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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 199, 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 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/199\hfil Fractional porous medium equation]
{Fractional porous medium  and mean field equations in Besov spaces}

\author[X. Zhou, W. Xiao, J. Chen \hfil EJDE-2014/199\hfilneg]
{Xuhuan Zhou, Weiliang Xiao, Jiecheng Chen}  % in alphabetical order

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

\address{Weiliang Xiao \newline
Department of Mathematics, Zhejiang University, 310027 Hangzhou,
China}
\email{xwltc123@163.com}

\address{Jiecheng Chen \newline
Department of Mathematics, Zhejiang Normal University, 321004 Jinhua,
China}
\email{jcchen@zjnu.edu.cn}

\thanks{Submitted April 10, 2014. Published September 23, 2014.}
\subjclass[2000]{35K55, 35K65, 76S05}
\keywords{Fractional porous medium equation; mean field equation;
\hfill\break\indent  local  solution; Besov space}

\begin{abstract}
 In this article, we consider the evolution model
 $$
 \partial_t{u} -\nabla\cdot(u\nabla Pu)=0,\quad
 Pu=(-\Delta)^{-s}u, \quad 0< s\leq 1,\; x\in\mathbb{R}^d,\; t>0.
 $$
 We show that when $s\in[1/2,1)$, $\alpha>d+1$, $d\geq 2$, the equation
 has a unique local in time solution for any initial data in $B^\alpha_{1,\infty}$.
 Moreover, in the critical case $s=1$, the solution exists in $B^\alpha_{p,\infty}$,
 $2\leq p\leq\infty$, $\alpha> d/p$, $d\geq3$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

Let $0< s\leq 1$, we consider the evolution equation
\begin{equation} \label{e1.1}
\partial_t{u} -\nabla\cdot(u\nabla Pu)=0,\quad Pu=(-\Delta)^{-s}u,\quad
u(x,0)=u_0(x).
\end{equation}
where $x\in\mathbb{R}^d$, $t>0$. $u=u(x,t)$ is a real-valued function,
representing the density or concentration. $P$ represents the pressure.

When $0<s<1$, we refer the equation as a fractional porous medium equation.
It was first introduced  by Caffarelli and V\'azquez \cite{c2}.
They proved the existence of a weak solution when $u_0$ is a bounded function
and has exponential decay at infinity.  Caffarelli, Soria,
and V\'azquez \cite{c1} studied its regularity theory of the weak solution with
$u_0 \in L^1\cap L^\infty$ and the continuity of bounded solutions.

When $s=1$, the equation leads to a mean field equation
\begin{equation} \label{e1.2}
u_t=\nabla\cdot(u\nabla Pu),\quad Pu=(-\Delta)^{-1}u, \quad u(x,0)=u_0(x).
\end{equation}
Equation that was first studied by Lin and Zhang \cite{l2}.
They proved the existence and uniqueness of positive $L^\infty$ solution
in two dimensions. When $d\geq 3$, V\'azquez and Serfaty \cite{v3} studied the
existence and uniqueness of the weak solution in $L^\infty$ spaces for \eqref{e1.2}
by taking the limit $s\to 1$ in the weak solutions of \eqref{e1.1}.
Since papers in literature only addressed the weak solution of \eqref{e1.2},
in this article we show that the strong solution exists in
 $B^\alpha_{p,\infty}$, $\alpha>0$.

In this article, we are interested in finding the strong solutions of \eqref{e1.1}
in the Besov spaces $B^\alpha_{p,\infty}$. We will show that when $d\geq2$,
$s\in[1/2,1)$, $\alpha> d+1$, equation \eqref{e1.1} has a unique local
in time solution for any initial data in $B^\alpha_{1,\infty}$.
 When $s=1$, the solution extends to $B^\alpha_{p,\infty}$,
$\alpha> d/p$, $2\leq p\leq\infty$, $d\geq3$. The idea of our proof is inspired
by the methods used in \cite{c3,w1}, where the authors studied the quasi-geostrophic
equation. In our proof, we construct two commutators, give three estimates of them,
and construct a function sequence fitting our equation.

The rest of this article is divided in four parts.
Section 2 recalls the definition and some properties of the Besov spaces,
the Bernstein inequality involving both integer and fractional derivatives,
as well as some properties of the fractional Laplacian.
In Section 3 we prove three estimates about the constructed commutators and
a priori estimate of the solution. Section 4 proves the existence and uniqueness
of the fractional porous medium equation.
Section 5 is devoted to the mean field equation. The main results are the
following two theorems.

\begin{theorem} \label{thm1.1}
Let $d\geq2,s\in[1/2,1],\alpha>d+1$. Assume that the initial data $u_0
\in B^\alpha_{1,\infty}$. Then we can find $T=T(\|u_0\|_{B^\alpha_{1,\infty}})$, 
such that a unique solution $u$ to  \eqref{e1.1}
 on $[0,T]\times \mathcal {\mathbb{R}}^d$ exists. And the solution belongs to
$C^1([0,T];B^{\alpha+2s-2}_{1,\infty})\cap L^\infty([0,T];B^{\beta}_{1,\infty})$, 
and $\beta\in[\alpha+2s-2,\alpha]$.
\end{theorem}

\begin{theorem} \label{thm1.2}
Let $d\geq3,\alpha> d/p$ when $2\leq p\leq\infty$. Assume that the initial data
$u_0 \in B^\alpha_{p,\infty}$. Then we can find
$T=T(\|u_0\|_{B^\alpha_{p,\infty}})$, such that a unique solution $u$ to the
given mean field equation \eqref{e1.2} on $[0,T]\times \mathcal {\mathbb{R}}^d$ exists. And the solution belongs to
$C^1([0,T];B^{\beta}_{p,\infty})\cap L^\infty([0,T];B^{\alpha}_{p,\infty})$,
 $\beta\in(d/p,\alpha)$, $2\leq p\leq\infty$.
\end{theorem}

\section{Preliminaries}

In this section, we recall the definition of the Besov space.
 We start with a dyadic decomposition of $\mathcal {\mathbb{R}}^d$.

Suppose $\chi \in C_0^\infty(\mathcal {\mathbb{R}}^d),
\phi\in C_0^\infty(\mathcal {\mathbb{R}}^d\setminus \{0\})$ satisfying
\begin{gather*}
\operatorname{supp}\chi \subset \{\xi\in {\mathbb{R}}^d:|\xi|\leq \frac 43\},\\
\operatorname{supp}\phi \subset \{\xi\in {\mathbb{R}}^d:\frac 34<|\xi|< \frac 83\},\\
\chi(\xi)+\sum_{j\geq 0}\phi(2^{-j}\xi)=1,\quad \xi \in \mathcal {\mathbb{R}}^d,\\
\sum_{j\in \mathbb{Z}}\phi(2^{-j}\xi)=1,\quad \xi \in \mathcal {\mathbb{R}}^d\backslash\{0\}.
\end{gather*}
Define the operators
\begin{equation} \label{e2.1}
 \begin{gathered}
 \Delta_ju=\phi(2^{-j}D)u=2^{jd}\int h(2^jy)u(x-y)\,dy, \\
 S_jf=\sum_{k\leq j-1}\Delta_kf=\chi(2^{-j}D)u=2^{jd}\int g(2^jy)u(x-y)\,dy,
\end{gathered}
\end{equation}
where $g=\chi^\vee$ and $h=\phi^\vee$ are the inverse Fourier transform of
$\chi$ and $\phi$, respectively.
It can be easily verified that with our choice of $\phi$,
\begin{equation} \label{e2.2}
 \Delta_j\Delta_k f\equiv 0, \ if \ |j-k|\geq 2,\quad
 \Delta_j(S_{k-1}f\Delta_kf)\equiv 0,\quad\text{if } |j-k|\geq 5.
\end{equation}

