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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 146, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/146\hfil Local well-posedness ]
{Local well-posedness for density-dependent incompressible Euler equations}

\author[Z. Wei \hfil EJDE-2013/146\hfilneg]
{Zhiqiang Wei}  % in alphabetical order

\address{Zhiqiang Wei \newline
School of Mathematics and Information Sciences \\
North China University of Water Resources and Electric Power \\
Zhengzhou 450011, China}
\email{wei.zhiqiang@yahoo.com}

\thanks{Submitted May 24, 2013. Published June 25, 2013.}
\subjclass[2000]{35B30, 35Q35, 46E35}
\keywords{Local well-posedness; density dependent Euler equation; Besov space}

\begin{abstract}
  In this article, we establish the local well-posedness for
  density-dependent incompressible Euler equations in critical Besov spaces.
\end{abstract}

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


\section{Introduction}
We consider the following  density-dependent incompressible Euler
equations in $\mathbb{R}^N$, $N\geq 2$,
\begin{gather*}
\partial_t \varrho+(v \cdot \nabla) \varrho=0,\\
\partial_t (\varrho v)+(v \cdot \nabla)(\varrho v)+\nabla P=\varrho f,\\
\operatorname{div} v=0,\\
(\varrho,v)|_{t=0}=(\varrho_0,v_0),
\end{gather*}
where $0<m<\varrho_0(x)<M<\infty$ and
$\lim_{x \to \pm \infty} \varrho_0(x)=\bar{\varrho}$,
without loss of generality, assume that $\bar{\varrho}=1$.
Suppose that $f=0$, just for
simplicity, then one can rewrite the above equations as
\begin{equation} \label{1.1}
\begin{gathered}
\partial_t \rho+(v \cdot \nabla) \rho=0,\\
\partial_t v+(v \cdot \nabla) v+(1+\rho) \nabla P=0,\\
\operatorname{div} v=0,\\
\rho=\frac{1}{\varrho} -1,\quad (\rho,v)|_{t=0}=(\rho_0,v_0).
\end{gathered}
\end{equation}
If $\varrho_0(x)=\bar{\varrho} \equiv 1$, then \eqref{1.1} is the
standard incompressible Euler equations. For this Euler model, we
mention the following local well-posedness results.
Given $v_0 \in H^m(\mathbb{R}^N)$, $m>N/2+1$, Kato~\cite{Ka} proved
local existence and uniqueness for a solution belonging to
$C([0,T];H^m(\mathbb{R}^N))$ with $T=T(\|v_0\|_{H^m})$.
Later on, many various function spaces
(see~\cite{Ch,KP,Vi,Zhou}) are used to establish the local existence
and uniqueness for the incompressible Euler equations. For example,
$W^{s,p}(\mathbb{R}^N)$ with $s>N/p+1$, $1<p<\infty$ is used
in~\cite{KP} and $F_{p,q}^s$ for $s>N/p+1$, $1<p<\infty$,
$1<q<\infty$ is used in~\cite{Ch}.

For the density-dependent Euler equations, it is worth noting the
following results. Beir\~ao da Veiga and Valli~\cite{BV4} discussed
the local existence and uniqueness for \eqref{1.1} in a bounded
domain with various condition. In unbounded domain, Itoh~\cite{It}
proved the local existence and uniqueness for \eqref{1.1} with
initial data in $H^3(\mathbb{R}^3)$.

In this work we establish the local existence and uniqueness for
the system \eqref{1.1} with initial data in the critical (with
respect to the scaling invariance) Besov spaces.

\begin{theorem} \label{thm1.1}
Let $p \in (1,\infty)$. There exists a constant $c$ depending only
on $N$, such that for any given
$\rho_0 \in B_{p,1}^{N/p+1}(\mathbb{R}^N)$ and
$v_0 \in B_{p,1}^{N/p+1}(\mathbb{R}^N)$, $\operatorname{div}v_0=0$ with
\begin{equation} \label{1.2}
\|\rho_0\|_{B_{p,1}^{N/p+1}} \leq c,
\end{equation}
there exists a
$T=T\big(p,\|\rho_0\|_{B_{p,1}^{N/p+1}},\|v_0\|_{B_{p,1}^{N/p+1}}\big)$,
the system \eqref{1.1} has a unique solution $(\rho,v,\nabla P)$
with $\rho \in C([0,T];B_{p,1}^{N/p+1})$, $v \in
C([0,T];B_{p,1}^{N/p+1})$ and $\nabla P \in L^1
(0,T;B_{p,1}^{N/p+1})$.
\end{theorem}

We remark that Theorem 1.1 gives a local existence
and uniqueness theorem for \eqref{1.1} under a small perturbation of
an initial constant density state. We wish to discuss the
well-posedness for problem \eqref{1.1} without the restriction
\eqref{1.2}, in other words, a perturbation of any initial density
in the future. It is worth to point out that local well-posedness is
established in~\cite{ZXF} for the periodic case without \eqref{1.2}.
 For the case of supercritical
Besov spaces, we refer to~\cite{Zhou2}.

For the standard 2-D incompressible Euler equations in the
critical (borderline) Besov spaces $B_{p,1}^{2/p+1}(\mathbb{R}^2)$,
Vishik~\cite{Vi} showed the (global) well-posedness recently. Just
as he said in his paper, it is of great interest to establish
local well-posedness for high dimensional Euler equations.
Obviously, Theorem 1.1 is true for the standard incompressible
Euler equations ($\rho_0(x) \equiv 0$, \eqref{1.2} automatically
holds). In other words, we recover the following local
well-posedness theorem for incompressible Euler equations in the
critical Besov space $B_{p,1}^{N/p+1}(\mathbb{R}^N)$~\cite{Zhou}.

\begin{corollary}[\cite{Zhou}] \label{coro}
Given any $v_0 \in B_{p,1}^{N/p+1}(\mathbb{R}^N)$, $1<p<\infty$,
there exists a $T=T(\|v_0\|_{B_{p,1}^{N/p+1}})$ and a unique
solution $(v,\nabla P)$ to the system
\begin{gather*}
\frac{\partial v}{\partial t}+(v \cdot \nabla) v +\nabla P=0,\\
\operatorname{div} v=0,\\
v(x,t=0)=v_0(x),
\end{gather*}
such that
\begin{align*}
v(x,t) \in C([0,T];B_{p,1}^{N/p+1})\quad \text{and}\quad
\nabla P \in L^1 (0,T;B_{p,1}^{N/p+1}).
\end{align*}
\end{corollary}

In Theorem 1.1, if $p=2$, then the smallness assumption on
$\rho_0$ can be removed. More precisely, we have

\begin{theorem} \label{thm1.3}
Assume $\rho_0 \in B_{2,1}^{N/2+1}$ and $v_0 \in B_{2,1}^{N/2+1}$,
$\operatorname{div}v_0=0$. Then there exists
$T=T(\|\rho_0\|_{B_{2,1}^{N/2+1}},\|v_0\|_{B_{2,1}^{N/2+1}})$ such
that the system \eqref{1.1} has a uniqueness solution
$(\rho,v,\nabla P)$ with $\rho \in C([0,T];B_{2,1}^{N/2+1})$, $v
\in C([0,T];B_{2,1}^{N/2+1})$ and $\nabla P \in L^1
(0,T;B_{2,1}^{N/2+1})$.
\end{theorem}

\begin{remark} \label{rmk1.2}\rm
After completing this article, the author
was informed that Theorem 1.3 already was proved
in~\cite{CL}. However, the proof of Theorem 1.3 in
Section 4 is different from proof in \cite{CL}.
 Another purpose is to
investigate the difference of space $B_{p,1}^{N/p+1}$ between
general $p \neq 2$ and $p=2$. We hope we can get ride of smallness
restriction \eqref{1.2} in a future work.
\end{remark}

\section{Littlewood-Paley decomposition and Besov spaces}

We start by recalling the Littlewood-Paley decomposition of
temperate distributions. Let $\mathcal{S}$ be the class of
Schwartz class of rapidly decreasing functions. Given $f \in
\mathcal{S}$, the Fourier transform is defined as
\begin{align*}
\mathcal{F}(f)=\hat{f}=\frac{1}{(2 \pi)^{n/2}} \int_{\mathbb{R}^N}
e^{- i x \cdot \xi} f(x) dx.
\end{align*}
One can extend $\mathcal{F}$ and $\mathcal{F}^{-1}$ to
$\mathcal{S}'$ in the usual way, where $\mathcal{S}'$ denotes the
set of all tempered distributions. Let $\phi \in \mathcal{S}$
satisfying
\begin{align*}
\operatorname{supp} \hat{\phi} \subset \big\{ \xi : \frac{5}{6} \leq
|\xi| \leq \frac{12}{5} \big\} \quad\text{and}\quad
\sum_{j \in \mathbb{Z}} \hat{\phi} (2^{-j} \xi)=1,
\end{align*}
for $\xi \neq 0$. Setting $\hat{\phi}_j=\hat{\phi}(2^{-j} \xi)$,
in other words, $\phi_j(x)=2^{jN} \phi(2^j x)$, for any
$f \in \mathcal{S}'$, we define
\begin{equation} \label{2.3}
\Delta_j f=\phi_j \ast f\quad\text{and}\quad
S_j f=\sum_{k \leq j-1} \phi_k \ast f.
\end{equation}
The homogeneous Besov semi-norm $\|f\|_{\dot{B}_{p,q}^s}$ and
Triebel-Lizorkin semi-norm $\|f\|_{\dot{F}_{p,q}^s}$ are defined
next.

\begin{definition}[\cite{RS,Tr}] \label{def2.1} \rm
 For $-\infty <s<\infty$,
$0 < p \leq \infty$, $0<q \leq \infty$, set
\begin{gather*}
 \|f\|_{\dot{B}_{p,q}^s}=  \begin{cases}
 \big( \sum_{j } 2^{j q s} \|\Delta_j f\|_{L^p}^q\big)^{1/q},
                 & \text{if } q \in (0,\infty),\\
 \sup_j 2^{j s} \|\Delta_j f\|_{L^p},    & \text{if }q=\infty.
                 \end{cases}
 \\
\|f\|_{\dot{F}_{p,q}^s}=  \begin{cases}
   \| \big(\sum_{j} 2^{j q s} |\Delta_j f|^q\big)^{1/q}\|_{L^p},
                 & \text{if } q \in (0,\infty),\\
   \|\sup_j (2^{j s} |\Delta_j f|)\|_{L^p},    & \text{if }q=\infty.
                 \end{cases}
\end{gather*}
\end{definition}

The spaces $\dot{B}_{p,q}^s$ and $\dot{F}_{p,q}^s$ are quasi-normed
spaces with the above quasi-norm given by Definition 2.1. For
$s>0$, $(p,q) \in (1,\infty) \times [1,\infty]$, we define the
inhomogeneous Besov space norm $\|f\|_{B_{p,q}^s}$ and
inhomogeneous Triebel-Lizorkin space norm $\|f\|_{F_{p,q}^s}$ of
$f \in \mathcal{S}'$ as
\begin{equation}
\|f\|_{B_{p,q}^s}=\|f\|_{L^p}+\|f\|_{\dot{B}_{p,q}^s},\quad
\|f\|_{F_{p,q}^s}= \|f\|_{L^p}+\|f\|_{\dot{F}_{p,q}^s}.
\end{equation}
The inhomogeneous Besov and Triebel-Lizorkin spaces are Banach
spaces equipped with the norm $\|f\|_{B_{p,q}^s}$ and
$\|f\|_{F_{p,q}^s}$ respectively.

