^{n}\otimes
u^{n}+b^{n}\otimes b^{n})\|_{L^{1}_{T}\dot{B}^{\frac{d}{p}-1}_{p,1}}$
$\displaystyle\leq\|u_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}+\|\mathbb{P}div(-u^{n}\otimes
u^{n}+b^{n}\otimes b^{n})\|_{L^{1}_{T}\dot{B}^{\frac{d}{p}-1}_{p,1}}$
$\displaystyle\leq E_{0}+C_{1}4a^{2}+36C_{1}E^{2}_{0}T\leq 3E_{0}.$ (3.5)

and

$\displaystyle\|b^{n+1}\|_{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})}$
$\displaystyle\leq
e^{C_{2}a}(\|b_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\|div(u^{n}\otimes
b^{n})\|_{L^{1}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})}$ $\displaystyle\leq
e^{C_{2}a}(E_{0}+12aE_{0})$ $\displaystyle\leq 3E_{0}.$ (3.6)

This implies $(H_{1})-(H_{2})$ hold true for $n+1$.

Finally, we have to obtain the relationship between the existence time $T$ and
the initial data via (3.3). It is easy to deduce that

$T\leq T_{0}:=\min\\{\frac{a}{72C_{1}E^{2}_{0}},\frac{1}{36C_{1}E_{0}}\\}.$

Now we turn to study the condition $\|e^{t\Delta}u_{0}\|_{A_{T}}\leq a$ of
(3.3). For this purpose, we have to classify the initial data.
(1) For
$\|u_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}\leq\bar{c}=\min\\{\frac{1}{4C_{1}},c\\}$,
we let $a:=\min\\{\sqrt{\frac{E_{0}}{4C_{1}}},c\\}$, which implies (3.2).

Then we have

$\|e^{t\Delta}u_{0}\|_{A_{T}}\leq\|u_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}\leq\min\\{\sqrt{\frac{\|u_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}}{4C_{1}}},c\\}\leq
a.$

(2) For
$\|u_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}>\bar{c}=\min\\{\frac{1}{4C_{1}},c\\}$,
we let
$a:=\min\\{\sqrt{\frac{\bar{c}}{4C_{1}}},c\\}\leq\min\\{\sqrt{\frac{E_{0}}{4C_{1}}},c\\}$,
which also implies (3.2).

Since $u_{0}\in\dot{B}^{\frac{d}{p}-1}_{p,1}$, there exists an integer $j_{0}$
such that ($j_{0}$ may not be unique):

$\displaystyle\sum_{|j|\geq
j_{0}}\|\dot{\Delta}_{j}u_{0}\|_{L^{p}}2^{(\frac{d}{p}-1)j}<\frac{a}{4}.$
(3.7)

Defining that
$T_{1}:=\frac{a}{4}\frac{1}{2^{2j_{0}}\|u_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}}$
and
$T_{2}:=\frac{a^{2}}{4^{2}}\frac{1}{2^{2j_{0}}\|u_{0}\|^{2}_{\dot{B}^{\frac{d}{p}-1}_{p,1}}}$,
we get

$\displaystyle\|e^{t\Delta}u_{0}\|_{L^{1}_{T_{1}}(\dot{B}^{\frac{d}{p}+1}_{p,1})}$
$\displaystyle\leq\sum_{|j|\leq
j_{0}}\int_{0}^{T_{1}}\|e^{t\Delta}\dot{\Delta}_{j}u_{0}\|_{L^{p}}2^{(\frac{d}{p}+1)j}dt+\sum_{|j|>j_{0}}\int_{0}^{T_{1}}e^{-t2^{2j}}\|\dot{\Delta}_{j}u_{0}\|_{L^{p}}2^{(\frac{d}{p}+1)j}dt$
$\displaystyle\leq 2^{2j_{0}}\sum_{|j|\leq
j_{0}}\int_{0}^{T_{1}}\|\dot{\Delta}_{j}u_{0}\|_{L^{p}}2^{(\frac{d}{p}-1)j}dt+\sum_{|j|>j_{0}}\int_{0}^{T_{1}}e^{-t2^{2j}}\|\dot{\Delta}_{j}u_{0}\|_{L^{p}}2^{(\frac{d}{p}+1)j}dt$
$\displaystyle\leq
2^{2j_{0}}T_{1}\|u_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}+\sum_{|j|>j_{0}}(1-e^{-T_{2}2^{2j}})\|\dot{\Delta}_{j}u_{0}\|_{L^{p}}2^{(\frac{d}{p}-1)j}$
$\displaystyle\leq
2^{2j_{0}}T_{1}\|u_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}+\sum_{|j|>j_{0}}\|\dot{\Delta}_{j}u_{0}\|_{L^{p}}2^{(\frac{d}{p}-1)j}\leq\frac{1}{2}a,$
(3.8)

and

$\displaystyle\|e^{t\Delta}u_{0}\|_{L^{2}_{T_{2}}(\dot{B}^{\frac{d}{p}}_{p,1})}$
$\displaystyle\leq\sum_{|j|\leq
j_{0}}[\int_{0}^{T_{2}}\|e^{t\Delta}\dot{\Delta}_{j}u_{0}\|^{2}_{L^{p}}dt]^{\frac{1}{2}}2^{\frac{d}{p}j}+\sum_{|j|>j_{0}}[\int_{0}^{T_{2}}(e^{-t2^{2j}}\|\dot{\Delta}_{j}u_{0}\|_{L^{p}})^{2}dt]^{\frac{1}{2}}2^{\frac{d}{p}j}$
$\displaystyle\leq
2^{j_{0}}T^{\frac{1}{2}}_{2}\|u_{0}\|_{\dot{B}^{\frac{d}{p}-1}}+\sum_{|j|>j_{0}}(1-e^{-T_{2}2^{2j}})^{\frac{1}{2}}\|\dot{\Delta}_{j}u_{0}\|_{L^{p}}2^{(\frac{d}{p}-1)j}$
$\displaystyle\leq
2^{j_{0}}T^{\frac{1}{2}}_{2}\|u_{0}\|_{\dot{B}^{\frac{d}{p}-1}}+\sum_{|j|>j_{0}}\|\dot{\Delta}_{j}u_{0}\|_{L^{p}}2^{(\frac{d}{p}-1)j}\leq\frac{1}{2}a.$
(3.9)

Letting $T=\min\\{T_{0},T_{1},T_{2}\\}$, we get

$\|e^{t\Delta}u_{0}\|_{A_{T}}\leq a.$

Finally, if we choose $T$ to satisfy that

$T=\begin{cases}T_{0},&\|u_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}\leq\frac{1}{4C_{1}},\\\
\min\\{T_{0},T_{1},T_{2}\\},&\|u_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}>\frac{1}{4C_{1}},\end{cases}$
(3.10)

then (3.3) holds true. For this $T$, we have the approximate sequence
$(u^{n},b^{n})$ is uniformly bounded in $E^{p}_{T}$.

######  Remark 3.1.

By (3.10), we know that if the initial data is small, the local existence time
$T$ depends only on $E_{0}$. However, for large initial data, the local
existence time $T$ depends on both $E_{0}$ and the $j_{0}$ which satisfies
(3.7).

Step 3: Existence of a solution.

