\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 192, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2016/192\hfil Logarithmically improved blow-up criteria]
{Logarithmically improved blow-up criteria for the 3D
nonhomogeneous incompressible Navier-Stokes equations with vacuum}
\author[Q. Hou, X. Xu, Z. Ye \hfil EJDE-2016/192\hfilneg]
{Qianqian Hou, Xiaojing Xu, Zhuan Ye}
\address{Qianqian Hou \newline
Department of Applied Mathematics,
Hong Kong Polytechnic University,
Hung Hom, Kowloon, Hong Kong}
\email{qianqian.hou@connect.polyu.hk}
\address{Xiaojing Xu \newline
School of Mathematical Sciences,
Beijing Normal University,
Laboratory of Mathematics and Complex Systems,
Ministry of Education,
Beijing 100875, China}
\email{ xjxu@bnu.edu.cn}
\address{Zhuan Ye (corresponding author)\newline
Department of Mathematics and Statistics,
Jiangsu Normal University,
101 Shanghai Road, Xuzhou 221116, Jiangsu, China}
\email{yezhuan815@126.com}
\thanks{Submitted April 29, 2015. Published July 14, 2016.}
\subjclass[2010]{35Q30, 35B40, 76D03, 76D05}
\keywords{Nonhomogeneous Navier-Stokes equations; blow-up criterion;
\hfill\break\indent strong solution; vacuum}
\begin{abstract}
This article is devoted to the study of the nonhomogeneous incompressible
Navier-Stokes equations in space dimension three. By making use of the
``weakly nonlinear'' energy estimate approach introduced by Lei and Zhou
in \cite{LZ}, we establish two logarithmically improved blow-up criteria
of the strong or smooth solutions subject to vacuum for the 3D
nonhomogeneous incompressible Navier-Stokes equations in the whole
space $\mathbb{R}^3$. This results extend recent regularity
criterion obtained by Kim (2006) \cite{KIM}.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\section{Introduction}\label{intro}
In this article we study a blow-up criterion of strong solutions to the
3D nonhomogeneous incompressible Navier-Stokes equation
in the whole space $\mathbb{R}^3$,
\begin{equation}\label{NS}
\begin{gathered}
\rho_t+{\rm div} (\rho u)=0, \\
(\rho u)_t+{\rm div} (\rho u\otimes u)-\Delta u+\nabla \pi=0,\\
\operatorname{div} u=0,\\
(\rho, \rho u)|_{t=0}=(\rho_0,\rho_0u_0),
\end{gathered}
\end{equation}
where $u=u(x, t)=(u_{1}(x, t)$, $u_2(x, t)$, $u_3(x, t))$,
$\rho=\rho(x, t)$ and $\pi=\pi(x, t)$ denote the unknown velocity,
density and pressure, respectively.
The system \eqref{NS} describes a fluid which is obtained by mixing
two miscible fluids that are incompressible and that have different densities.
It may also describe a fluid containing a melted substance.
One may check \cite{Li1998} for the detailed derivation.
In the past decades, there has been a lot of literature about the
well-posedness theory of the incompressible Navier-Stokes equations \eqref{NS}.
When the initial density is strictly positive, there has been proved that
there is a unique strong solution to the problem \eqref{NS} in dimension three,
which is locally defined for large initial
data, while globally defined for the case of small data
(see for example \cite{Ab2007,AGZ2012,AKM1990,Dan2003,GHZ2011,Ka1974,LS1978}).
On the other hand, for initial data which permits regions of vacuum, i.e.
regions where the density $\rho$ vanishes on some set, the problem becomes
much more complicated. The global existence of weak solutions of the
system \eqref{NS} has been established (see \cite{KIM1987,Li1998,Si1990}).
However, the problem of uniqueness and regularity of such weak solutions is
full of challenge and remains open. Very recently, Craig-Huang-Wang \cite{CHW2013}
proved the global existence of strong solution with vacuum of the system \eqref{NS}
under the assumption that the initial data $\|u_0\|_{\dot{H}^{\frac{1}{2}}}$
is small enough.
We refer the interested readers to \cite{CK2003,CK2004,CK2006,HW2013,De1997,ZZZ}
for many more results.
Recently, Choe and Kim \cite{CK2003} established an existence result
on strong solutions with nonnegative densities for the system \eqref{NS}.
