\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 $0N/2+1$, 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 $s>N/p+1$, $1N/p+1$, $10$, $(p,q) \in (1,\infty) \times [1,\infty]$, 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 $$\|f\|_{B_{p,q}^s}=\|f\|_{L^p}+\|f\|_{\dot{B}_{p,q}^s},\quad \|f\|_{F_{p,q}^s}= \|f\|_{L^p}+\|f\|_{\dot{F}_{p,q}^s}.$$ 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 $q_1,q_2 \in [1,\infty]$, $q_10$. 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}, $$\label{3.2} \begin{gathered} \partial_t \rho^{n+1}+v^n \cdot \nabla \rho^{n+1}=0,\\ \partial_t v^{n+1}+v^n \cdot \nabla v^{n+1}+\nabla P^{n+1}=-\rho^n \nabla P^n,\\ \operatorname{div} v^{n+1}=\operatorname{div} v^n=0, \\ (\rho^{n+1},v^{n+1})|_{t=0}=(\rho^{n+1}(0),v^{n+1}(0))=(S_{n+1} \rho_0,S_{n+1} v_0), \end{gathered}$$ 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 $1 0$ and $C_1$ is sufficiently small. Then the following inequalities hold $$\label{3.19} \begin{gathered} \|\rho^{n+1}\|_{L^{\infty}(0,T_*;B_{p,1}^{N/p+1})} \leq C_1,\\ \|v^{n+1}\|_{L^{\infty}(0,T_*;B_{p,1}^{N/p+1})} \leq C_2,\\ \|\nabla P^{n+1}\|_{L^1(0,T_*;B_{p,1}^{N/p+1})} \leq C_3, \end{gathered}$$ 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 $$\label{3.20} \begin{gathered} \partial_t w^{n+1}+v^n \cdot \nabla w^{n+1}+u^n \cdot \nabla \rho^n=0,\\ \partial_t u^{n+1}+v^n \cdot \nabla u^{n+1}+u^n \cdot \nabla v^n+\nabla \Pi^{n+1} =-w^n \nabla P^n-\rho^{n-1} \nabla \Pi^n,\\ \operatorname{div} u^{n+1}=\operatorname{div} v^n=0, \\ (w^{n+1},u^{n+1})|_{t=0}=(w^{n+1}(0),u^{n+1}(0))=(\Delta_n \rho_0,\Delta_n v_0), \end{gathered}$$ 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 \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} Applying $\Delta_j$ on both sides of the first equation of \eqref{3.20}, we have $$\label{3.22} \partial_t \Delta_j w^{n+1}+v^n \cdot \Delta_j w^{n+1}+\Delta_j (u^n \cdot \nabla \rho^n)=[v^n,\Delta_j] \nabla w^{n+1}.$$ 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 $$\label{3.23} \frac{d}{dt} \|w^{n+1}\|_{\dot{B}_{p,1}^{N/p}} \leq C \|u^n\|_{\dot{B}_{p,1}^{N/p}} \|\rho^n\|_{\dot{B}_{p,1}^{N/p+1}}+C \|v^n\|_{\dot{B}_{p,1}^{N/p+1}} \|w^{n+1}\|_{\dot{B}_{p,1}^{N/p}}.$$ Combining \eqref{3.21} and \eqref{3.23}, it follows that $$\label{3.023} \frac{d}{dt}\|w^{n+1}\|_{B_{p,1}^{N/p}} \leq C \|\rho^n\|_{B_{p,1}^{N/p+1}} \|u^n\|_{B_{p,1}^{N/p}}+C \|v^n\|_{B_{p,1}^{N/p+1}} \|w^{n+1}\|_{B_{p,1}^{N/p}}.$$ 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 \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} Applying $\Delta_j$ on the second equation of \eqref{3.20}, we obtain \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} 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 \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} Then applying $2^{jN/p}$ on \eqref{3.27} and taking summation yields \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} Combining \eqref{3.25} and \eqref{3.28}, we have \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} 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 \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} where we used the embedding in Lemma 2.3 and the product