This step is similar to the process of [3, 13, 6], we also use the compactness
argument in Besov spaces for the approximate sequence $(u^{n},b^{n})$ to get
some solution $(u,b)$ of (1.2). Since $(u^{n},b^{n})$ is uniformly bounded in
$E^{p}_{T}$, the interpolation inequality yields that $u^{n+1}$ is also
uniformly bounded in $L^{q}_{T}(\dot{B}^{\frac{d}{2}-1+\frac{2}{q}}_{p,1})$
for $1\leq q\leq\infty$. Then, by Lemma 2.6-2.7, after some calculations, we
can easily get that (for fixed $0<\epsilon<\frac{d}{p}$):

$\partial_{t}u^{n+1}\text{is uniformly bounded in
}L^{\frac{2}{2-\epsilon}}_{T}(\dot{B}^{\frac{d}{p}-1-\epsilon}_{p,1}+\dot{B}^{\frac{d}{p}-1}_{p,1}),$
$\partial_{t}b^{n+1}\text{is uniformly bounded in
}L^{2}_{T}(\dot{B}^{\frac{d}{p}-1}_{p,1}).$

Let $\\{\chi_{j}\\}_{j\in\mathbb{N}}$ be a sequence of smooth functions with
value in $[0,1]$ supported in the ball $B(0,j+1)$ and equal to 1 on $B(0,j)$.
The above argument ensures that $u^{n+1}$ is uniformly bounded in
$C^{\sigma(\epsilon)}_{T}(\dot{B}^{\frac{d}{p}-1-\epsilon}_{p,1}+\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
C_{T}(\dot{B}^{\frac{d}{p}-1}_{p,1})$ ($\sigma(\epsilon)>0$ for fixed
$\epsilon>0$ small enough), and $b^{n+1}$ is uniformly bounded in
$C^{\frac{1}{2}}_{T}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
C_{T}(\dot{B}^{\frac{d}{p}}_{p,1})$. Then by Lemma 2.3 with
$\epsilon_{1}=2\epsilon$ ($d\geq 2$), since the embedding
$\dot{B}^{\frac{d}{p}-1-2\epsilon}_{p,1}\cap\dot{B}^{\frac{d}{p}-1}_{p,1}\hookrightarrow\dot{B}^{\frac{d}{p}-1-\epsilon}_{p,1}$
and
$\dot{B}^{\frac{d}{p}-1-2\epsilon}_{p,1}\cap\dot{B}^{\frac{d}{p}}_{p,1}\hookrightarrow
B^{\frac{d}{p}-1}_{p,1}$ are locally compact, by applying Ascoli’s theorem and
Cantor’s diagonal process, there exist some functions $(u_{j},b_{j})$ such
that for any $j\in\mathbb{N}$, $\chi_{j}u^{n}$ tends to $u_{j}$, and
$\chi_{j}b^{n}$ tends to $b_{j}$. As $\chi_{j}\chi_{j+1}=\chi_{j}$, we have
$u_{j}=\chi_{j}u_{j+1}$ and $b_{j}=\chi_{j}b_{j+1}$. From that, we can easily
deduce that there exists $(u,b)$ such that for all $\chi\in D(R^{d})$,

$\left\\{\begin{array}[]{l}\chi u^{n}\rightarrow\chi u\quad in\quad
C_{T}(\dot{B}^{\frac{d}{p}-1-\epsilon}_{p,1}),\\\ \chi b^{n}\rightarrow\chi
b\quad in\quad C_{T}(\dot{B}^{\frac{d}{p}-1}_{p,1}),\end{array}\right.$ (3.11)

as n tends to $\infty$ (up to a subsequence). By interpolation, we have

$\left\\{\begin{array}[]{l}\chi u^{n}\rightarrow\chi u\quad in\quad
L^{1}_{T}(\dot{B}^{\frac{d}{p}+1-\delta}_{p,1}),\quad
0<\epsilon<1+\epsilon,\\\ \chi b^{n}\rightarrow\chi b\quad in\quad
C_{T}(\dot{B}^{\frac{d}{p}-\delta}_{p,1}),\quad 0<\delta<1.\end{array}\right.$
(3.12)

Note that $(u^{n},b^{n})$ is uniformly bounded in $E^{p}_{T}$. By the Fatou
property, we readily get

$(u,b)\in(\widetilde{L}^{\infty}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
L^{1}(\dot{B}^{\frac{d}{p}+1}_{p,1}))^{d}\times(L^{\infty}(\dot{B}^{\frac{d}{p}}_{p,1}))^{d}.$

Finally, it is a routine process to verify that $(u,b)$ satisfies the system
(1.2). Following the argment of Theorem 3.19 in [3], we have $(u,b)\in
E^{p}_{T}$.

Step 4: Uniqueness.

The proof of the uniqueness of (1.2) is similar to [13] with $p\leq 2d$, we
omit it here.

##  4 Continuous dependence

Before proving the continuous dependence of solutions to (1.2), firstly we
need to prove that let $T$ be a lifespan corresponding to the initial data
$u_{0}$ by (3.10), if $(u^{n}_{0},b^{n}_{0})$ tends to $(u_{0},b_{0})$ in
$\dot{B}^{\frac{d}{p}-1}_{p,1}\times\dot{B}^{\frac{d}{p}}_{p,1}$, then there
exists a lifespan $T^{n}$ corresponding to $(u^{n}_{0},b^{n}_{0})$ such that
$T^{n}\rightarrow T$. This implies that $T-\delta$ (for some small $\delta$)
is a common lifespan both for $u^{n}$ and $u$ when $n$ is sufficiently large.
We first give a useful lemma:

######  Lemma 4.1.

Let
$(u_{0},b_{0})\in\dot{B}^{\frac{d}{p}-1}_{p,1}\times\dot{B}^{\frac{d}{p}}_{p,1}$
be the initial data of (1.2) with $p\leq 2d$, if there exists another initial
data
$(u^{n}_{0},b^{n}_{0})\in\dot{B}^{\frac{d}{p}-1}_{p,1}\times\dot{B}^{\frac{d}{p}}_{p,1}$
such that
$\|u^{n}_{0}-u_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}},\|b^{n}_{0}-b_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\rightarrow
0\quad(n\rightarrow\infty)$, then we can construct a lifespan $T^{n}$
corresponding to $(u^{n}_{0},b^{n}_{0})$ such that

$T^{n}\rightarrow T,\quad\quad n\rightarrow\infty,$

where the lifespan $T$ correspondsto $(u_{0},b_{0})$.

###### Proof.

By virtue of Remark 3.1, we only consider the large initial data. Thus, we
need to prove that $T^{n}\rightarrow T$, when
$\|u_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}>\frac{1}{4C_{1}}$. For convenience,
we write down the definitions of $T_{0},T_{1},T_{2}$ here:

$T_{0}=\min\\{\frac{a}{72C_{1}E^{2}_{0}},\frac{1}{36C_{1}E_{0}}\\},\quad
T_{1}=\frac{a}{4}\frac{1}{2^{2j_{0}}\|u_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}},\quad
T_{2}=\frac{a^{2}}{4^{2}}\frac{1}{2^{2j_{0}}\|u_{0}\|^{2}_{\dot{B}^{\frac{d}{p}-1}}},$

where $j_{0}$ is a fixed integer such that

$\sum_{|j|\geq
j_{0}}\|\dot{\Delta}_{j}u_{0}\|_{L^{p}}2^{(\frac{d}{p}-1)j}<\frac{a}{4}.$

Since $u_{0}\in\dot{B}^{\frac{d}{p}-1}_{p,1}$, we can suppose that $j_{0}$ is
the smallest integer such that the above inequality holds true. Since
$E^{n}_{0}\rightarrow E_{0}$, it follows that $T^{n}_{0}\rightarrow T_{0}$. In
order to prove that $T^{n}_{1}\rightarrow T_{1}$ and $T^{n}_{2}\rightarrow
T_{2}$, it is sufficient to show that there exists a corresponding sequence
$j^{n}_{0}$ satisfying

$\sum_{|j|\geq
j^{n}_{0}}\|\dot{\Delta}_{j}u^{n}_{0}\|_{L^{p}}2^{(\frac{d}{p}-1)j}<\frac{a}{4},$

and $j^{n}_{0}\rightarrow j_{0}$.

