\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 261, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/261\hfil Porous medium equation in Besov spaces]
{Well-posedness and blowup criterion of generalized porous medium
 equation in \\ Besov spaces}

\author[X. Zhou, W. Xiao, T. Zheng \hfil EJDE-2015/261\hfilneg]
{Xuhuan Zhou, Weiliang Xiao, Taotao Zheng}

\address{Xuhuan Zhou \newline
 Department of Mathematics, 
Zhejiang University, Hangzhou 310027, China}
\email{zhouxuhuan@163.com}

\address{Weiliang Xiao \newline
chool of Applied Mathematics, 
Nanjing University of Finance and Economics,
Nanjing 210023, China}
\email{xwltc123@163.com}

\address{Taotao Zheng \newline
Department of Mathematics, 
Zhejiang University of Science and Technology,
Hangzhou 310023, China}
\email{taotzheng@126.com}

\thanks{Submitted June 25, 2015. Published October 7, 2015.}
\subjclass[2010]{35K15, 35K55, 35Q35, 76S05}
\keywords{Generalized porous medium equation;  well-posedness;
 \hfill\break\indent blowup criterion; Besov spaces}

\begin{abstract}
 We study the generalized porous medium equation of the form
 $u_t+\nu \Lambda^{\beta}u=\nabla\cdot(u\nabla Pu)$
 where $P$ is an abstract operator. We obtain the local well-posedness
 in Besov spaces for large initial data, and show the solution becomes
 global if the initial data is small.
 Also, we prove a blowup criterion for the solution.
\end{abstract}

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

\section{Introduction}

 In this article, we study the equation in $\mathbb{R}^n$ of the form
 \begin{equation}\label{pme}
 \begin{gathered}
 u_t+\nu \Lambda^{\beta}u=\nabla\cdot(u\nabla Pu); \\
 u(0,x)=u_0.
 \end{gathered}
 \end{equation}
Here $u=u(t,x)$ ia a real-valued function, represents a density or concentration.
The dissipative coefficient $\nu>0$ corresponds to the viscous case, while $\nu=0$
corresponds to the inviscid case. The fractional operator $\Lambda^{\beta}$
is defined by Fourier transform as $(\Lambda^{\beta}u)^{\wedge}=|\xi|^{\beta}\hat{u}$,
 and $P$ is an abstract operator.

The general form of equation \eqref{pme} has good suitability in many cases.
 The simplest case $\nu=0$, $Pu=u$ comes from
a model in groundwater infiltration, that is, $u_t=\Delta u^2$
(see \cite{bear01,vazquez01}). We call \eqref{pme}
generalized porous medium equation (GPME) inspired by
Caffarelli and V\'azquez \cite{caffarelli01}, in which they introduced the
fractional porous medium flow (FPME) when $\nu=0$ and
$Pu=\Lambda^{-2s}u,0<s<1$. When $Pu=\Lambda^{-2}u$, it is the mean field equation,
which is first studied by Lin and Zhang \cite{lin01}.
Some results on the well-posedness and regularity on those equations can be seen
\cite{biler01,biler02,caffarelli02,caffarelli03,serfaty01,stan01,vazquez02,zhou01}
and the references therein.

Another similar model occurs in the aggregation equation, which is an important
 equation arising in physics, biology,
chemistry, population dynamics, etc.(\cite{carrillo01,blanchet01,holm01,topaz01}).
In this model, the operator $P$ is a convolution operator with kernel $K$;
 that is, $Pu=K*u$. The typical kernels are $|x|^{\gamma}$,
see \cite{biler03,huang01,li03}, and $-e^{-|x|}$,
see \cite{bertozzi01,bertozzi04,li01,li02}.
For more results on this equation, we refer to
\cite{bertozzi02,bertozzi03,bodnar01,karch01,laurent01} and the references therein.
Besides, if we rewrite the equation \eqref{pme} with same initial data as
 \begin{equation}\label{pme12}
 \begin{gathered}
 u_t+\nu \Lambda^{\beta}u+v\cdot \nabla u=-u(\nabla\cdot v); \\
 v=-\nabla Pu,
 \end{gathered}
 \end{equation}
then it is a kind of special transport type equation.
Furthermore, if we assume that $v$ is divergence-free vector function
($\nabla\cdot v=0$),
the form \eqref{pme12} can contain the $2-D$ quasi-geostrophic (Q-G)
 equation \cite{chen01,chen02,constantin01,wang01}.

In this article we study the well-posedness of equation \eqref{pme}
 in homogenerous Besov spaces under a general condition
 \begin{equation} \label{estimate11}
 \|\nabla Pu\|_{\dot{B}_{p,q}^s}\leq C\|u\|_{\dot{B}_{p,q}^{s+\sigma}}.
 \end{equation}
It is widespread adopted in the case of FPME, Q-G equation, or aggregation
equation with its usual kernel $|x|^{\gamma}$
and it plays somewhat key role in the well-posedness and regularity of those
equations in Besov spaces or Sobolev spaces.
Based on the ideas used in \cite{miao01,wu01,yamazaki01,yuan01},
 we prove the following theorem.

 \begin{theorem} \label{thm1.1}
 Assume $P$ satisfies \eqref{estimate11}, $\beta\in(0,2]$, $p\geq 1$ and
$\sigma+1<\beta<\sigma+n\min(2/p,1)$. Then for any initial data
 $u_0\in\dot{B}_{p,1}^{\frac np+\sigma-\beta}\cap \dot{B}_{p,1}^{\frac np
+\sigma-\beta+1}$, the Cauchy problem \eqref{pme}
 admits a unique solution
 $$
u\in C([0,T);\dot{B}_{p,1}^{\frac np+\sigma-\beta}\cap
\dot{B}_{p,1}^{\frac np+\sigma-\beta+1})
 \cap L^1([0,T);\dot{B}_{p,1}^{\frac np+\sigma}\cap
\dot{B}_{p,1}^{\frac np+\sigma+1}).
$$
 Moreover, if $T^*$ denotes the maximal time of existence of $u$,
 \begin{itemize}
 \item[(i)] there exists a constant $C_0>0$ such that if
$\|u_0\|_{\dot{B}_{p,1}^{\frac np+\sigma-\beta+1}}< C_0$, then $T^*=\infty$;
 \item[(ii)] if $T^*<\infty$, then
$\int_0^{T^*}\|u(t)\|_{\dot{B}_{p,1}^{n/p+\sigma+1}}dt=\infty$.
 \end{itemize}
 \end{theorem}

\begin{remark} \label{rmk1.1} \rm
 In the case of aggregation equation, Wu and Zhang \cite{wu01} proved a
similar result under the condition $\nabla K\in W^{1,1}$, $\beta\in (0,1)$.
 Corresponding to their case we prove same result
for $\sigma=0$, that is $\nabla K\in L^1$, $\beta\in (1,2)$, and also
a similar result  for $\sigma=-1$; that is,
 $\nabla K\in \dot{W}^{1,1}$, $\beta\in (0,1)$.
\end{remark}

Throughout this article, $C$ denotes a positive constant that may differ
line by line, the notation $A\lesssim B$ means $A\leq CB$,
and $A\sim B$ denotes $A\lesssim B$ and $B\lesssim A$.