estimate. Similarly, \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} Combining \eqref{3.29} and \eqref{3.30}, it follows that \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} Therefore, if we add \eqref{3.023}, \eqref{3.028} and \eqref{3.31} together, then we obtain, \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} 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 \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} 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 $$\label{3.34} \begin{gathered} \partial_t \rho+v_1 \cdot \nabla \rho+v \cdot \nabla \rho_2=0,\\ \partial_t v+v_1 \cdot \nabla v+v \cdot \nabla v_2+\nabla P =-\rho_1 \nabla P-\rho \nabla P_2,\\ \operatorname{div} v_1=\operatorname{div} v_2=0, \\ (\rho,v)|_{t=0}=(0,0). \end{gathered}$$ 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 $$\label{4.1} \begin{gathered} \partial_t \rho^{n+1}+v^n \cdot \nabla \rho^{n+1}=0,\\ \partial_t v^{n+1}+v^n \cdot \nabla v^{n+1}+(1+\rho^n)\nabla P^{n+1}=0,\\ \operatorname{div} v^{n+1}=\operatorname{div} v^n=0, \\ (\rho^{n+1},v^{n+1})|_{t=0}=(\rho^{n+1}(0),v^{n+1}(0))=(S_{n+1} \rho_0,S_{n+1} v_0), \end{gathered}$$ with the corresponding linear system $$\label{4.01} \begin{gathered} \partial_t v+w \cdot \nabla v+(1+\rho) \nabla P=0,\\ \operatorname{div} v=0, \\ v(x,t=0)=v_0(x). \end{gathered}$$ 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 $$\label{4.2} \sup_{0 \leq t \leq T} \|\rho^{n+1} (.,t)\|_{B_{p,1}^s} \leq \|\rho^{n+1} (0)\|_{B_{p,1}^s} \exp\Big( \int_0^T C \|v^n(.,t)\|_{\dot{B}_{p,1}^{N/p+1}} dt \Big).$$ Multiplying the second equation of \eqref{4.1} by $v^{n+1}$ and integrating over $\mathbb{R}^N$, we obtain $$\label{4.3} \frac{d}{dt} \|v^{n+1}(.,t)\|_{L^2} \leq \|1+\rho^n\|_{L^{\infty}} \|\nabla P^{n+1}\|_{L^2} \leq C(1+\|\rho^n\|_{\dot{B}_{2,1}^{N/2}}) \|\nabla P^{n+1}\|_{L^2}.$$ Applying $\Delta_j$ on the second equation of \eqref{3.2}, we obtain $$\label{4.4} \partial_t \Delta_j v^{n+1} +v^n \cdot \nabla \Delta_j v^{n+1}=[v^n,\Delta_j] \nabla v^{n+1}-\Delta_j \big((1+\rho^n) \nabla P^n\big).$$ Multiplying \eqref{4.4} by $\Delta_j v^{n+1}$ and taking the divergence free property of $v^n$ into account, we have \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} Applying $2^{j (N/2+1)}$ on \eqref{4.5} and taking summation, and using the product estimate, we have \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} 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 $$\label{4.7} \operatorname{div} \left((1+\rho^n) \nabla P^{n+1}\right)=-\operatorname{div} (v^n \cdot \nabla v^{n+1}).$$ 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 $$\label{4.8} \|\nabla P^{n+1}\|_{L^2} \leq C \|v^n \cdot \nabla v^{n+1}\|_{L^2} \leq C \|v^n\|_{L^2} \|v^{n+1}\|_{\dot{B}_{2,1}^{N/2+1}}.$$ Applying $\Delta_j$ on \eqref{4.7}, we have $$\label{4.9} -\operatorname{div}\left((1+\rho^n) \nabla \Delta_j P^{n+1}\right) = \operatorname{div}\left([\Delta_j,(1+\rho^n)] \nabla P^{n+1}\right)+\Delta_j \operatorname{div} \left(v^n \cdot \nabla v^{n+1}\right).