# Inhomogeneous incompressible viscous flows in vacuum: the whole-space case

Christophe Prange
Laboratoire de Mathématiques AGM, UMR CNRS 8088, Cergy Paris Université, 2
Avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France
<EMAIL_ADDRESS>and  Jin Tan
Laboratoire de Mathématiques AGM, UMR CNRS 8088, Cergy Paris Université, 2
Avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France<EMAIL_ADDRESS>
###### Abstract.

In this paper, we are concerned with the regularity and uniqueness issue of
Lions’ finite-energy weak solutions for the Inhomogeneous incompressible
Navier-Stokes equations in the presence of vacuum in the whole-space case
${\mathbb{R}}^{d}$, $d=2$ or $3$. We handle bounded and nonnegative initial
density and assume regularity only on the initial velocity. We address in
particular the case of the so-called _far-field vacuum_ or _density patch_ ,
which is particularly involved and was left open in previous studies.
Moreover, our work allows us to study the evolution of a vacuum bubble
embedded in an incompressible fluid, as well as a patch of a homogeneous fluid
embedded in the vacuum, which provides an answer to an open question of Lions
in the whole-space case.

###### Key words and phrases:

Inhomogeneous incompressible Navier-Stokes system; bounded density; vacuum;
far-field vacuum; vacuum bubbles; density patch; uniqueness; weak-strong
uniqueness; regularity of free-boundaries; Lion’s conjecture; whole-space

###### 2010 Mathematics Subject Classification:

35A02, 35Q30, 35R35, 76D05, 76D27

##  1\. Introduction

In the present paper, we are concerned with the following Inhomogeneous
incompressible Navier-Stokes equations (INS) in the whole-space
${\mathbb{R}}^{d}$ (with $d=2,3$)

(INS) $\left\\{\begin{aligned} &\partial_{t}\rho+\nabla\cdot(\rho u)=0,\\\
&\partial_{t}(\rho u)+\nabla\cdot(\rho u\otimes u)+\nabla P=\nu\Delta u,\\\
&\nabla\cdot u=0.\end{aligned}\right.$

in the presence of vacuum. The unknowns are the velocity field $u=u(t,x),$ the
density $\rho=\rho(t,x)$ and the pressure $P=P(t,x)$ and $\nu>0$ is the
viscosity constant.

The initial values are prescribed as follows:

(1.1) $\rho|_{t=0}=\rho_{0},\quad\rho u|_{t=0}=m_{0}.$

It will be assumed that the initial density $\rho_{0}$ is nonnegative and
bounded in $L^{\infty}({\mathbb{R}}^{d})$, but not necessarily bounded from
below by a positive constant. This allows us to study, for instance, the
evolution of a vacuum bubble embedded in an incompressible fluid, as well as
of a patch of a homogeneous fluid embedded in the vacuum. Let us mention that
this problem may be reformulated as a free-boundary problem. Lions in [42,
page 34] raised the following question about the persistence of the interface
regularity through the evolution:

> [In the case of a density patch $\rho=\mathbf{1}_{D(t)}$, the system] (INS)
> can be reformulated as a somewhat complicated free boundary problem. It is
> also very natural to ask whether the regularity of $D$ is preserved by the
> time evolution.

The question asked by Lions was solved by Danchin and Mucha in [16] in the
case of bounded domains or the Torus. These authors mention the whole-space
case as an open problem:

> [T]he generalization to unbounded domains (even the whole space) within our
> approach [is] unclear as regards global-in-time results.

One of the main motivations in this paper is to address that question. We give
a positive answer, see Theorem C below.