 \section{Preliminaries}

Let us recall some basic knowledge on Littlewood-Paley theory and Besov spaces.
Let $\mathscr{S}(\mathbb{R}^n)$ be the Schwartz class and $\mathscr{S}'$
be its dual space. Given $f\in\mathscr{S}(\mathbb{R}^n)$,
 we use its Fourier transform $\mathcal{F}f=\hat{f}$ as
 \[
 \hat{f}=(2\pi)^{-n/2}\int_{\mathbb{R}^n}e^{-ix\cdot\xi}f(x)dx.
 \]
Let $\varphi\in C_c^{\infty}(\mathbb{R}^n)$ be a radial real-valued
smooth function such that $0\leq\varphi(\xi)\leq1$ and
 $$
\operatorname{supp} \varphi\subset\{\xi\in \mathbb{R}^n:\frac34
\leq|\xi|\leq\frac83\},\quad
 \sum_{j\in Z}\varphi(2^{-j}\xi)=1\ \text{for any}\ \xi\neq0.
$$
We denote $\varphi_{_j}(\xi)=\varphi(2^{-j}\xi)$ and $\mathbb{P}$
the set of all polynomials. Setting $h=\mathcal{F}^{-1}\varphi$,
we define the frequency localization operators as follows:
 \[
 \Delta_ju=\varphi(2^{-j}D)u=2^{jn}\int_{\mathbb{R}^n}h(2^jy)u(x-y)dy,
 \quad S_jf=\sum_{k\leq j-1}\Delta_ku.
 \]

\begin{definition}\label{def21} \rm
 For $s\in \mathbb{R},p,q\in[1,\infty]$, we define the homogeneous Besov
space $\dot{B}_{p,q}^s$ as
 \begin{align*}
 \dot{B}_{p,q}^s=\{f\in \mathscr{S}'/\mathbb{P}:\|f\|_{\dot{B}_{p,q}^s}
 =\Big(\sum_{j\in\mathbb{Z}}2^{jsq}\|\Delta_jf\|_{L^p}^q\Big)^{1/q}<\infty\}.
 \end{align*}
 Here the norm changes normally when $p=\infty$ or $q=\infty$.
\end{definition}

\begin{definition}\label{def22} \rm
 In this paper, we need two kinds of mixed time-space norm defined as follows:
For $s\in \mathbb{R},1\leq p,q\leq\infty,I=[0,T),T\in(0,\infty]$,
 and $X$ a Banach space with norm $\|\cdot\|_X$ where
 \begin{gather*}
 \|f(t,x)\|_{L^r(I;X)}:=\Big(\int_I\|f(\tau,\cdot)\|^r_{X}d\tau\Big)^{1/r}, \\
 \|f(t,x)\|_{\mathcal{L}^r(I;\dot{B}_{p,q}^s)}
:=\Big(\sum_{j\in\mathbb{Z}}2^{jsq}\|\Delta_jf\|^q_{L^r(I;L^p)}\big)^{1/q}.
 \end{gather*}
 By Minkowski' inequality, there holds
 \begin{equation} \label{estimate21}
\begin{gathered}
L^r(I;\dot{B}_{p,q}^s)\hookrightarrow \mathcal{L}^r(I;\dot{B}_{p,q}^s),\quad
 \text{if } r\leq q, \\
 \mathcal{L}^r(I;\dot{B}_{p,q}^s)\hookrightarrow L^r(I;\dot{B}_{p,q}^s),\quad
 \text{if } r\geq q.
 \end{gathered}
\end{equation}
\end{definition}

Now we state some basic properties about the homogeneous Besov spaces.

\begin{proposition}[\cite{bahouri01}] \label{prop01}
 For $s\in\mathbb{R},1\leq p,q\leq\infty$, the following hold:
\begin{itemize}
 \item[(i)] Let $\beta\in \mathbb{R}$, we have the equivalence of norms:
 $\|\Lambda^\beta f\|_{\dot{B}_{p,q}^s}\sim \|f\|_{\dot{B}_{p,q}^{s+\beta}}$.

 \item[(ii)] If $p_1\leq p_2,q_1\leq q_2$, then
 $\dot{B}_{p_1,q_1}^s\hookrightarrow \dot{B}_{p_2,q_2}^{s-n({1}/{p_1}-{1}/{p_2})}$.

 \item[(iii)] Let $1\leq p,q\leq\infty,s_1,s_2<\frac{n}{p}$ when
$q>1$ (or $s_1,s_2\leq\frac{n}{p}$ when $q=1$),
 and $s_1+s_2>0$, there holds
 $$
\|uv\|_{\dot{B}_{p,q}^{s_1+s_2-\frac{n}{p}}}
 \leq C\|u\|_{\dot{B}_{p,q}^{s_1}}\|v\|_{\dot{B}_{p,q}^{s_2}},
$$
 where $C>0$ be a constant depending on $s_1,s_2,p,q,n$.
\end{itemize}
\end{proposition}

\begin{lemma}[Bernstein's inequalities \cite{chen01}]\label{bernstein}
 Set $\mathcal {B}$ to be a ball and $\mathcal {C}$ to be an annulus,
and let $1\leq p\leq q\leq \infty$, $\alpha\in (\{0\}\cup\mathbb{N})^n$,
 then the following estimates hold:
 \begin{itemize}
\item[(i)] If $\operatorname{supp}\widehat{f}\in 2^j\mathcal{B}$,
 $\beta+|\alpha|\geq0$, then
 $$
\|\Lambda^{\beta}D^{\alpha}f\|_{L^q}\leq C2^{j(\beta+|\alpha|+n(1/p-1/q))}\|f\|_{L^p}.
$$

\item[(ii)] If $\operatorname{supp}\,\widehat{f}\in 2^j\mathcal{C}$, then
 \begin{gather*}
 C2^{j(\beta+|\alpha|)}\|f\|_{L^p}\leq\|\Lambda^{\beta}D^{\alpha}f\|_{L^p}
 \leq C'2^{j(\beta+|\alpha|)}\|f\|_{L^p},
 \end{gather*}
 where $C\leq C'$ are positive constants independent of $j$.
\end{itemize}
\end{lemma}

\begin{lemma}[\cite{bahouri01}]
Let $\mathcal{C}$ ba an annulus.
If $\operatorname{supp}\,\widehat{f}\subset2^j\mathcal{C}$,
then positive constant $c>0$ exists such
 that for any $t>0$, there holds
 $$
\|e^{-t\Lambda^{\beta}}f\|_{L^p}\leq Ce^{-ct2^{j\beta}}\|f\|_{L^p}.
$$
\end{lemma}

\begin{lemma}[\cite{miao01}]\label{lem21}
 Let $s\in \mathbb{R}$ and $1\leq p\leq p_1\leq \infty$.
Set $R_j:=(S_{j-1}v-v)\cdot \nabla \Delta_ju-[\Delta_j,v\cdot\nabla]u$.
 There exists a constant $C=C(n,s)$ such that
 \begin{align*}
2^{js}\|R_j\|_{L^{p}}
&\leq C\Big(\sum_{|j-j'|\leq4}\|S_{j'-1}\nabla v\|_{L^{\infty}}2^{j's}
\|\Delta_{j'}u\|_{L^p} \\
&\quad +\sum_{j'\geq j-3}2^{j-j'}\|\Delta_{j'}\nabla v\|_{L^{\infty}}2^{js}
 \|\Delta_{j}u\|_{L^p} \\
&\quad +\sum_{\substack{|j'-j|\leq4\\j''\leq j'-2}}2^{(j-j'')
 (s-1-\frac{n}{p_1})}2^{j'\frac{n}{p_1}}
 \|\Delta_{j'}\nabla v\|_{L^{p_1}}2^{j''s}\|\Delta_{j''}u\|_{L^{p}} \\
&\quad  +\sum_{\substack{j'\geq j-3\\|j''-j'|\leq1}}2^{(j-j')
 (s+n\min(\frac{1}{p_1},\frac{1}{p'}))}
 2^{j'\frac{n}{p_1}}(2^{j-j'}\|\Delta_{j'}\nabla v\|_{L^{p_1}}\\
&\quad +\|\Delta_{j'}\nabla\cdot v\|_{L^{p_1}}) 2^{j''s}\|\Delta_{j''}u\|_{L^p}\Big).
 \end{align*}
\end{lemma}