Let us now state some classical results.

\begin{lemma}[Bernstein's Lemma \cite{RS,Tr}] \label{lem2.2}
Assume that $k \in \mathbb{Z}^+$, $f \in L^p$, $1 \leq p \leq \infty$, and
$\operatorname{supp}\hat{f} \subset \{ 2^{j-2} \leq |\xi|<2^{j}\}$, then there
exists a constant $C(k)$ such that the following inequality holds.
\begin{align*}
C(k)^{-1} 2^{j k} \|f\|_{L^p} \leq \|D^k f\|_{L^p} \leq C(k) 2^{j k}
\|f\|_{L^p}.
\end{align*}
For any $k \in \mathbb{Z}^+$, there exists a constant $C(k)$ such that
the following inequalities are true:
\begin{gather} \label{2.4}
C(k)^{-1} \|D^k f\|_{\dot{B}_{p,q}^s} \leq
\|f\|_{\dot{B}_{p,q}^{s+k}} \leq C(k) \|f\|_{\dot{B}_{p,q}^s},
\\
 \label{2.04}
C(k)^{-1} \|D^k f\|_{\dot{F}_{p,q}^s} \leq
\|f\|_{\dot{F}_{p,q}^{s+k}} \leq C(k) \|f\|_{\dot{F}_{p,q}^s}.
\end{gather}
\end{lemma}

\begin{lemma}[Embeddings \cite{RS,Tr}]
(I) Let $s \in \mathbb{R}$, $p \in (1,\infty)$, $\epsilon >0$ and
$q_1,q_2 \in [1,\infty]$, $q_1<q_2$, then
\[
\dot{B}_{p,1}^s \hookrightarrow \dot{F}_{p,2}^s \hookrightarrow
\dot{B}_{p,\infty}^s,\quad \dot{B}_{p,q_1}^{s+\epsilon}
\hookrightarrow \dot{B}_{p,q_2}^s.
\]
(II) Let $p \in (1,\infty)$, then
\[
\dot{B}_{p,1}^{N/p} \hookrightarrow
L^{\infty},\quad B_{p,1}^{N/p} \hookrightarrow L^{\infty}.
\]
\end{lemma}

\begin{proposition}[Product] \label{prop2.4}
  If $s \geq N/p$, and suppose $f,g \in \dot{B}_{p,1}^{N/p} \cap
\dot{B}_{p,1}^{s}$, then $f g \in \dot{B}_{p,1}^{s}$ and
\begin{equation} \label{2.6}
\|f g\|_{\dot{B}_{p,1}^{s}} \leq C \Big(\|f\|_{\dot{B}_{p,1}^{s}}
\|g\|_{\dot{B}_{p,1}^{N/p}}+\|g\|_{\dot{B}_{p,1}^{s}}
\|f\|_{\dot{B}_{p,1}^{N/p}}\Big).
\end{equation}
\end{proposition}

We will prove this proposition in the appendix.

\begin{lemma}[Commutator \cite{Da1}] \label{lemm2.5}
Suppose that $s \in (-N/p-1,N/p]$. Then for $f \in \dot{B}_{p,1}^{N/p+1}$
and $g \in \dot{B}_{p,1}^s$, we have
\begin{align*}
\|[f,\Delta_j] g\|_{L^p} \leq C_j 2^{-j (s+1)}
\|f\|_{\dot{B}_{p,1}^{N/p+1}} \|g\|_{\dot{B}_{p,1}^s},
\end{align*}
with $\sum_j C_j \leq 1$.
\end{lemma}

\begin{lemma}[Interpolation \cite{RS,Tr}]
Let $1 \leq p_1,q_1,p_2 \leq \infty$ and $1 \leq q_2 <\infty$. Then
\[
\|f\|_{\dot{B}_{p,q}^s} \leq
\|f\|_{\dot{B}_{p_1,q_1}^{s_1}}^{\theta}
\|f\|_{\dot{B}_{p_2,q_2}^{s_2}}^{1-\theta},
\]
holds for all $f \in \dot{B}_{p_1,q_1}^{s_1} \cap
\dot{B}_{p_2,q_2}^{s_2}$ provided that
\[
s=\theta s_1+(1-\theta)s_2,\quad
\frac{1}{p}=\frac{\theta}{p_1}+\frac{1-\theta}{p_2},\quad
\frac{1}{q}=\frac{\theta}{q_1}+\frac{1-\theta}{q_2}.
\]
\end{lemma}

\section{Proof of Theorem 1.1}

In this section, we establish the existence and uniqueness of the
solutions to \eqref{1.1} (Theorem 1.1). In the sequel, $C$ denotes
a absolute constant, which maybe different from line to line.

Consider the  linear system
\begin{equation} \label{3.1}
\begin{gathered}
\partial_t v+ (w \cdot \nabla) v+\nabla P=f,\\
\operatorname{div} v=0, \\
v(x,t=0)=v_0(x).
\end{gathered}
\end{equation}
We have easily the existence of a local solution for \eqref{3.1}.

\begin{proposition} \label{prop3.1}
Assume that $\operatorname{div}w=0$,
$w \in L^{\infty}(0,T;B_{p,1}^{N/p+1})$,
$f \in L^1(0,T;\\B_{p,1}^{N/p+1})$, for some $T>0$. Then for any
$v_0 \in B_{p,1}^{N/p+1}$, $\operatorname{div}v_0=0$, there exists a unique
solution $v \in C (0,T;B_{p,1}^{N/p+1})$ to \eqref{3.1}, and then
$\nabla P$ can be determined uniquely.
\end{proposition}

The above proposition will be showed in the appendix.
To prove the existence, we consider the following
approximate linear iteration system for \eqref{1.1},
\begin{equation} \label{3.2}
\begin{gathered}
\partial_t \rho^{n+1}+v^n \cdot \nabla \rho^{n+1}=0,\\
\partial_t v^{n+1}+v^n \cdot \nabla v^{n+1}+\nabla P^{n+1}=-\rho^n
\nabla P^n,\\
\operatorname{div} v^{n+1}=\operatorname{div} v^n=0, \\
(\rho^{n+1},v^{n+1})|_{t=0}=(\rho^{n+1}(0),v^{n+1}(0))=(S_{n+1}
\rho_0,S_{n+1} v_0),
\end{gathered}
\end{equation}
where $(\rho^0,v^0,P^0)=(0,0,0)$. If we have the uniform estimate
for the sequence $(\rho^n,v^n,\nabla P^n)$ by induction, which
satisfies the conditions in Proposition 3.1, then the second
equation of \eqref{3.2} can be solved with $v^{n+1}$ and $\nabla
P^{n+1}$. While $\rho^{n+1}$ can be obtained easily by solving the
linear transport equation. So we establish uniform estimates
first.