For any $0<\epsilon<\frac{a}{4}$, there exists $N_{\epsilon}$ such that for
$n\geq N_{\epsilon}$ we have

$\|u^{n}_{0}-u_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}\leq\epsilon.$

For this $\epsilon$, we define that $j^{\epsilon}_{0}$ is the smallest integer
such that

$\sum_{|j|\geq
j^{\epsilon}_{0}}\|\dot{\Delta}_{j}u_{0}\|_{L^{p}}2^{(\frac{d}{p}-1)j}<\frac{a}{4}-\epsilon.$

By the definition of $j_{0}$, we have $j_{0}\leq j^{\epsilon}_{0}$.

Replacing $\epsilon$ by $\frac{\epsilon}{m}~{}(m\in\mathbb{N}^{+})$, we can
find $N_{\frac{\epsilon}{m}}$ such that for $n\geq N_{\frac{\epsilon}{m}}$,

$\|u^{n}_{0}-u_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}\leq\frac{\epsilon}{m}.$

For this $\frac{\epsilon}{m}$, we define that $j^{\frac{\epsilon}{m}}_{0}$ is
the smallest integer such that

$\sum_{|j|\geq
j^{\frac{\epsilon}{m}}_{0}}\|\dot{\Delta}_{j}u_{0}\|_{L^{p}}2^{(\frac{d}{p}-1)j}<\frac{a}{4}-\frac{\epsilon}{m}.$

Since $\frac{a}{4}-\frac{\epsilon}{m}>\frac{a}{4}-\frac{\epsilon}{m-1}$, it
follows that

$j_{0}\leq j^{\frac{\epsilon}{m}}_{0}\leq j^{\frac{\epsilon}{m-1}}_{0}.$

Now letting $\bar{j}^{m}_{0}:=j^{\frac{\epsilon}{m}}_{0},m=1,2,3,...$, we
deduce that

$\displaystyle\sum_{|j|\geq\bar{j}^{m}_{0}}\|\dot{\Delta_{j}}u^{n}_{0}\|_{L^{p}}2^{(\frac{d}{p}-1)j}\leq\|u^{n}_{0}-u_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}+\sum_{|j|>\bar{j}^{m}_{0}}\|\dot{\Delta}_{j}u_{0}\|_{L^{p}}2^{(\frac{d}{p}-1)j}<\frac{\epsilon}{m}+\frac{a}{4}-\frac{\epsilon}{m}=\frac{a}{4},\quad
n\geq N_{\frac{\epsilon}{m}}.$ (4.1)

Since $\bar{j}^{m}_{0}$ is a monotone and bounded sequence, we deduce that
$\bar{j}^{m}_{0}\rightarrow\bar{j}_{0}$ $(m\rightarrow\infty)$ for some
integer $\bar{j}_{0}\geq j_{0}$. For any $0<\bar{\epsilon}<1$ there exists $N$
such that if $m\geq N$

$|\bar{j}^{m}_{0}-\bar{j}_{0}|\leq\bar{\epsilon}<1,$

Note that $\bar{j}^{m}_{0},\bar{j}_{0}$ are integers, we deduce that
$\bar{j}_{0}=\bar{j}^{m}_{0}$ when $m\geq N$ and $\bar{j}_{0}$ is the smallest
integer such that

$\sum_{|j|\geq\bar{j}_{0}}\|\dot{\Delta}_{j}u_{0}\|_{L^{p}}2^{(\frac{d}{p}-1)j}=\sum_{|j|\geq\bar{j}^{m}_{0}}\|\dot{\Delta}_{j}u_{0}\|_{L^{p}}2^{(\frac{d}{p}-1)j}<\frac{a}{4}-\frac{\epsilon}{m}.$

We claim that $\bar{j}_{0}=j_{0}$. Otherwise, if $\bar{j}_{0}>j_{0}$, we
deduce from the above inequality that

$\sum_{|j|\geq
j_{0}}\|\dot{\Delta}_{j}u_{0}\|_{L^{p}}2^{(\frac{d}{p}-1)j}\geq\frac{a}{4}-\frac{\epsilon}{m},\quad\forall
m\geq N.$

Since the left hand-side of the above inequality is independent of $m$, we
have

$\sum_{|j|\geq
j_{0}}\|\dot{\Delta}_{j}u_{0}\|_{L^{p}}2^{(\frac{d}{p}-1)j}\geq\frac{a}{4}.$

This contradicts the definition of $j_{0}$. So we have
$\bar{j}^{m}_{0}\rightarrow\bar{j}_{0}=j_{0}$ $(m\rightarrow\infty)$.

Finally, taking $\epsilon=\frac{a}{8}<\frac{a}{4}$, we can construct a
sequence $\\{j^{n}_{0}\\}$ by $\\{\bar{j}^{m}_{0}\\}$ when $n\geq
N_{\epsilon}$:

$j^{n}_{0}:=\left\\{\begin{array}[]{l}\bar{j}^{1}_{0},\quad N_{\epsilon}\leq
n<N_{\frac{\epsilon}{2}},\\\ \bar{j}^{2}_{0},\quad N_{\frac{\epsilon}{2}}\leq
n<N_{\frac{\epsilon}{3}},\\\ ......\\\ \bar{j}^{m}_{0},\quad
N_{\frac{\epsilon}{m}}\leq n<N_{\frac{\epsilon}{m+1}},\\\ ......\\\
\end{array}\right.$ (4.2)

By virtue of (4.1), one can check that

$\sum_{|j|\geq
j^{n}_{0}}\|\dot{\Delta}_{j}u^{n}_{0}\|_{L^{p}}2^{(\frac{d}{p}-1)j}<\frac{a}{4}.$

Using the monotone bounded theorem, one can prove that $j^{n}_{0}\rightarrow
j_{0}(n\rightarrow\infty)$. Therefore, we have

$T^{n}_{1}\rightarrow T_{1},T^{n}_{2}\rightarrow T_{2}\quad\Longrightarrow
T^{n}\rightarrow T,\quad n\rightarrow\infty.$

This completes the proof of the lemma. ∎

######  Remark 4.2.

The sequence $\bar{j}^{m}_{0}$ we construct in the proof of Lemma 4.1 is only
a subsequence since $m\neq n$ but depends on $n$. And one can obtain a
subsequence $T^{n_{m}}$ of $T^{n}$ such that $T^{n_{m}}\to T$. This is much
weaker than previous one. Therefore , we have to construct the $j^{n}$ by
(4.2).