More precisely, it was proved that if the data $\rho_0$ and $u_0$ satisfy the
following regularity condition
\[
0\leq \rho_0 \in L^{3/2}\cap H^2,\quad
u_0 \in H_0^{1}\cap H^2
\]
and the compatibility condition
\[ %\\label{O2}
-\Delta u_0+\nabla \pi_0=\sqrt{\rho_0}g,\quad \operatorname{div} u_0=0,
\]
with $(\pi_0,\,g)\in H^{1}\times L^2$. Then there exist a time
$T_{\star}\in (0,T)$ and a unique strong solution $(\rho,u,\pi)$
to the system \eqref{NS} such that
\begin{gather*}
\rho\in L^{\infty}(0,T_{\star}; L^{\infty}\cap H^{1}),\quad
\nabla u,\pi\in L^{\infty}(0,T_{\star}; H^{1})\cap L^2(0,T_{\star}; W^{1,6}),\\
\rho_{t}\in L^{\infty}(0,T_{\star}; L^2),\quad
\sqrt{\rho}u_{t}\in L^{\infty}(0,T_{\star}; L^2),\quad
u_{t}\in L^2(0,T_{\star}; H_0^{1}),
\end{gather*}
Here we would like to emphasize that Kim \cite{KIM} established the so-called
Serrin type regularity criterion to the system \eqref{NS}, which reads:
If
$$
u\in L^{q}(0,T;L_{w}^{p}(\mathbb{R}^3)),\quad
\frac{3}{p}+\frac{2}{q}\leq1,\; 3
0$,
there exists $T_0=T_0(\epsilon)3$,
\begin{equation}\label{ZJt206}
\begin{aligned}
\|\Delta u\|_{L^{\frac{3p}{2p+3}}}
&\leq C(\|\rho u_{t}\|_{L^{\frac{3p}{2p+3}}}
+\|\rho u\cdot\nabla u\|_{L^{\frac{3p}{2p+3}}}) \\
&\leq C(\|\sqrt{\rho}\|_{L^{\frac{6p}{p+6}}}\|\sqrt{\rho } u_{t}\|_{L^2}
+\|\rho\|_{L^{p}}\|u\|_{L^{6}}\|\nabla u\|_{L^2}) \\
&\leq C(\|\sqrt{\rho_0}\|_{L^{\frac{6p}{p+6}}}\|\sqrt{\rho } u_{t}\|_{L^2}
+\|\rho_0\|_{L^{p}}\|\nabla u\|_{L^2}\|\nabla u\|_{L^2}) \\
&\leq C(\|\sqrt{\rho} u_{t}\|_{L^2}+\|\nabla u\|_{L^2}^2).
\end{aligned}
\end{equation}
Combining \eqref{t206} and \eqref{ZJt206} leads to
\begin{equation}\label{t207}
\|\Delta u\|_{L^2}+\|\Delta u\|_{L^{\frac{3p}{2p+3}}}
\leq C(\|\sqrt{\rho} u_{t}\|_{L^2}+\|\nabla u\|_{L^2}^2+\|\nabla u\|_{L^2}^3).
\end{equation}
Note that by \eqref{t205}, we obtain
\begin{equation}\label{t208}
\begin{aligned}
\int_{T_0}^{t}{\|\Delta u(s)\|_{L^2}^2\,ds}
&\leq C\int_{T_0}^{t}{(\|\sqrt{\rho} u_{t}\|_{L^2}^2
+\|\nabla u\|_{L^2}^{4}+\|\nabla u\|_{L^2}^{6})(s)\,ds} \\
&\leq C\big(e+y(t)\big)^{3A\epsilon}.
\end{aligned}
\end{equation}
Combining \eqref{t205} and \eqref{t208}, we obtain
\begin{equation}\label{tttt}
\|\nabla u(t)\|_{L^2}^{6}+\int_{T_0}^{t}{(\|\sqrt{\rho} u_{t}\|_{L^2}^2
+\|\Delta u\|_{L^2}^2)(s)\,ds}\leq C\big(e+y(t)\big)^{3A\epsilon}.
\end{equation}
Differentiating the momentum equation with respect to $t$, multiplying by $u_{t}$,
and then integrating over whole space, one can obtain that
\begin{equation}\label{t209}
\begin{aligned}
\frac{1}{2}\frac{d}{dt}\|\sqrt{\rho} u_{t}(t)\|_{L^2}^2+\|
\nabla u_{t}\|_{L^2}^2
&=-\int_{\mathbb{R}^2}{\rho_{t}u_{t}\cdot u_{t} \,dx}
-\int_{\mathbb{R}^2}{(\rho u)_{t}\cdot \nabla u\cdot u_{t} \,dx} \\
&:= J_{1}+J_2.