Now we recall a priori estimates needed in our proof.
Consider the  transport-diffusion equation
\begin{equation} \label{TDnubeta}
\partial_t u+v\cdot\nabla u+ \nu \Lambda^{\beta}u=f,\quad u(0,x)=u_0(x).
\end{equation}

 \begin{lemma}[\cite{miao01}] \label{transport}
 Let $1\leq r_1\leq r\leq\infty,1\leq p\leq p_1\leq\infty$ and
$1\leq q\leq\infty.$ Assume $s\in \mathbb{R}$
 satisfies the following conditions:
\begin{gather*}
s<1+\frac{n}{p_1}\quad (\text{or } s\leq \frac {n}{p_1},\text{if}\ q=1), \\
 s>-\min{(\frac{n}{p_1},\frac{n}{p'})}\quad
(\text{or } s>-1-\min{(\frac{n}{p_1},\frac{n}{p'})},
\text{if}\operatorname{div} u=0).
 \end{gather*}
There exists a positive constant $C=C(n,\beta,s,p,p_1,q)$ such that for any
smooth solution $u$ of \eqref{TDnubeta}
 with $\nu\geq0$, the following a priori estimate holds:
 $$
\nu^{1/r}\|u\|_{\mathcal{L}_T^r\dot{B}_{p,q}^{s+\beta/r}}
 \leq Ce^{CZ(T)}\big(\|u_0\|_{\dot{B}_{p,q}^s}
 +\nu^{1/{r_1}-1}\|f\|_{\mathcal{L}_T^{r_1}
\dot{B}_{p,q}^{s-\beta+\beta/{r_1}}}\big),
$$
 where $Z(T)=\int_0^T\|\nabla v(t)\|_{\dot{B}_{p_1,\infty}^{n/{p_1}}
\cap L^{\infty}}dt$.
 \end{lemma}


\section{Local and global well-posedness}

In this section we prove our main theorem. We first rewrite \eqref{pme} as
 \begin{gather*}
  u_t+\Lambda^{\beta}u+v\cdot\nabla u=u(\Delta Pu); \\
 v=-\nabla Pu; \\
 u(0,x)=u_0.
  \end{gather*}

 \subsection*{Step 1: Approximate solutions}
 In this step we construct approximate equations, and prove the boundedness 
of the approximate solutions.
 Set $u^0=e^{-t\Lambda^{\beta}}u_0(x)$ and let $u^{m+1}$ be the solution 
of the linear equation
 \begin{equation}\label{linear}
 \begin{gathered}
 u^{m+1}_t+\Lambda^{\beta}u^{m+1}+v^m\cdot\nabla u^{m+1}=u^m(\Delta Pu^m); \\
 v^m=-\nabla Pu^m; \\
 u^{m+1}(0,x)=u_0.
 \end{gathered}
 \end{equation}
 Set $X=\dot{B}_{p,1}^{\frac np+\sigma-\beta}\cap
 \dot{B}_{p,1}^{\frac np+\sigma-\beta+1}$
 and $Y=\dot{B}_{p,1}^{\frac np+\sigma}\cap \dot{B}_{p,1}^{\frac np+\sigma+1}$. 
It is easy to show
 $$
u^0\in \mathcal{L}^{\infty}(\mathbb{R}^+;X)
 \cap \mathcal{L}^{1}(\mathbb{R}^+;Y).
$$
 Now by induction, we deduce $u^m$ belongs to the same spaces.
 In fact, by Lemma \ref{transport}
 \begin{equation} \label{Linfty}
 \begin{split}
 &\|u^{m+1}\|_{\mathcal{L}_T^{\infty} \dot{B}_{p,1}^{n/p+\sigma-\beta+1}}
  +\|u^{m+1}\|_{\mathcal{L}_T^{1} \dot{B}_{p,1}^{n/p+\sigma+1}} \\
 &\lesssim e^{c\int_0^T\|\nabla v^m(t)\|_{\dot{B}_{p,\infty}^{n/p} \cap L^{\infty}}dt}
 \big(\|u_0\|_{\dot{B}_{p,1}^{n/p+\sigma-\beta+1}}\\
 &\quad +\|u^m(\Delta Pu^m)\|_{\mathcal{L}_T^{1}\dot{B}_{p,1}^{n/p+\sigma-\beta+1}}\big) \\
 &\lesssim e^{c\int_0^T\|\nabla v^m(t)\|_{\dot{B}_{p,1}^{n/p}}dt}
 \big(\|u_0\|_{\dot{B}_{p,1}^{n/p+\sigma-\beta+1}}\\
&\quad +\|u^m\|_{L_T^{\infty}\dot{B}_{p,1}^{n/p+\sigma-\beta+1}}
 \|\Delta Pu^m\|_{L_T^{1}\dot{B}_{p,1}^{n/p}}\big) \\
 &\lesssim e^{c\|u^m\|_{L_T^1\dot{B}_{p,1}^{n/p+\sigma+1}}}
 \big(\|u_0\|_{\dot{B}_{p,1}^{n/p+\sigma-\beta+1}}\\
&\quad  +\|u^m\|_{\mathcal{L}_T^{\infty}\dot{B}_{p,1}^{n/p+\sigma-\beta+1}}
 \|u^m\|_{\mathcal{L}_T^{1}\dot{B}_{p,1}^{n/p+\sigma+1}}\big).
 \end{split}
 \end{equation}
 Similarly, we conclude that
 \begin{equation} \label{estimate03}
 \begin{split}
 &\|u^{m+1}\|_{\mathcal{L}_T^{\infty} \dot{B}_{p,1}^{n/p+\sigma-\beta}}
 +\|u^{m+1}\|_{\mathcal{L}_T^{1} \dot{B}_{p,1}^{n/p+\sigma}} \\
 &\lesssim e^{c\|u^m\|_{L_T^1\dot{B}_{p,1}^{n/p+\sigma+1}}}
 \big(\|u_0\|_{\dot{B}_{p,1}^{n/p+\sigma-\beta}} 
  +\|u^m\|_{\mathcal{L}_T^{\infty}\dot{B}_{p,1}^{n/p+\sigma-\beta}}
 \|u^m\|_{\mathcal{L}_T^{1}\dot{B}_{p,1}^{n/p+\sigma+1}}\big).
 \end{split}
 \end{equation}
 Thus for all $m\in N$, we have
 $u^m\in \mathcal{L}^{\infty}(\mathbb{R}^+;X)\cap \mathcal{L}^{1}(\mathbb{R}^+;Y)$.

 \subsection*{Step 2: Uniform estimates}
 We prove the key uniform estimates during this step. 
Set $u_j:=\Delta_ju,\ f_j:=\Delta_j(u^m\Delta Pu^m)$.
Then we can obtain

 \begin{claim}\label{claim01} 
There exists $T_1\leq T$, such that for all $t\in[0,T_1]$
 \begin{align*}
 \|u^{m+1}\|_{\mathcal{L}_t^r\dot{B}_{p,1}^{s+\beta/r}}
&\lesssim \sum_{j\in \mathbb{Z}}
 \big(1-e^{-ctr2^{j\beta}}\big)^{1/r}2^{js} \|u_{0,j}\|_{L^p}\\
&\quad +\sum_{j\in \mathbb{Z}}\int_0^t2^{js}\big(\|f_j\|_{L^p}+\|R_j\|_{L^p}\big)d\tau,
 \end{align*}
 where $R_j:=(S_{j-1}v^m-v^m)\cdot \nabla u_j^{m+1}-[\Delta_j,v^m\cdot\nabla]u^{m+1}$.
 \end{claim}