######  Remark 4.3.

By Lemma 4.1, letting $T$ be the lifespan time of $(u^{\infty},b^{\infty})$,
then we can define a $T^{n}$ corresponding with $(u^{n},b^{n})$ such that
$T^{n}\rightarrow T,n\rightarrow\infty$. That is, for fixed any small
$\delta>0$, there exists an integer $N$, when $n\geq N$, we have

$|T^{n}-T|<\delta.$

Thus, we can consider $T_{n}:=\min\\{T^{n},T\\}$ as the common lifespan both
for $(u^{\infty},b^{\infty})$ and $(u^{n},b^{n})$. Then we still have

$T_{n}\rightarrow T,\quad n\rightarrow\infty.$

Roughly, we can choose $T-\delta$ as the common lifespan both for
$(u^{\infty},b^{\infty})$ and $(u^{n},b^{n})$, which is independent of $n$.

Now we begin to prove the continuous dependence.

######  Theorem 4.4.

Let $p\leq 2d$. Assume that $(u^{n},b^{n})_{n\in\mathbb{N}}$ be the solution
to the system (1.2) with the initial data
$(u^{n}_{0},b^{n}_{0})_{n\in\mathbb{N}}$. If $(u^{n}_{0},b^{n}_{0})$ tends to
$(u^{\infty}_{0},b^{\infty}_{0})$ in
$\dot{B}^{\frac{d}{p}-1}_{p,1}\times\dot{B}^{\frac{d}{p}}_{p,1}$, then there
exists a positive ${T}$ independent of $n$ such that $(u^{n},b^{n})$ tends to
$(u^{\infty},b^{\infty})$ in $C_{T}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
L^{1}_{T}(\dot{B}^{\frac{d}{p}+1}_{p,1})\times
C_{T}(\dot{B}^{\frac{d}{p}}_{p,1})$.

###### Proof.

Our aim is to estimate
$\|u^{n}-u^{\infty}\|_{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
L^{1}_{T}(\dot{B}^{\frac{d}{p}+1}_{p,1})}$ and
$\|b^{n}-b^{\infty}\|_{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})}$ when
$n\rightarrow\infty$. Note that

$\left\\{\begin{array}[]{l}\|u^{n}-u^{\infty}\|_{{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
L^{1}_{{T}}(\dot{B}^{\frac{d}{p}+1}_{p,1})}}\\\
\leq\|u^{n}-u^{n}_{j}\|_{{L^{\infty}_{{T}}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
L^{1}_{{T}}(\dot{B}^{\frac{d}{p}+1}_{p,1})}}+\|u^{n}_{j}-u^{\infty}_{j}\|_{{L^{\infty}_{{T}}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
L^{1}_{{T}}(\dot{B}^{\frac{d}{p}+1}_{p,1})}}+\|u^{\infty}_{j}-u^{\infty}\|_{{L^{\infty}_{{T}}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
L^{1}_{{T}}(\dot{B}^{\frac{d}{p}+1}_{p,1})}},\\\
\|b^{n}-b^{\infty}\|_{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})}\\\
\leq\|b^{n}-b^{n}_{j}\|_{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})}+\|b^{n}_{j}-b^{\infty}_{j}\|_{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})}+\|b^{\infty}_{j}-b^{\infty}\|_{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})},\end{array}\right.$
(4.3)

where

$(u^{n},b^{n})\text{ corresponds to the initial data
}(u^{n}_{0},b^{n}_{0}),\quad n\in\mathbb{N}\cup{\infty},$
$(u^{n}_{j},b^{n}_{j})\text{ corresponds to the initial data
}(\dot{S}_{j}u^{n}_{0},\dot{S}_{j}b^{n}_{0}),\quad
n\in\mathbb{N}\cup{\infty}.$

By Lemma 4.1, we find that $T-\delta$ (we still write it as $T$) is the common
lifespan for $(u^{n},b^{n})$, $(u^{n}_{j},b^{n}_{j})$,
$(u^{\infty},b^{\infty})$ and $(u^{\infty}_{j},b^{\infty}_{j})$ when $n,j$ are
large enough. By the argument as in Step 2, since
$(u^{n}_{0},b^{n}_{0})\rightarrow(u^{\infty}_{0},b^{\infty}_{0})$ and
$(\dot{S}_{j}u^{n}_{0},\dot{S}_{j}b^{n}_{0})\rightarrow(u^{n}_{0},b^{n}_{0})$
in $\dot{B}^{\frac{d}{p}-1}_{p,1}\times\dot{B}^{\frac{d}{p}}_{p,1}$, it
follows that for any large $n$ and $j$,

$\displaystyle\|u^{n},u^{n}_{j}\|_{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}-1}_{p,1})},\quad\|b^{n},b^{n}_{j}\|_{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})}\leq
C_{E_{0}},\quad\|u^{n}\|_{L^{p}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})\cap
L^{1}_{T}(\dot{B}^{\frac{d}{p}+1}_{p,1})}\leq 2a\leq\frac{1}{4C_{1}},$ (4.4)

where
$E^{n}_{0}:=\|u^{n}_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}+\|b^{n}_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}$,
$a$ is a small quantity satisfying (3.2). For any $t\in[0.T]$, we now divide
the estimations of (4.3) into 4 steps.

Step 1. Estimate
$\|u^{n}_{j}-u^{\infty}_{j}\|_{{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
L^{1}_{T}(\dot{B}^{\frac{d}{p}+1}_{p,1})}}$ and
$\|b^{n}_{j}-b^{\infty}_{j}\|_{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})}$
for fixed $j$.

Recall the equations of $(u^{n}_{j},b^{n}_{j})$,
$n\in\mathbb{N}\cup\\{\infty\\}$:

$\left\\{\begin{array}[]{lll}u^{n}_{jt}-\Delta
u^{n}_{j}=\mathbb{P}(b^{n}_{j}\nabla b^{n}_{j}+u^{n}_{j}\nabla u^{n}_{j}),\\\
b^{n}_{jt}+u^{n}_{j}\nabla b^{n}_{j}=b^{n}_{j}\nabla u^{n}_{j},\\\
(u^{n}_{0},b^{n}_{0}):=(\dot{S}_{j}u^{n}_{0},\dot{S}_{j}b^{n}_{0}).\end{array}\right.$
(4.5)

Multiplying both sides of the first equation in (4.5) by $\eta$ ($\eta$ is
determined later) and applying Lemmas 2.6-2.7 to (4.5), we have

$\displaystyle\eta(\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\|u^{n}_{j}\|_{L^{2}_{t}(\dot{B}^{\frac{d}{p}+1}_{p,1})}+\|u^{n}_{j}\|_{L^{1}_{t}(\dot{B}^{\frac{d}{2}+2}_{p,1})})$
$\displaystyle\leq\eta\|\dot{S}_{j}u^{n}_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\eta\int_{0}^{t}\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}+\|b^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\|b^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}ds$
$\displaystyle\leq
2^{j}\eta\|\dot{S}_{j}u^{n}_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}+\eta\int_{0}^{t}\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}+\|b^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\|b^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}ds$
(4.6)

and

$\displaystyle\|b^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}$
$\displaystyle\leq\|\dot{S}_{j}b^{n}_{0}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}+C_{E_{0}}\int_{0}^{t}\|b^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}+\|b^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+2}_{p,1}}ds$
$\displaystyle\leq
2^{j}\|\dot{S}_{j}b^{n}_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+C^{\prime}_{E_{0}}\|u^{n}_{j}\|_{L^{1}_{T}(\dot{B}^{\frac{d}{p}+2}_{p,1})}+C\int_{0}^{t}\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}\|b^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}ds,$
(4.7)

where we used the fact that
$\|\dot{S}_{j}g\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\leq
C2^{m}\|\dot{S}_{j}g\|_{\dot{B}^{\frac{d}{p}-m}_{p,1}},\quad m>0.$