\begin{definition} \label{def2.1} \rm
For any $\alpha\in \mathcal {\mathbb{R}}$, and $p,q \in [1,\infty]$,
the homogeneous Besov spaces $\dot{B}^r_{p,q}$ are defined as
\[
\dot{B}^\alpha_{p,q}= \{ f \in \mathcal {Z}'(\mathbb{R}^d):
 \|f\|_{\dot{B}^\alpha_{p,q}} < \infty\}.
\]
Here,
\[
\|f\|_{\dot{B}^\alpha_{p,q}}
= \Big[\sum_{j\in \mathbb{Z}}2^{j\alpha q}\|\Delta_j f\|_{L^p}^q\Big]^{1/q} ,
\quad\text{where } q<\infty.
\]
When $q=\infty$,
\[
\|f\|_{\dot{B}^\alpha_{p,\infty}}
= \sup_{j \in \mathbb{Z}} 2^{j\alpha}\|\Delta_j f\|_{L^p}.
\]
$\mathcal {Z}'(\mathbb{R}^d)$ denotes the dual space of
$\mathcal {Z}(\mathbb{R}^d)= \{f \in \mathcal {S}(\mathbb{R}^d):
\partial^\gamma \hat{f}(0)=0, \forall \gamma \in \mathbb{N}^d\}$
and can be identified by the quotient space of $\mathcal {S}'/ \mathcal {P}$
with the polynomials space $\mathcal {P}$.
\end{definition}

\begin{definition} \label{def2.2} \rm
For any $\alpha\in \mathcal {\mathbb{R}}$, and $p,q \in [1,\infty]$, the
inhomogeneous Besov space $B^r_{p,q}$ is defined as
 \[
B^\alpha_{p,q}= \{ f \in \mathcal {S}'(\mathbb{R}^d):
\|f\|_{B^\alpha_{p,q}} < \infty\}.
\]
Here,
\[
\|f\|_{B^\alpha_{p,q}}
= \Big(\sum_{j\geq 0}^\infty 2^{j\alpha q}\|\Delta_j f\|_{L^p}^q\Big)^{1/q}
+\|S_0(f)\|_{L^p} ,\quad\text{when } q<\infty.
\]
When $q=\infty$,
\[
\|f\|_{B^\alpha_{p,\infty}}
= \sup_{j \geq 0} 2^{j\alpha}\|\Delta_j f\|_{L^p}+\|S_0(f)\|_{L^p} .
\]
\end{definition}

Let us state some basic properties of the Besov spaces.

\begin{proposition} \label{prop2.1}
Let $s\in \mathbb{R}$, $1\leq p\leq \infty$, $1\leq q\leq \infty$.
\begin{itemize}
\item[(i)] If $\alpha>0$, then $B^\alpha_{p,q}=\dot{B}^\alpha_{p,q}\cap L^p$, and
$\|f\|_{B^\alpha_{p,q}}=\|f\|_{\dot{B}^{\alpha}_{p,q}}+\|f\|_{L^p}$;

\item[(ii)] If $\alpha_1\leq \alpha_2$, then
$ B^{\alpha_2}_{p,q} \subset B^{\alpha_1}_{p,q}$.
 If $1\leq q_1\leq q_2 \leq\infty$, then
$\dot{B}^{\alpha}_{p,q_1}\subset \dot{B}^{\alpha}_{p,q_2}$ and
$B^{\alpha}_{p,q_1}\subset B^{\alpha}_{p,q_2}$;

\item[(iii)] If $ \alpha> \frac dp$, then $B^\alpha_{p,q}\hookrightarrow L^\infty$.
If $p_1\leq p_2$, $\alpha_1-\frac d{p_1}>\alpha_2-\frac d{p_2}$, then
$B^{\alpha_1}_{p_1,q_1}\hookrightarrow B^{\alpha_2}_{p_2,q_2}$,
$B^\alpha_{p,\min(p,2)}\hookrightarrow H^\alpha_p\hookrightarrow
B^\alpha_{p,\max(p,2)}$;

\item[(iv)] If $\alpha>0$, $p\geq 1$, then
$\|uv\|_{B^\alpha_{p,\infty}}\leq C\|u\|_{L^\infty}\|v\|_{B^\alpha_{p,\infty}}
+C\|u\|_{B^\alpha_{p,\infty}}\|v\|_{L^\infty}$.
\end{itemize}
\end{proposition}

We now turn to Bernstein's inequalities. When the Fourier transform of a function
is supported on a ball or an annulus, the $L^p$-norms of the derivatives of
 the function can be bounded in terms of the $L^p$-norms of the function itself.
 And it also exists when one replaces the derivatives by the fractional derivatives
 (see \cite{l1,w3}).

\begin{proposition} \label{prop2.2}
Let $1\leq p \leq q\leq \infty$, $\gamma\in \mathbb{N}^d$.
(1) If $\alpha\geq 0$ and
$\operatorname{supp}\hat{f} \subset\{\xi\in \mathcal {\mathbb{R}}^d :
 |\xi|\leq K2^j \}$ for some $K>0$ and integer $j$, then
\[
\|(-\Delta)^\alpha D^\gamma f\|_{L^q}\leq C2^{j(2\alpha+|\gamma|)
+jd(\frac 1p -\frac 1q)}\|f\|_{L^p}.
\]

(2) If $\alpha\in \mathbb{R}$ and
$\operatorname{supp}\hat{f} \subset\{\xi\in \mathcal {\mathbb{R}}^d :
 K_12^j\leq|\xi|\leq K_22^j \}$ for some $K_1,K_2>0$ and integer $j$, then
\[
C2^{j(2\alpha+|\gamma|)+jd(\frac 1p -\frac 1q)}\|f\|_{L^p}
\leq \|(-\Delta)^\alpha D^\gamma f\|_{L^q}
\leq \widetilde{C}2^{j(2\alpha+|\gamma|)+jd(\frac 1p -\frac 1q)}\|f\|_{L^p},
\]
where $C$ and $\widetilde{C}$ are positive constants independent of $j$.
\end{proposition}

Next we state two pointwise inequalities which were proved in \cite{c5,w2}.

\begin{proposition} \label{prop2.3}
Let $0\leq \alpha\leq 1$, $f\in C^2(\mathbb{R}^d)$ decay sufficiently fast at
infinity. Then for any $x \in \mathbb{R}^d$,
\[
2f(x)(-\Delta)^\alpha f(x)\geq (-\Delta)^\alpha f^2(x).
\]
\end{proposition}

\begin{proposition} \label{prop2.4}
Let $0\leq \alpha\leq 1$, $p_1=\frac {k_1}{l_1}\geq0$,
$p_2=\frac {k_2}{l_2}\geq1$ be rational numbers with $l_1,l_2$ odd, and
$k_1l_1+k_2l_2$ even. Then for any $f\in C^2(\mathbb{R}^d)$ that decays
sufficiently fast at infinity, and for any $x \in \mathbb{R}^d$,
\[
(p_1+p_2)f^{p_1}(x)(-\Delta)^\alpha f^{p_2}(x)
\geq p_2(-\Delta)^\alpha f^{p_1+p_2}(x).
\]
\end{proposition}