\noindent\textbf{Uniform estimates.}
For the first equation of \eqref{3.2}, thanks to the divergence
free of $v^n$, it follows that for any $1<p \leq \infty$,
\begin{equation} \label{3.3}
\|\rho^{n+1}(.,t)\|_{L^p} \leq
\|\rho^{n+1}(0)\|_{L^p},\quad \text{for }t\geq 0.
\end{equation}
Applying the operator $\Delta_j$ on the both sides of the linear
transport equation, we obtain
\begin{equation} \label{3.4}
\partial_t \Delta_j \rho^{n+1} + (v^n \cdot \nabla )\Delta_j
\rho^{n+1}=[v^n,\Delta_j] \nabla \rho^{n+1}.
\end{equation}
Multiply \eqref{3.4} by $|\Delta_j \rho^{n+1}|^{p-2} \Delta_j
\rho^{n+1}$, and integrate over $\mathbb{R}^N$, due to the divergence
free of $v^n$, then one has
\begin{align*}
\frac{1}{p} \frac{d}{dt} \|\Delta_j \rho^{n+1}\|_{L^p}^p
&\leq \|[v^n,\Delta_j] \nabla \rho^{n+1}\|_{L^p} \|\Delta_j
\rho^{n+1}\|_{L^p}^{p-1} \\
&\leq C C_j 2^{-js} \|v^n\|_{\dot{B}_{p,1}^{N/p+1}}
\|\rho^{n+1}\|_{\dot{B}_{p,1}^s} \|\Delta_j \rho^{n+1}
\|_{L^p}^{p-1},
\end{align*}
where we used the commutator estimate for $s \leq \frac{N}{p}+1$
and H\"older's inequality. Thus
\begin{equation} \label{3.5}
\frac{d}{dt} \|\Delta_j \rho^{n+1}\|_{L^p} \leq C C_j 2^{-js}
\|v^n\|_{\dot{B}_{p,1}^{N/p+1}} \|\rho^{n+1}\|_{\dot{B}_{p,1}^s}.
\end{equation}
Multiplying \eqref{3.5} by $2^{js}$ and taking summation over $j$,
we have
\begin{equation} \label{3.6}
\frac{d}{dt} \|\rho^{n+1} \|_{\dot{B}_{p,1}^s} \leq C
\|v^n\|_{\dot{B}_{p,1}^{N/p+1}} \|\rho^{n+1}\|_{\dot{B}_{p,1}^s}.
\end{equation}
By Gronwall inequality and \eqref{3.3}, we have the estimate for
$\rho^{n+1}$,
\begin{equation} \label{3.7}
\sup_{0 \leq t \leq T} \|\rho^{n+1} (.,t)\|_{B_{p,1}^s} \leq
\|\rho^{n+1} (0)\|_{B_{p,1}^s} \exp\Big( \int_0^T C
\|v^n(.,t)\|_{\dot{B}_{p,1}^{N/p+1}} dt \Big).
\end{equation}
Multiplying each coordinate in second equation of \eqref{3.2}  by
 $|v_l^{n+1}|^{p-2} v_l^{n+1}$, where $v_l^{n+1}$
is the $l$-th coordinate of the vector field $v^{n+1}$, thanks to
H\"older's inequality, we have
\begin{align*}
\frac{1}{p} \frac{d}{dt} \|v_l^{n+1}\|_{L^p}^p
&\leq  \|\rho^n
\nabla P^n\|_{L^p} \|v_l^{n+1}\|_{L^p}^{p-1}+\|\nabla
P^{n+1}\|_{L^p} \|v_l^{n+1}\|_{L^p}^{p-1} \\
&\leq  C \|\rho^n\|_{\dot{B}_{p,1}^{N/p}} \|\nabla P^n\|_{L^p}
\|v_l^{n+1}\|_{L^p}^{p-1}+\|\nabla P^{n+1}\|_{L^p}
\|v_l^{n+1}\|_{L^p}^{p-1}.
\end{align*}
So
\begin{equation} \label{3.9}
\begin{aligned}
\sup_{0 \leq t \leq T} \|v^{n+1}\|_{L^p}
&\leq \|v^{n+1}(0)\|_{L^p}+C \int_0^T \|\rho^n\|_{\dot{B}_{p,1}^{N/p}}
\|\nabla P^n(.,t)\|_{L^p} dt \\
&\quad +\int_0^T \|\nabla P^{n+1}(.,t)\|_{L^p} dt.
\end{aligned}
\end{equation}
Now taking $\Delta_j$ on the second equation of \eqref{3.2}, we
obtain
\begin{equation} \label{3.10}
\partial_t \Delta_j v^{n+1} +v^n \cdot \nabla \Delta_j
v^{n+1}+\nabla \Delta_j P^{n+1}=[v^n,\Delta_j] \nabla
v^{n+1}-\Delta_j (\rho^n \nabla P^n).
\end{equation}
Multiplying \eqref{3.10} coordinate by coordinate with $|\Delta_j
v_l^{n+1}|^{p-2} \Delta_j v_l^{n+1}$, and integrating over
$\mathbb{R}^N$, we have
\begin{equation} \label{3.12}
\begin{aligned}
&\frac{1}{p} \frac{d}{dt} \|\Delta_j v_l^{n+1}\|_{L^p}^p \\
&\leq  C \left\|[v^n,\Delta_j] \nabla v_l^{n+1}\right\|_{L^p} \|\Delta_j
v_l^{n+1}\|_{L^p}^{p-1}  \\
&\quad +\|\Delta_j (\rho^n \nabla P^n)\|_{L^p} \|\Delta_j
v_l^{n+1}\|_{L^p}^{p-1}+\|\Delta_j \nabla P^{n+1}\|_{L^p}
\|\Delta_j v_l^{n+1}\|_{L^p}^{p-1}   \\
&\leq  C C_j 2^{-j(N/p+1)} \|v^n\|_{\dot{B}_{p,1}^{N/p+1}}
\|v^{n+1} \|_{\dot{B}_{p,1}^{N/p+1}} \|\Delta_j
v_l^{n+1}\|_{L^p}^{p-1}  \\
&\quad + \|\Delta_j (\rho^n \nabla P^n)\|_{L^p} \|\Delta_j
v_l^{n+1}\|_{L^p}^{p-1}
+\|\Delta_j \nabla P^{n+1}\|_{L^p} \|\Delta_j
v_l^{n+1}\|_{L^p}^{p-1}.
\end{aligned}
\end{equation}
Then applying $2^{j(N/p+1)}$ on \eqref{3.12} and taking summation
yields
\begin{align*}
\frac{d}{dt} \|v^{n+1}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}
&\leq  C \|v^n\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}} \|v^{n+1}
\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}+C \|\rho^n \nabla P^n
\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}+\|\nabla P^{n+1}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}\\
&\leq  C \|v^n\|_{\dot{B}_{p,1}^{N/p+1}} \|v^{n+1}
\|_{\dot{B}_{p,1}^{N/p+1}}+C \|\rho^n\|_{\dot{B}_{p,1}^{N/p}}
\|\nabla P^n\|_{\dot{B}_{p,1}^{N/p+1}}\\
&\quad +C \|\rho^n\|_{\dot{B}_{p,1}^{N/p+1}} \|\nabla
P^n\|_{\dot{B}_{p,1}^{N/p}}+\|\nabla
P^{n+1}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}.
\end{align*}
By Gronwall's inequality and \eqref{3.9}, for all $0 \leq t \leq T$,
we have
\begin{equation} \label{3.13}
\begin{aligned}
\|v^{n+1}\|_{B_{p,1}^{N/p+1}}
&\leq \|v^{n+1}(0)\|_{B_{p,1}^{N/p+1}} \exp\Big(C \int_0^T
\|v^n(.,t)\|_{\dot{B}_{p,1}^{N/p+1}} dt\Big)  \\
&\quad +\int_0^T A_n(t) \exp\Big(C\int_t^T
\|v^n(.,\tau)\|_{\dot{B}_{p,1}^{N/p+1}} d \tau \Big) dt \\
&\quad + C \int_0^T \|\rho^n\|_{\dot{B}_{p,1}^{N/p}} \|\nabla
P^n(.,t)\|_{L^p} dt,
\end{aligned}
\end{equation}
where
\begin{equation} \label{3.14}
A_n(t)=C \|\rho^n\|_{\dot{B}_{p,1}^{\frac{N}{p}}} \|\nabla
P^n\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}+C
\|\rho^n\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}} \|\nabla
P^n\|_{\dot{B}_{p,1}^{\frac{N}{p}}}+\|\nabla
P^{n+1}\|_{B_{p,1}^{\frac{N}{p}+1}}.
\end{equation}
Now we give estimates for the pressure. Taking divergence on both
sides of the second equation of \eqref{3.2}, we have
\begin{align*}
-\Delta P^{n+1}=\operatorname{div}(v^n \cdot \nabla
v^{n+1})+\operatorname{div} (\rho^n \nabla P^n);
\end{align*}
thus
\begin{equation} \label{3.15}
\partial_i \partial_j P^{n+1}=R_i R_j\operatorname{div}(v^n \cdot \nabla
v^{n})+R_i R_j \operatorname{div} (\rho^n \nabla P^n).
\end{equation}
For $1<p<\infty$, in \cite{RS,Tr}, it was proved that
$\dot{F}_{p,2}^0=L^p$ and $R_i$ is bounded from $\dot{F}_{p,q}^s$
into itself~\cite{Fr}. Due to Bernstein's lemma, we have
\begin{equation} \label{3.16}
\begin{aligned}
\|\nabla P^{n+1}\|_{L^p}
&=  \|\nabla P^{n+1}\|_{\dot{F}_{p,2}^0}
\leq C \sum_{i,j=1}^N \|\partial_i
\partial_j P^{n+1}\|_{\dot{F}_{p,2}^{-1}}   \\
&\leq  C \|\operatorname{div}(v^n \cdot \nabla
v^{n+1})\|_{\dot{F}_{p,2}^{-1}}+C \|\operatorname{div} (\rho^n \nabla
P^n)\|_{\dot{F}_{p,2}^{-1}}   \\
&\leq  C \|v^n \cdot \nabla v^{n+1}\|_{L^p}+C\|\rho^n \nabla
P^n\|_{L^p}  \\
&\leq  C \|v^n\|_{\dot{B}_{p,1}^{N/p}}
\|v^{n+1}\|_{\dot{B}_{p,1}^1}+C
\|\rho^n\|_{\dot{B}_{p,1}^{N/p}}\|\nabla P^n\|_{L^p},
\end{aligned}
\end{equation}
where we used the embedding Lemma 2.3.  From
\eqref{3.15} it follows that
\begin{equation} \label{3.17}
\begin{aligned}
\|\nabla P^{n+1}\|_{\dot{B}_{p,1}^{N/p+1}}
&\leq  C \sum_{i,j=1}^N
\|\partial_i \partial_j P^{n+1}\|_{\dot{B}_{p,1}^{N/p}} \\
&\leq  C \sum_{i,j,k,l=1}^N \|R_i R_j \partial_k v_l^n \partial_l
v_k^{n+1}\|_{\dot{B}_{p,1}^{N/p}}+C \|\rho^n \nabla
P^n\|_{\dot{B}_{p,1}^{N/p+1}}  \\
&\leq  C \|v^n\|_{\dot{B}_{p,1}^{N/p+1}}
\|v^{n+1}\|_{\dot{B}_{p,1}^{N/p+1}}+C \|\rho^n
\|_{\dot{B}_{p,1}^{N/p}} \|\nabla P^n\|_{\dot{B}_{p,1}^{N/p+1}}
  \\
&\quad +C \|\rho^n \|_{\dot{B}_{p,1}^{N/p+1}}
\|\nabla P^n\|_{\dot{B}_{p,1}^{N/p}}.
\end{aligned}
\end{equation}
Combining \eqref{3.16} and \eqref{3.17}, one has
\begin{equation} \label{3.18}
\begin{aligned}
&\int_0^T \|\nabla P^{n+1}(.,t)\|_{B_{P,1}^{N/p+1}} \\
&\leq C\|\rho^n\|_{L^{\infty}(0,T;B_{p,1}^{N/p+1})}
\int_0^T \|\nabla P^n(.,t)\|_{B_{p,1}^{N/p+1}} dt \\
&\quad +C T \|v^n\|_{L^{\infty}(0,T;B_{p,1}^{\frac{N}{p}+1})}
\|v^{n+1}\|_{L^{\infty}(0,T;B_{p,1}^{\frac{N}{p}+1})}.
\end{aligned}
\end{equation}
Note that although the above constants $C$ maybe depend on $N,m,M$
and $p$, it is nothing to do with $n$, therefore we can obtain
uniform estimates by induction.

In fact, suppose that initial data $\rho_0$ and $v_0$ satisfies
\[
\|\rho_0\|_{B_{p,1}^{N/p+1}} \leq \frac{C_1}{2},\quad
\|v_0\|_{B_{p,1}^{N/p+1}} \leq \frac{C_2}{2},
\]
for some $C_1,C_2>0$ and $C_1$ is sufficiently small. Then the
following inequalities hold
\begin{equation} \label{3.19}
\begin{gathered}
\|\rho^{n+1}\|_{L^{\infty}(0,T_*;B_{p,1}^{N/p+1})} \leq C_1,\\
\|v^{n+1}\|_{L^{\infty}(0,T_*;B_{p,1}^{N/p+1})} \leq C_2,\\
\|\nabla P^{n+1}\|_{L^1(0,T_*;B_{p,1}^{N/p+1})} \leq C_3,
\end{gathered}
\end{equation}
for all $n \geq 0$ and some $C_3< C_2/(8C C_1)$, provided that
$T_*$ (independent on $n$) is sufficiently small.