Then setting $\eta>4C^{\prime}_{E_{0}}$, combining (4), (4) and the Gronwall
inequality, we thus have

$\displaystyle\frac{\eta}{2}(\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\|u^{n}_{j}\|_{L^{2}_{t}(\dot{B}^{\frac{d}{p}+1}_{p,1})}+\|u^{n}_{j}\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}+2}_{p,1})})+\|b^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}$
$\displaystyle\leq
2^{j}(\|b^{n}_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\|\dot{S}_{j}u^{n}_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}})$
(4.8)
$\displaystyle\quad+C_{E_{0}}\int_{0}^{t}\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}+\|b^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\|b^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}+\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}\|b^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}ds$
$\displaystyle\leq
C_{E_{0},j}[\|b^{n}_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\|u^{n}_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}]$
(4.9)
$\displaystyle\quad+\int_{0}^{t}\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\|b^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\|b^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}+\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}\|b^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}ds$
$\displaystyle\leq
C^{\prime}_{E_{0},j}(\|b^{n}_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\|u^{n}_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}),$
(4.10)

which along with the Gronwall inequality leads to

$\displaystyle\frac{\eta}{2}(\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\|u^{n}_{j}\|_{L^{2}_{t}\dot{B}^{\frac{d}{p}+1}_{p,1}}+\|u^{n}_{j}\|_{L^{1}_{t}\dot{B}^{\frac{d}{p}+2}_{p,1}})+\|b^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}\leq
C^{\prime}_{E_{0},j}(\|b^{n}_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\|u^{n}_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}).$
(4.11)

For fixed $j$, letting $\delta^{n}u=u^{n}_{j}-u^{\infty}_{j}$ and
$\delta^{n}b=b^{n}_{j}-b^{\infty}_{j}$, we have

$\left\\{\begin{array}[]{lll}\delta^{n}u_{t}-\Delta\delta^{n}u+u^{n}_{j}\nabla\delta^{n}u+\delta^{n}u\nabla
u^{\infty}_{j}+\nabla(P^{n}_{j}-P^{\infty}_{j})=b^{n}_{j}\nabla\delta^{n}b+\delta^{n}b\nabla
b^{\infty}_{j},\\\
\delta^{n}b_{t}+u^{n}_{j}\nabla\delta^{n}b+\delta^{n}u\nabla
b^{\infty}_{j}=b^{n}_{j}\nabla\delta^{n}u+\delta^{n}b\nabla u^{\infty}_{j},\\\
(\delta^{n}u,\delta^{n}b)|_{t=0}=(\dot{S}_{j}u^{n}_{0},\dot{S}_{j}b^{n}_{0}).\end{array}\right.$
(4.12)

Multiplying both sides of the first equation in (4.12) by $\lambda_{j}$
($\lambda_{j}$ is determined later) and applying Lemma 2.6 for (4.12), we have

$\displaystyle\lambda_{j}(\|\delta^{n}u\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}+\|\delta^{n}u\|_{L^{2}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}+\|\delta^{n}u\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}+1}_{p,1})})$
$\displaystyle\leq\lambda_{j}\|\dot{S}_{j}(u^{n}_{0}-u^{\infty}_{0})\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}+C\lambda_{j}\|u^{n}_{j},u^{\infty}_{j}\|_{L^{2}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\|\delta^{n}u\|_{L^{2}_{t}\dot{B}^{\frac{d}{p}}_{p,1}}+C\lambda_{j}\int_{0}^{t}\|b^{n}_{j},b^{\infty}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\|\delta^{n}b\|_{\dot{B}^{\frac{d}{p}}_{p,1}}ds$
$\displaystyle\leq\lambda_{j}\|u^{n}_{0}-u^{\infty}_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}+\frac{\lambda_{j}}{2}\|\delta^{n}u\|_{L^{2}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}+C\lambda_{j}\int_{0}^{t}C_{E_{0}}\|\delta^{n}b\|_{\dot{B}^{\frac{d}{p}}_{p,1}}ds,$
(4.13)

where $\|u^{n}_{j},u^{\infty}_{j}\|_{L^{2}_{t}\dot{B}^{\frac{d}{p}}_{p,1}}\leq
4a\leq\frac{1}{2C}$ by (3.2). Taking advantage of Lemma 2.7, we get

$\displaystyle\|\delta^{n}b\|_{\dot{B}^{\frac{d}{p}}_{p,1}}$
$\displaystyle\leq\|\dot{S}_{j}(b^{n}_{0}-b^{\infty}_{0})\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\int_{0}^{t}\|\delta^{n}u\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\|b^{\infty}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}+\|b^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\|\delta^{n}u\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}+\|u^{\infty}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}\|\delta^{n}b\|_{\dot{B}^{\frac{d}{p}}_{p,1}}ds$
$\displaystyle\leq\|b^{n}_{0}-b^{\infty}_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+C_{E_{0},j}(\|\delta^{n}u\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}+1}_{p,1})}+\|\delta^{n}u\|_{L^{2}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})})+\int_{0}^{t}\|u^{\infty}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}\|\delta^{n}b\|_{\dot{B}^{\frac{d}{p}}_{p,1}}ds.$
(4.14)

Combining (4) and (4), selecting $\lambda_{j}$ large enough such that
$\lambda_{j}>4(C_{E_{0},j}+1)$, for fixed $j$ we obtain that

$\displaystyle\frac{\lambda_{j}}{4}\|\delta^{n}u\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}\cap
L^{2}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})\cap
L^{1}_{t}(\dot{B}^{\frac{d}{p}+1}_{p,1})}+\|\delta^{n}b\|_{{\dot{B}^{\frac{d}{p}}_{p,1}}}$
$\displaystyle\leq
C_{E_{0},j}(\|b^{n}_{0}-b^{\infty}_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\|u^{n}_{0}-u^{\infty}_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}})+\int_{0}^{t}C_{E_{0},j}\|\delta^{n}b\|_{\dot{B}^{\frac{d}{p}}_{p,1}}ds\rightarrow
0,\quad n\rightarrow\infty,$ (4.15)

where the last inequality is based on the Gronwall inequality. This implies
that for any fixed $j$, we have

$\displaystyle\|u^{n}_{j}-u^{\infty}_{j}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
L^{1}_{t}(\dot{B}^{\frac{d}{p}+1}_{p,1})}+\|b^{n}_{j}-b^{\infty}_{j}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\rightarrow
0,\quad n\rightarrow\infty.$ (4.16)

Step 2. Estimate
$\|u^{n}-u^{n}_{j}\|_{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
L^{1}_{T}(\dot{B}^{\frac{d}{p}+1}_{p,1})}$ for any
$n\in\mathbb{N}\cup\\{\infty\\}$ .