\section{A priori estimate}

\begin{proposition} \label{prop3.1}
Let $\alpha>0$, $s\in(0,1)$, $p \in [1,\infty]$ be given. Assume $r>d/p $.
Then there exists some constant $C$ such that
\begin{equation} \label{e3.1}
2^{j\alpha}\|[\Delta_j,\partial_i(-\Delta)^{-s}u]\partial_iv\|_{L^p}
\leq C\|v\|_{B^{r+1}_{p,\infty}}\|u\|_{B^{\alpha+1-2s}_{p,\infty}}
+C\|u\|_{B^{r+2-2s}_{p,\infty}}\|v\|_{B^{\alpha}_{p,\infty}},
\end{equation}
where the brackets $[,]$ represents the commutator, namely
\[
[\Delta_j,\partial_i(-\Delta)^{-s}u]\partial_iv
=\Delta_j(\partial_i(-\Delta)^{-s}u)\partial_iv)
-\partial_i(-\Delta)^{-s}u\Delta_j(\partial_iv).
\]
\end{proposition}

\begin{proof}
Using Bony's para-product decomposition, we have
\[
[\Delta_j,\partial_i(-\Delta)^{-s}u]\partial_iv=L_1+L_2+L_3+L_4+L_5,
\]
where
\begin{gather*}
 L_1=\sum_{|k-j|\leq 4}\Delta_j[S_{k-1}(\partial_i(-\Delta)^{-s}u)\Delta_k(\partial_iv)]-
S_{k-1}(\partial_i(-\Delta)^{-s}u)\Delta_k(\Delta_j(\partial_iv)),\\
 L_2=\sum_{|k-j|\leq 4}\Delta_j[S_{k-1}(\partial_iv)\Delta_k(\partial_i(-\Delta)^{-s}u)],\\
 L_3=\sum_{k\geq j-2}\Delta_j(\Delta_k(\partial_i(-\Delta)^{-s}u)\widetilde{\Delta}_k(\partial_iv)),\\
 L_4=\sum_{k}S_{k-1}(\Delta_j(\partial_iv))\Delta_k(\partial_i(-\Delta)^{-s}u),\\
 L_5=\sum_{|j'-j''|\leq 1}\Delta_{j'}(\Delta_j(\partial_iv))\Delta_{j''}(\partial_i(-\Delta)^{-s}u).
\end{gather*}
We shall estimate the above terms separately. First observe
\begin{align*}
L_1&=\sum_{|k-j|\leq4}2^{jd}\int h(2^j(x-y))\big[S_{k-1}(\partial_i(-\Delta)^{-s}u)(y)\\
&\quad -S_{k-1}(\partial_i(-\Delta)^{-s}u)(x)\big]\Delta_k(\Delta_j(\partial_iv))(y)\,dy.
\end{align*}
By Young's inequality and Bernstein's inequality,
 \begin{align*}
 \|L_1\|_{L^p}
 &\leq C\sum_{|k-j|\leq4}2^{-j}\|\nabla
 \partial_i(-\Delta)^{-s}u\|_{L^\infty}\|\Delta_j(\partial_iv)\|_{L^p}\int|y||h(y)|\,dy \\
 &\leq C2^{-j}2^j\|(-\Delta)^{1-s}u\|_{L^\infty}\|\Delta_jv\|_{L^p}\\
 &\leq C\|\Delta_jv\|_{L^p}\|u\|_{B^{r+2-2s}_{p,\infty}}.
\end{align*}
Similarly,
\[
 \|L_2\|_{L^p}\leq C2^{j(1-2s)}\|\nabla v\|_{L^\infty}\|\Delta_ju\|_{L^p}
\leq C2^{-j\alpha}\|v\|_{B^{r+1}_{p,\infty}}\|u\|_{B^{\alpha+1-2s}_{p,\infty}}.
\]
We can also estimate,
 \begin{align*}
 \|L_3\|_{L^p}
&\leq C\sum_{k\geq j-2}\|\Delta_k(\partial_i(-\Delta)^{-s}u)\|_{L^p}
 \|\nabla v\|_{L^\infty}\\
 &\leq C2^{-j\alpha}\sum_{k\geq j-2}2^{(j-k)\alpha}2^{k(\alpha+1-2s)}
\|\Delta_ku\|_{L^p}\|v\|_{B^{r+1}_{p,\infty}}\\
 &\leq C2^{-j\alpha}\|u\|_{B^{\alpha+1-2s}_{p,\infty}}
\|v\|_{B^{r+1}_{p,\infty}}.
\end{align*}
To estimate $L_4$, by the definition of $S_j$ and $\Delta_j$, we can observe
that only $k$ satisfying $k\geq j$ survive. Thus
\begin{align*}
 \|L_4\|_{L^p}
 &\leq \sum_{k\geq j} C\|\nabla v\|_{L^\infty}2^{k(1-2s)}\|\Delta_ku\|_{L^p}\\
 &\leq C2^{-j\alpha}\sum_{k\geq j}2^{(j-k)\alpha}\|v\|_{B^{r+1}_{p,\infty}}\|u\|_{B^{\alpha+1-2s}_{p,\infty}}\\
 &\leq C2^{-j\alpha}\|v\|_{B^{r+1}_{p,\infty}}\|u\|_{B^{\alpha+1-2s}_{p,\infty}}.
\end{align*}
Since $\Delta_k\Delta_j=0$, for $|j-k|\geq 2$, we have
\[
 \|L_5\|_{L^p}
 \leq C\|\nabla v\|_{L^\infty}2^{j(1-2s)}\|\Delta_ju\|_{L^p}
\leq C2^{-j\alpha}\|v\|_{B^{r+1}_{p,\infty}}\|u\|_{B^{\alpha+1-2s}_{p,\infty}}.
\]
Collecting the estimates above, we obtain
\begin{align*}
& 2^{j\alpha}\|[\Delta_j,\partial_i(-\Delta)^{-s}u]\partial_iv\|_{L^p}\\
& \leq C2^{j\alpha}\|\Delta_jv\|_{L^p}\|u\|_{B^{r+2-2s}_{p,\infty}}
+2^{j(\alpha+1-2s)}\|v\|_{B^{r+1}_{p,\infty}}\|\Delta_ju\|_{L^p}
+\|u\|_{B^{\alpha+1-2s}_{p,\infty}}\|v\|_{B^{r+1}_{p,\infty}}\\
&\leq \|v\|_{B^{\alpha}_{p,\infty}}\|u\|_{B^{r+2-2s}_{p,\infty}}+\|u\|_{B^{\alpha+1-2s}_{p,\infty}}\|v\|_{B^{r+1}_{p,\infty}}.
\end{align*}
This completes the proof.
\end{proof}

\begin{proposition} \label{prop3.2}
Let $\alpha,s,p,r$ be as in Proposition \ref{prop3.1}. Then there exists a constant $C$
such that
\begin{equation} \label{e3.2}
2^{j\alpha}\|[\Delta_j,\partial_i(-\Delta)^{-s}u]\partial_iv\|_{L^p}
\leq C\|v\|_{B^{r}_{p,\infty}}\|u\|_{B^{\alpha+2-2s}_{p,\infty}}+C\|u\|_{B^{r+2-2s}_{p,\infty}}\|v\|_{B^{\alpha}_{p,\infty}}.
\end{equation}
When $p=\infty$, this inequality becomes: for any $r>0$,
\begin{equation} \label{e3.3}
2^{j\alpha}\|[\Delta_j,\partial_i(-\Delta)^{-s}u]\partial_iv\|_{L^\infty}
\leq C\|v\|_{B^{r}_{\infty,\infty}}\|u\|_{B^{\alpha+2-2s}_{\infty,\infty}}
+C\|u\|_{B^{r+2-2s}_{\infty,\infty}}\|v\|_{B^{\alpha}_{\infty,\infty}}.
\end{equation}
\end{proposition}