\end{aligned}
\end{equation}
By the mass equation, we derive
\begin{equation}\label{t210}
\begin{aligned}
J_{1}&= \int_{\mathbb{R}^2}{{\rm div} (\rho u)u_{t}\cdot u_{t} \,dx} \\
&\leq 2\Big|\int_{\mathbb{R}^2}{\rho u \nabla u_{t}\cdot u_{t} \,dx}\Big|
\\
&\leq C \|u\|_{L^{6}}\|\nabla u_{t}\|_{L^2}\|\sqrt{\rho}u_{t}\|_{L^3} \\
&\leq C \|\nabla u\|_{L^2}\|\nabla u_{t}\|_{L^2}\|\sqrt{\rho}u_{t}\|_{L^2}^{\frac{1}{2}}
\|\sqrt{\rho}u_{t}\|_{L^{6}}^{\frac{1}{2}} \\
&\leq C \|\nabla u\|_{L^2}\|\nabla u_{t}\|_{L^2}\|\sqrt{\rho}u_{t}\|_{L^2}^{\frac{1}{2}}
\|u_{t}\|_{L^{6}}^{\frac{1}{2}} \\
&\leq C \|\nabla u\|_{L^2}\|\nabla u_{t}\|_{L^2}\|\sqrt{\rho}u_{t}\|_{L^2}^{\frac{1}{2}}
\|\nabla u_{t}\|_{L^2}^{\frac{1}{2}} \\
&\leq \frac{1}{8}\|\nabla u_{t}\|_{L^2}^2+C \|\nabla u\|_{L^2}^{4}
\|\sqrt{\rho}u_{t}\|_{L^2}^2.
\end{aligned}
\end{equation}
Again we resort to the mass equation to obtain
\begin{align*}
J_2&= -\int_{\mathbb{R}^2}{\rho u_{t}\cdot \nabla u\cdot u_{t} \,dx}
-\int_{\mathbb{R}^2}{\rho_{t} u\cdot \nabla u\cdot u_{t} \,dx} \\
&= -\int_{\mathbb{R}^2}{\rho u_{t}\cdot \nabla u\cdot u_{t} \,dx}
+\int_{\mathbb{R}^2}{{\rm div} (\rho u) u\cdot \nabla u\cdot u_{t} \,dx}
\\
&= -\int_{\mathbb{R}^2}{\rho u_{t}\cdot \nabla u\cdot u_{t} \,dx}
-\int_{\mathbb{R}^2}{(\rho u) \nabla (u\cdot \nabla u\cdot u_{t}) \,dx} \\
&= J_{21}+J_{22}.
\end{align*}
The Young inequality and Sobolev embedding theorem entail us to obtain
\begin{align*}
J_{21}&\leq C \|\sqrt{\rho}u_{t}\|_{L^{4}}^2\|\nabla u\|_{L^2} \\
&\leq C (\|\sqrt{\rho}u_{t}\|_{L^2}^{\frac{1}{4}}\|\sqrt{\rho}u_{t}\|_{L^{6}}^{\frac{3}{4}})^2
\|\nabla u\|_{L^2} \\&\leq C \|\sqrt{\rho}u_{t}\|_{L^2}^{\frac{1}{2}}\|u_{t}\|_{L^{6}}^{3/2}
\|\nabla u\|_{L^2} \\
&\leq C \|\sqrt{\rho}u_{t}\|_{L^2}^{\frac{1}{2}}\|\nabla u_{t}\|_{L^2}^{3/2}
\|\nabla u\|_{L^2} \\
&\leq \frac{1}{8} \|\nabla u_{t}\|_{L^2}^2+C\|\nabla u\|_{L^2}^{4}\|\sqrt{\rho}u_{t}\|_{L^2}^2.
\end{align*}
Similarly, we obtain by using Young inequality and Sobolev embedding theorem
\begin{align*}
J_{22}
&\leq \big|\int_{\mathbb{R}^2}{(\rho u) \nabla u\cdot \nabla u\cdot u_{t} \,dx}\big|
+\big|\int_{\mathbb{R}^2}{(\rho u) u\cdot \nabla^2 u\cdot u_{t} \,dx}\big|\\
&\quad +\big|\int_{\mathbb{R}^2}{(\rho u) u\cdot \nabla u\cdot \nabla u_{t} \,dx}\big| \\
&\leq C \|u\|_{L^{6}}\|\nabla u\|_{L^3}^2\|u_{t}\|_{L^{6}}+ C\|u\|_{L^{6}}^2\|\Delta u\|_{L^2}\|u_{t}\|_{L^{6}}+ C\|u\|_{L^{6}}^2\|\nabla u\|_{L^{6}}\|\nabla u_{t}\|_{L^2} \\
&\leq C \|\nabla u\|_{L^2}^2\|\Delta u\|_{L^2}\|\nabla u_{t}\|_{L^2} \\
&\leq \frac{1}{8} \|\nabla u_{t}\|_{L^2}^2+C \|\nabla u\|_{L^2}^{4}
\|\Delta u\|_{L^2}^2.