Letting $\delta_{j}u=u^{n}-u^{n}_{j}$ and $\delta_{j}b=b^{n}-b^{n}_{j}$, then
we have

$\left\\{\begin{array}[]{lll}\delta_{j}u_{t}-\Delta\delta_{j}u+u^{n}\nabla\delta_{j}u+\delta_{j}u\nabla
u^{n}_{j}+\nabla(P^{n}-P^{n}_{j})=b^{n}\nabla\delta_{j}b+\delta_{j}b\nabla
b^{n}_{j},\\\ \delta_{j}b_{t}+u^{n}\nabla\delta_{j}b+\delta_{j}u\nabla
b^{n}_{j}=b^{n}\nabla\delta_{j}u+\delta_{j}b\nabla u^{n}_{j},\\\
(\delta_{j}u_{0},\delta_{j}b_{0})|_{t=0}=((Id-S_{j})u^{n}_{0},(Id-
S_{j})b^{n}_{0}).\end{array}\right.$ (4.17)

By Lemma 2.4, for $p\leq 2d$ we have
$\|fg\|_{\dot{B}^{\frac{d}{p}-1}_{p,\infty}}\leq\|f\|_{\dot{B}^{\frac{d}{p}-1}_{p,\infty}}\|g\|_{\dot{B}^{\frac{d}{p}}_{p,1}}.$
Using Lemmas 2.6-2.7 to (4.17), we have

$\displaystyle\|\delta_{j}u\|_{\dot{B}^{\frac{d}{2}-2}_{p,\infty}}+\|\delta_{j}u\|_{\widetilde{L}^{2}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,\infty})}+\|\delta_{j}u\|_{\widetilde{L}^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,\infty})}$
$\displaystyle\leq\|(Id-\dot{S}_{j})u^{n}_{0}\|_{\dot{B}^{\frac{d}{2}-2}_{p,1}}+C\|u^{n}_{j},u^{n}\|_{\widetilde{L}^{2}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\|\delta_{j}u\|_{\widetilde{L}^{2}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,\infty})}+\int_{0}^{t}\|b^{n}_{j},b^{n}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\|\delta_{j}b\|_{\dot{B}^{\frac{d}{p}-1}_{p,\infty}}ds$
$\displaystyle\leq 2^{-j}\|(Id-
S_{j})u^{n}_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}+\frac{1}{2}\|\delta_{j}u\|_{\widetilde{L}^{2}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,\infty})}+\int_{0}^{t}C_{E_{0}}\|\delta_{j}b\|_{\dot{B}^{\frac{d}{p}-1}_{p,\infty}}ds,$
(4.18)

where we used the fact that
$\|u^{n}_{j},u^{n}\|_{L^{2}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})}\leq
4a\leq\frac{1}{2C}$, and

$\displaystyle\|\delta_{j}b\|_{\dot{B}^{\frac{d}{p}-1}_{p,\infty}}$
$\displaystyle\leq\|(Id-
S_{j})b^{n}_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}+\int_{0}^{t}[\|b^{n}_{j},b^{n}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\|\delta_{j}u\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}\|\delta_{j}b\|_{\dot{B}^{\frac{d}{p}-1}_{p,\infty}}ds$
$\displaystyle\leq 2^{-j}\|(Id-
S_{j})b^{n}_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+C_{E_{0}}\|\delta_{j}u\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})},$
(4.19)

where we used the fact that $\|(Id-
S_{j})v\|_{\dot{B}^{\frac{d}{p}-m}_{p,1}}\leq C\|(Id-
S_{j})v\|_{\dot{B}^{\frac{d}{p}}_{p,1}}2^{-m},m>0$, and the last inequality is
based on the Gronwall inequality.

By interpolation, it follows that

$\displaystyle\|\delta_{j}u\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\leq
C\|\delta_{j}u\|_{\widetilde{L}^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,\infty})}ln(e+\frac{\|\delta_{j}u\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,1})}+\|\delta_{j}u\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}+1}_{p,1})}}{\|\delta_{j}u\|_{\widetilde{L}^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,\infty})}}),$
(4.20)

which together with (4) and (4) yields that

$\displaystyle\|\delta_{j}u\|_{\dot{B}^{\frac{d}{2}-2}_{p,\infty}}+\|\delta_{j}u\|_{\widetilde{L}^{2}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,\infty})}+\|\delta_{j}u\|_{\widetilde{L}^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,\infty})}$
$\displaystyle\leq C_{E_{0}}(\|(Id-
S_{j})u^{n}_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}+\|(Id-
S_{j})b^{n}_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}})+C_{E_{0}}\int_{0}^{t}\|\delta_{j}u\|_{\widetilde{L}^{1}_{s}(\dot{B}^{\frac{d}{p}}_{p,\infty})}ln(e+\frac{C_{E_{0}}}{\|\delta_{j}u\|_{\widetilde{L}^{1}_{s}(\dot{B}^{\frac{d}{p}}_{p,\infty})}})ds.$
(4.21)

By Lemma 2.12 with $\mu(r)=rln(e+\frac{C_{E_{0}}}{r})$, $\gamma(s)=C_{E_{0}}$,
we obtain

$\displaystyle\|\delta_{j}u\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{2}-2}_{p,\infty})}+\|\delta_{j}u\|_{\widetilde{L}^{2}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,\infty})}+\|\delta_{j}u\|_{\widetilde{L}^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,\infty})}$
$\displaystyle\leq C_{E_{0}}(\|(Id-
S_{j})u^{n}_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}+\|(Id-
S_{j})b^{n}_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}})$ $\displaystyle\rightarrow
0,\quad j\rightarrow\infty,\quad\forall n\in\mathbb{N}\cup\\{\infty\\}.$
(4.22)

Thus, by (4) and (4.20) we have

$\displaystyle\|\delta_{j}b\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,\infty})},\|\delta_{j}u\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\rightarrow
0,\quad j\rightarrow\infty,\quad\forall n\in\mathbb{N}\cup\\{\infty\\}.$
(4.23)

Next we estimate
$\|\delta_{j}u\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap(L^{1}_{t}\dot{B}^{\frac{d}{p}+1}_{p,1})}$.
Similarly, we have

$\displaystyle\|\delta_{j}u\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,1})}+\|\delta_{j}u\|_{L^{2}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}+\|\delta_{j}u\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}+1}_{p,1})}$
$\displaystyle\leq\|(Id-\dot{S}_{j})u^{n}_{0}\|_{B^{\frac{d}{p}-1}_{p,1}}+C\|u^{n}_{j},u^{n}\|_{L^{2}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\|\delta_{j}u\|_{L^{2}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}+C\int_{0}^{t}\|b^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\|\delta_{j}b\|_{\dot{B}^{\frac{d}{p}}_{p,1}}ds$
$\displaystyle\leq\|(Id-\dot{S}_{j})u^{n}_{0}\|_{B^{\frac{d}{p}-1}_{p,1}}+\frac{1}{2}\|\delta_{j}u\|_{L^{2}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}+\int_{0}^{t}C_{E_{0}}\|\delta_{j}b\|_{\dot{B}^{\frac{d}{p}}_{p,1}}ds,$
(4.24)

which implies that

$\displaystyle\|\delta_{j}u\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
L^{1}_{t}(\dot{B}^{\frac{d}{p}+1}_{p,1})}\leq
C\|(Id-\dot{S}_{j})u^{n}_{0}\|_{B^{\frac{d}{p}-1}_{p,1}}+C_{E_{0}}\int_{0}^{t}\|\delta_{j}b\|_{\dot{B}^{\frac{d}{p}}_{p,1}}ds,\quad\forall
n\in\mathbb{N}\cup\\{\infty\\}.$ (4.25)

Thus, we must combine the estimation of (4.25) with
$\|\delta_{j}b\|_{\dot{B}^{\frac{d}{p}}_{p,1}}$ to prove the continuous
dependence of $(\delta_{j}u,\delta_{j}b)$.