\begin{proof}
We want to give a new estimate of the commutator in Proposition \ref{prop3.1}.
 Following the above proof, the estimate of $L_1$ unchanged, we give different
bounds for $L_2,L_3,L_4,L_5$.
First,
\begin{align*}
L_2&=\sum_{|k-j|\leq4}2^{jd}\int h(2^j(x-y))(S_{k-1}\partial_iv)(y)
 \Delta_k(\partial_i(-\Delta)^{-s}u)(y)\,dy\\
&=\sum_{|k-j|\leq4}2^{jd}\int \partial_ih(2^j(x-y))2^j(S_{k-1}v)(y)\Delta_k(\partial_i(-\Delta)^{-s}u)(y)\,dy\\
&\quad -\sum_{|k-j|\leq4}2^{jd}\int h(2^j(x-y))(S_{k-1}v)(y)\Delta_k(\partial_{ii}(-\Delta)^{-s}u)(y)\,dy.
\end{align*}
So we obtain
\begin{align*}
 \|L_2\|_{L^p}&\leq C2^{2-2s}(\|\partial_ih\|_{L^1}+\|h\|_{L^1})
 \|v\|_{L^\infty}\|\Delta_ju\|_{L^p}\\
&\leq C2^{-j\alpha}\|u\|_{B^{\alpha+2-2s}_{p,\infty}}\|v\|_{B^{r}_{p,\infty}}.
\end{align*}
Similarly,
\begin{align*}
 \|L_3\|_{L^p}
&\leq C\sum_{k\geq j-2}2^{jd}\int \partial_ih(2^j(x-y))2^j
 \Delta_k(\partial_i(-\Delta)^{-s}u)(y)\Delta_kv(y)\,dy\\
&\quad -2^{jd}\int h(2^j(x-y))\Delta_k(\partial_{ii}
 (-\Delta)^{-s}u)(y)\Delta_kv(y)\,dy.
\end{align*}
Hence we obtain,
\begin{align*}
 \|L_3\|_{L^p}
&\leq C\sum_{k\geq j-2}(2^{(j-k)(\alpha+1)}+2^{(j-k)\alpha})2^{k(\alpha+2-2s)}
 \|v\|_{L^\infty}\|\Delta_ku\|_{L^p}\\
&\leq C2^{-j\alpha}\|u\|_{B^{\alpha+2-2s}_{p,\infty}}\|v\|_{B^{r}_{p,\infty}}.
 \end{align*}
Also,
\begin{align*}
\|L_4\|_{L^p}
 &\leq \sum_{k\geq j} C2^j\|\Delta_jv\|_{L^\infty}2^{k(1-2s)}\|\Delta_ku\|_{L^p}\\
 &\leq C2^{-j\alpha}\sum_{k\geq j}2^{(j-k)(\alpha+1)}2^{k(\alpha+2-2s)}
 \|v\|_{B^{r}_{p,\infty}}\|\Delta_ku\|_{L^p}\\
 &\leq C2^{-j\alpha}\|v\|_{B^{r}_{p,\infty}}\|u\|_{B^{\alpha+2-2s}_{p,\infty}}.
\end{align*}
Finally,
\[
 \|L_5\|_{L^p}
 \leq C\|\Delta_jv\|_{L^\infty}2^{j(2-2s)}\|\Delta_ju\|_{L^p}
\leq C2^{-j\alpha}\|v\|_{B^{r}_{p,\infty}}\|u\|_{B^{\alpha+2-2s}_{p,\infty}}.
\]
Collecting the estimates above, we can obtain
\begin{align*}
 2^{j\alpha}\|[\Delta_j,\partial_i(-\Delta)^{-s}u]\partial_iv\|_{L^p}
&\leq C2^{j\alpha}\|\Delta_jv\|_{L^p}\|u\|_{B^{r+2-2s}_{p,\infty}}
+\|v\|_{B^{r}_{p,\infty}}\|u\|_{B^{\alpha+2-2s}_{p,\infty}}\\
&\leq \|v\|_{B^{\alpha}_{p,\infty}}\|u\|_{B^{r+2-2s}_{p,\infty}}+\|u\|_{B^{\alpha+2-2s}_{p,\infty}}\|v\|_{B^{r}_{p,\infty}}.
\end{align*}
This completes the proof.
\end{proof}

\begin{proposition} \label{prop3.3}
Let $\alpha>0$, $s\in(0,1)$, $p \in [1,\infty]$. Assume $r>\frac dp $.
Then there exists a constant $C$ such that
\begin{equation} \label{e3.4}
2^{j\alpha}\|[\Delta_j,v](-\Delta)^{1-s}u\|_{L^p}
\leq C\|v\|_{B^{r+1}_{p,\infty}}\|u\|_{B^{\alpha+1-2s}_{p,\infty}}
+C\|u\|_{B^{r+2-2s}_{p,\infty}}\|v\|_{B^{\alpha}_{p,\infty}},
\end{equation}
where the brackets $[,]$ represents the commutator,
\begin{equation} \label{e3.5}
[\Delta_j,v](-\Delta)^{1-s}u=\Delta_j(v(-\Delta)^{1-s}u)
-v\Delta_j((-\Delta)^{1-s}u).
\end{equation}
\end{proposition}

\begin{proof}
This proposition is proved similarly to Proposition \ref{prop3.1}.
 We start the proof by writing
\[
[\Delta_j,v](-\Delta)^{1-s}u=I_1+I_2+I_3+I_4+I_5,
\]
where
\begin{gather*}
 I_1=\sum_{|k-j|\leq 4}\Delta_j(S_{k-1}v\Delta_k(-\Delta)^{1-s}u)-
S_{k-1}v\Delta_k(\Delta_j(-\Delta)^{1-s}u),\\
 I_2=\sum_{|k-j|\leq 4}\Delta_j(S_{k-1}(-\Delta)^{1-s}u)\Delta_kv),\\
 I_3=\sum_{k\geq j-2}\Delta_j(\Delta_kv)\widetilde{\Delta}_k(-\Delta)^{1-s}u),\\
 I_4=\sum_{k}S_{k-1}(\Delta_j(-\Delta)^{1-s}u)\Delta_kv,\\
 I_5=\sum_{|j'-j''|\leq 1}\Delta_{j'}v\Delta_{j''}(\Delta_k(-\Delta)^{1-s}u).
\end{gather*}
Similar to the proof for Proposition \ref{prop3.1}, first we observe that
\begin{align*}
\|I_1\|_{L^p}
 &=\|\sum_{|k-j|\leq4}2^{jd}\int h(2^j(x-y))(S_{k-1}v(y)
 -S_{k-1}v(x))\Delta_k(-\Delta)^{1-s}u)(y)dy\|_{L^p}\\
 &\leq C\sum_{|k-j|\leq4}2^{-j}\|\nabla
 v\|_{L^\infty}\|\Delta_j(-\Delta)^{1-s}u\|_{L^p}\int|y||h(y)|\,dy \\
 &\leq C2^{-j}2^{2j(1-s)}\|\nabla v\|_{L^\infty}\|\Delta_ju\|_{L^p}\\
 &\leq C2^{j(1-2s)}\|\Delta_ju\|_{L^p}\|v\|_{B^{r+1}_{p,\infty}}.
\end{align*}
Also we obtain
\begin{gather*}
 \|I_2\|_{L^p}
 \leq C\|(-\Delta)^{1-s}u\|_{L^\infty}\|\Delta_jv\|_{L^p}
 \leq C\|u\|_{B^{r+2-2s}_{p,\infty}}\|\Delta_jv\|_{L^p}, \\
\begin{aligned}
 \|I_3\|_{L^p}
&\leq C\sum_{k\geq j-2}\|\Delta_kv\|_{L^p}2^{k(2-2s)}\|\Delta_ku\|_{L^\infty}\\
&\leq C2^{-j\alpha}\sum_{k\geq j-2}2^{(j-k)\alpha}2^{k\alpha}
 \|\Delta_kv\|_{L^p}2^{k(2-2s)}\|\Delta_ku\|_{L^\infty}\\
&\leq C2^{-j\alpha}\|v\|_{B^{\alpha}_{p,\infty}}\|u\|_{B^{r+2-2s}_{p,\infty}}.
\end{aligned}
\end{gather*}
Similarly, we estimate
\begin{align*}
 \|I_4\|_{L^p}
&\leq C\sum_{k\geq j}\|(-\Delta)^{1-s}u\|_{L^\infty}\|\Delta_kv\|_{L^p}\\
 &\leq C2^{-j\alpha}\sum_{k\geq j}2^{(j-k)\alpha}
 \|(-\Delta)^{1-s}u\|_{L^\infty}2^{k\alpha}\|\Delta_jv\|_{L^p}\\
 &\leq C2^{-j\alpha}\|u\|_{B^{r+2-2s}_{p,\infty}}\|v\|_{B^{\alpha}_{p,\infty}},
\end{align*}
\[
 \|I_5\|_{L^p}
 \leq C\|(-\Delta)^{1-s}u\|_{L^\infty}\|\Delta_jv\|_{L^p}\leq C\|u\|_{B^{r+2-2s}_{p,\infty}}\|\Delta_jv\|_{L^p}.
 \]
Collecting the estimates above, we obtain
\begin{align*}
2^{j\alpha}\|[\Delta_j,v](-\Delta)^{1-s}u\|_{L^p}
&\leq C2^{j(\alpha+1-2s)}\|\Delta_ju\|_{L^p}\|v\|_{B^{r+1}_{p,\infty}}\\
&\quad + C2^{j\alpha}\|u\|_{B^{r+2-2s}_{p,\infty}}\|\Delta_jv\|_{L^p}
 +C\|v\|_{B^{\alpha}_{p,\infty}}\|u\|_{B^{r+2-2s}_{p,\infty}}\\
&\leq C\|v\|_{B^{r+1}_{p,\infty}}\|u\|_{B^{\alpha+1-2s}_{p,\infty}}
+C\|v\|_{B^{\alpha}_{p,\infty}}\|u\|_{B^{r+2-2s}_{p,\infty}}.
\end{align*}
This completes the proof.
\end{proof}