###  1.1. A brief state of the art

##### Global-in-time finite-energy weak solutions

Since the pioneering works by Leray [35] and Ladyzhenskaya [33] on the
incompressible homogeneous Navier-Stokes equations, the existence and
uniqueness issues of solutions for the inhomogeneous system (INS) has been
intensively investigated. Finite-energy weak solutions in the spirit of ‘Leray
solutions’ were built first by Kazhikov et al. [32, 5], in the case when
$\rho_{0}$ is bounded away from $0$. This result was extended by Simon [47]
allowing $\rho_{0}$ to vanish, in the case of bounded domains. Then, Lions
[42] (see also Desjardins [22, 23]) considered the so-called density-dependent
Navier-Stokes equations, i.e. the case where viscosity $\nu$ depends on
density $\rho,$ by using general results on transport equations obtained in
DiPerna and Lions’ work [25]. As a consequence, in the whole-space
${\mathbb{R}}^{d}$, $d=2$ or $3$, it is proved in Lions’s book [42], see
Theorem Theorem below, that global weak solutions such that $\rho$ is bounded
and $u\to u^{\infty}\quad{\rm as}~{}|x|\to\infty,\quad{\rm for~{}all}~{}t\geq
0$, exist provided that

(1.2) $\left\\{\begin{aligned} &\rho_{0}\geq 0\quad{\rm
a.e.~{}in}~{}{\mathbb{R}}^{d},\quad\rho_{0}\in
L^{\infty}({\mathbb{R}}^{d}),\\\ &m_{0}\in L^{2}({\mathbb{R}}^{d}),\quad
m_{0}=0\quad{\rm a.e.~{}on}~{}\\{\rho_{0}=0\\},\\\ &|m_{0}|^{2}/\rho_{0}\in
L^{1}({\mathbb{R}}^{d}).\end{aligned}\right.$

Because of the fact that the momentum equation is degenerate when there is
vacuum, Lions needs to assume in addition to (1.2), one of the following three
conditions:
_almost No Vaccum_

(aNV) $(1/\rho_{0})\,\mathbf{1}_{\rho_{0}<\delta_{0}}\in
L^{1}({\mathbb{R}}^{d}),\quad{\rm for~{}some}~{}\delta_{0}>0,$

or _Vacuum Bubble_

(VB) $(\underline{\rho}-\rho_{0})_{+}\in L^{p}({\mathbb{R}}^{d}),\quad{\rm
for~{}some}~{}\underline{\rho}\in(0,\infty),\,\,p\in(d/2,\infty),$

or _Far-Field Vacuum_

(FFV-1) $\displaystyle\left\\{\begin{aligned} &{\rm
if}~{}d=2,\quad\int_{{\mathbb{R}}^{2}}\rho_{0}^{p}\,\langle
x\rangle^{2(p-1)}(\log\langle x\rangle)^{r}\,dx<\infty\\\ &\hskip
56.9055pt\quad{\rm for~{}some}~{}p\in(1,+\infty]~{}{\rm and}~{}r>2p-1\\\ &{\rm
if}~{}d=3,\quad\rho_{0}\in
L^{\frac{3}{2},\infty}({\mathbb{R}}^{3}),\end{aligned}\right.$

where $\langle x\rangle:=(e+|x|^{2})^{1/2}.$

It is important to note that above three conditions allow several physically
interesting cases. For example, condition (aNV) allows the density to vanish
on zero-measure sets, condition (VB) allows vacuum bubbles, while condition
(FFV-1) allows density patches.

Let us give a precise definition of a weak solution to the Cauchy problem
associated to system (INS). The following definition and result are stated in
the book [42] of Lions.

######  Definition 1.1 ([42, Chapter 2]).

Let $T>0$. We say that $(\rho,{u})$ is a finite-energy weak solution of system
(INS) with the initial conditions (1.2), if $(\rho,{u})$ satisfies the
following properties:

$\displaystyle\rho\geq 0,\quad\rho\in
L^{\infty}((0,T)\times{\mathbb{R}}^{d}),\quad\rho\in\mathcal{C}([0,T];L^{p}_{\rm
loc}({\mathbb{R}}^{d}))~{}~{}\,\,{\rm for~{}all~{}}1\leq p<\infty,$
$\displaystyle\rho|{u}|^{2}\in L^{\infty}(0,T;L^{1}({{\mathbb{R}}^{d}})),$
$\displaystyle u\in
L^{2}(0,T;\widetilde{\mathcal{D}}^{1,2}{({\mathbb{R}}^{2})})~{}{\rm
if}~{}d=2\quad{\rm{or}}\quad u\in
L^{2}(0,T;\mathcal{D}^{1,2}({\mathbb{R}}^{3}))~{}{\rm if}~{}d=3,$