Step 3. Estimate
$\|b^{n}-b^{n}_{j}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}$ for any
$n\in\mathbb{N}\cup\\{\infty\\}$ .

Define that $b^{n}_{\infty}:=b^{n}$, $u^{n}_{\infty}:=u^{n}$ and recall the
equations of $b^{n}_{j}$ with $n,j\in\mathbb{N}\cup\\{\infty\\}$:

$\left\\{\begin{array}[]{l}\frac{d}{dt}b^{n}_{j}+u^{n}_{j}\nabla
b^{n}_{j}=b^{n}_{j}\nabla u^{n}_{j},\\\
b^{n}_{j}(0,x)=\dot{S}_{j}b^{n}_{0}.\end{array}\right.$ (4.26)

We let $b^{n}_{j}:=w^{n}_{j}+z^{n}_{j}$ such that

$\left\\{\begin{array}[]{lll}\frac{d}{dt}w^{n}_{j}+u^{n}_{j}\nabla
w^{n}_{j}=F^{\infty},\\\ w^{n}_{j}|_{t=0}=b^{n}_{0},\end{array}\right.$ (4.27)

and

$\left\\{\begin{array}[]{lll}\frac{d}{dt}z^{n}_{j}+u^{n}_{j}\nabla
z^{n}_{j}=F^{j}-F^{\infty},\\\
z^{n}_{j}|_{t=0}=\dot{S}_{j}b^{n}_{0}-b^{n}_{0},\end{array}\right.$ (4.28)

where $F^{j}:=b^{n}_{j}\nabla u^{n}_{j}$ and $F^{\infty}:=b^{n}_{\infty}\nabla
u^{n}_{\infty}$.

Since $F^{\infty},F^{j}$ are bounded in
$L^{1}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})\cap
L^{2}_{T}(\dot{B}^{\frac{d}{p}-1}_{p,1})$, by Remark 2.10, we deduce that
(4.27) and (4.28) have a unique solution $w^{n}_{j},z^{n}_{j}\in
C_{T}(\dot{B}^{\frac{d}{p}}_{p,1})$.

Our main idea is to verify that
$(w^{n}_{j},z^{n}_{j})\rightarrow(w^{n}_{\infty},0)\text{ in
}\dot{B}^{\frac{d}{p}}_{p,1}$ for any $n\in\mathbb{N}\cup\\{\infty\\}$, which
implies that $b^{n}_{j}\rightarrow b^{n}_{\infty}\text{ in
}\dot{B}^{\frac{d}{p}}_{p,1}.$ For this purpose, we divide the verification
into the following three small parts.

Firstly, we estimate
$\|w^{n}_{j}-w^{n}_{\infty}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}$.
Similarly to (4.3), we see that

$\displaystyle\|w^{n}_{j}-w^{n}_{\infty}\|_{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})}\leq\|w^{n}_{j}-w^{n}_{jk}\|_{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})}+\|w^{n}_{jk}-w^{n}_{\infty
k}\|_{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})}+\|w^{n}_{\infty
k}-w^{n}_{\infty}\|_{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})},$ (4.29)

where

$\left\\{\begin{array}[]{lll}\frac{d}{dt}w^{n}_{jk}+u^{n}_{j}\nabla(w^{n}_{jk})=\dot{S}_{k}F_{\infty},\\\
w^{n}_{jk}|_{t=0}=\dot{S}_{k}b^{n}_{0}.\end{array}\right.$ (4.30)

i. Estimate $\|w^{n}_{jk}-w^{n}_{\infty
k}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}$ for fixed $k$ .
From (4.30) we deduce that:

$\left\\{\begin{array}[]{lll}\frac{d}{dt}(w^{n}_{jk}-w^{n}_{\infty
k})+u^{n}_{j}\nabla(w^{n}_{jk}-w^{n}_{\infty
k})=-(u^{n}_{j}-u^{n}_{\infty})\nabla w^{n}_{\infty k},\\\
(w^{n}_{jk}-w^{n}_{\infty k})|_{t=0}=0.\end{array}\right.$ (4.31)

By Lemma 2.7 we have

$\|w^{n}_{\infty
k}\|_{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}+1}_{p,1})}\leq\|\dot{S}_{k}b^{\infty}_{0}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}+\|\dot{S}_{k}F^{\infty}\|_{L^{1}_{T}(\dot{B}^{\frac{d}{p}+1}_{p,1})}\leq
2^{k}\|b^{\infty}_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+2^{k}\|F^{\infty}\|_{L^{1}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})}\leq
2^{k}C_{E_{0}},$

and

$\displaystyle\|w^{n}_{jk}-w^{n}_{\infty
k}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}$
$\displaystyle\leq\int_{0}^{t}\|u^{n}_{j}-u^{n}_{\infty}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\|w^{n}_{\infty
k}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}ds$
$\displaystyle\leq\int_{0}^{t}\|u^{n}_{j}-u^{n}_{\infty}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}(2^{k}C_{E_{0}})ds$
$\displaystyle\leq
2^{k}C_{E_{0}}\|u^{n}_{j}-u^{n}_{\infty}\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}$
$\displaystyle\rightarrow 0,\quad j\rightarrow\infty,$ (4.32)

where the last inequality is based on (4.23).
ii. Estimate
$\|w^{n}_{j}-w^{n}_{jk}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}$ for
any $j\in\mathbb{N}\cup\\{\infty\\}$.
From (4.27) and (4.30), we obtain

$\left\\{\begin{array}[]{lll}\frac{d}{dt}(w^{n}_{j}-w^{n}_{jk})+u^{n}_{j}\nabla(w^{n}_{j}-w^{n}_{jk})=(Id-\dot{S}_{k})F_{\infty},\\\
(w^{i}_{j}-w^{i}_{jk})|_{t=0}=(Id-\dot{S}_{k})b_{0}.\end{array}\right.$ (4.33)

By Lemma 2.7, we have

$\displaystyle\|w^{n}_{j}-w^{n}_{jk}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}$
$\displaystyle\leq\|(Id-\dot{S}_{k})b_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\int_{0}^{t}\|(Id-\dot{S}_{k})F_{\infty}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}ds$
$\displaystyle\rightarrow 0,\quad k\rightarrow\infty.$ (4.34)

By (4), for any $\epsilon>0$, there exists $N$ (independent on $j$ and $n$)
such that when $k\geq N$ we have

$\|w^{n}_{j}-w^{n}_{jk}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\leq\frac{\epsilon}{3},\quad\|w^{n}_{\infty}-w^{n}_{\infty
k}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\leq\frac{\epsilon}{3}.$

For this $\epsilon$ and $k\geq N$, by (4), there exists $M$ (independent on
$n$) such that when $j\geq M$ we have

$\|w^{n}_{jk}-w^{n}_{\infty
k}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\leq\frac{\epsilon}{3}.$

Thus we get

$\displaystyle\|w^{n}_{j}-w^{n}_{\infty}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\leq\|w^{n}_{j}-w^{n}_{jk}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}+\|w^{n}_{\infty}-w^{n}_{\infty
k}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}+\|w^{n}_{jk}-w^{n}_{\infty
k}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\leq\epsilon,$ (4.35)

that is

$\displaystyle\|w^{n}_{j}-w^{n}_{\infty}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\rightarrow
0,\quad j\rightarrow\infty,\quad\forall n\in\mathbb{N}\cup\\{\infty\\}.$
(4.36)