\begin{proposition} \label{prop3.4}
Let $s\in[1/2,1]$, $p=k/l$ be a rational number with $k$ even, $l$ odd, and
 $\alpha>\frac dp+1$. Assume that $u(x,t)\in B_{p,\infty}^\alpha$ is a solution
of  \eqref{e1.1} with $u_0\in B_{p,\infty}^\alpha$ for
$t\in [0,T]$. Then, when $u(x,t)\geq0$, we can find some $C=C(p,\alpha)$,
that for any $t\leq T$,
\begin{equation} \label{e3.6}
 \|u\|_{B^{\alpha}_{p,\infty}}
 \leq C\|u_{0}\|_{B^{\alpha}_{p,\infty}}\exp
\{C\int_0^t\|u\|_{B^{\alpha}_{p,\infty}}d\tau\}.
\end{equation}
\end{proposition}

\begin{proof}
 Applying $\Delta_j$ on \eqref{e1.1}, we obtain
\begin{align*}
\partial_t\Delta_ju&=\sum[\Delta_j,\partial_i(-\Delta)^{-s}u]\partial_iu
+\nabla(-\Delta)^{-s}u\Delta_j(\nabla u)\\
&\quad -[\Delta_j,u](-\Delta)^{1-s}u-u\Delta_j((-\Delta)^{1-s}u.
\end{align*}
Multiplying both sides by $p\Delta_ju|\Delta_ju|^{p-2}$ and
integrating over $\mathbb{R}^d$, the equation becomes
\begin{align*}
&\frac d{dt}\|\Delta_ju\|^p_{L^p}\\
&=\sum\int p\Delta_ju|\Delta_ju|^{p-2}[\Delta_j,\partial_i(-\Delta)^{-s}u]\partial_iu
+\int p\Delta_ju|\Delta_ju|^{p-2}\nabla(-\Delta)^{-s}u\Delta_j(\nabla u)\\
&\quad -\int p\Delta_ju|\Delta_ju|^{p-2}[\Delta_j,u](-\Delta)^{1-s}u
-\int p\Delta_ju|\Delta_ju|^{p-2}u\Delta_j((-\Delta)^{1-s}u\\
&=J_1+J_2+J_3+J_4.
\end{align*}
From Propositions \ref{prop3.1} and \ref{prop3.3}, we obtain the estimates
\begin{gather*}
J_1\leq C2^{-j\alpha}\|\Delta_ju\|^{p-1}_{L^p}\|u\|_{B^{\alpha}_{p,\infty}}
\|u\|_{B^{\alpha+1-2s}_{p,\infty}}
\leq C2^{-j\alpha}\|\Delta_ju\|^{p-1}_{L^p}\|u\|^2_{B^{\alpha}_{p,\infty}},
\\
J_3\leq C2^{-j\alpha}\|\Delta_ju\|^{p-1}_{L^p} \|u\|_{B^{\alpha}_{p,\infty}}
\|u\|_{B^{\alpha+1-2s}_{p,\infty}}
\leq C2^{-j\alpha}\|\Delta_ju\|^{p-1}_{L^p}\|u\|^2_{B^{\alpha}_{p,\infty}}.
\end{gather*}
It is easy to see that
\begin{align*}
 J_2&=\int\nabla(-\Delta)^{-s}u)\nabla(|\Delta_j(u)|^p)
=\int (-\Delta)^{1-s}u|\Delta_ju|^pdx\\
&\leq C\|(-\Delta)^{1-s}u\|_{B^{\alpha-1}_{p,\infty}}\|\Delta_ju\|^{p}_{L^p}
\leq C\|u\|_{B^{\alpha+1-2s}_{p,\infty}}\|\Delta_ju\|^{p-1}_{L^p}\|\Delta_ju\|_{L^p}\\
&\leq C2^{-j\alpha} \|u\|_{B^{\alpha+1-2s}_{p,\infty}}
\|u\|_{B^{\alpha}_{p,\infty}}\|\Delta_ju\|^{p-1}_{L^p}.
\end{align*}
Using the fact that $u\geq0$ and Propositions \ref{prop2.3} and \ref{prop2.4},
 we estimate
\begin{align*}
 J_4&\leq- \int p|\Delta_ju|^{p-2}u(-\Delta)^{1-s}|\Delta_ju|^2
 \leq -\int u(-\Delta)^{1-s}|\Delta_ju|^p\\
 &=-\int (-\Delta)^{1-s}u|\Delta_ju|^p
\leq C\|(-\Delta)^{1-s}u\|_{B^{\alpha-1}_{p,\infty}}\|\Delta_ju\|^{p}_{L^p}\\
 &\leq C2^{-j\alpha}\|u\|_{B^{\alpha+1-2s}_{p,\infty}}
\|u\|_{B^{\alpha}_{p,\infty}}\|\Delta_ju\|^{p-1}_{L^p}.
 \end{align*}
 Combining the above four estimates, we set
$ \frac d{dt}\|u\|_{\dot{B}^{\alpha}_{p,\infty}}
\leq C\|u\|_{B^{\alpha}_{p,\infty}}^2$. Since $u\geq 0$, it follows
\begin{align*}
 \frac d{dt}\|u\|_{L^p}
&=\frac d{dt}\int u^p dx=p\int u^{p-1}u_tdx\\
&=p\int u^{p-1}\nabla\cdot(u\nabla(-\Delta)^{-s}u)
 =-(p-1)\int\nabla u^{p}\cdot(\nabla(-\Delta)^{-s}u)\\
&=-(p-1)\int u^{p}((-\Delta)^{1-s}u)\\
&\leq C\|u\|_{L^p}^{p-1}\|u\|_{B^{\alpha}_{p,\infty}}
\|u\|_{B^{\alpha+1-2s}_{p,\infty}}.
\end{align*}
This implies
\[
\frac d{dt}\|u\|_{B^{\alpha}_{p,\infty}}
\leq C\|u\|_{B^{\alpha}_{p,\infty}}\|u\|_{B^{\alpha}_{p,\infty}},
\]
which with the Gronwall's inequality yield \eqref{e3.6}.
\end{proof}