We show \eqref{3.19} by mathematical induction.
Note that \eqref{3.19} holds obviously for $n=0$.
Suppose \eqref{3.19} is true for $n$, we want
to prove \eqref{3.19} holds for $n+1$. From \eqref{3.7},
\eqref{3.13}, \eqref{3.14} and \eqref{3.18}, we have
\begin{gather*}
\|\rho^{n+1}\|_{L^{\infty}(0,T;B_{p,1}^{N/p+1})} \leq
\frac{C_1}{2} \exp(TC_2),\\
\begin{aligned}
&\|v^{n+1}\|_{L^{\infty}(0,T;B_{p,1}^{N/p+1})}\\
&\leq \frac{C_2}{2} \exp(CTC_2)+C C_1 C_3
+C(C_1C_3+C_2 \|v^{n+1}\|_{L^{\infty}(0,T;B_{p,1}^{N/p+1})}
T)\exp(CTC_2),
\end{aligned}\\
\|\nabla P^{n+1}\|_{L^1(0,T;B_{p,1}^{N/p+1})}
\leq C C_1 C_3+C T C_2 \|v^{n+1}\|_{L^{\infty}(0,T;B_{p,1}^{N/p+1})},
\end{gather*}
So one can choose $T_*$ sufficient small, such that
\begin{align*}
C C_2 T_* \exp(CT_*C_2) \leq \frac{1}{4}.
\end{align*}
Moreover, $T_*$ satisfies
\begin{gather*}
\exp(T_* C_2) \leq 2, \\
2 C_2 \exp(C T_* C_2)+4 C C_1C_3 \exp(CT_*C_2) \leq 5 C_2, \\
C T_* C_2^2 \leq \frac{C_3}{2},
\end{gather*}
provided that $C_1 < C/2$. Then by induction, \eqref{3.19} holds for
$n+1$-th step. Hence we get
the uniform estimate for each $n$.