 We postpone the proof of this claim to the appendix. 
Taking $s=\frac np+\sigma-\beta+1$ and $\rho$ large enough such that 
$\sigma+1+\beta/{\rho}\leq\beta$.
 Then by Proposition \ref{prop01} with 
$s_1=n/p+\sigma+1-\beta+\beta/r,s_2=n/p-\beta/r$, there holds
\[
 \sum_{j\in \mathbb{Z}}\int_0^t2^{j(n/p+\sigma+1-\beta)}\|f_j\|_{L^p}d\tau
 \lesssim \|u^m\|_{L_t^{\rho}\dot{B}_{p,1}^{n/p+\sigma+1-\beta+\beta/{\rho}}}
 \|u^m\|_{L_t^{\rho'}\dot{B}_{p,1}^{n/p+\sigma+1-\beta/{\rho}}}.
\]
 Taking $\rho$ large enough and using Lemma \ref{lem21} with
$s=n/p+1+\sigma-\beta,p_1=p$,
 \begin{align*}
&\sum_{j\in \mathbb{Z}}\int_0^t2^{j(n/p+\sigma+1-\beta)}\|R_j\|_{L^p}d\tau\\
&\lesssim \|u^{m+1}\|_{L_t^{\rho}\dot{B}_{p,1}^{n/p+\sigma+1-\beta+\beta/{\rho}}}
 \|u^m\|_{L_t^{\rho'}\dot{B}_{p,1}^{n/p+\sigma+1-\beta/{\rho}}}.
\end{align*}
 Hence  for $r\geq1$ and $\rho$ large enough, we have
 \begin{equation}  \label{key}
 \begin{split}
&\|u^{m+1}\|_{\mathcal{L}_t^r\dot{B}_{p,1}^{n/p+\sigma+1-\beta+\beta/r}}\\
&\lesssim \sum_{j\in \mathbb{Z}}
 \big(1-e^{-ctr2^{j\beta}}\big)^{1/r}2^{j(n/p+\sigma+1-\beta)} \|u_{0,j}\|_{L^p} \\
&\quad +\|u^{m}\|_{L_t^{\rho}\dot{B}_{p,1}^{n/p+\sigma+1-\beta+\beta/{\rho}}}
 \|u^m\|_{L_t^{\rho'}\dot{B}_{p,1}^{n/p+\sigma+1-\beta/{\rho}}} \\
&\quad +\|u^{m+1}\|_{L_t^{\rho}\dot{B}_{p,1}^{n/p+\sigma+1-\beta+\beta/{\rho}}}
 \|u^m\|_{L_t^{\rho'}\dot{B}_{p,1}^{n/p+\sigma+1-\beta/{\rho}}}.
 \end{split}
\end{equation}
Now by \eqref{key} with $r=\rho$ and the fact that
 $\big(1-e^{-ct\rho'2^{j\beta}}\big)^{1/\rho'}
\leq\big(1-e^{-ct\rho2^{j\beta}}\big)^{1/\rho}$
 for $\rho$ large,
 \begin{align*}
&\|u^{m+1}\|_{\mathcal{L}_t^{\rho}\dot{B}_{p,1}^{n/p+\sigma+1-\beta+\beta/\rho}}
 +\|u^{m+1}\|_{\mathcal{L}_t^{\rho'}\dot{B}_{p,1}^{n/p+\sigma+1-\beta/\rho}} \\
&\lesssim \sum_{j\in \mathbb{Z}}
 \big(1-e^{-ct\rho2^{j\beta}}\big)^{1/\rho}2^{j(n/p+\sigma+1-\beta)}
 \|u_{0,j}\|_{L^p} \\
&\quad +\|u^m\|_{\mathcal{L}_t^{\rho}\dot{B}_{p,1}^{n/p+\sigma+1-\beta+\beta/\rho}}
 \|u^m\|_{\mathcal{L}_t^{\rho'}\dot{B}_{p,1}^{n/p+\sigma+1-\beta/\rho}} \\
&\quad +\|u^{m+1}\|_{\mathcal{L}_t^{\rho}\dot{B}_{p,1}^{n/p+\sigma+1-\beta+\beta/\rho}}
 \|u^m\|_{\mathcal{L}_t^{\rho'}\dot{B}_{p,1}^{n/p+\sigma+1-\beta/\rho}}.
 \end{align*}
By Lebesgue dominated convergence theorem,  for $\rho<\infty$, we have
 \begin{align*}
 \lim_{t\mapsto0^+}\sum_{j\in \mathbb{Z}}
 \big(1-e^{-ct\rho2^{j\beta}}\big)^{1/\rho}2^{j(n/p+\sigma+1-\beta)} \|u_{0,j}\|_{L^p}=0.
 \end{align*}
Now we set
 \begin{align*}
 T=\sup\big\{t>0:c\sum_{j\in \mathbb{Z}}
 \big(1-e^{-ct\rho2^{j\beta}}\big)^{1/\rho}2^{j(n/p+\sigma+1-\beta)}
 \|u_{0,j}\|_{L^p}\leq \eta\big\},
 \end{align*}
for some $\eta>0$ sufficiently small. By definition of $u^0,\forall t\leq T$,
we have
 \begin{align*}
 \|u^0\|_{\mathcal{L}_t^{\rho}\dot{B}_{p,1}^{n/p+\sigma+1-\beta+\beta/\rho}}\leq \eta,\quad
 \|u^0\|_{\mathcal{L}_t^{\rho'}\dot{B}_{p,1}^{n/p+\sigma+1-\beta/\rho}}\leq\eta.
 \end{align*}
Choosing $\eta$ small enough such that $c\eta\leq1/2$,
 \begin{equation} \label{u^1}
 \|u^{1}\|_{\mathcal{L}_t^{\rho}\dot{B}_{p,1}^{n/p+\sigma+1-\beta+\beta/\rho}}
 +\|u^{1}\|_{\mathcal{L}_t^{\rho'}\dot{B}_{p,1}^{n/p+\sigma+1-\beta/\rho}}
 \leq3\eta.
\end{equation}
 If we assume that \eqref{u^1} holds for $u^m$ and further take $\eta$
small enough such that
 $3c\eta\leq1/3$, we obtain
 \begin{equation} \label{u^m}
 \|u^{m+1}\|_{\mathcal{L}_t^{\rho}\dot{B}_{p,1}^{n/p+\sigma+1-\beta+\beta/\rho}}
 +\|u^{m+1}\|_{\mathcal{L}_t^{\rho'}\dot{B}_{p,1}^{n/p+\sigma+1-\beta/\rho}}
 \leq3\eta.
\end{equation}
 Thus by induction, we prove the uniform boundedness \eqref{u^m} for some
suitable $\eta$  and $\forall t\leq T$. Let $r=1$ in \eqref{key}, since
 $\big(1-e^{-ct2^{j\beta}}\big)\leq\big(1-e^{-ct\rho2^{j\beta}}\big)^{1/\rho}$,
we have
 \begin{align*}
 \|u^{m+1}\|_{\mathcal{L}_t^1\dot{B}_{p,1}^{n/p+\sigma+1}}
 &\lesssim \sum_{j\in \mathbb{Z}}
 \big(1-e^{-ct\rho2^{j\beta}}\big)^{1/\rho}2^{j(n/p+\sigma+1-\beta)} \|u_{0,j}\|_{L^p} \\
 &\quad +\|u^m\|_{\mathcal{L}_t^{\rho}\dot{B}_{p,1}^{n/p+\sigma+1-\beta+\beta/\rho}}
 \|u^m\|_{\mathcal{L}_t^{\rho'}\dot{B}_{p,1}^{n/p+\sigma+1-\beta/\rho}} \\
 &\quad +\|u^{m+1}\|_{\mathcal{L}_t^{\rho}\dot{B}_{p,1}^{n/p+\sigma+1-\beta+\beta/\rho}}
 \|u^m\|_{\mathcal{L}_t^{\rho'}\dot{B}_{p,1}^{n/p+\sigma+1-\beta/\rho}}.
 \end{align*}
This and \eqref{u^m} imply
 \begin{align} \label{ul1}
 \|u^{m+1}\|_{\mathcal{L}_t^1\dot{B}_{p,1}^{n/p+\sigma+1}}
 \leq \eta+18c\eta^2\leq3\eta.
 \end{align}
Next we prove that $u^m$ are uniformly bounded in
$\mathcal{L}_t^{\infty}\dot{B}_{p,1}^{n/p+\sigma+1-\beta}$.
 In fact, by \eqref{Linfty}
 \begin{align*}
 \|u^{m+1}\|_{\mathcal{L}_t^{\infty} \dot{B}_{p,1}^{n/p+\sigma+1-\beta}}
 \leq ce^{1/3}\|u_0\|_{\dot{B}_{p,1}^{n/p+\sigma+1-\beta}}
 +\frac13e^{1/3}\|u^m\|_{\mathcal{L}_t^{\infty}\dot{B}_{p,1}^{n/p+\sigma+1-\beta}}.
 \end{align*}
By induction, we conclude
 \begin{equation} \label{estimate22}
\begin{split}
&\|u^{m}\|_{\mathcal{L}_t^{\infty} \dot{B}_{p,1}^{n/p+\sigma+1-\beta}}\\
 &\leq \frac{1-(\frac{1}{3}e^{1/3})^m}{1-\frac{1}{3}e^{1/3}}c e^{1/3}
 \|u_0\|_{\dot{B}_{p,1}^{n/p+\sigma+1-\beta}}
 +\frac{e^{m/3}}{3^m}\|u^0\|_{\mathcal{L}_t^{\infty}\dot{B}_{p,1}^{n/p+\sigma+1-\beta}} \\
 &\leq (3ce^{1/3}+1)\|u_0\|_{\dot{B}_{p,1}^{n/p+\sigma+1-\beta}}.
 \end{split}
\end{equation}
 Thus  approximate solutions are uniformly bounded in the space
 $\mathcal{L}_T^{\infty}\dot{B}_{p,1}^{n/p+\sigma
+1-\beta}\cap\mathcal{L}_T^1\dot{B}_{p,1}^{n/p+\sigma+1}$.
 Now we return to \eqref{estimate03} and by the uniform
 estimate \eqref{ul1}
 \begin{equation} \label{estimate04}
 \begin{split}
 &\|u^{m+1}\|_{\mathcal{L}_T^{\infty} \dot{B}_{p,1}^{n/p+\sigma-\beta}}
 +\|u^{m+1}\|_{\mathcal{L}_T^{1} \dot{B}_{p,1}^{n/p+\sigma}} \\
 &\lesssim e^{c\|u^m\|_{L_T^1\dot{B}_{p,1}^{n/p+\sigma+1}}}
 \big(\|u_0\|_{\dot{B}_{p,1}^{n/p+\sigma-\beta}}
 +\|u^m\|_{\mathcal{L}_T^{\infty}\dot{B}_{p,1}^{n/p+\sigma-\beta}}
 \|u^m\|_{\mathcal{L}_T^{1}\dot{B}_{p,1}^{n/p+\sigma+1}}\big) \\
 &\leq ce^{1/3}\|u_0\|_{\dot{B}_{p,1}^{n/p+\sigma-\beta}}
 +\frac13e^{1/3}\|u^m\|_{\mathcal{L}_t^{\infty}\dot{B}_{p,1}^{n/p+\sigma-\beta}}.
 \end{split}
\end{equation}
 Hence, similarly to \eqref{estimate22}, by induction again,
 \begin{align*}
 \|u^{m}\|_{\mathcal{L}_t^{\infty} \dot{B}_{p,1}^{n/p+\sigma-\beta}}
 \leq (3ce^{1/3}+1)\|u_0\|_{\dot{B}_{p,1}^{n/p+\sigma-\beta}}.
 \end{align*}
Substituting  this into \eqref{estimate04} we conclude
 \begin{align*}
 \|u^{m}\|_{\mathcal{L}_t^{1} \dot{B}_{p,1}^{n/p+\sigma}}
 &\leq (4ce^{1/3}+1)\|u_0\|_{\dot{B}_{p,1}^{n/p+\sigma-\beta}}.
 \end{align*}
Thus we conclude that $(u^m)_{m\in\mathbb{N}}$ is uniformly bounded in
 $\mathcal{L}_T^{\infty}X\cap \mathcal{L}_T^{1}Y$.