\section{Proof of Theorem \ref{thm1.1}}

From the definition of Riesz potentials
$(-\Delta)^{-s}u=c(n,s)|x|^{-d+2s}\ast u$, $0<2s<d$.
So when $d\geq 2$, $\frac 12\leq s<1$,
 we define $P_\epsilon u=c(n,s) (|x|^{-d+2s}\ast\sigma_\epsilon)\ast u
=(-\Delta)^{-s}(\sigma_\epsilon\ast u)=(-\Delta)^{-s}u_\epsilon$,
$u_\epsilon=\sigma_\epsilon\ast u$ . Here $\sigma\in C_c^\infty$,
is nonnegative, radially symmetric and decreasing, $\int \sigma=1$,
$\sigma_\epsilon=\varepsilon^{-d}\sigma(x/\epsilon)$.
Then we construct a sequence $\{u^{(n)}\}$, defined recursively by solving
the following equations
\begin{equation} \label{e4.1}
\begin{gathered}
u^{(1)}=S_2(u_0)\\
\partial_tu^{(n+1)}=\nabla\cdot(u^{(n+1)}\nabla(P_\epsilon u^{(n)}))\\
u^{(n+1)}(x,0)=u_0^{n+1}=S_{n+2}u_0.
\end{gathered}
\end{equation}
Since $u^{(n)}$ solves the linear system, we can always find the sequence.
Similarly to the proof of Proposition \ref{prop3.4}, taking $\Delta_j$ on \eqref{e4.1},
 we obtain
\begin{equation} \label{e4.2}
\begin{split}
\partial_t\Delta_ju^{(n+1)}
&=\sum[\Delta_j,\partial_i(-\Delta)^{-s}u_\epsilon^{(n)}]\partial_iu^{(n+1)}
+\sum\partial_i(-\Delta)^{-s}u_\epsilon^{(n)}\Delta_j(\partial_iu^{(n+1)})\\
&\quad -[\Delta_j,u^{(n+1)}](-\Delta)^{1-s}u_\epsilon^{(n)}-u^{(n+1)}
\Delta_j((-\Delta)^{1-s}u_\epsilon^{(n)}).
\end{split}
\end{equation}
Multiplying both sides by $\frac {\Delta_ju^{(n+1)}}{|\Delta_ju^{(n+1)}|}$,
integrating over $\mathbb{R}^d$, we denote each corresponding part in the right
side by $J'_1,J'_2,J'_3,J'_4$. Now we obtain the estimates
\begin{equation} \label{e4.3}
 \begin{gathered}
 J'_1\leq C2^{-j\alpha} \|u^{(n+1)}\|_{B^{\alpha}_{1,\infty}}
 \|u^{(n)}\|_{B^{\alpha+1-2s}_{1,\infty}},\\
 J'_3\leq C2^{-j\alpha} \|u^{(n+1)}\|_{B^{\alpha}_{1,\infty}}
\|u^{(n)}\|_{B^{\alpha+1-2s}_{1,\infty}},
 \end{gathered}
\end{equation}
where we used the fact that
$\|u_\epsilon^{(n)}\|_{B^{\alpha+1-2s}_{1,\infty}}
\leq \|u^{(n)}\|_{B^{\alpha+1-2s}_{1,\infty}}$. Also, we have
\begin{equation} \label{e4.4}
 \begin{split}
 J'_2&=\int\nabla(-\Delta)^{-s}u_\epsilon^{(n)})\nabla(|\Delta_j(u^{(n+1)})|)\\
&=\int (-\Delta)^{1-s}u_\epsilon^{(n)}|\Delta_ju^{(n+1)}|dx\\
&\leq C2^{-j\alpha} \|u^{(n+1)}\|_{B^{\alpha}_{1,\infty}}
 \|u^{(n)}\|_{B^{\alpha+1-2s}_{1,\infty}},
 \end{split}
\end{equation}
and
\begin{equation} \label{e4.5}
 \begin{split}
 J'_4&=-\int\frac {\Delta_ju^{(n+1)}}{|\Delta_ju^{(n+1)}|}u^{(n+1)}
\Delta_j((-\Delta)^{1-s}u_\epsilon^{(n)})\\
&=\int\frac {\Delta_ju^{(n+1)}}{|\Delta_ju^{(n+1)}|}u^{(n+1)}
\Delta_j(\Delta(-\Delta)^{-s}u_\epsilon^{(n)})\\
&=-\int\frac {\Delta_ju^{(n+1)}}{|\Delta_ju^{(n+1)}|}\nabla u^{(n+1)}
 \Delta_j(\nabla(-\Delta)^{-s}u_\epsilon^{(n)}) \\
&\leq C2^{-j\alpha} \|u^{(n)}\|_{B^{\alpha+1-2s}_{1,\infty}}
 \|u^{(n+1)}\|_{B^{\alpha}_{1,\infty}}.
 \end{split}
\end{equation}
On the other hand,
\[
\frac d{dt}\int |u^{(n+1)}| dx
=\int \frac{u^{(n+1)}}{|u^{(n+1)}|}u^{(n+1)}_tdx
=\int\nabla\cdot(|u^{(n+1)}|\nabla(-\Delta)^{-s}u_\epsilon^{(n)})=0.
\]
Collecting the estimates above, when $s\geq 1/2$, we obtain
\begin{equation} \label{e4.6}
 \begin{split}
 \frac d{dt}\|u^{(n+1)}\|_{B^{\alpha}_{1,\infty}}
&\leq C\|u^{(n+1)}\|_{B^{\alpha}_{1,\infty}}\|u^{(n)}
 \|_{B^{\alpha+1-2s}_{1,\infty}}\\
&\leq C\|u^{(n+1)}\|_{B^{\alpha}_{1,\infty}}\|u^{(n)}\|_{B^{\alpha}_{1,\infty}}.
 \end{split}
\end{equation}
Now from Gronwall's inequality we obtain
\begin{align*}
 \|u^{(n+1)}\|_{B^{\alpha}_{1,\infty}}
 &\leq \|u^{(n+1)}_{0}\|_{B^{\alpha}_{1,\infty}}
\exp\{C\int_0^t\|u^{(n)}\|_{B^{\alpha}_{1,\infty}})d\tau\}\\
 &\leq C\|u_{0}\|_{B^{\alpha}_{1,\infty}}
\exp\{C\int_0^t\|u^{(n)}\|_{B^{\alpha}_{1,\infty}})d\tau\},
 \end{align*}
with $C$ independent of $n$.
Defining $X_T:=C([0,T];B^{\alpha}_{1,\infty})$, we have
\[
\|u^{(n+1)}\|_{X_T}\leq C\|u_{0}\|_{B^{\alpha}_{1,\infty}}\exp(CT\|u^{(n)}\|_{X_T}).
\]
Thus, by the standard induction argument we find
\begin{equation} \label{e4.7}
\sup_{0\leq t\leq T_0}\|u^{(n)}\|_{B^{\alpha}_{1,\infty}}
\leq 2C\|u_{0}\|_{B^{\alpha}_{1,\infty}},
\end{equation}
for all $n\geq1$, if $\exp(2CT_0\|u_{0}\|_{B^{\alpha}_{1,\infty}})\leq2$; namely,
 $T_0\leq\frac {\ln2}{2C_0(1+\|u_{0}\|_{B^{\alpha}_{1,\infty}})}$.
Using Proposition \ref{prop3.2}, \ref{prop3.3}, and \eqref{e4.4}, \eqref{e4.5}, we see
\begin{equation} \label{e4.8}
\frac d{dt}\|u^{(n+1)}\|_{B^{\alpha+2s-2}_{1,\infty}}
\leq C\|u^{(n+1)}\|_{B^{\alpha+2s-2}_{1,\infty}}\|u^{(n)}\|_{B^{\alpha}_{1,\infty}}.
\end{equation}
Thus, from the uniform estimate \eqref{e4.7}, we have
\begin{equation} \label{e4.9}
\sup_{0\leq t\leq T_0}\|\frac \partial {\partial t}u^{(n)}
\|_{B^{\alpha+2s-2}_{1,\infty}}
\leq C\|u_0\|_{B^\alpha_{1,\infty}}^2.
\end{equation}
Let $Y_T:=C([0,T];B^{\alpha+2s-2}_{1,\infty})$. We will prove that the sequence
$\{u^{(n)}\}$ is Cauchy in $Y_{T_1}$
for some $T_1 \in (0,T_0)$. Considering the difference $u^{(n+1)}-u^{(n)}$,
\begin{align*}
 &\partial_t(u^{(n+1)}-u^{(n)})\\
&=\nabla u^{(n+1)}\cdot\nabla(-\Delta)^{-s}u_\epsilon^{(n)}-u^{(n+1)}
(-\Delta)^{1-s}u_\epsilon^{(n)}
 -\nabla u^{(n)}\cdot\nabla(-\Delta)^{-s}u_\epsilon^{(n-1)}\\
&\quad +u^{(n)}(-\Delta)^{1-s}u_\epsilon^{(n-1)}\\
&=\nabla\cdot((u^{(n+1)}-u^{(n)})\nabla(-\Delta)^{-s}u_\epsilon^{(n)})
 +\nabla\cdot(u^{(n)}\nabla(-\Delta)^{-s}(u_\epsilon^{(n)}-u_\epsilon^{(n-1)})),
\end{align*}
with initial datum $(u^{(n+1)}-u^{(n)})(x,0)=\Delta_{n+1}u_0$.
Proceeding as in the proof of \eqref{e4.6}, we obtain the estimate
\[
 \|\nabla\cdot(u^{(n)}\nabla(-\Delta)^{-s}(u_\epsilon^{(n)}-u_\epsilon^{(n-1)}))\|_{B^{\alpha+2s-2}_{1,\infty}}\leq
 C\|u^{(n)}-u^{(n-1)}\|_{B^{\alpha-1}_{1,\infty}}
\|u^{(n)}\|_{B^{\alpha}_{1,\infty}}.
\]
Proceeding as in the proof of \eqref{e4.8}, we obtain the estimate
\[
 \|\nabla\cdot((u^{(n+1)}-u^{(n)})
\nabla(-\Delta)^{-s}u_\epsilon^{(n)})\|_{B^{\alpha+2s-2}_{1,\infty}}
\leq C\|u^{(n+1)}-u^{(n)}\|_{B^{\alpha+2s-2}_{1,\infty}}
\|u^{(n)}\|_{B^{\alpha}_{1,\infty}}\,.
\]
Then we obtain
\begin{align*}
 \frac d {dt}\|u^{(n+1)}-u^{(n)}\|_{B^{\alpha+2s-2}_{1,\infty}}
&\leq C\|u^{(n+1)}-u^{(n)}\|_{B^{\alpha+2s-2}_{1,\infty}}
\|u^{(n)}\|_{B^{\alpha}_{1,\infty}}\\
&\quad + C\|u^{(n)}-u^{(n-1)}\|_{B^{\alpha+2s-2}_{1,\infty}}
\|u^{(n)}\|_{B^{\alpha}_{1,\infty}},
\end{align*}
and
\begin{align*}
&\|(u^{(n+1)}-u^{(n)})(x,0)\|_{B^{\alpha+2s-2}_{1,\infty}}\\
&=\|\Delta_{n+1}u_0\|_{B^{\alpha+2s-2}_{1,\infty}}=
 \sup_j2^{j(\alpha+2s-2)}\|\Delta_j\Delta_{n+1}u_0\|_{L^1}\\
 &\leq C2^{n(2s-2)}\sup_{n\leq j\leq n+2}2^{j\alpha}\|\Delta_ju_0\|_{L^1}\\
 &\leq C2^{n(2s-2)}\|u_0\|_{B^\alpha_{1,\infty}}.
\end{align*}
The above inequalities and Gronwall's inequality imply that
\begin{equation} \label{e4.10}
 \begin{split}
&\|u^{(n+1)}(t)-u^{(n)}(t)\|_{B^{\alpha+2s-2}_{1,\infty}}\\
&\leq \|(u^{(n+1)}-u^{(n)})(x,0)\|_{B^{\alpha+2s-2}_{1,\infty}}
\exp(C\|u_0\|_{B^\alpha_{1,\infty}})\\
&\quad +\int_0^t \exp ({C\|u_0\|_{B^\alpha_{1,\infty}}(t-s)})
 \|u^{(n)}-u^{(n-1)}\|_{B^{\alpha+2s-2}_{1,\infty}}(s)ds\\
&\leq C'2^{n(2s-2)}+C'\|u^{(n)}-u^{(n-1)}\|_{Y_{T_1}}(\exp{C'T_1}-1),
\end{split}
\end{equation}
where $T_1 \in [0,T_0]$, and the constant $C'=C'(\|u_0\|_{B^\alpha_{1,\infty}})$.
Thus, if $C'(\exp{T_1}-1)<\frac12$, we can deduce that ${u^{(n)}}$ converges
to $u\in L^\infty ([0,T_1];B^{\alpha+2s-2}_{1,\infty})$ in $Y_{T_1}$.
By the well-known interpolation inequality in the Besov spaces we have
$u^n\to u$ in $L^\infty ([0,T_1];B^{\beta}_{1,\infty})$ for all
$\beta\in[\alpha+2s-2,\alpha]$. Moreover, the estimate \eqref{e4.8} allows us
 to conclude that $u \in C^1([0,T_1];B^{\alpha+2s-2}_{1,\infty})$.
Letting $\epsilon\to 0,n\to \infty,$ we find that $u$ is a solution to
 \eqref{e1.1} in $B^{\beta}_{1,\infty}$.