where the spaces

$\displaystyle\widetilde{\mathcal{D}}^{1,2}({\mathbb{R}}^{2}):=\\{z\in
H^{1}_{\rm loc}({\mathbb{R}}^{2});\nabla z\in
L^{2}({\mathbb{R}}^{2})\\},\quad{\mathcal{D}}^{1,2}({\mathbb{R}}^{3}):=\\{z\in
L^{6}({\mathbb{R}}^{3});\nabla z\in L^{2}({\mathbb{R}}^{3})\\},$

are endowed with the norms

$\|z\|_{\widetilde{\mathcal{D}}^{1,2}({\mathbb{R}}^{2})}:=\|\nabla
z\|_{L^{2}({\mathbb{R}}^{2})},\quad\|z\|_{\mathcal{D}^{1,2}({\mathbb{R}}^{3})}:=\|z\|_{L^{6}({\mathbb{R}}^{3})}+\|\nabla
z\|_{L^{2}({\mathbb{R}}^{3})}.$

If (aNV) or (VB) hold, we require in addition that $u\in
L^{2}((0,T)\times{\mathbb{R}}^{d}).$ Moreover, $(\rho,u)$ satisfies system
(INS) in the sense of distributions in $(0,\infty)\times{\mathbb{R}}^{d}$ and
the following energy inequality for a.e. $t\in(0,T)$

(1.3)
$\int_{{\mathbb{R}}^{d}}\rho(t,x)|u(t,x)|^{2}\,dx+2\nu\int_{0}^{t}\int_{{\mathbb{R}}^{d}}|\nabla
u(s,x)|^{2}\,dxds\leq\int_{{\mathbb{R}}^{d}}\frac{|m_{0}(x)|^{2}}{\rho_{0}(x)}\,dx.$

The solution $(\rho,{u})$ is a _global-in-time_ finite-energy weak solution of
system (INS) if the properties stated above hold for all $T\in(0,\infty)$.

######  Theorem ([42, Theorem 2.1]).

Assume that the initial data $(\rho_{0},u_{0})$ satisfies condition (1.2) and
one of the conditions (aNV)-(FFV-1), then there exists a global weak solution
$(\rho,u)$ of system (INS) in the sense of Definition 1.1. Furthermore, one
has for all $0\leq\alpha_{0}\leq\beta_{0}<\infty$

(1.4) $\displaystyle{\rm
meas}\\{x\in{\mathbb{R}}^{d}\,|\,\,\alpha_{0}\leq\rho(t,x)\leq\beta_{0}\\}~{}\,~{}{\rm
is~{}independent~{}of}~{}t\geq 0.$

And if $\rho_{0}-\rho^{\infty}\in L^{p}({\mathbb{R}}^{d})$ for some $1\leq
p<\infty,\,\rho^{\infty}\in[0,\infty),$ then
$\rho-\rho^{\infty}\in\mathcal{C}([0,\infty);L^{p}({\mathbb{R}}^{d}))$ and
$\|\rho-\rho^{\infty}\|_{L^{p}({\mathbb{R}}^{d})}$ is independent of $t\geq
0.$ Also, if $\rho\equiv\bar{\rho}$ for some $\bar{\rho}\in[0,\infty)$, then
$\rho_{0}\equiv\bar{\rho}$ on $[0,\infty)\times{\mathbb{R}}^{d}.$

The uniqueness of global weak solutions for system (INS) remains an open
question even in the 2D case, although several weak-strong uniqueness results
were established e.g. in Lions [42] and Germain [29].

##### Uniqueness in the absence of vacuum

Let us recall some recent developments on the unique solvability of the
inhomogeneous incompressible Navier-Stokes equations (INS) in the absence of
vacuum. Since the density is bounded below away from zero, this case is rather
close to the homogeneous flows (i.e. density is constant). Strong solutions
were first considered by Ladyzhenskaya and Solonnikov [34] in the bounded
domain case, whenever initial velocity and density are smooth enough and away
from vacuum.