\noindent\textbf{Convergence.}

To prove the convergence, it is sufficient to estimate the
difference of the iteration. Take the difference between the
equation \eqref{3.2} for the $(n+1)$-th step and the $n$-th step,
and set
\[
w^{n+1}=\rho^{n+1}-\rho^n,\quad
u^{n+1}=v^{n+1}-v^n,,\quad
\Pi^{n+1}=P^{n+1}-P^n,
\]
then we obtain the equation
\begin{equation} \label{3.20}
\begin{gathered}
\partial_t w^{n+1}+v^n \cdot \nabla w^{n+1}+u^n \cdot \nabla \rho^n=0,\\
\partial_t u^{n+1}+v^n \cdot \nabla u^{n+1}+u^n \cdot \nabla v^n+\nabla \Pi^{n+1}
=-w^n \nabla P^n-\rho^{n-1} \nabla \Pi^n,\\
\operatorname{div} u^{n+1}=\operatorname{div} v^n=0, \\
(w^{n+1},u^{n+1})|_{t=0}=(w^{n+1}(0),u^{n+1}(0))=(\Delta_n
\rho_0,\Delta_n v_0),
\end{gathered}
\end{equation}
First, we do the estimate for $w^{n+1}$. Multiplying
$|w^{n+1}|^{p-2} w^{n+1}$ on both sides of the first equation of
\eqref{3.20} and integrating over $\mathbb{R}^N$, we have
\begin{equation} \label{3.21}
\begin{aligned}
\frac{d}{dt} \|w^{n+1}\|_{L^p}
&\leq  \|u^n \cdot \nabla \rho^n \|_{L^p}
 \leq \|u^n\|_{L^{\infty}} \|\nabla \rho^n\|_{L^p}  \\
&\leq  C \|u^n\|_{\dot{B}_{p,1}^{N/p}} \|\nabla
\rho^n\|_{\dot{F}_{p,2}^0} \\
&\leq \|u^n\|_{\dot{B}_{p,1}^{N/p}} \| \rho^n\|_{\dot{B}_{p,1}^1}.
\end{aligned}
\end{equation}
Applying $\Delta_j$ on both sides of the first equation of
\eqref{3.20}, we have
\begin{equation} \label{3.22}
\partial_t \Delta_j w^{n+1}+v^n \cdot \Delta_j w^{n+1}+\Delta_j
(u^n \cdot \nabla \rho^n)=[v^n,\Delta_j] \nabla w^{n+1}.
\end{equation}
Multiplying \eqref{3.22} by
$|\Delta_j w^{n+1}|^{p-2} \Delta_j w^{n+1}$ and integrating over
$\mathbb{R}^N$, we have
\begin{align*}
\frac{1}{p} \frac{d}{dt} \|\Delta_j w^{n+1} \|_{L^p}^p
&\leq
\|\Delta_j (u^n \cdot \nabla \rho^n)\|_{L^p} \|\Delta_j w^{n+1}
\|_{L^p}^{p-1}\\
&\quad +C_j 2^{-j N/p} \|v^n\|_{\dot{B}_{p,1}^{N/p+1}} \|\nabla
w^{n+1}\|_{\dot{B}_{p,1}^{N/p-1}} \|\Delta_j w^{n+1}\|_{L^p}^{p-1}.
\end{align*}
Then applying $2^{j N/p}$ and taking summation, we obtain
\begin{equation} \label{3.23}
\frac{d}{dt} \|w^{n+1}\|_{\dot{B}_{p,1}^{N/p}}
 \leq C \|u^n\|_{\dot{B}_{p,1}^{N/p}} \|\rho^n\|_{\dot{B}_{p,1}^{N/p+1}}+C
\|v^n\|_{\dot{B}_{p,1}^{N/p+1}} \|w^{n+1}\|_{\dot{B}_{p,1}^{N/p}}.
\end{equation}
Combining \eqref{3.21} and \eqref{3.23}, it follows that
\begin{equation} \label{3.023}
\frac{d}{dt}\|w^{n+1}\|_{B_{p,1}^{N/p}} \leq C
\|\rho^n\|_{B_{p,1}^{N/p+1}} \|u^n\|_{B_{p,1}^{N/p}}+C
\|v^n\|_{B_{p,1}^{N/p+1}} \|w^{n+1}\|_{B_{p,1}^{N/p}}.
\end{equation}
Just as what done for $v^{n+1}$, multiplying the second equation
of \eqref{3.20} coordinate by coordinate with $|u_l^{n+1}|^{p-2}
u_l^{n+1}$, where $u_l^{n+1}$ is the $l$-th coordinate of the
vector field $u^{n+1}$. Thanks to H\"older's inequality, we have
\begin{equation} \label{3.25}
\begin{aligned}
\frac{d}{dt} \|u^{n+1}\|_{L^p} &\leq   \|u^n \cdot \nabla
v^n\|_{L^p}+\|w^n \nabla P^n\|_{L^p} \\
&\quad +\|\rho^{n-1} \nabla
\Pi^n\|_{L^p}+\|\nabla \Pi^{n+1}\|_{L^p}  \\
&\leq  C \|u^n\|_{\dot{B}_{p,1}^{N/p}}
\|v^n\|_{\dot{B}_{p,1}^1}+C\|w^n\|_{\dot{B}_{p,1}^{N/p}} \|\nabla
P^n\|_{L^p} \\
&\quad +C\|\rho^{n-1}\|_{\dot{B}_{p,1}^{N/p}} \|\nabla
\Pi^n\|_{L^p}+\|\nabla \Pi^{n+1}\|_{L^p}.
\end{aligned}
\end{equation}
Applying $\Delta_j$ on the second equation of \eqref{3.20}, we obtain
\begin{equation} \label{3.26}
\begin{aligned}
&\partial_t \Delta_j u^{n+1} +v^n \cdot \nabla \Delta_j
u^{n+1}+\nabla \Delta_j \Pi^{n+1} \\
&= [v^n,\Delta_j] \nabla u^{n+1} -\Delta_j(u^n \cdot \nabla
v^n)-\Delta_j (w^n \nabla P^n+\rho^{n-1} \nabla \Pi^n).
\end{aligned}
\end{equation}
Multiplying each coordinate  with $|\Delta_j
u_l^{n+1}|^{p-2} \Delta_j u_l^{n+1}$, and integrating over
$\mathbb{R}^N$, we have
\begin{equation} \label{3.27}
\begin{aligned}
&\frac{d}{dt} \|\Delta_j u^{n+1}\|_{L^p} \\
&\leq  C C_j 2^{-jN/p}
\|v^n\|_{\dot{B}_{p,1}^{N/p+1}} \|u^{n+1} \|_{\dot{B}_{p,1}^{N/p}}
+\|\Delta_j (u^n \cdot \nabla
v^n)\|_{L^p}  \\
&\quad+ \|\Delta_j (w^n \nabla P^n)\|_{L^p}+\|\Delta_j (\rho^{n-1}
\nabla \Pi^n)\|_{L^p} +\|\Delta_j\nabla \Pi^{n+1}\|_{L^p}.
\end{aligned}
\end{equation}
Then applying $2^{jN/p}$ on \eqref{3.27} and taking summation yields
\begin{equation} \label{3.28}
\begin{aligned}
&\frac{d}{dt} \|u^{n+1}\|_{\dot{B}_{p,1}^{N/p}} \\
&\leq  C \|v^n\|_{\dot{B}_{p,1}^{N/p+1}} \|u^{n+1}
\|_{\dot{B}_{p,1}^{N/p}}+C \|u^n \cdot \nabla v^n\|_{\dot{B}_{p,1}^{N/p}} \\
&\quad + \|w^n \nabla P^n
\|_{\dot{B}_{p,1}^{N/p}}+\|\rho^{n-1} \nabla \Pi^n\|_{\dot{B}_{p,1}^{N/p}}
+\|\nabla \Pi^{n+1}\|_{\dot{B}_{p,1}^{N/p}} \\
&\leq  C \|v^n\|_{\dot{B}_{p,1}^{N/p+1}} \|u^{n+1}
\|_{\dot{B}_{p,1}^{N/p}}+ C\|u^n\|_{\dot{B}_{p,1}^{N/p}}
\|v^n\|_{\dot{B}_{p,1}^{N/p+1}} \\
&\quad + C \|w^n\|_{\dot{B}_{p,1}^{N/p}} \|\nabla
 P^n\|_{\dot{B}_{p,1}^{N/p}}+C \|\rho^{n-1}\|_{\dot{B}_{p,1}^{N/p}}
 \|\nabla \Pi^n\|_{\dot{B}_{p,1}^{N/p}}
+\|\nabla \Pi^{n+1}\|_{\dot{B}_{p,1}^{N/p}}.
\end{aligned}
\end{equation}
Combining \eqref{3.25} and \eqref{3.28}, we have
\begin{equation} \label{3.028}
\begin{aligned}
&\frac{d}{dt} \|u^{n+1}\|_{B_{p,1}^{N/p}} \\
&\leq  C \|v^n\|_{B_{p,1}^{N/p+1}} \|u^{n+1} \|_{B_{p,1}^{N/p}}+C
\|v^n\|_{B_{p,1}^{N/p+1}} \|u^n\|_{B_{p,1}^{N/p}}  \\
&\quad +C \|\nabla P^n\|_{B_{p,1}^{N/p}} \|w^n\|_{B_{p,1}^{N/p}}+C
\|\rho^{n-1}\|_{B_{p,1}^{N/p}} \|\nabla
\Pi^n\|_{B_{p,1}^{N/p}}
+\|\nabla \Pi^{n+1}\|_{B_{p,1}^{N/p}}.
\end{aligned}
\end{equation}
Now we give estimates for $\nabla \Pi^{n+1}$. Applying the operator
$\operatorname{div}$ on both
sides of the second equation of \eqref{3.20}, we have
\[
-\Delta \Pi^{n+1}=\operatorname{div}(v^{n} \cdot \nabla
u^{n+1})+\operatorname{div} (u^n \cdot \nabla v^n)+\operatorname{div} (w^n
\nabla P^n+\rho^{n-1} \nabla \Pi^n);
\]
thus
\begin{align*}
\partial_i \partial_j \Pi^{n+1}
&= R_i R_j\operatorname{div}(v^{n} \cdot \nabla
u^{n+1})+R_i R_j \operatorname{div}(u^n \cdot \nabla v^n)\\
&\quad +R_i R_j \operatorname{div} (w^n \nabla P^n)+R_i R_j \operatorname{div}
(\rho^{n-1} \nabla \Pi^n).
\end{align*}
Thanks to the divergence free of $v^n$, we have
\[
\operatorname{div}(v^n \cdot \nabla u^{n+1})=\sum_{k,l=1}^N
\partial_{k}(v_l^n \partial_l u_k^{n+1})=\sum_{k,l=1}^N
\partial_{k} \partial_l (v_l^n u_k^{n+1})=\sum_{k,l=1}^N
\partial_{l}( \partial_k v_l^n u_k^{n+1}).
\]
Due to Bernstein's lemma, we have
\begin{equation} \label{3.29}
\begin{aligned}
\|\nabla \Pi^{n+1}\|_{L^p}
&=  \|\nabla \Pi^{n+1}\|_{\dot{F}_{p,2}^0} \leq C \sum_{i,j=1}^N \|\partial_i
\partial_j \Pi^{n+1}\|_{\dot{F}_{p,2}^{-1}}   \\
&\leq  C \|\operatorname{div}(v^{n} \cdot \nabla
u^{n+1})\|_{\dot{F}_{p,2}^{-1}}+C \|\operatorname{div}(u^n \cdot \nabla
v^n)\|_{\dot{F}_{p,2}^{-1}}  \\
&\quad +C \|\operatorname{div} (w^n \nabla P^n)\|_{\dot{F}_{p,2}^{-1}}+C
\|\operatorname{div} (\rho^{n-1} \nabla
\Pi^n)\|_{\dot{F}_{p,2}^{-1}}   \\
&\leq  C \sum_{k,l=1}^N \|\partial_k v_l^{n} u_k^{n+1}\|_{L^p}+C
\|u^n \cdot \nabla
v^n\|_{L^p}  \\
&\quad +C \|w^n \nabla P^n\|_{L^p}+C \|\rho^{n-1} \nabla
\Pi^n\|_{L^p}   \\
&\leq  C \|v^{n}\|_{\dot{B}_{p,1}^{N/p+1}} \|u^{n+1}\|_{L^p}+C
\|u^n\|_{L^p}
\|v^{n}\|_{\dot{B}_{p,1}^{N/p+1}}  \\
&\quad+C \|w^n\|_{\dot{B}_{p,1}^{N/p}}\|\nabla P^n\|_{L^p}+C
\|\rho^{n-1}\|_{\dot{B}_{p,1}^{N/p}}\|\nabla \Pi^n\|_{L^p},
\end{aligned}
\end{equation}
where we used the embedding in Lemma 2.3 and the product estimate.
Similarly,
\begin{equation} \label{3.30}
\begin{aligned}
\|\nabla \Pi^{n+1}\|_{\dot{B}_{p,1}^{N/p}}
&\leq  C \sum_{i,j=1}^N
\|\partial_i \partial_j \Pi^{n+1}\|_{\dot{B}_{p,1}^{N/p-1}} \\
&\leq  C \sum_{i,j,k,l=1}^N \|R_i R_j \partial_l (\partial_k
v_l^{n} u_k^{n+1})\|_{\dot{B}_{p,1}^{N/p-1}}+C \|w^n \nabla
P^n\|_{\dot{B}_{p,1}^{N/p}}  \\
&\quad +C \|u^n \cdot \nabla v^n\|_{\dot{B}_{p,1}^{N/p}}+C \|\rho^{n-1}
\nabla
\Pi^n\|_{\dot{B}_{p,1}^{N/p}}  \\
&\leq  C \sum_{k,l=1}^N \|\partial_k v_l^{n}
u_k^{n+1}\|_{\dot{B}_{p,1}^{N/p}}+C \|w^n \nabla
P^n\|_{\dot{B}_{p,1}^{N/p}}  \\
&\quad +C \|u^n \cdot \nabla v^n\|_{\dot{B}_{p,1}^{N/p}}+C \|\rho^{n-1}
\nabla
\Pi^n\|_{\dot{B}_{p,1}^{N/p}}  \\
&\leq  C \|v^{n}\|_{\dot{B}_{p,1}^{N/p+1}}
\|u^{n+1}\|_{\dot{B}_{p,1}^{N/p}}+C \|w^n \|_{\dot{B}_{p,1}^{N/p}}
\|\nabla P^n\|_{\dot{B}_{p,1}^{N/p}}  \\
&\quad +C \|v^n\|_{\dot{B}_{p,1}^{N/p+1}}
\|u^{n}\|_{\dot{B}_{p,1}^{N/p}}+C \|\rho^{n-1}
\|_{\dot{B}_{p,1}^{N/p}} \|\nabla \Pi^n\|_{\dot{B}_{p,1}^{N/p}}.
\end{aligned}
\end{equation}
Combining \eqref{3.29} and \eqref{3.30}, it follows that
\begin{equation} \label{3.31}
\begin{aligned}
\|\nabla \Pi^{n+1}\|_{B_{p,1}^{N/p}}
 &\leq  C \|v^{n}\|_{B_{p,1}^{N/p+1}} \|u^{n+1}\|_{B_{p,1}^{N/p}}+C \|\nabla
P^n\|_{B_{p,1}^{N/p}} \|w^n \|_{B_{p,1}^{N/p}}  \\
&\quad +C\|v^n\|_{B_{p,1}^{N/p+1}} \|u^{n}\|_{B_{p,1}^{N/p}}+C
\|\rho^{n-1} \|_{B_{p,1}^{N/p}} \|\nabla \Pi^n\|_{B_{p,1}^{N/p}}.
\end{aligned}
\end{equation}
Therefore, if we add \eqref{3.023}, \eqref{3.028} and \eqref{3.31}
together, then we obtain,
\begin{equation} \label{3.32}
\begin{aligned}
&\frac{d}{dt}
\|w^{n+1}\|_{B_{p,1}^{\frac{N}{p}}}+\frac{d}{dt}
\|u^{n+1}\|_{B_{p,1}^{\frac{N}{p}}}+\|\nabla
\Pi^{n+1}\|_{B_{p,1}^{\frac{N}{p}}}  \\
&\leq C_4\Big(\|w^{n+1}\|_{B_{p,1}^{\frac{N}{p}}}+\|u^{n+1}\|_{B_{p,1}^{\frac{N}{p}}}
+\|u^n\|_{B_{p,1}^{\frac{N}{p}}}\Big) \\
&\quad +C \|\nabla P^n\|_{B_{p,1}^{N/p}} \|w^n \|_{B_{p,1}^{N/p}}+C
\|\rho^{n-1} \|_{B_{p,1}^{N/p}} \|\nabla \Pi^n\|_{B_{p,1}^{N/p}},
\end{aligned}
\end{equation}
where $C_4$ is a constant depending on the uniform bounds of
$\|\rho^{n}\|_{L^{\infty}(0,T_*;B_{p,1}^{N/p+1})}$,
$\|v^{n-1}\|_{L^{\infty} (0,T_*;B_{p,1}^{N/p+1})}$ and
$\|v^{n}\|_{L^{\infty} (0,T_*;B_{p,1}^{N/p+1})}$. Then integrate
\eqref{3.32} on the time interval $(0,T_1) \subset [0,T_*]$, $T_1$
sufficiently small, such that
\begin{align*}
C_4 T_1 \leq \frac{1}{4},\quad
C \|\nabla P^n\|_{L^1 (0,T_1;B_{p,1}^{N/p})} \leq \frac{1}{4}.
\end{align*}
Then \eqref{3.32} yields
\begin{equation} \label{3.33}
\begin{aligned}
&\|w^{n+1}\|_{L^{\infty}(0,T_1;B_{p,1}^{N/p})}+
\|u^{n+1}\|_{L^{\infty}(0,T_1;B_{p,1}^{N/p})}+\|\nabla
\Pi^{n+1}\|_{L^1 (0,T_1;B_{p,1}^{N/p})}  \\
&\leq  \frac{4}{3}
\Big(\|w^{n+1}(0)\|_{B_{p,1}^{N/p}}+\|u^{n+1}(0)\|_{B_{p,1}^{N/p}}
\Big)+ \frac{1}{3} \|w^{n}\|_{L^{\infty}(0,T_1;B_{p,1}^{N/p})}
 \\
&\quad +\frac{1}{3} \|u^{n}\|_{L^{\infty}(0,T_1;B_{p,1}^{N/p})}+
\frac{4}{3} C \|\rho^{n-1}\|_{L^{\infty}(0,T_1;B_{p,1}^{N/p})}
\|\nabla \Pi^n\|_{L^1 (0,T_1;B_{p,1}^{N/p})}.
\end{aligned}
\end{equation}
Due to the smallness of $C_1$, say $C C_1 \leq 1/4$,
from \eqref{3.33} it follows that
\begin{align*}
\|w^{n+1}\|_{L^{\infty}(0,T_1;B_{p,1}^{N/p})}+
\|u^{n+1}\|_{L^{\infty}(0,T_*;B_{p,1}^{N/p})}+\|\nabla
\Pi^{n+1}\|_{L^1(0,T_1;B_{p,1}^{N/p})} \to 0,
\end{align*}
as $n$ tends to infinity.