 \subsection*{Step 3: Strong convergence}

 Let $(m,k)\in\mathbb{N}^2,m>k$ and $u^{m,k}=u^m-u^k,v^{m,k}=v^m-v^k$.
 Then $u^{m,k}$ satisfies the equation
 \begin{gather*}
\begin{aligned}
&u^{m+1,k+1}_t+\Lambda^{\beta}u^{m+1,k+1}+v^k\cdot\nabla u^{m+1,k+1}\\
&=u^{m,k}(\Delta Pu^m)  +u^{k}(\Delta Pu^{m,k})-v^{m,k}\cdot\nabla u^{m+1};
\end{aligned} \\
 v^{m,k}=-\nabla Pu^{m,k}; \\
 u^{m+1,k+1}(0,x)=0.
 \end{gather*}
 Set $U^{m+1,k+1}(T)=\|u^{m+1,k+1}\|_{\mathcal{L}_T^{\rho}
\dot{B}_{p,1}^{n/p+\sigma-\beta/\rho'}}
 +\|u^{m+1,k+1}\|_{\mathcal{L}_T^{\rho'}\dot{B}_{p,1}^{n/p+\sigma-\beta/\rho}}$.
 Applying Lemma \ref{transport} with $s=n/p+\sigma-\beta$, there holds
 \begin{align} \label{umk}
 \begin{split}
 &U^{m+1,k+1}(T) \\
 &\lesssim e^{c\|\nabla v^k\|_{L_T^1\dot{B}_{p,1}^{n/p}}}
 \big(\|u^{m,k}(\Delta Pu^m)\|_{\mathcal{L}_T^{1}\dot{B}_{p,1}^{n/p+\sigma-\beta}} \\
 &\quad +\|u^{k}(\Delta Pu^{m,k})\|_{\mathcal{L}_T^{1}\dot{B}_{p,1}^{n/p
 +\sigma-\beta}}
 +\|v^{m,k}\cdot\nabla u^{m+1}\|_{\mathcal{L}_T^{1}\dot{B}_{p,1}^{n/p
 +\sigma-\beta}}\big).
 \end{split}
 \end{align}
 Now applying Proposition \ref{prop01} with 
$s_1=n/p-\beta/\rho,s_2=n/p+\sigma-{\beta}/{\rho'}$,
 \[
 \|v^{m,k}\cdot\nabla u^{m+1}\|_{\mathcal{L}_T^{1}\dot{B}_{p,1}^{n/p+\sigma-\beta}}
 \lesssim \|u^{m,k}\|_{\mathcal{L}_T^{\rho'} \dot{B}_{p,1}^{n/p+\sigma-{\beta}/{\rho}}}
 \|u^{m+1}\|_{\mathcal{L}_T^{\rho}\dot{B}_{p,1}^{n/p+\sigma+1-{\beta}/{\rho'}}}.
 \]
 Similarly, for $\rho$ large enough such that $\sigma+1+\beta/{\rho}\leq\beta$, we conclude
 \begin{align*}
U^{m+1,k+1}(T)
 &\lesssim e^{c\|\nabla v^k\|_{L_T^1\dot{B}_{p,1}^{n/p}}}
 \big(\|u^{m,k}\|_{\mathcal{L}_T^{\rho'} \dot{B}_{p,1}^{n/p+\sigma-{\beta}/{\rho}}}
 \|u^{m+1}\|_{\mathcal{L}_T^{\rho}\dot{B}_{p,1}^{n/p+\sigma+1-{\beta}/{\rho'}}} \\
 &\quad +\|u^{k}\|_{\mathcal{L}_T^{\rho}\dot{B}_{p,1}^{n/p+\sigma+1-{\beta}/{\rho'}}}
 \|u^{m,k}\|_{\mathcal{L}_T^{\rho'} \dot{B}_{p,1}^{n/p+\sigma-{\beta}/{\rho}}}\\
 &\quad +\|u^{m,k}\|_{\mathcal{L}_T^{\rho'}\dot{B}_{p,1}^{n/p+\sigma-{\beta}/{\rho}}}
 \|u^{m}\|_{\mathcal{L}_T^{\rho}\dot{B}_{p,1}^{n/p+\sigma+1-{\beta}/{\rho'}}}\big) \\
 &\leq c\eta U^{m,k}(T).
\end{align*}
 Choosing $\eta$ small enough such that $c\eta<1$ and by induction, 
we conclude that
 $\{u^m\}$ is a Cauchy sequence in
 ${\mathcal{L}_T^{\rho} \dot{B}_{p,1}^{n/p+\sigma-\beta/{\rho'}}}
\cap{\mathcal{L}_T^{\rho'} \dot{B}_{p,1}^{n/p+\sigma-\beta/{\rho}}}$.
 So $u^m$ hence converges strongly to some $u$ in it. Now by taking $r=1$ and 
$r=\infty$ in \eqref{key}, respectively, and
 by passing to the limit into the approximation equation, we can get a solution to in
 $\mathcal{L}_T^{\infty}X\cap \mathcal{L}_T^{1}Y$.