\end{align*}
Plugging the above estimates into inequality \eqref{t209} we arrive at
\begin{equation}\label{t211}
\frac{d}{dt}\|\sqrt{\rho} u_{t}(t)\|_{L^2}^2+\|\nabla u_{t}\|_{L^2}^2
\leq C\|\nabla u\|_{L^2}^{4}(\|\sqrt{\rho}u_{t}\|_{L^2}^2+\|\Delta u\|_{L^2}^2).
\end{equation}
Integrating above differential inequality and using the estimate \eqref{tttt},
it gives
\begin{equation}\label{t212}
\begin{aligned}
&\|\sqrt{\rho} u_{t}(t)\|_{L^2}^2+\int_{T_0}^{t}{\|
\nabla u_{t}(s)\|_{L^2}^2\,ds} \\
&\leq C\int_{T_0}^{t}{\|\nabla u\|_{L^2}^{4}(
\|\sqrt{\rho}u_{t}\|_{L^2}^2+\|\Delta u\|_{L^2}^2)\,ds} \\
&\leq C\int_{T_0}^{t}{\big(e+y(s)\big)^{2A\epsilon}(
\|\sqrt{\rho}u_{t}\|_{L^2}^2+\|\Delta u\|_{L^2}^2)\,ds} \\
&\leq C\big(e+y(t)\big)^{2A\epsilon}\int_{T_0}^{t}{(
\|\sqrt{\rho}u_{t}\|_{L^2}^2+\|\Delta u\|_{L^2}^2)\,ds} \\
&\leq C\big(e+y(t)\big)^{5A\epsilon}.
\end{aligned}
\end{equation}
Next, we split the range $30$, there exists $T_0=T_0(\epsilon)0, \quad (\text{see, e.g., \cite{YZ}}) \\
\|f\|_{\dot{H}^{1}}\approx \|f\|_{\dot{B}_{2,2}^{1}}\quad \text{and}\quad
\|u\|_{\dot{B}_{2,2}^{1+\delta}}
\leq C\|\nabla u\|_{L^2}^{1-\delta}\|\Delta u\|_{L^2}^{\delta},\quad 0<\delta<1.
\end{gather*}
Applying Stokes theorem once again gives
\begin{equation}\label{ZJt301}
\begin{aligned}
\|\Delta u\|_{L^{\frac{3}{2+\delta}}}
&\leq C(\|\rho u_{t}\|_{L^{\frac{3}{2+\delta}}}
+\|\rho u\cdot\nabla u\|_{L^{\frac{3}{2+\delta}}}) \\
&\leq C(\|\sqrt{\rho}\|_{L^{\frac{6}{1+2\delta}}}\|\sqrt{\rho } u_{t}\|_{L^2}
+\|\rho\|_{L^{\frac{3}{\delta}}}\|u\|_{L^{6}}\|\nabla u\|_{L^2}) \\
&\leq C(\|\sqrt{\rho} u_{t}\|_{L^2}+\|\nabla u\|_{L^2}^2).
\end{aligned}
\end{equation}
Thus, one deduces from \eqref{t206} and \eqref{ZJt301} that
\begin{equation}\label{t302}
\|\Delta u\|_{L^2}+\|\Delta u\|_{L^{\frac{3}{2+\delta}}}
\leq C(\|\sqrt{\rho} u_{t}\|_{L^2}+\|\nabla u\|_{L^2}^2+\|\nabla u\|_{L^2}^3).