Therefore, from the uniform estimates, we find that there exists a
limit $(\rho,v,\nabla P)$ belonging to $C(0,T;B_{p,1}^{N/p+1})
\times C(0,T;B_{p,1}^{N/p+1}) \times L^1 (0,T;B_{p,1}^{N/p})$,
which is the solution to \eqref{1.1}, for sufficient small $T$.

This complete the proof of local existence theorem.
Next we turn our attention to the uniqueness of solutions.

\noindent\textbf{Uniqueness.}
 Suppose $(\rho_1,v_1,\nabla P_1)$ and
$(\rho_2,v_2,\nabla P_2)$ are two solutions to \eqref{1.1} with
the same initial data. If we set $\rho=\rho_1-\rho_2$, $v=v_1-v_2$
and $P=P_1-P_2$, then we get a similar system as \eqref{3.20} as
\begin{equation} \label{3.34}
\begin{gathered}
\partial_t \rho+v_1 \cdot \nabla \rho+v \cdot \nabla \rho_2=0,\\
\partial_t v+v_1 \cdot \nabla v+v \cdot \nabla v_2+\nabla P
=-\rho_1 \nabla P-\rho \nabla P_2,\\
\operatorname{div} v_1=\operatorname{div} v_2=0, \\
(\rho,v)|_{t=0}=(0,0).
\end{gathered}
\end{equation}
Just as in the convergence part for the sequences, we
can treat \eqref{3.34} as \eqref{3.20}, and obtain
\begin{align*}
&\|\rho\|_{L^{\infty}(0,T_1;B_{p,1}^{N/p})}+
\|v\|_{L^{\infty}(0,T_1;B_{p,1}^{N/p})}+\|\nabla
P\|_{L^1 (0,T_1;B_{p,1}^{N/p})}  \\
&\leq  \frac{1}{4}
\Big(\|\rho\|_{L^{\infty}(0,T_1;B_{p,1}^{N/p})}+
\|v\|_{L^{\infty}(0,T_1;B_{p,1}^{N/p})}+\|\nabla P\|_{L^1
(0,T_1;B_{p,1}^{N/p})}\Big),
\end{align*}
provided that $T$ is sufficiently small and
$\|\rho_0\|_{B_{p,1}^{N/p+1}}$ is sufficiently small. This implies
the uniqueness.

\section{Proof of Theorem 1.3}

Now we use the following iteration system
\begin{equation} \label{4.1}
\begin{gathered}
\partial_t \rho^{n+1}+v^n \cdot \nabla \rho^{n+1}=0,\\
\partial_t v^{n+1}+v^n \cdot \nabla v^{n+1}+(1+\rho^n)\nabla P^{n+1}=0,\\
\operatorname{div} v^{n+1}=\operatorname{div} v^n=0, \\
(\rho^{n+1},v^{n+1})|_{t=0}=(\rho^{n+1}(0),v^{n+1}(0))=(S_{n+1}
\rho_0,S_{n+1} v_0),
\end{gathered}
\end{equation}
with the corresponding linear system
\begin{equation} \label{4.01}
\begin{gathered}
\partial_t v+w \cdot \nabla v+(1+\rho) \nabla P=0,\\
\operatorname{div} v=0, \\
v(x,t=0)=v_0(x).
\end{gathered}
\end{equation}
First, we have the following existence and uniqueness result, which
will be proved in the appendix.

\begin{proposition} \label{prop4.1}
Assume that $\operatorname{div}w=0$,
$w \in L^{\infty}(0,T;B_{2,1}^{N/2+1})$,
$\rho \in L^{\infty}(0,T;\\B_{2,1}^{N/2+1})$, for some $T>0$. Then for any
$v_0 \in B_{2,1}^{N/2+1}$, $\operatorname{div}v_0=0$, there exists a
unique solution $v \in C (0,T;B_{2,1}^{N/2+1})$ to \eqref{4.01}.
Consequently, $\nabla P$ can be uniquely determined.
\end{proposition}

Now, we go to the proof for Theorem 1.3.

\noindent\textbf{Uniform estimates.}

As in \eqref{3.7}, for $\rho^{n+1}$, we have the estimate
\begin{equation} \label{4.2}
\sup_{0 \leq t \leq T} \|\rho^{n+1} (.,t)\|_{B_{p,1}^s} \leq
\|\rho^{n+1} (0)\|_{B_{p,1}^s} \exp\Big( \int_0^T C
\|v^n(.,t)\|_{\dot{B}_{p,1}^{N/p+1}} dt \Big).
\end{equation}
Multiplying the second equation of \eqref{4.1} by $v^{n+1}$ and
integrating over $\mathbb{R}^N$, we obtain
\begin{equation} \label{4.3}
\frac{d}{dt} \|v^{n+1}(.,t)\|_{L^2} \leq \|1+\rho^n\|_{L^{\infty}}
\|\nabla P^{n+1}\|_{L^2} \leq C(1+\|\rho^n\|_{\dot{B}_{2,1}^{N/2}})
\|\nabla P^{n+1}\|_{L^2}.
\end{equation}
Applying $\Delta_j$ on the second equation of \eqref{3.2}, we obtain
\begin{equation} \label{4.4}
\partial_t \Delta_j v^{n+1} +v^n \cdot \nabla \Delta_j
v^{n+1}=[v^n,\Delta_j] \nabla v^{n+1}-\Delta_j \big((1+\rho^n)
\nabla P^n\big).
\end{equation}
Multiplying \eqref{4.4} by $\Delta_j v^{n+1}$ and taking the
divergence free property of $v^n$ into account, we have
\begin{equation}\label{4.5}
\begin{aligned}
\frac{d}{dt}\|\Delta_j v^{n+1}\|_{L^2}
&\leq  C C_j 2^{-j (N/2+1)}\|v^n\|_{\dot{B}_{2,1}^{N/2+1}}
\|v^{n+1}\|_{\dot{B}_{2,1}^{N/2+1}} \\
&\quad +\|\Delta_j\big((1+\rho^n) \nabla
P^{n+1}\big)\|_{L^2}.
\end{aligned}
\end{equation}
Applying $2^{j (N/2+1)}$ on \eqref{4.5} and taking summation, and using
the product estimate, we have
\begin{equation} \label{4.6}
\begin{aligned}
\frac{d}{dt} \|v^{n+1}\|_{\dot{B}_{2,1}^{N/2+1}}
&\leq  C \|v^n\|_{\dot{B}_{2,1}^{N/2+1}} \|v^{n+1}
\|_{\dot{B}_{2,1}^{N/2+1}}+C \|(1+\rho^n) \nabla P^{n+1}
\|_{\dot{B}_{2,1}^{N/2+1}} \\
&\leq  C \|v^n\|_{\dot{B}_{2,1}^{N/2+1}} \|v^{n+1}
\|_{\dot{B}_{2,1}^{N/2+1}}+C(1+ \|\rho^n\|_{\dot{B}_{2,1}^{N/2}})
\|\nabla P^{n+1}\|_{\dot{B}_{2,1}^{N/2+1}} \\
&\quad +C (1+\|\rho^n\|_{\dot{B}_{2,1}^{N/2+1}}) \|\nabla
P^{n+1}\|_{\dot{B}_{2,1}^{N/2}}.
\end{aligned}
\end{equation}
So the remaining thing is to give an estimate for the pressure. For
this purpose, we apply the operator div on both sides of the second equation of
the system \eqref{4.1}, and get
\begin{equation} \label{4.7}
\operatorname{div} \left((1+\rho^n) \nabla P^{n+1}\right)=-\operatorname{div}
(v^n \cdot \nabla v^{n+1}).
\end{equation}
Since $1+\rho=\frac{1}{\varrho}$ is bounded away from 0, we can
assume that $1+\rho^n$ bounded away from 0, without loss of
generality (otherwise, we take $\rho^n(0)=S_{n+m} \rho_0$, such
that $1+\rho^n$ bounded away from 0 for sufficiently large integer
$m$).