 \subsection*{Step 4: Uniqueness}

 Let $u_1,u_2\in \mathcal{L}_T^{\infty}X\cap \mathcal{L}_T^{1}Y$
 be two solutions of \eqref{pme} with the same initial data. Let $u_{1,2}=u_1-u_2$, 
then
 \begin{gather*}
 \partial_tu_{1,2}+\Lambda^{\beta}u_{1,2}+v_2\cdot\nabla u_{1,2}=u_{1,2}(\Delta Pu_1)
 +u_{2}(\Delta Pu_{1,2})-v_{1,2}\cdot\nabla u_{1}; \\
 v_{1,2}=-\nabla Pu_{1,2}; \\
 u_{1,2}(0,x)=0.
 \end{gather*}
According to Lemma \ref{transport},
 \begin{align*}
&\|u_{1,2}\|_{\mathcal{L}_t^{\infty} \dot{B}_{p,\rho}^{n/p+\sigma-\beta}}\\
&\lesssim e^{c\|u_2\|_{L_t^1\dot{B}_{p,1}^{n/p+\sigma+1}}}
 \big(\|u_{1,2}(\Delta Pu_1)\|_{\mathcal{L}_t^{\rho}\dot{B}_{p,\rho}^{n/p+\sigma-2\beta+\beta/\rho}} \\
&\quad+\|u_{2}(\Delta Pu_{1,2})\|_{\mathcal{L}_t^{\rho}\dot{B}_{p,\rho}^{n/p+\sigma-2\beta+\beta/\rho}}
 +\|v_{1,2}\cdot\nabla u_{1}\|_{\mathcal{L}_t^{\rho}\dot{B}_{p,\rho}^{n/p+\sigma-2\beta+\beta/\rho}}\big).
\end{align*}
 By a similar argument as in Step 3, we have
 \begin{align*}
&\|u_{1,2}\|_{\mathcal{L}_t^{\infty} \dot{B}_{p,\rho}^{n/p+\sigma-\beta}}^{\rho}\\
&\lesssim e^{c\rho\|u_2\|_{L_T^1\dot{B}_{p,1}^{n/p+\sigma+1}}}
 \int_0^t\|u_{1,2}\|_{L_{\tau}^{\infty} \dot{B}_{p,\rho}^{n/p+\sigma-\beta}}^{\rho}
 \big(\|u_{1}(\tau)\|_{\dot{B}_{p,\rho}^{n/p+\sigma+1-\beta+\beta/\rho}}^{\rho} \\
&\quad+\|u_{2}(\tau)\|_{\dot{B}_{p,\rho}^{n/p+\sigma+1-\beta+\beta/\rho}}^{\rho}\big)d\tau.
\end{align*}
 Since the inclusion $\dot{B}_{p,1}^s\subset \dot{B}_{p,\rho}^s$ holds for any
 $\rho\in[1,\infty]$.
 Thus the Minkowski's inequality and Gronwall's inequality give that 
$u_1=u_2,\forall t\in[0,T]$.

 \subsection*{Step 5: Continuity in time}

 For all $t,t'\in[0,T)$, we have
\begin{align*}
&\|u(t)-u(t')\|_{\dot{B}_{p,1}^{n/p+\sigma-\beta}}\\
&\leq \sum_{j\leq N}2^{j(n/p+\sigma-\beta)}\|u_j(t)-u_j(t')\|_{L^p}
 +2 \sum_{j>N}2^{j(n/p+\sigma-\beta)}\|u_j\|_{L_T^{\infty}L^p}.
\end{align*}
 Since $u\in L_T^{\infty}\dot{B}_{p,1}^{n/p+\sigma-\beta}$, for any $\epsilon>0$, 
we can choose $N$ large enough such that
\[
 \sum_{j>N}2^{j(n/p+\sigma-\beta)}\|u_j\|_{L_T^{\infty}L^p}\leq \frac{\epsilon}{4}.
\]
 On the other hand,
 \begin{align*}
 \sum_{j\leq N}2^{j(n/p+\sigma-\beta)}\|u_j(t)-u_j(t')\|_{L^p}
 &\leq |t-t'|\sum_{j\leq N}2^{j(n/p+\sigma-\beta)}\|\partial_tu_j\|_{L_T^{\infty}L^p}
 \\
 &\leq |t-t'|2^{N\beta}\|\partial_tu\|_{\mathcal{L}_T^{\infty}
\dot{B}_{p,1}^{n/p+\sigma-2\beta}}.
 \end{align*}
 Now write
 \begin{align*}
 u_t=-\Lambda^{\beta}u-v\cdot\nabla u+u(\Delta Pu)
 \quad \text{and} \quad
 v=-\nabla Pu.
 \end{align*}
 Obviously, we have
\[
 \|\Lambda^{\beta}u\|_{\mathcal{L}_T^{\infty}\dot{B}_{p,1}^{n/p+\sigma-2\beta}}
 \leq \|u\|_{\mathcal{L}_T^{\infty}\dot{B}_{p,1}^{n/p+\sigma-\beta}}.
\]
 Applying Proposition \ref{prop01} with $s_1=n/p-\beta,s_2=n/p+\sigma-\beta$,
 \begin{align*}
 \|v\cdot\nabla u\|_{\mathcal{L}_T^{\infty}\dot{B}_{p,1}^{n/p+\sigma-2\beta}}
 \leq \|u\|_{\mathcal{L}_T^{\infty}\dot{B}_{p,1}^{n/p+\sigma-\beta}}
 \|u\|_{\mathcal{L}_T^{\infty}\dot{B}_{p,1}^{n/p+\sigma+1-\beta}}.
 \end{align*}
 Similarly, we have
 \begin{align*}
 \|u(\Delta Pu)\|_{\mathcal{L}_T^{\infty}\dot{B}_{p,1}^{n/p+\sigma-2\beta}}
 \leq \|u\|_{\mathcal{L}_T^{\infty}\dot{B}_{p,1}^{n/p+\sigma-\beta}}
 \|u\|_{\mathcal{L}_T^{\infty}\dot{B}_{p,1}^{n/p+\sigma+1-\beta}}.
 \end{align*}
Thus for $|t-t'|\leq (2^{N\beta}
\|\partial_tu\|_{\mathcal{L}_T^{\infty}\dot{B}_{p,1}^{n/p+\sigma-2\beta}})^{-1}
\frac{\epsilon}{2}$,
 we conclude
\[
 \|u(t)-u(t')\|_{\dot{B}_{p,1}^{n/p+\sigma-\beta}}
 \leq \epsilon.
\]
 Hence $u\in C([0,T);\dot{B}_{p,1}^{n/p+\sigma-\beta})$. 
Similarly we obtain $u\in C([0,T);\dot{B}_{p,1}^{n/p+\sigma+1-\beta})$.