\end{equation}
Multiplying the second equation of \eqref{NS} by $u_{t}$ and integrating
over whole space, one can obtain that for any $0<\delta<1$,
\begin{equation}\label{t303}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\|\nabla u(t)\|_{L^2}^2+\|\sqrt{\rho} u_{t}\|_{L^2}^2 \\
&\leq \big|\int_{\mathbb{R}^3}{\rho u\cdot \nabla u\cdot u_{t} \,dx}\big| \\
&\leq C\|\sqrt{\rho}\|_{L^{\infty}}\|u\cdot \nabla u\|_{L^2}\|\sqrt{\rho} u_{t}\|_{L^2}
\\
&\leq C\|\nabla\cdot (u\otimes u)\|_{L^2}\|\sqrt{\rho} u_{t}\|_{L^2}\quad \big(\operatorname{div} u=0\big)
\\
&\leq C\|uu\|_{\dot{B}_{2,2}^{1}}\|\sqrt{\rho} u_{t}\|_{L^2}
\\
&\leq C\|u\|_{\dot{B}_{\infty,\infty}^{-\delta}}\|u\|_{\dot{B}_{2,2}^{1+\delta}}\|\sqrt{\rho} u_{t}\|_{L^2} \\
&\leq C\|u\|_{\dot{B}_{\infty,\infty}^{-\delta}}\|\nabla u\|_{L^2}^{1-\delta}\|\Delta u\|_{L^2}^{\delta}\|\sqrt{\rho} u_{t}\|_{L^2} \\
&\leq C\|u\|_{\dot{B}_{\infty,\infty}^{-\delta}}\|\nabla u\|_{L^2}^{1-\delta}\big(\|\sqrt{\rho} u_{t}\|_{L^2}+\|u\|_{\dot{B}_{\infty,\infty}^{-\delta}}^{\frac{1}{1-\delta}}\|\nabla u\|_{L^2}\big)^{\delta}\|\sqrt{\rho} u_{t}\|_{L^2} \\
&\leq C\|u\|_{\dot{B}_{\infty,\infty}^{-\delta}}\|\nabla u\|_{L^2}^{1-\delta}\|\sqrt{\rho} u_{t}\|_{L^2}^{1+\delta}+C\|u\|_{\dot{B}_{\infty,\infty}^{-\delta}}^{\frac{1}{1-\delta}}\|\nabla u\|_{L^2}\|\sqrt{\rho} u_{t}\|_{L^2}
\\
&\leq
\frac{1}{2}\|\sqrt{\rho} u_{t}\|_{L^2}^2+C\|u\|_{\dot{B}_{\infty,\infty}^{-\delta}}
^{\frac{2}{1-\delta}}\|\nabla u\|_{L^2}^2.
\end{aligned}
\end{equation}
For any $t\in(T_0,T)$,
we denote
$$
y(t):=\max_{\tau\in[T_0,\, t]}
\|\Lambda^{\frac{3}{2}-\delta} u(\tau)\|_{L^2}.
$$
Applying Gronwall inequality to \eqref{t303}, we conclude
\begin{equation}\label{t304}
\begin{aligned}
&\|\nabla u(t)\|_{L^2}^2+\int_{T_0}^{t}{\|\sqrt{\rho} u_{t}(s)\|_{L^2}^2\,ds} \\
&\leq \|\nabla u(T_0)\|_{L^2}^2\exp
\Big[A\int_{T_0}^{t}{\|u(s)\|_{\dot{B}_{\infty,\infty}^{-\delta}}
^{\frac{2}{1-\delta}} \,ds}\Big] \\
&\leq C\|\nabla u(T_0)\|_{L^2}^2\exp
\Big[A\int_{T_0}^{t}{\frac{\|u(s)\|_{\dot{B}_{\infty,\infty}
^{-\delta}}^{\frac{2}{1-\delta}}}{\ln \big(e+\|u(s)\|_{\dot{B}_{\infty,\infty}
^{-\delta}}\big)}\ln \big(e+\|u(s)\|_{\dot{B}_{\infty,\infty}^{-\delta}}\big)\,ds}
\Big] \\
&\leq C\|\nabla u(T_0)\|_{L^2}^2\exp\Big[A\int_{T_0}^{t}
{\frac{\|u(s)\|_{\dot{B}_{\infty,\infty}^{-\delta}}^{\frac{2}{1-\delta}}
}{\ln \big(e+\|u(s)\|_{\dot{B}_{\infty,\infty}^{-\delta}}\big)}
\ln \big(e+\|\Lambda^{\frac{3}{2}-\delta}(s)\|_{L^2}\big)\,ds}\Big] \\
&\leq C\|\nabla u(T_0)\|_{L^2}^2\exp\Big[A\int_{T_0}^{t}
{\frac{\|u(s)\|_{\dot{B}_{\infty,\infty}^{-\delta}}^{\frac{2}{1-\delta}}
}{\ln \big(e+\|u(s)\|_{\dot{B}_{\infty,\infty}^{-\delta}}\big)}\ln
\big(e+y(s)\big)\,ds}\Big] \\
&\leq C\|\nabla u(T_0)\|_{L^2}^2\exp \Big[A\int_{T_0}^{t}
{\frac{\|u(s)\|_{\dot{B}_{\infty,\infty}^{-\delta}}^{\frac{2}{1-\delta}}
}{\ln \big(e+\|u(s)\|_{\dot{B}_{\infty,\infty}^{-\delta}}\big)}\,ds}
\cdot\ln \big(e+y(t)\big)\Big] \\
&\leq C\big(e+y(t)\big)^{A\epsilon},
\end{aligned}
\end{equation}
where we have used
$$
\|u\|_{\dot{B}_{\infty,\infty}^{-\delta}(\mathbb{R}^3)}
\leq C\|\Lambda^{\frac{3}{2}-\delta}u\|_{L^2(\mathbb{R}^3)},
$$
which can be easily derived by the Littlewood-Paley technique
with the Berstein inequality.