Multiplying \eqref{4.7} by $P^{n+1}$ and integrating by parts, one
has
\begin{equation} \label{4.8}
\|\nabla P^{n+1}\|_{L^2} \leq C \|v^n \cdot \nabla v^{n+1}\|_{L^2}
\leq C \|v^n\|_{L^2} \|v^{n+1}\|_{\dot{B}_{2,1}^{N/2+1}}.
\end{equation}
Applying $\Delta_j$ on \eqref{4.7}, we have
\begin{equation} \label{4.9}
-\operatorname{div}\left((1+\rho^n) \nabla \Delta_j P^{n+1}\right)
= \operatorname{div}\left([\Delta_j,(1+\rho^n)] \nabla
P^{n+1}\right)+\Delta_j \operatorname{div} \left(v^n \cdot \nabla
v^{n+1}\right).
\end{equation}
Multiplying \eqref{4.9} by $\Delta_j P^{n+1}$ and integrating over
$\mathbb{R}^N$, due to the bounds of $1+\rho^n$, we obtain that
\begin{equation} \label{4.10}
\begin{aligned}
\|\Delta_j \nabla P^{n+1}\|_{L^2}^2
&\leq  C \|[\Delta_j,(1+\rho^n)]
\nabla P^{n+1}\|_{L^2} \|\Delta_j \nabla P^{n+1}\|_{L^2} \\
&\quad +C 2^{-j} \|\Delta_j (v^n \cdot \nabla v^{n+1})\|_{L^2}
\|\Delta_j \nabla P^{n+1}\|_{L^2},
\end{aligned}
\end{equation}
Multiplying by $2^{j N/2}$ on \eqref{4.10} and taking summation, we
have
\begin{equation} \label{4.11}
\begin{aligned}
&\|\nabla P^{n+1}\|_{\dot{B}_{2,1}^{N/2}} \\
&\leq C(1+\|\rho^n\|_{\dot{B}_{2,1}^{N/2+1}}) \|\nabla
P^{n+1}\|_{\dot{B}_{2,1}^{N/2-1}}+C\|v^n\|_{\dot{B}_{2,1}^{N/2}}
\|v^{n+1}\|_{\dot{B}_{2,1}^{N/2+1}} \\
&\leq  C (1+\|\rho^n\|_{\dot{B}_{2,1}^{N/2+1}}) \|\nabla
P^{n+1}\|_{\dot{B}_{2,1}^{N/2}}^{(N-1)/(N+1)} \|\nabla
P^{n+1}\|_{\dot{B}_{2,1}^{-1/2}}^{2/(N+1)}  \\
&\quad +C\|v^n\|_{\dot{B}_{2,1}^{N/2}}
\|v^{n+1}\|_{\dot{B}_{2,1}^{N/2+1}} \\
&\leq   C (1+\|\rho^n\|_{\dot{B}_{2,1}^{N/2+1}}) \|\nabla
P^{n+1}\|_{\dot{B}_{2,1}^{N/2}}^{(N-1)/(N+1)} \|\nabla
P^{n+1}\|_{L^2}^{2/(N+1)}  \\
&\quad +C\|v^n\|_{\dot{B}_{2,1}^{N/2}}
\|v^{n+1}\|_{\dot{B}_{2,1}^{N/2+1}},
\end{aligned}
\end{equation}
where we used the interpolation and embedding lemmas, which listed
in Section 2. So thanks to Young's inequality and \eqref{4.8}, it
follows from \eqref{4.11} that
\begin{equation} \label{4.12}
\begin{aligned}
\|\nabla P^{n+1}\|_{\dot{B}_{2,1}^{N/2}}
&\leq C(1+\|\rho^n\|_{\dot{B}_{2,1}^{N/2+1}}^{(N+1)/2}) \|v^n\|_{L^2}
\|v^{n+1}\|_{\dot{B}_{2,1}^{N/2+1}}  \\
&\quad +C\|v^n\|_{\dot{B}_{2,1}^{N/2}}
\|v^{n+1}\|_{\dot{B}_{2,1}^{N/2+1}}.
\end{aligned}
\end{equation}
If we apply $2^{j(N/2+1)}$ on \eqref{4.10}, similarly, we obtain
\begin{align*}
\|\nabla P^{n+1}\|_{\dot{B}_{2,1}^{N/2+1}}
&\leq C(1+\|\rho^n\|_{\dot{B}_{2,1}^{N/2+1}}) \|\nabla
P^{n+1}\|_{\dot{B}_{2,1}^{N/2}}+C\|\operatorname{div}(v^n \cdot \nabla
v^{n+1})\|_{\dot{B}_{2,1}^{N/2}} \\
&\leq  C (1+\|\rho^n\|_{\dot{B}_{2,1}^{N/2+1}}) \|\nabla
P^{n+1}\|_{\dot{B}_{2,1}^{N/2+1}}^{(N+1)/(N+3)} \|\nabla
P^{n+1}\|_{\dot{B}_{2,1}^{-1/2}}^{2/(N+3)}  \\
&\quad +C\|\partial_k v_l^n \partial_l
v_k^{n+1}\|_{\dot{B}_{2,1}^{N/2}}
 \\
&\leq   C (1+\|\rho^n\|_{\dot{B}_{2,1}^{N/2+1}}) \|\nabla
P^{n+1}\|_{\dot{B}_{2,1}^{N/2+1}}^{(N+1)/(N+3)} \|\nabla
P^{n+1}\|_{L^2}^{2/(N+3)}  \\
&\quad +C\|v^n\|_{\dot{B}_{2,1}^{N/2+1}}
\|v^{n+1}\|_{\dot{B}_{2,1}^{N/2+1}},
\end{align*}
where we used interpolation, embedding and product lemmas. Hence
\begin{equation} \label{4.13}
\begin{aligned}
\|\nabla P^{n+1}\|_{\dot{B}_{2,1}^{N/2+1}}
&\leq C(1+\|\rho^n\|_{\dot{B}_{2,1}^{N/2+1}}^{(N+3)/2}) \|v^n\|_{L^2}
\|v^{n+1}\|_{\dot{B}_{2,1}^{N/2+1}}  \\
&\quad +C\|v^n\|_{\dot{B}_{2,1}^{N/2+1}}
\|v^{n+1}\|_{\dot{B}_{2,1}^{N/2+1}}.
\end{aligned}
\end{equation}
Now combining \eqref{4.3}, \eqref{4.6} \eqref{4.12} and
\eqref{4.13}, we obtain
\begin{equation} \label{4.14}
\begin{aligned}
&\frac{d}{dt} \|v^{n+1}\|_{B_{2,1}^{\frac{N}{2}+1}}\\
&\leq  C \|v^n\|_{L^2} (1+\|\rho^n\|_{\dot{B}_{2,1}^{\frac{N}{2}}})
(1+\|\rho^n\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^{\frac{N+1}{2}}+
\|\rho^n\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^{\frac{N+3}{2}})
  \|v^{n+1}\|_{B_{2,1}^{\frac{N}{2}+1}}  \\
&\quad +C (1+\|\rho^n\|_{\dot{B}_{2,1}^{N/2}})
\|v^n\|_{\dot{B}_{2,1}^{N/2+1}} \|v^{n+1}\|_{B_{2,1}^{N/2+1}}
 \\
&\quad +C (1+\|\rho^n\|_{\dot{B}_{2,1}^{N/2+1}})
\|v^n\|_{\dot{B}_{2,1}^{N/2}}  \|v^{n+1}\|_{B_{2,1}^{N/2+1}}.
\end{aligned}
\end{equation}
Apply Gronwall's inequality on \eqref{4.14}, we obtain
\begin{equation} \label{4.15}
\sup_{0\leq t\leq T} \|v^{n+1}\|_{B_{2,1}^{N/2+1}} \leq
\|v^{n+1}(0)\|_{\dot{B}_{2,1}^{N/2+1}} \exp \Big(\int_0^T B_n(t)
dt\Big),
\end{equation}
where $B_n(t)$ is the coefficient of
$\|v^{n+1}\|_{B_{2,1}^{N/2+1}}$ in \eqref{4.14}; i.e.,
\begin{equation} \label{4.16}
\begin{aligned}
B_n(t)
&= C (1+\|\rho^n\|_{\dot{B}_{2,1}^{N/2}})
\Big(\|v^n\|_{\dot{B}_{2,1}^{N/2+1}}+\|v\|_{L^2}
 (1+\|\rho^n\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^{\frac{N+1}{2}}\\
&\quad +\|\rho^n\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^{\frac{N+3}{2}})\Big)
+C (1+\|\rho^n\|_{\dot{B}_{2,1}^{N/2+1}})
\|v^n\|_{\dot{B}_{2,1}^{N/2}}.
\end{aligned}
\end{equation}
It is clear that the uniform estimates follows from \eqref{4.2},
\eqref{4.15} and \eqref{4.16}.

\noindent\textbf{Convergence.}
Let $w^{n+1}$, $u^{n+1}$ and $\Pi^{n+1}$ be the same sequences as
those in Section 3. The system reads
\begin{equation} \label{4.17}
\begin{gathered}
\partial_t w^{n+1}+v^n \cdot \nabla w^{n+1}+u^n \cdot \nabla \rho^n=0,\\
\partial_t u^{n+1}+v^n \cdot \nabla u^{n+1}+u^n \cdot \nabla v^n+(1+\rho^n)
\nabla \Pi^{n+1}+w^n \nabla P^n=0,\\
\operatorname{div} w^{n+1}=\operatorname{div} v^n=0, \\
(w^{n+1},u^{n+1})|_{t=0}=(w^{n+1}(0),u^{n+1}(0))=(\Delta_n
\rho_0,\Delta_n v_0).
\end{gathered}
\end{equation}
The estimates for $w^{n+1}$, $u^{n+1}$ and $\Pi^{n+1}$ are similar
to those in Section 3, so we just write down the estimates directly.
\begin{gather} \label{4.18}
\frac{d}{dt}\|w^{n+1}\|_{B_{2,1}^{N/2}} \leq C
\|\rho^n\|_{B_{2,1}^{N/2+1}} \|u^n\|_{B_{2,1}^{N/2}}+C
\|v^n\|_{B_{2,1}^{N/2+1}} \|w^{n+1}\|_{B_{2,1}^{N/2}}.
\\
 \label{4.19} \begin{aligned}
\frac{d}{dt} \|u^{n+1}\|_{L^2}
&\leq  C \|u^n\|_{\dot{B}_{2,1}^{N/2}}
\|v^n\|_{\dot{B}_{2,1}^{1}}+C(1+\|\rho^n\|_{\dot{B}_{2,1}^{N/2}})
\|\nabla \Pi^n\|_{L^2} \\
&\quad +C\|w^{n}\|_{\dot{B}_{2,1}^{N/2}} \|\nabla P^n\|_{L^2}.
\end{aligned} \\
 \label{4.20} \begin{aligned}
\frac{d}{dt} \|u^{n+1}\|_{\dot{B}_{2,1}^{N/2}}
&\leq C \|v^n\|_{\dot{B}_{2,1}^{N/2+1}} \|u^{n+1}
\|_{\dot{B}_{2,1}^{N/2}}+ C (1+\|\rho^{n}\|_{\dot{B}_{2,1}^{N/2}})
\|\nabla \Pi^n\|_{\dot{B}_{2,1}^{N/2}} \\
&\quad + C \|w^n\|_{\dot{B}_{2,1}^{N/2}} \|\nabla
P^n\|_{\dot{B}_{2,1}^{N/2}}+C\|u^n\|_{\dot{B}_{2,1}^{N/2}}
\|v^n\|_{\dot{B}_{2,1}^{N/2+1}}.
\end{aligned} \\
 \label{4.21} \begin{aligned}
\|\nabla \Pi^{n+1}\|_{L^2}
&\leq  C \|v^n\|_{\dot{B}_{2,1}^{N/2}}
\|u^{n+1}\|_{\dot{B}_{2,1}^1}+C \|u^n\|_{\dot{B}_{2,1}^{N/2}}
\|v^{n}\|_{\dot{B}_{2,1}^1}  \\
&\quad +C \|w^n\|_{\dot{B}_{2,1}^{N/2}}\|\nabla P^n\|_{L^2}.
\end{aligned} \\
 \label{4.22} \begin{aligned}
\|\nabla \Pi^{n+1}\|_{\dot{B}_{2,1}^{N/2}}
 &\leq  C (1+\|\rho^n\|_{\dot{B}_{2,1}^{N/2+1}}) \|\nabla
\Pi^{n+1}\|_{L^2}+C \|v^n\|_{\dot{B}_{2,1}^{N/2+1}}
\|u^{n+1}\|_{\dot{B}_{2,1}^{N/2}} \\
&\quad +C \|w^n \|_{\dot{B}_{2,1}^{N/2}} \|\nabla
P^n\|_{\dot{B}_{2,1}^{N/2}}+C \|v^n\|_{\dot{B}_{2,1}^{N/2+1}}
\|u^{n}\|_{\dot{B}_{2,1}^{N/2}}.
\end{aligned}
\end{gather}
It follows from \eqref{4.8} and \eqref{4.12} that the estimate for
$\nabla \Pi^{n+1}$ can be represented in term of $\rho^{n-1}$,
$v^{n-1}$ and $v^n$. Due to the uniform estimates and
\eqref{4.18}-\eqref{4.22}, then the convergence follows from the
same argument as that in Section 3.

The uniqueness follows from the analogous argument and estimates as
that in Section 3 and \eqref{4.18}-\eqref{4.22}.