Next we prove that the solution is unique. Suppose that $u$, $v$
in $L^\infty([0,T_1];B^{\beta}_{1,\infty})$ are two solutions of \eqref{e1.1}
associated with the initial condition $u_0$, $v_0$. Then $u-v$ satisfies the
equation
\begin{gather*}
u-v=\nabla\cdot[(u-v)\nabla(-\Delta)^{-s}u]+\nabla\cdot[v\nabla(-\Delta)^{-s}(u-v)],\\
(u-v)(x,0)=u_0-v_0.
\end{gather*}
Working similarly with \eqref{e4.10}, we can obtain
\begin{align*}
\|(u-v)(t)\|_{B^{\beta}_{1,\infty}}
&\leq M2^{j(\beta-\alpha)}\|(u-v)(x,0)\|_{B^{\beta}_{1,\infty}}\\
&\quad +M\|u-v\|_{C([0,T_1];B^{\beta}_{1,\infty})}(\exp{C''T_1}-1),
\end{align*}
for $M=M(\|(u_0\|_{B^{\beta}_{1,\infty}},\|v_0\|_{B^{\beta}_{1,\infty}})$.
If $M(\exp{C''T_1}-1)<1$, then we get the uniqueness of solutions
in $C([0,T_1];B^{\beta}_{1,\infty})$.
The proof is complete.