By \eqref{t302}, it is easy to see that
\[
\int_{T_0}^{t}{\|\Delta u\|_{L^2}^2(s)\,ds}
\leq C\int_{T_0}^{t}{(\|\sqrt{\rho} u_{t}\|_{L^2}^2+\|\nabla u\|_{L^2}^{6})(s)\,ds}
\leq C\big(e+y(t)\big)^{3A\epsilon},
\]
which together with \eqref{t304} imply
\begin{equation}
\|\nabla u(t)\|_{L^2}^{6}+\int_{T_0}^{t}{(\|\sqrt{\rho}u_{t}\|_{L^2}^2
+\|\Delta u\|_{L^2}^2)(s)\,ds}\leq C\big(e+y(t)\big)^{3A\epsilon}.
\end{equation}
With the same argument as in Section 2, one can infer that
$$
\frac{d}{dt}\|\sqrt{\rho} u_{t}(t)\|_{L^2}^2+\|
\nabla u_{t}\|_{L^2}^2
\leq C\|\nabla u\|_{L^2}^{4}(
\|\sqrt{\rho}u_{t}\|_{L^2}^2+\|\Delta u\|_{L^2}^2).
$$
Thus, integrating the above inequality over $[T_0,t]$ results in
(see also \eqref{t212})
\begin{equation}
\|\sqrt{\rho} u_{t}(t)\|_{L^2}^2+\int_{T_0}^{t}{\|
\nabla u_{t}(s)\|_{L^2}^2\,ds}
\leq C\big(e+y(t)\big)^{5A\epsilon},
\end{equation}
which along with \eqref{t302} give
\begin{equation}\label{t305}
\|\Delta u\|_{L^2}^2+\|\Delta u\|_{L^{\frac{3}{2+\delta}}}^2
\leq C(\|\sqrt{\rho} u_{t}\|_{L^2}^2+\|\nabla u\|_{L^2}^{4}
+\|\nabla u\|_{L^2}^{6})\leq C\big(e+y(t)\big)^{5A\epsilon}.
\end{equation}
Note the interpolation inequality
\begin{equation}\label{t306}
\|\Lambda^{\frac{3}{2}-\delta}u\|_{L^2(\mathbb{R}^3)}
\leq C\|\Delta u\|_{L^{\frac{3}{2+\delta}}(\mathbb{R}^3)},
\quad 0< \delta< 1.
\end{equation}
Thus, we conclude the following by combining the inequalities \eqref{t305}
and \eqref{t306}
$$
y(t)\leq C\big(e+y(t)\big)^{5A\epsilon}.
$$
The remainder proof is the same as the previous section.
Thus, this completes the proof of Theorem \ref{thm2}.
\section{roof of Theorem \ref{thm3}}
As above, we only establish several a priori estimates for the strong solutions.
Now we recall the following bilinear estimate which is an easy consequence
of \cite[Lemma 1]{YZ},
\begin{equation}\label{t401}
\|ff\|_{\dot{B}_{2,2}^{1}}\leq C\|f\|_{\dot{B}_{\infty,\infty}^{-1}}
\|f\|_{\dot{B}_{2,2}^2}.