 \subsection*{Step 6: Blowup criterion}

 We give a blowup criterion as follows:

 \begin{proposition} \label{prop3.1}
 Let $T^*$ denote the maximal time of existence of a solution $u$ in
 $C([0,T^*);X)\cap L^1([0,T^*);Y)$. If $T^*<\infty$, then
\[
 \int_0^{T^*}\|u(t)\|_{\dot{B}_{p,1}^{n/p+\sigma+1}}dt=\infty.
\]
 \end{proposition}

 \begin{proof}
 Supposing  $T^*<\infty$ and 
$\int_0^{T^*}\|u(t)\|_{\dot{B}_{p,1}^{n/p+\sigma+1}}dt<\infty$,
and using Lemma \ref{transport} with $\rho=\infty$, we have
\begin{align*}
 \|u\|_{\mathcal{L}_{T^*}^{\infty}\dot{B}_{p,1}^{n/p+\sigma+1-\beta}} 
 &\lesssim e^{c\int_0^{T^*}\|u(t)\|_{\dot{B}_{p,1}^{n/p+\sigma+1}}dt}
 \big(\|u_0\|_{\dot{B}_{p,1}^{n/p+\sigma+1-\beta}}\\
&\quad +\int_0^{T^*}\|u(t)\|_{\dot{B}_{p,1}^{n/p+\sigma+1-\beta}}
 \|u(t)\|_{\dot{B}_{p,1}^{n/p+\sigma+1}}dt\big).
\end{align*}
 Hence by Gronwall's inequality we have
 \begin{align} \label{blowup01}
 \|u\|_{\mathcal{L}_{T^*}^{\infty}\dot{B}_{p,1}^{n/p+\sigma+1-\beta}}
 \lesssim e^{c\int_0^{T^*}\|u(t)\|_{\dot{B}_{p,1}^{n/p+\sigma+1}}dt}
 \|u_0\|_{\dot{B}_{p,1}^{n/p+\sigma+1-\beta}}
 < \infty.
 \end{align}
By a similar argument there also holds
 \begin{align} \label{blowup02}
 \|u\|_{\mathcal{L}_{T^*}^{\infty}\dot{B}_{p,1}^{n/p+\sigma-\beta}}
 \lesssim e^{c\int_0^{T^*}\|u(t)\|_{\dot{B}_{p,1}^{n/p+\sigma+1}}dt}
 \|u_0\|_{\dot{B}_{p,1}^{n/p+\sigma-\beta}}
 < \infty.
 \end{align}
 From Step 5, for all $t,t'\in[0,T^*)$, we have
 \begin{align*}
 \|u(t)-u(t')\|_{X}\to 0\quad \text{as}\quad t\to t'.
 \end{align*}
 This means that $u(t)$ satisfies the Cauchy criterion at $T^*$.
So there exists an element $u^*$ in
 $X$ such that $u(t)\to u^*$ in $X$ as $t\to T^*$. Now set
 $u(T^*)=u^*$ and consider the equation with initial data $u^*$. 
By the well-posedness we obtain a solution
 existing on a larger time interval than $[0,T^*)$, which is a contradiction.
 \end{proof}

 \subsection*{Step 7: Global solution}

 To obtain global well-posedness for small initial data, it is sufficient to bound
 \begin{align*}
 F(t):=\int_0^t\|u(\tau)\|_{\dot{B}_{p,1}^{n/p+\sigma+1}}d\tau.
 \end{align*}
 Lemma \ref{transport} gives
\[
 \|u\|_{\mathcal{L}_{t}^{1}\dot{B}_{p,1}^{n/p+\sigma+1}}
 \lesssim e^{cF(t)}
 \big(\|u_0\|_{\dot{B}_{p,1}^{n/p+\sigma+1-\beta}}
 +\int_0^{t}\|u(\tau)\|_{\dot{B}_{p,1}^{n/p+\sigma+1-\beta}}
 \|u(\tau)\|_{\dot{B}_{p,1}^{n/p+\sigma+1}}d\tau\big).
\]
A similar argument with \eqref{blowup01} gives
\[
 \|u\|_{\mathcal{L}_{t}^{\infty}\dot{B}_{p,1}^{n/p+\sigma+1-\beta}}
 \lesssim e^{cF(t)}
 \|u_0\|_{\dot{B}_{p,1}^{n/p+\sigma+1-\beta}}.
\]
 Hence we conclude
\[
 F(t)\leq Ce^{CF(t)}(1+F(t))\|u_0\|_{\dot{B}_{p,1}^{n/p+\sigma+1-\beta}}.
\]
 Since $F(t)$ is continuous and $F(0)=0$, we obtain that: if the initial data
satisfies  $\|u_0\|_{\dot{B}_{p,1}^{n/p+\sigma+1-\beta}}<\frac{e^{-C}}{1+C}$,
 then
\[
 F(t)\leq Ce^{C^2}(1+C)\|u_0\|_{\dot{B}_{p,1}^{n/p+\sigma+1-\beta}}.
\]
 By the blow-up criterion, the solution is global.

\section{Appendix}

We now give the proof of Claim \ref{claim01}.
 We list some lemmas which will be used in our proof.

\begin{lemma}[\cite{danchin01}] \label{lemma42}
 Let $v$ be a smooth vector field, and $\psi_t$ be the solution to
 \[
 \psi_t(x)=x+\int_0^tv(\tau,\psi_{\tau}(x))d\tau.
 \]
 Then for all $t\in \mathbb{R}^+$, the flow $\psi_t$ is a $C^1$ diffeomorphism 
over $\mathbb{R}^n$ and there holds
 \begin{gather*}
 \|\nabla\psi_t^{\pm1}\|_{L^{\infty}}\leq e^{V(t)}, \\
 \|\nabla\psi_t^{\pm1}-Id\|_{L^{\infty}}\leq e^{V(t)}-1, \\
 \|\nabla^2\psi_t^{\pm1}\|_{L^{\infty}}\leq e^{V(t)}
\int_0^t\|\nabla^2v(\tau)\|_{L^{\infty}}e^{V(\tau)}d\tau,
 \end{gather*}
where $V(t)=\int_0^t\|\nabla v(\tau)\|_{L^{\infty}}d\tau$.
\end{lemma}

\begin{lemma}[\cite{danchin01}]\label{lemma43}
 Let $\chi\in \mathscr{S}(\mathbb{R}^n)$. There exists a constant $C=C(\chi,n)$ 
such that for all $C^2$ diffeomorphism
 $\psi$ over $\mathbb{R}^n$ with inverse $\phi$, and for all 
$u\in\mathscr{S}',p\in[1,+\infty],(j,j')\in \mathbb{Z}^2$,
 \begin{align*}
&\|\chi(2^{-j'}D)(\Delta_ju\circ\psi)\|_{L^p}\\
&\leq C\|J_{\phi}\|^{1/p}_{L^{\infty}}\|\Delta_ju\|_{L^p}(2^{-j}\|DJ_{\phi}\|_{L^{\infty}}\|J_{\psi}\|_{L^{\infty}}
 +2^{j'-j}\|D\phi\|_{L^{\infty}}).
 \end{align*}
\end{lemma}

\begin{lemma}[\cite{cannone01}] \label{lemma41}
 Let $v\in L_{loc}^1(\mathbb{R}^+;Lip)$ be a fixed vector field. 
For $j\in \mathbb{Z}$, set $u_j=\Delta_ju$,
 $\psi_j$ be the flow of the regularized vector field $S_{j-1}v$. 
Then for $u\in\dot{B}_{p,\infty}^{\beta}$
 with $\beta\in[0,2)$, $p\in[1,\infty]$, there holds
 $$
\|\Lambda^{\beta}(u_j\circ\psi_j)-(\Lambda^{\beta}u_j)\circ\psi_j\|_{L^p}
\leq Ce^{CV(t)}V^{1-\frac{\beta}{2}}(t)2^{j\beta}\|u_j\|_{L^p},
$$
 and when $\beta=2$,
$$
\|\Lambda^{2}(u_j\circ\psi_j)-(\Lambda^{2}u_j)\circ\psi_j\|_{L^p}
\leq Ce^{CV(t)}V(t)2^{2j}\|u_j\|_{L^p},
$$
 where $V(t)=\int_0^t\|\nabla v(\tau)\|_{L^{\infty}}d\tau$ and $C=C(\beta,p)>0$.
\end{lemma}