$$ 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 \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} Multiplying by $2^{j N/2}$ on \eqref{4.10} and taking summation, we have \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} 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 \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} 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 \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} Now combining \eqref{4.3}, \eqref{4.6} \eqref{4.12} and \eqref{4.13}, we obtain \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} Apply Gronwall's inequality on \eqref{4.14}, we obtain $$\label{4.15} \sup_{0\leq t\leq T} \|v^{n+1}\|_{B_{2,1}^{N/2+1}} \leq \|v^{n+1}(0)\|_{\dot{B}_{2,1}^{N/2+1}} \exp \Big(\int_0^T B_n(t) dt\Big),$$ where $B_n(t)$ is the coefficient of $\|v^{n+1}\|_{B_{2,1}^{N/2+1}}$ in \eqref{4.14}; i.e., \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} 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 $$\label{4.17} \begin{gathered} \partial_t w^{n+1}+v^n \cdot \nabla w^{n+1}+u^n \cdot \nabla \rho^n=0,\\ \partial_t u^{n+1}+v^n \cdot \nabla u^{n+1}+u^n \cdot \nabla v^n+(1+\rho^n) \nabla \Pi^{n+1}+w^n \nabla P^n=0,\\ \operatorname{div} w^{n+1}=\operatorname{div} v^n=0, \\ (w^{n+1},u^{n+1})|_{t=0}=(w^{n+1}(0),u^{n+1}(0))=(\Delta_n \rho_0,\Delta_n v_0). \end{gathered}$$ 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 $s>0$, $10$, $1\leq p,q,p_1,r_2 \leq \infty$, $g \in L^{r_1} \cap \dot{B}_{p_2,q}^s$, $1 \leq r_1, p_2 \leq \infty$ 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 $$\label{5.5} \begin{gathered} \partial_t v + w \cdot \nabla v+\nabla P=f,\\ -\Delta P=\operatorname{div}(w \cdot \nabla v)-\operatorname{div}f,\\ v(x,t=0)=v_0(x),\quad \operatorname{div}v_0=0. \end{gathered}$$ So we approximate \eqref{5.5} by the linear transport equations $$\label{5.6} \begin{gathered} \partial_t v^{n+1} + w \cdot \nabla v^{n+1}+\nabla P^n=f,\\ -\Delta P^n=\operatorname{div}(w \cdot \nabla v^n)-\operatorname{div}f,\\ v^{n+1}(x,t=0)=S_{n+1} v_0(x). \end{gathered}$$ 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 $$\label{5.7} \frac{d}{dt} \|v(.,t)\|_{B_{p,1}^{N/p+1}} \leq C \|w\|_{\dot{B}_{p,1}^{N/p+1}} \|v\|_{B_{p,1}^{N/p+1}}+\|f\|_{B_{p,1}^{N/p+1}}+\|\nabla P\|_{B_{p,1}^{N/p+1}}.$$ 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 $$\label{5.8} \frac{d}{dt} \|v(.,t)\|_{B_{p,1}^{N/p+1}} \leq C \|w\|_{\dot{B}_{p,1}^{N/p+1}} \|v\|_{B_{p,1}^{N/p+1}}+C\|f\|_{B_{p,1}^{N/p+1}}.$$ Apply Gronwall inequality on \eqref{5.8}, \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} 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 $$\label{5.10} \begin{gathered} \partial_t v + w \cdot \nabla v+(1+\rho)\nabla P=0,\\ -\operatorname{div}\left((1+\rho)\nabla P\right) =\operatorname{div}(w \cdot \nabla v),\\ v(x,t=0)=v_0(x),\quad \operatorname{div}v_0=0. \end{gathered}$$ The linear transport approximate system is $$\label{5.11} \begin{gathered} \partial_t v^{n+1} + w \cdot \nabla v^{n+1}+(1+\rho)\nabla P^n=0,\\ -\operatorname{div}\left((1+\rho)\nabla P^n\right) =\operatorname{div}(w \cdot \nabla v^n),\\ v^{n+1}(x,t=0)=S_{n+1} v_0(x). \end{gathered}$$ 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. 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Nonlinear Anal. 73 (2010), no. 3, 750--766. \end{thebibliography} \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}