\end{equation}
Applying the Stokes theorem (or \eqref{t206}) yields
\begin{equation}\label{t402}
\begin{aligned}
\|\Delta u\|_{L^2}
&\leq (\|\rho u_{t}\|_{L^2}+\|\rho u\cdot\nabla u\|_{L^2}) \\
&\leq (\|\rho u_{t}\|_{L^2}+\|\rho\|_{L^{\infty}}\|u\cdot\nabla u\|_{L^2}) \\
&\leq C(\|\sqrt{\rho} u_{t}\|_{L^2}+\|\nabla\cdot (u\otimes u)\|_{L^2})\quad
\big(\operatorname{div} u=0\big) \\
&\leq C(\|\sqrt{\rho} u_{t}\|_{L^2}+\|uu\|_{\dot{H}^{1}}) \\
&\leq C(\|\sqrt{\rho} u_{t}\|_{L^2}+\|uu\|_{\dot{B}_{2,2}^{1}}) \\
&\leq C(\|\sqrt{\rho} u_{t}\|_{L^2}+\|u\|_{\dot{B}_{\infty,\infty}^{-1}}
\|u\|_{\dot{B}_{2,2}^2})\quad \big(\text{see \eqref{t401}}\big)
\\
&\leq C\|\sqrt{\rho} u_{t}\|_{L^2}+C\|u\|_{\dot{B}_{\infty,\infty}^{-1}}
\|\Delta u\|_{L^2}.
\end{aligned}
\end{equation}
Thanks to condition \eqref{Small}, one has
$$
C\|u\|_{\dot{B}_{\infty,\infty}^{-1}}\leq\frac{1}{2},
$$
which leads to
\begin{equation}\label{t403}
\|\Delta u\|_{L^2}\leq C\|\sqrt{\rho} u_{t}\|_{L^2}.
\end{equation}
As a consequence, this gives
\begin{equation}\label{t404}
\begin{aligned}
\frac{1}{2}\frac{d}{dt}\|\nabla u(t)\|_{L^2}^2+\|\sqrt{\rho} u_{t}\|_{L^2}^2
&\leq \Big|\int_{\mathbb{R}^3}{\rho u\cdot \nabla u\cdot u_{t} \,dx}\Big| \\
&\leq \|\sqrt{\rho}\|_{L^{\infty}}\|u\cdot \nabla u\|_{L^2}\|\sqrt{\rho} u_{t}\|_{L^2}
\\
&\leq \|\nabla\cdot (u\otimes u)\|_{L^2}\|\sqrt{\rho} u_{t}\|_{L^2}\quad \big(\operatorname{div} u=0\big)
\\
&\leq C\|uu\|_{\dot{B}_{2,2}^{1}}\|\sqrt{\rho} u_{t}\|_{L^2}
\\
&\leq C\|u\|_{\dot{B}_{\infty,\infty}^{-1}}\|u\|_{\dot{B}_{2,2}^2}
\|\sqrt{\rho} u_{t}\|_{L^2} \\
&\leq C\|u\|_{\dot{B}_{\infty,\infty}^{-1}}\|\Delta u\|_{L^2}
\|\sqrt{\rho} u_{t}\|_{L^2} \\
&\leq C\|u\|_{\dot{B}_{\infty,\infty}^{-1}}\|\sqrt{\rho} u_{t}\|_{L^2}^2 \\
&\leq \frac{1}{2}\|\sqrt{\rho} u_{t}\|_{L^2}^2,
\end{aligned}
\end{equation}
which implies
$$
\frac{d}{dt}\|\nabla u(t)\|_{L^2}^2+\|\sqrt{\rho} u_{t}\|_{L^2}^2\leq0.
$$
Thus
$$
\|\nabla u(t)\|_{L^2}^2+\int_0^{t}{\|\sqrt{\rho} u_{t} (s)\|_{L^2}^2\,ds}
\leq \|\nabla u_0\|_{L^2}^2
\leq C<\infty
$$
for any $0\leq t< T$. As in proving Theorem \ref{thm1}, we
get the desired result. The proof of Theorem \ref{thm3} is complete.
\subsection*{Acknowledgements}
The authors would like to express their hearty thanks to the anonymous
referees for their insightful comments and many valuable suggestions,
which greatly improved the exposition of the manuscript.
\begin{thebibliography}{00}
\bibitem{Ab2007} H. Abidi;
\emph{\'Equation de Navier-Stokes avec densit\'e et viscosit\'e variables
dans l'espace critique}, Rev. Mat. Iberoam 23 (2007) 537-586.
\bibitem{AGZ2012} H. Abidi, G. Gui, P. Zhang;
\emph{On the wellposedness of three-dimensional inhomogeneous Navier-Stokes
equations in the critical spaces}, Arch. Rational Mech. Anal. 204 (2012) 189-230.