\begin{proof}[Proof of Claim \ref{claim01}]
Applying $\Delta_j$ to \eqref{linear} we have
 \begin{align}\label{uniform01}
 \partial_t u_j^{m+1}+S_{j-1}v^m\cdot \nabla u_j^{m+1}+\Lambda^{\beta}u_j^{m+1}
=f_j+R_j,
 \end{align}
where $R_j:=(S_{j-1}v^m-v^m)\cdot \nabla u_j^{m+1}-[\Delta_j,v^m\cdot\nabla]u^{m+1}$.
Let $\psi_j$ be the flow of the regularized vector field $S_{j-1}v^m$.
 Denote $\bar{u}_j:=u_j\circ\psi_j$,
 then \eqref{uniform01} becomes
 \begin{equation} \label{u_j}
 \partial_t\bar{u}_j^{m+1}+\Lambda^{\beta}\bar{u}_j^{m+1}=\bar{f}_j+\bar{R}_j+G_j,
\end{equation}
 where $G_j:=\Lambda^{\beta}(u_j^{m+1}\circ\psi_j)
-(\Lambda^{\beta}u_j^{m+1})\circ\psi_j.$
 \par Applying $\Delta_k$ on the equivalent integral equation of \eqref{u_j},
 we have
 \begin{equation} \label{integ}
 \begin{split}
&\|\Delta_k\bar{u}_j^{m+1}(t)\|_{{L^p}}\\
&\lesssim e^{-ct2^{k\beta}}\|\Delta_k u_{0,j}\|_{L^p} \\
&\quad +\int_0^te^{-c(t-\tau)2^{k\beta}}\big(\|\Delta_k \bar{f}_j\|_{L^p}
+\|\Delta_k \bar{R}_j\|_{L^p}  +\|\Delta_k G_j\|_{L^p}\big)d\tau.
\end{split}
\end{equation}
 Lemma \ref{lemma41} implies
\[
 \|\Delta_k G_j(t)\|_{L^p}\lesssim e^{cV(t)}V^{1-\beta/2}(t)
2^{j\beta}\|u_j^{m+1}\|_{L^p},
\]
 with $V(t)=\int_0^t\|\nabla v^m(\tau)\|_{L^{\infty}} d\tau$.
From Bernstein inequality and Lemma \ref{lemma42}
\[
 \|\Delta_k \bar{f}_j\|_{L^p}\lesssim 2^{-k}\|\nabla\Delta_k\bar{f}_j\|_{L^p}
 \lesssim 2^{-k}\|(\nabla f_j)\circ \psi_j\|_{L^p} \|\nabla \psi_j\|_{L^{\infty}}
 \lesssim 2^{j-k}e^{cV(t)}\|f_j\|_{L^p}.
\]
A similarly argument implies
\[
 \|\Delta_k\bar{R}_j(t)\|_{L^p}
 \lesssim 2^{j-k}e^{cV(t)}\|R_j\|_{L^p}.
\]
 Taking the $L^r$ norm over $[0,t]$ on \eqref{integ} and plugging the above
estimates give
 \begin{equation}  \label{Lr norm 1}
 \begin{split}
2^{j(s+\beta/r)}\|\Delta_k \bar{u}_j^{m+1}\|_{L_t^rL^p}
 &\lesssim 2^{(j-k)\beta/r} (1-e^{-ctr2^{k\beta}})^{1/r} 2^{js} \|\Delta_k u_{0,j}\|_{L^p} \\
 &+2^{j(s+\beta/r)} 2^{(j-k)\beta} e^{cV(t)} V^{1-\beta/2}(t) \|u_j^{m+1}\|_{L_t^rL^p} \\
 &+2^{(j-k)(1+\beta/r)} e^{cV(t)} \int_0^t2^{js}\big(\|f_j\|_{L^p}+\|R_j\|_{L^p}\big)d\tau.
 \end{split}
\end{equation}
 Let $M_0\in \mathbb{Z}$ to be fixed later. Decomposing
\[
 u_j^{m+1}=S_{j-M_0}\bar{u}_j^{m+1} \circ \psi_j^{-1}
+ \sum_{k\geq j-M_0}\Delta_k\bar{u}_j^{m+1} \circ \psi_j^{-1}.
\]
 For all $t\in[0,T]$, there holds
\begin{equation}  \label{decomp}
 \|u_j^{m+1}\|_{L_t^rL^p}\lesssim e^{cV(t)}\big(\|S_{j-M_0}\bar{u}_j^{m+1}\|_{L_t^rL^p}
 +\sum_{k\geq j-M_0}\|\Delta_k\bar{u}_j^{m+1}\|_{L_t^rL^p}\big).
\end{equation}
 By Lemma \ref{lemma42} and Lemma \ref{lemma43},
\begin{equation} \label{Lr norm 2}
 \|S_{j-M_0}\bar{u}_j^{m+1}\|_{L_t^rL^p}
 \lesssim e^{cV(t)}\big(e^{cV(t)}-1+2^{-M_0}\big)\|u_j^{m+1}\|_{L_t^rL^p}.
\end{equation}
 Since $\Delta_ku_{0,j}=0$ for $|k-j|\geq2$ and
$e^{cV(t)}-1+2^{-M_0}\lesssim e^{-c'V(t)}2^{-M_0}$,
 multiplying \eqref{decomp} by $2^{j(s+\beta/r)}$ and using \eqref{Lr norm 1}
and \eqref{Lr norm 2}, we obtain
 \begin{align*}
 2^{j(s+\beta/r)}\|u_j^{m+1}\|_{L_t^rL^p}
 &\lesssim e^{cV(t)} 2^{M_0\beta/r} \big(1-e^{-ctr2^{j\beta}}\big)^{1/r} 2^{js}
  \|u_{0,j}\|_{L^p} \\
 &\quad +e^{cV(t)} 2^{j(s+\beta/r)} \big(2^{-M_0}+2^{M_0\beta}
 V^{1-\beta/2}(t)\big)\|u_j^{m+1}\|_{L_t^rL^p} \\
 &\quad +e^{cV(t)} 2^{M_0(1+\beta/r)} \int_0^t2^{js}
 \big(\|f_j\|_{L^p}+\|R_j\|_{L^p}\big)d\tau.
 \end{align*}
 Now we choose $M_0$ to be the unique integer such that $2c2^{-M_0}\in(1/8,1/4]$
and $T_1\leq T$ be the largest real number such that
\[
 cV(T_1)\leq \min \big(\ln 2,
(\frac{2^{-M_0\beta}}{8c^{\beta/2}})^{\frac{2}{2-\beta}}\big).
\]
 Thus for $t\in[0,T_1]$,
\[
 2^{j(s+\beta/r)}\|u_j^{m+1}\|_{L_t^rL^p}
 \lesssim \big(1-e^{-ctr2^{j\beta}}\big)^{1/r}2^{js}\|u_{0,j}\|_{L^p}
 +\int_0^t2^{js}\big(\|f_j\|_{L^p}+\|R_j\|_{L^p}\big)d\tau.
\]
 Taking the $l^1$-norm we conclude that
 \begin{align*}
& \|u^{m+1}\|_{\mathcal{L}_t^r\dot{B}_{p,1}^{s+\beta/r}}\\
&\lesssim \sum_{j\in \mathbb{Z}}
 \big(1-e^{-ctr2^{j\beta}}\big)^{1/r}2^{js} \|u_{0,j}\|_{L^p}
 +\sum_{j\in \mathbb{Z}}\int_0^t2^{js}\big(\|f_j\|_{L^p}+\|R_j\|_{L^p}\big)d\tau.
\end{align*}
\end{proof}

\subsection{Acknowledgments}
This research was supported by the NNSF of China under grant 11271330.


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\end{document}