\section{Appendix}

To prove the second part of Proposition 2.4, we show the
following lemma, which clearly implies Proposition 2.4.

\begin{lemma} \label{lem5.1}
Let $s>0$, $1<p<\infty$. If $f$ and $g$ belong to
$\dot{B}_{p,1}^{s} \cap L^{\infty}$, then $f g$ is in
$\dot{B}_{p,1}^s$, and
\begin{equation} \label{5.1}
\|fg\|_{\dot{B}_{p,1}^{s}} \leq C\Big(\|f\|_{L^{\infty}}
\|g\|_{\dot{B}_{p,1}^s}+\|g\|_{L^{\infty}}
\|f\|_{\dot{B}_{p,1}^s}\Big).
\end{equation}
\end{lemma}

\begin{proof}
We use Bony's decomposition~\cite{Bo} to represent the product as
\[
f g=T_f g+T_g f+R(f,g),
\]
where
\[
T_f g =\sum_{j} S_{j-1}f \Delta_j g,\quad
R(f,g)=\sum_{j} \Delta_j f (\Delta_{j-1}+\Delta_j+\Delta_{j+1}) g.
\]
By compactness of the supports of the series of Fourier transform,
for any $u,v$,
\[
\Delta_k \Delta_l u \equiv 0,\quad
|k-l|\geq 2,\quad \Delta_k (S_{q-1} u \Delta_q
v)=0,\quad\text{if }|k-q|\geq 5.
\]
It follows that
\begin{equation} \label{5.2}
\begin{aligned}
\sum_{j \in \mathbb{Z}} 2^{js} \|\Delta_j T_f g\|_{L^p}
 &=  \sum_{j \in \mathbb{Z}} 2^{js}
\sum_{|j-j'|\leq 4}\|\Delta_j (S_{j'-1} f \Delta_{j'} g)\|_{L^p}  \\
&\leq  C \sup_{q}\|S_{q} f\|_{L^{\infty}} \sum_{j' \in \mathbb{Z}}
2^{js}
\|\Delta_{j'} g\|_{L^p}   \\
&\leq  C \|g\|_{L^{\infty}} \|f\|_{\dot{B}_{p,1}^s}.
\end{aligned}
\end{equation}
Similarly,
\begin{equation} \label{5.3}
\|T_g f\|_{\dot{B}_{p,1}^s} \leq C \|f\|_{L^{\infty}}
\|g\|_{\dot{B}_{p,1}^s}.
\end{equation}
It follows from Bony's formula that
\begin{align*}
\Delta_j R(f,g)
&= \sum_{\max\{i',j'\} \geq j-3,|i'-j'| \leq 1}
\Delta_j (\Delta_{i'} f \Delta_{j'} g)\\
&= \sum_{j' \geq j-4} \sum_{|i'-j'| \leq 1}\Delta_j (\Delta_{i'} f
\Delta_{j'} g).
\end{align*}
Therefore, by Minkowski inequality, we have
\begin{equation} \label{5.4}
\begin{aligned}
\sum_{j \in \mathbb{Z}} 2^{js}
&\leq  \sum_{k \geq -4} \sum_{m=-1}^{1} \sum_{j' \in \mathbb{Z}}
2^{(j'-k)s}\|\Delta_{j'-k}(\Delta_{j'-m} f \Delta_{j'} g)\|_{L^p}
 \\
&\leq  C \sum_{k \geq -4} 2^{-ks} \sum_{m=-1}^{1} \sum_{j'\in
\mathbb{Z}} 2^{j's} \|\Delta_{j'-m} f \Delta_{j'} g\|_{L^p}
 \\
&\leq  C \sup_{q} \|\Delta_{q} f\|_{L^{\infty}} \sum_{j' \in
\mathbb{Z}} 2^{j's} \|\Delta_{j'} g\|_{L^p}  \\
&\leq  C \|f\|_{L^{\infty}} \|g\|_{\dot{B}_{p,1}^s}.
\end{aligned}
\end{equation}
Then \eqref{5.1} follows from \eqref{5.2}, \eqref{5.3} and
\eqref{5.4}.
\end{proof}

\begin{remark} \label{rmk4.1}\rm
Actually, we can prove the Moser type inequality
\begin{align*}
\|fg\|_{\dot{B}_{p,q}^{s}} \leq C\Big(\|f\|_{L^{p_1}}
\|g\|_{\dot{B}_{p_2,q}^s}+\|g\|_{L^{r_1}}
\|f\|_{\dot{B}_{r_2,q}^s}\Big),
\end{align*}
provided that $f \in L^{p_1} \cap \dot{B}_{r_2,q}^s$, $s>0$,
$1\leq p,q,p_1,r_2 \leq \infty$,
$g \in L^{r_1} \cap \dot{B}_{p_2,q}^s$, $1 \leq r_1, p_2 \leq \infty$ and
\begin{align*}
\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{r_1}+\frac{1}{r_2}.
\end{align*}
\end{remark}

\begin{proof}[Proof of Proposition 3.1]
The idea is to approximate \eqref{3.1} by linear transport equations.
First we find that \eqref{3.1} is equivalent to the system
\begin{equation} \label{5.5}
\begin{gathered}
\partial_t v + w \cdot \nabla v+\nabla P=f,\\
-\Delta P=\operatorname{div}(w \cdot \nabla v)-\operatorname{div}f,\\
v(x,t=0)=v_0(x),\quad \operatorname{div}v_0=0.
\end{gathered}
\end{equation}
So we approximate \eqref{5.5} by the linear transport equations
\begin{equation} \label{5.6}
\begin{gathered}
\partial_t v^{n+1} + w \cdot \nabla v^{n+1}+\nabla P^n=f,\\
-\Delta P^n=\operatorname{div}(w \cdot \nabla v^n)-\operatorname{div}f,\\
v^{n+1}(x,t=0)=S_{n+1} v_0(x).
\end{gathered}
\end{equation}
The existence theorem for \eqref{5.6} is well-known for each $n$.
Just as the proof of Theorem 1.1, we should give a uniform estimates
for the sequence $v^{n+1}$ and the convergence of the corresponding
sequence. In order to do so, we only need to do a priori estimates
for the equivalent system \eqref{5.5}. First, we have
\begin{equation} \label{5.7}
\frac{d}{dt} \|v(.,t)\|_{B_{p,1}^{N/p+1}} \leq C
\|w\|_{\dot{B}_{p,1}^{N/p+1}}
\|v\|_{B_{p,1}^{N/p+1}}+\|f\|_{B_{p,1}^{N/p+1}}+\|\nabla
P\|_{B_{p,1}^{N/p+1}}.
\end{equation}
The estimate for the pressure is easy now,  it reads
\begin{align*}
\|\nabla P\|_{B_{p,1}^{N/p+1}} \leq C \|w\|_{\dot{B}_{p,1}^{N/p+1}}
\|v\|_{\dot{B}_{p,1}^{N/p+1}}+C\|f\|_{B_{p,1}^{N/p+1}}.
\end{align*}
Therefore,  from \eqref{5.7} it follows that
\begin{equation} \label{5.8}
\frac{d}{dt} \|v(.,t)\|_{B_{p,1}^{N/p+1}} \leq C
\|w\|_{\dot{B}_{p,1}^{N/p+1}}
\|v\|_{B_{p,1}^{N/p+1}}+C\|f\|_{B_{p,1}^{N/p+1}}.
\end{equation}
Apply Gronwall inequality on \eqref{5.8},
\begin{equation} \label{5.9}
\begin{aligned}
\|v(.,t)\|_{B_{p,1}^{N/p+1}}
&\leq  \|v_0\|_{B_{p,1}^{N/p+1}}
\exp\Big(\int_0^t C\|w(.,s)\|_{\dot{B}_{p,1}^{N/p+1}} d
s\Big) \\
&\quad +\int_0^t \|f(.,\tau)\|_{B_{p,1}^{\frac{N}{p}+1}}
\exp\Big(\int_{\tau}^t C \|w(.,s)\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}
d s\Big) d \tau.
\end{aligned}
\end{equation}
Since we have the a priori estimate \eqref{5.9}, the existence and
uniqueness of solutions for the system \eqref{5.5} can be obtained
by the approximate sequence $v^{n+1}$, solutions to \eqref{5.6}.
This completes the proof.
\end{proof}


\begin{proof}[Proof of Proposition 4.1]
Just as for Proposition 3.1, note that \eqref{4.2} is equivalent to the linear
system
\begin{equation} \label{5.10}
\begin{gathered}
\partial_t v + w \cdot \nabla v+(1+\rho)\nabla P=0,\\
-\operatorname{div}\left((1+\rho)\nabla P\right)
=\operatorname{div}(w \cdot \nabla v),\\
v(x,t=0)=v_0(x),\quad \operatorname{div}v_0=0.
\end{gathered}
\end{equation}
The linear transport approximate system is
\begin{equation} \label{5.11}
\begin{gathered}
\partial_t v^{n+1} + w \cdot \nabla v^{n+1}+(1+\rho)\nabla P^n=0,\\
-\operatorname{div}\left((1+\rho)\nabla P^n\right)
=\operatorname{div}(w \cdot \nabla v^n),\\
v^{n+1}(x,t=0)=S_{n+1} v_0(x).
\end{gathered}
\end{equation}
It is easy to establish a priori estimates for the system
\eqref{5.10}, then we can prove the existence and uniqueness of the
solution, which is a limit of the iteration sequence. We would like
to skip the details of the proof, for conciseness.
\end{proof}

\subsection*{Acknowledgments}
The author would like to thank the anonymous referees for their
helpful comments.

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\section*{Addendum posted on September 5, 2013}

After publication, the author received the following comments.

The smallness assumption on initial data was removed in:
Rapha\"el Danchin; On the well-posedness of the incompressible 
density-dependent Euler equations in the $L^p$ framework. 
J. Differential Equations 248 (2010), 8, 2130--2170.

The case $p=\infty$ was treated in:
Rapha\"el Danchin, Francesco Fanelli; The well-posedness issue for 
the density-dependent Euler equations in endpoint Besov spaces. 
J. Math. Pures Appl. (9) 96 (2011), 3, 253--278.

The author wants to thank the anonymous reader for sending this information.

It was also commented that a result similar to Theorem 1.1 was obtained in:
Young Zhou; Local well-posedness and regularity criterion for the 
density dependent incompressible Euler equations. 
Nonlinear Anal. 73 (2010), no. 3, 750--766.

Our article studies the critical case $s=p/n+1$, in the space 
$B_{p,1}^{p/n+1}$;
while the above reference studies the super-critical case $s>p/n+1$,
in the space $B_{p,q}^s$.

End of addendum.
\end{document}