\bibitem{AKM1990} S. Antontesv, A. Kazhikov, V. Monakhov;
\emph{Boundary Value Problems in Mechanics of Nonhomogeneous Fluids},
North-Holland, Amsterdam (1990).
\bibitem{CK2003} H. Choe, H. Kim;
\emph{Strong solutions of the Navier-Stokes equations for nonhomogeneous
incompressible fluids}, Comm. Partial Differential Equations, 28 (2003) 1183-1201.
\bibitem{CK2004} Y. Cho, H. Kim;
\emph{Unique solvability for the density-dependent Navier-Stokes equations},
Nonlinear Anal., 59 (2004) 465-489.
\bibitem{CK2006} Y. Cho, H. Kim;
\emph{On classical solutions of the compressible Navier-Stokes equations with
nonnegative initial densities}, Manuscripta Math. 120 (2006) 91-129.
\bibitem{CHW2013} W. Craig, X. Huang, Y. Wang;
\emph{Global wellposedness for the 3D inhomogeneous incompressible Navier-Stokes
equations}, J. Math. Fluid Mech. 15 (2013) 747-758.
\bibitem{Dan2003} R. Danchin;
\emph{Density-dependent incompressible viscous fluids in critical spaces},
Proc. R. Soc. Edinburgh Sect. A 133 (2003) 1311-1334.
\bibitem{De1997} B. Desjardins;
\emph{Regularity results for two-dimensional flows of multiphase viscous fluids},
Arch. Rational Mech. Anal., 137 (1997) 135-158.
\bibitem{GHZ2011} G. Gui, J. Huang, P. Zhang;
\emph{Large global solutions to 3-D inhomogeneous Navier-Stokes equations
slowly varying in one variable}, J. Funct. Anal. 261 (2011) 3181-3210.
\bibitem{HW2013} X. Huang, Y. Wang;
\emph{Global strong solution to the 2D nonhomogeneous incompressible MHD
system}, Journal of Differential Equations, 254 (2013) 511-527.
\bibitem{Ka1974} A. Kazhikov;
\emph{Solvability of the initial-boundary value problem for the equations
of the motion of an inhomogeneous viscous incompressible fluid}, (Russian).
Dokl. Akad. Nauk SSSR 216 (1974) 1008-1010.
\bibitem{KIM} H. Kim;
\emph{A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes
equations}, SIAM J. Math. Anal. 37 (2006) 1417-1434.
\bibitem{KIM1987} J. Kim;
\emph{Weak solutions of an initial boundary value problem for an incompressible
viscous fluid with nonnegative density}, SIAM J. Math. Anal. 18 (1987) 8-96.
\bibitem{LS1978} O. Ladyzhenskaya, V. Solonnikov;
\emph{Unique solvability of an initial and boundary value problem for viscous
incompressible non-homogeneous fluids}, J. Soviet Math. 9 (1978) 697-749.
\bibitem{LZ} Z. Lei, Y. Zhou;
\emph{BKM's criterion and global weak solutions for
magnetohydrodynamics with zero viscosity}, Discrete Contin. Dyn.
Syst. 25 (2009), 575-583.
\bibitem{Li1998} P. Lions;
\emph{Mathematical topics in fluid mechanics. Incompressible models}.
Oxford Lecture Series in Mathematics and its Applications, 3. Oxford
Science Publications, vol. 1. Clarendon Press/Oxford University Press,
New York (1996).
\bibitem{Si1990} J. Simon;
\emph{Nonhomogeneous viscous incompressible fluids: Existence of velocity,
density, and pressure}, SIAM J. Math. Anal. 21 (1990) 1093-1117.
\bibitem{YX} Z. Ye, X. Xu;
\emph{A note on blow-up criterion of strong solutions for the 3D inhomogeneous
incompressible Navier-Stokes equations with vacuum}, Math. Phys. Anal. Geom.
18 (2015), no. 1, Art. 14, 10 pp.
\bibitem{YZ} B. Yuan, B. Zhang;
\emph{Blow-up criterion of strong solutions
to the Navier-Stokes equations in Besov spaces
with negative indices}, J. Differential Equations 242 (2007) 1-10.
\bibitem{ZZZ} P. Zhang, C. Zhao, J. Zhang;
\emph{Global regularity of the three-dimensional equations for
nonhomogeneous incompressible fluids}, Nonlinear Anal., 110 (2014) 61-76.
\end{thebibliography}
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