\section{Proof of Theorem \ref{thm1.2}}

 When $d\geq3$, we know $(-\Delta)^{-1}u=c(n,s)|x|^{-d+2}\ast u$. Let
$P_\epsilon u=c(n,s) (|x|^{-d+2}\ast\sigma_\epsilon)\ast u
=(-\Delta)^{-1}(\sigma_\epsilon\ast u)=(-\Delta)^{-1}u_\epsilon$, where
 $\sigma$ is defined in the beginning of Section 4. Consider the successive
approximation sequence $\{u^{(n)}\}$ satisfying
\begin{equation} \label{e5.1}
\begin{gathered}
u^{(1)}=S_2(u_0)\\
\partial_tu^{(n+1)}=\nabla\cdot(u^{(n+1)}
\nabla(P_\epsilon u_\epsilon^{(n)}))=\nabla u^{(n+1)}\cdot
\nabla(-\Delta)^{-1}u_\epsilon^{(n)}-u^{(n+1)}u_\epsilon^{(n)}\\
u^{(n+1)}(x,0)=u_0^{n+1}=S_{n+2}u_0.
\end{gathered}
\end{equation}
Appling $\Delta_j$ on \eqref{e5.1},
\begin{equation} \label{e5.2}
\begin{split}
\partial_t\Delta_ju^{(n+1)}
&=\sum[\Delta_j,\partial_i(-\Delta)^{-s}u_\epsilon^{(n)}]\partial_iu^{(n+1)}\\
&\quad+\nabla(-\Delta)^{-s}u_\epsilon^{(n)}\nabla\Delta_j(u^{(n+1)})
-\Delta_j(u^{(n+1)}u_\epsilon^{(n)}).
\end{split}
\end{equation}
When $p\geq2$, multiplying both sides of \eqref{e5.2} by
$p\Delta_ju^{(n+1)}|\Delta_ju^{(n+1)}|^{p-2}$ and integrating over $\mathbb{R}^d$,
 we obtain
\begin{align*}
\frac d{dt}\|\Delta_ju^{(n+1)}\|^p_{L^p}
&=\sum\int p\Delta_ju^{(n+1)}|\Delta_ju^{(n+1)}|^{p-2}
 [\Delta_j,\partial_i(-\Delta)^{-1}u_\epsilon^{(n)}]\partial_iu^{(n+1)}\\
&+\int p\Delta_ju^{(n+1)}|\Delta_ju^{(n+1)}|^{p-2}
 \nabla(-\Delta)^{-1}u_\epsilon^{(n)}\nabla\Delta_j(u^{(n+1)})\\
&-\int p\Delta_ju^{(n+1)}|\Delta_ju^{(n+1)}|^{p-2}
 \Delta_j(u^{(n+1)}u_\varepsilon^{(n)})=J''_1+J''_2+J_3''.
\end{align*}
By H\"older's inequality,
\[
J_1''\leq C\|\Delta_ju^{(n+1)}\|^{p-1}_{L^p} \|[\Delta_j,\partial_i
(-\Delta)^{-1}u_\epsilon^{(n)}]\partial_iu^{(n+1)}\|_{L^p}.
\]
From the proof of Proposition \ref{prop2.1} (iv),
\[
 J_3''\leq \|\Delta_ju^{(n+1)}\|^{p-1}_{L^p}[\|u^{(n)}\|_{L^\infty}
\|\Delta_ju^{(n+1)}\|_{L^p}+\|u^{(n+1)}\|_{L^\infty}\|\Delta_ju^{(n)}\|_{L^p}].
\]
Also we obtain
\begin{align*}
 J_2''&=\int\nabla(-\Delta)^{-1}u_\epsilon^{(n)}\nabla(|\Delta_j( u^{(n+1)})|^p)\\
 &=\int u_\epsilon^{(n)}|\Delta_ju^{(n+1)}|^pdx
\leq \|u^{(n)}\|_{L^\infty}\|\Delta_ju^{(n+1)}\|^p_{L^p}.
 \end{align*}
Now we obtain
\begin{align*}
\frac d{dt}\|\Delta_ju^{(n+1)}\|_{L^p}&\leq C\|[\Delta_j,\partial_i(-\Delta)^{-1}u^{(n)}]\partial_iu^{(n+1)}\|_{L^p}\\
&\quad +C\|u^{(n)}\|_{L^\infty}\|\Delta_ju^{(n+1)}\|_{L^p}+C\|u^{(n+1)}\|_{L^\infty}\|\Delta_ju^{(n)}\|_{L^p}.
\end{align*}
When, $\alpha>d/p$,
\[
 \frac d{dt}\|u^{(n+1)}\|_{\dot{B}^{\alpha}_{p,\infty}}
 \leq C\|u^{(n+1)}\|_{B^{\alpha}_{p,\infty}}\|u^{(n)}\|_{B^{\alpha}_{p,\infty}}.
\]
Letting $p\to\infty$,
\begin{align*}
\frac d{dt}\|\Delta_ju^{(n+1)}\|_{L^\infty}&\leq C\|[\Delta_j,\partial_i(-\Delta)^{-1}u^{(n)}]\partial_iu^{(n+1)}\|_{L^\infty}\\
&\quad+C\|u^{(n)}\|_{L^\infty}\|\Delta_ju^{(n+1)}\|_{L^\infty}+C\|\Delta_j(u^{(n+1)}u^{(n)})\|_{L^\infty}.
\end{align*}
So for any $\alpha>0$,
\[
 \frac d{dt}\|u^{(n+1)}\|_{\dot{B}^{\alpha}_{\infty,\infty}}
 \leq C\|u^{(n+1)}\|_{B^{\alpha}_{\infty,\infty}}\|u^{(n)}\|_{B^{\alpha}_{\infty,\infty}}.
\]
It is easy to prove that when $p=1$,
\begin{align*}
 \frac d{dt}\int |u^{(n+1)}| dx
&=\int \frac{u^{(n+1)}}{|u^{(n+1)}|}u^{(n+1)}_tdx\\
&=\int \nabla\cdot(|u^{(n+1)}|(|x|^{-d+2}\ast\sigma_\epsilon)\ast u^{(n)})=0.
\end{align*}
Noting $u^{(n+1)}(x,0)=u_0^{n+1}=S_{n+2}u_0$, we know $\|u^{(n+1)}\|_{L^1}\leq C$.

When $p\geq 2$,
\begin{align*}
 \frac d{dt}\int |u^{(n+1)}|^p dx
&=p\int |u^{(n+1)}|^{p-2}u^{(n+1)}u^{(n+1)}_tdx\\
 &=p\int |u^{(n+1)}|^{p-2}u^{(n+1)}\nabla\cdot(u^{(n+1)}\nabla(-\Delta)^{-1}u_\epsilon^{(n)})\\
 &=-(p-1)\int\nabla|u^{(n+1)}|^{p}\cdot(\nabla(-\Delta)^{-1}u_\epsilon^{(n)})\\
 &=(p-1)\int|u^{(n+1)}|^{p}u_\epsilon^{(n)}\leq C\|u^{(n+1)}\|_{L^p}^{p}\|u^{(n)}\|_{L^\infty}.
 \end{align*}
It means
\[
 \frac d{dt}\|u^{(n+1)}\|_{L^p}\leq\|u^{(n+1)}\|_{L^p}\|u^{(n)}\|_{L^\infty}
\leq C\|u^{(n+1)}\|_{B^{\alpha}_{p,\infty}}\|u^{(n)}\|_{B^{\alpha}_{p,\infty}}.
\]
Again letting $p\to\infty$, we obtain
\[
 \frac d{dt}\|u^{(n+1)}\|_{L^\infty}\leq C\|u^{(n+1)}\|_{B^{\alpha}_{\infty,\infty}}\|u^{(n)}\|_{B^{\alpha}_{\infty,\infty}}.
\]
Collecting the estimates above, we now obtain
\begin{gather*}
 \frac d{dt}\|u^{(n+1)}\|_{B^{\alpha}_{p,\infty}}
 \leq C\|u^{(n+1)}\|_{B^{\alpha}_{p,\infty}}\|u^{(n)}\|_{B^{\alpha}_{p,\infty}},
\quad p<\infty, \; \alpha>\frac dp,
\\
 \frac d{dt}\|u^{(n+1)}\|_{B^{\alpha}_{\infty,\infty}}
 \leq C\|u^{(n+1)}\|_{B^{\alpha}_{\infty,\infty}}\|u^{(n)}
\|_{B^{\alpha}_{\infty,\infty}},\quad p=\infty,\; \alpha>0.
\end{gather*}
For any $\beta\in(d/p,\alpha)$, $p<\infty$, or $\beta\in(0,\alpha)$, $p=\infty$
we have
\[
 \frac d{dt}\|u^{(n+1)}\|_{B^{\beta}_{p,\infty}}
 \leq \|u^{(n+1)}\|_{B^{\beta}_{p,\infty}}\|u^{(n)}\|_{B^{\beta}_{p,\infty}},
\]
and
\begin{align*}
 \frac d{dt}\|u^{(n+1)}-u^{(n)}\|_{B^{\beta}_{p,\infty}}
 &\leq \|u^{(n+1)}-u^{(n)}\|_{B^{\beta}_{p,\infty}}\|u^{(n)}
 \|_{B^{\beta}_{p,\infty}}\\
 &\quad+ \|u^{(n)}-u^{(n-1)}\|_{B^{\beta}_{p,\infty}}
\|u^{(n)}\|_{B^{\beta}_{p,\infty}}.
 \end{align*}
The rest of the proof is similar to the proof of Theorem \ref{thm1.1},
we omit it.

\subsection*{Acknowledgments}
This research was supported by the National Natural Science Foundation of China 
(11201103).


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