Next, we estimate
$\|z^{n}_{j}\|_{L^{\infty}_{T}(\dot{B}^{\frac{d}{p}}_{p,1})}$. Recall that

$\left\\{\begin{array}[]{lll}\frac{d}{dt}z^{n}_{j}+u^{n}_{j}\nabla
z^{n}_{j}=F^{j}-F^{\infty},\\\
z^{i}_{j}|_{t=0}=(\dot{S}_{j}-Id)b_{0},\end{array}\right.$ (4.37)

where $F^{j}:=b^{n}_{j}\nabla u^{n}_{j}$. By the Bony decomposition, we have

$\displaystyle\|F^{j}-F^{\infty}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}$
$\displaystyle\leq\|(b^{n}_{j}-b^{n}_{\infty})\nabla
u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\|b^{n}_{\infty}\nabla(u^{n}_{j}-u^{n}_{\infty})\|_{\dot{B}^{\frac{d}{p}}_{p,1}}$
$\displaystyle\leq\|b^{n}_{j}-b^{n}_{\infty}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}+\|u^{n}_{j}-u^{n}_{\infty}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}\|b^{n}_{\infty}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}$
$\displaystyle\leq(\|z^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\|w^{n}_{j}-w^{n}_{\infty}\|_{\dot{B}^{\frac{d}{p}}_{p,1}})\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}+\|u^{n}_{j}-u^{n}_{\infty}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}\|b^{n}_{\infty}\|_{\dot{B}^{\frac{d}{p}}_{p,1}},$
(4.38)

where the last inequality is based on
$\|b^{n}_{j}-b^{n}_{\infty}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\leq\|w^{n}_{j}-w^{n}_{\infty}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\|z^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}$.
Combining (4.25) , (4) and (4.37), we have

$\displaystyle\|z^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}$ $\displaystyle\leq$
$\displaystyle\|(Id-\dot{S}_{j})b^{n}_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+C\int_{0}^{t}(\|z^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\|w^{n}_{j}-w^{n}_{\infty}\|_{\dot{B}^{\frac{d}{p}}_{p,1}})\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}ds+C_{E_{0}}\|u^{n}_{j}-u^{n}_{\infty}\|_{L^{1}_{t}\dot{B}^{\frac{d}{p}+1}_{p,1}}$
$\displaystyle\leq$ $\displaystyle
C_{E_{0}}(\|(Id-\dot{S}_{j})b^{n}_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\|(Id-
S_{j})u^{n}_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}+\|w^{n}_{j}-w^{n}_{\infty}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})})(C_{E_{0}}+\|u^{n}_{j}\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}+1}_{p,1})})$
$\displaystyle+\int_{0}^{t}C_{E_{0}}(\|u^{n}_{j}\|_{\dot{B}^{\frac{d}{p}+1}_{p,1}}+1)\|z^{n}_{j}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}ds.$
(4.39)

Applying the Gronwall inequality and (4.36), we obtain

$\displaystyle\|z^{n}_{j}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\leq$
$\displaystyle C_{E_{0}}(\|(Id-
S_{j})b^{n}_{0}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\|(Id-
S_{j})u^{n}_{0}\|_{\dot{B}^{\frac{d}{p}-1}_{p,1}}+\|w^{n}_{j}-w^{n}_{\infty}\|_{\dot{B}^{\frac{d}{p}}_{p,1}})$
$\displaystyle\rightarrow$ $\displaystyle 0,\quad
j\rightarrow\infty,\quad\forall n\in\mathbb{N}\cup\\{\infty\\}.$ (4.40)

Finally, combining (4.36) and (4), we have

$\displaystyle\|b^{n}_{j}-b^{n}_{\infty}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\rightarrow
0,\quad j\rightarrow\infty,\quad\forall n\in\mathbb{N}\cup\\{\infty\\},$
(4.41)

and

$\displaystyle\|u^{n}_{j}-u^{n}_{\infty}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
L^{1}_{t}(\dot{B}^{\frac{d}{p}+1}_{p,1})}$ $\displaystyle\leq
C\|(Id-\dot{S}_{j})u^{n}_{0}\|_{B^{\frac{d}{p}-1}_{p,1}}+C_{E_{0}}\int_{0}^{t}\|b^{n}_{j}-b^{n}_{\infty}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}ds$
$\displaystyle\rightarrow 0,\quad j\rightarrow\infty,\quad\forall
n\in\mathbb{N}\cup\\{\infty\\}.$ (4.42)

Thus, we complete the estimations of
$\|b^{n}_{j}-b^{n}_{\infty}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}$
and
$\|u^{n}_{j}-u^{n}_{\infty}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
L^{1}_{t}(\dot{B}^{\frac{d}{p}+1}_{p,1})}$.

Step 4. Proof of the continuous dependence

Finally, combining (4) and (4.41), we obtain

$\displaystyle\|u^{n}_{j}-u^{n}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
L^{1}_{t}(\dot{B}^{\frac{d}{p}+1}_{p,1})}+\|b^{n}_{j}-b^{n}_{\infty}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\rightarrow
0\quad,j\rightarrow\infty,\forall n\in\mathbb{N}\cup\\{\infty\\}.$ (4.43)

By (4.43), for any $\epsilon>0$, there exists $N$ (independent of $n$) such
that when $j\geq N$ we have

$\|u^{n}-u^{n}_{j}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
L^{1}_{t}(\dot{B}^{\frac{d}{p}+1}_{p,1})}\leq\frac{\epsilon}{3}\quad\forall
n\in\mathbb{N}^{+}\cup\\{\infty\\}.$

For this $\epsilon$ and $j\geq N$, by (4.16) there exists $M$ such that when
$n\geq M$, we get

$\|u^{n}_{j}-u^{\infty}_{j}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
L^{1}_{t}(\dot{B}^{\frac{d}{p}+1}_{p,1})}\leq\frac{\epsilon}{3}.$

Thus we deduce

$\displaystyle\quad\|u^{n}-u^{\infty}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,1}\cap
L^{1}_{t}\dot{B}^{\frac{d}{p}+1}_{p,1})}$
$\displaystyle\leq\|u^{n}-u^{n}_{j}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
L^{1}_{t}(\dot{B}^{\frac{d}{p}+1}_{p,1})}+\|u^{n}_{j}-u^{\infty}_{j}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,1})\cap
L^{1}_{t}(\dot{B}^{\frac{d}{p}+1}_{p,1})}+\|u^{\infty}_{j}-u^{\infty}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}-1}_{p,1}\cap
L^{1}_{t}\dot{B}^{\frac{d}{p}+1}_{p,1})}$ $\displaystyle\leq\epsilon.$ (4.44)

Similarly, we have

$\displaystyle\|b^{n}-b^{\infty}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\leq\|b^{n}-b^{n}_{j}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}+\|b^{n}_{j}-b^{\infty}_{j}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}+\|b^{\infty}_{j}-b^{\infty}\|_{L^{\infty}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\leq\epsilon.$
(4.45)

This completes the proof of the continuous dependence in $t\in$ $[0,T]$. ∎

Acknowledgements. This work was partially supported by National Natural
Science Foundation of China [grant number 11671407 and 11701586], the Macao
Science and Technology Development Fund (grant number 0091/2018/A3), Guangdong
Special Support Program (grant number 8-2015), and the key project of NSF of
Guangdong province (grant number 2016A030311004).

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