After these early works, a number of papers were devoted to the study of
strong solutions to system (INS), with particular interest in classes of
initial data generating regular unique solutions. Here, an important feature
for system (INS) is the scaling invariance: if $(\rho,u,P)$ is a solution
associated to the initial data $(\rho_{0},u_{0})$, then $(\rho,\lambda
u,\lambda^{2}P)(\lambda^{2}t,\lambda x)$ is a solution associated to
$(\rho_{0},\lambda u_{0})(\lambda x),$ for all $\lambda>0.$ In the
critical111A functional space for the data $(\rho_{0},u_{0})$ or for the
solution $(\rho,u)$ is said to be critical if its norm is invariant under the
natural scaling of (INS). regularity framework, the local and global existence
results were first obtained by Danchin [12] in the case when initial density
has small variation, but still not including patches of density. Abidi [1] and
Abidi and Paicu [3] extended these results to the case with variable viscosity
in critical Besov spaces. Then, many efforts focused on removing these
smallness assumptions on the density, see for example [13, 7, 4, 48, 2].

For discontinuous densities, Danchin and Mucha [14] first proved well-
posedness results for data including initial density patches (that have small
variation) by a Lagrangian approach. Later, Paicu, Zhang and Zhang [46] (see
also [15]) established global unique solvability with only bounded initial
density, in addition bounded from below; see also further developments [31,
19, 49, 21] on initial velocity in critical functional spaces. These results
enable initial density of the type
$\rho_{0}=\rho_{1}\,\mathbf{1}_{\Omega}+\rho_{2}\,\mathbf{1}_{\Omega^{c}},$
where $\rho_{1},\rho_{2}$ are positive constants, $\Omega$ is a bounded domain
in ${\mathbb{R}}^{d}$ and $\mathbf{1}_{\Omega}$ is the characteristic function
of $\Omega.$ In connection with Lions’ question on the persistence of boundary
regularity of $\Omega,$ we refer to the works [20, 27, 39, 40, 41, 28, 45].

##### Uniqueness in the presence of vacuum

Concerning strong solutions allowing vacuum, local well-posedness was proved
by Choe and Kim [10] and Cho and Kim [9] under compatibility conditions. Later
that condition was removed by Li [36]. Recently, the work of Craig, Huang and
Wang [11] and Lü, Shi and Zhong [43] established global strong solutions. All
these results allow compactly supported initial densities, but still need to
be smooth enough so that discontinuous initial densities are not allowed.

A breakthrough on the global unique solvability for system (INS) with only
bounded and nonnegative initial density was made very recently by Danchin and
Mucha [16] in the case where the fluid domain is either bounded or the torus.
We also mention the stability result [18] of the density patches problem after
the paper [16].

###  1.2. Main results

The main goal of the present paper is to prove regularity and uniqueness
results of weak solutions for system (INS) with only bounded and nonnegative
initial density. Hence our results are extensions to the whole-space of the
results of Danchin and Mucha [16] (bounded domains or the Torus). We assume
the additional regularity $\widetilde{\mathcal{D}}^{1,2}({\mathbb{R}}^{2})$ or
$\mathcal{D}^{1,2}({\mathbb{R}}^{3})$ for the initial velocity.

Let us now state our main results. We first address the two-dimensional case.

######  Theorem A (existence and uniqueness in 2D).

Consider any initial data $(\rho_{0},u_{0})$ satisfying (1.2) such that
$\nabla u_{0}\in L^{2}({\mathbb{R}}^{2})$ and $\nabla\cdot u_{0}=0.$ Assume
that for some constant $\rho_{*}>0,$

(1.5) $0\leq\rho_{0}\leq\rho_{*}.$

Then, there are two cases.

$\triangleright$ _Case 1: $\rho_{0}$ satisfies either (aNV) or (VB)._
There exists a _unique global-in-time solution_ $(\rho,u)$ for the Cauchy
problem of system (INS) supplemented with the initial data $(\rho_{0},u_{0})$,
in the sense of Definition 1.1, satisfying in addition the following
regularity properties:

$\displaystyle 0\leq\rho\leq\rho_{*},\quad u\in
L^{\infty}({\mathbb{R}}_{+};H^{1}({\mathbb{R}}^{2})),\quad\sqrt{\rho}u_{t},~{}\Delta
u,~{}\nabla P\in L^{2}({\mathbb{R}}_{+}\times{\mathbb{R}}^{2}),$
$\displaystyle{\rm and~{}for~{}any}~{}T>0,~{}\sqrt{\rho t}u_{t}\in
L^{\infty}(0,T;L^{2}({\mathbb{R}}^{2})),~{}\nabla(\sqrt{t}u_{t})\in
L^{2}((0,T)\times{\mathbb{R}}^{2}),$
$\displaystyle\nabla^{2}(\sqrt{t}u),~{}\nabla(\sqrt{t}P)\in
L^{q}(0,T;L^{r}({\mathbb{R}}^{2})),~{}{\rm
for~{}all}~{}r\in[2,\infty),~{}q\in[2,{2r}{/(r-2)}).$

Moreover, we have222This $L^{1}_{t}Lip_{x}$ regularity is essential not only
for the uniqueness part of the statement, but also for Theorem C below.
$\nabla u\in L^{1}_{\rm loc}({\mathbb{R}}_{+};L^{\infty}({\mathbb{R}}^{2}))$.

$\triangleright$ _Case 2: neither ( aNV) nor (VB) are satisfied._
There still exists a _unique global-in-time solution_ $(\rho,u)$ for the
Cauchy problem of system (INS) supplemented with the initial data
$(\rho_{0},u_{0})$, in the sense of Definition 1.1, which satisfies333Here and
elsewhere in the paper $\dot{u}$ denotes the convective derivative, i.e.
$\dot{u}=(\partial_{t}+u\cdot\nabla)u$.

$\displaystyle 0\leq\rho\leq\rho_{*},~{}u\in L^{\infty}_{\rm
loc}({\mathbb{R}}_{+}\times{\mathbb{R}}^{2}),~{}\nabla u\in
L^{\infty}({\mathbb{R}}_{+};L^{2}({\mathbb{R}}^{2})),$
$\displaystyle\sqrt{\rho}\dot{u},~{}\Delta u,~{}\nabla P\in
L^{2}({\mathbb{R}}_{+}\times{\mathbb{R}}^{2}),$ $\displaystyle\sqrt{\rho
t}\dot{u}\in
L^{\infty}({\mathbb{R}}_{+};L^{2}({\mathbb{R}}^{2})),~{}\nabla(\sqrt{t}\dot{u})\in
L^{2}({\mathbb{R}}_{+}\times{\mathbb{R}}^{2}),$

and for any $T>0$,

$\displaystyle\nabla^{2}(\sqrt{t}u),~{}\nabla(\sqrt{t}P)\in
L^{q}(0,T;L^{r}({\mathbb{R}}^{2})),~{}{\rm
for~{}all}~{}r\in[2,\infty),~{}q\in[2,{2r}{/(r-2)}),$ $\displaystyle{\rm
and}~{}~{}\sqrt{\rho}\dot{u},~{}\Delta u,~{}\nabla P\in
L^{\infty}(0,T;L^{2}({\mathbb{R}}^{2})),~{}\nabla\dot{u}\in
L^{2}(0,T;{\mathbb{R}}^{2}),$

provided that $(\rho_{0},u_{0})$ additionally satisfies the following _Far-
Field Vacuum_ conditions444Notice that the parameter $\alpha>1$ in (FFV-2) is
fixed throughout the paper.

(FFV-2) $\displaystyle\bar{x}^{\alpha}\rho_{0}\in L^{1}({\mathbb{R}}^{2})\cap
L^{\infty}({\mathbb{R}}^{2})\quad{\rm for~{}some}~{}\alpha>1\quad{\rm
with}\quad\bar{x}:=\langle x\rangle\,(\ln\langle x\rangle)^{2}$

and the _Compatibility_ condition

(Compa) $\displaystyle-\Delta u_{0}+\nabla P_{0}=\sqrt{\rho_{0}}g,\quad{\rm
for}~{}~{}u_{0}\in L^{2}({\mathbb{R}}^{2}),\ \nabla u_{0}\in
L^{1}({\mathbb{R}}^{2}),~{}g\in L^{2}({\mathbb{R}}^{2}).$

Moreover, we have $\nabla u\in L^{1}_{\rm
loc}({\mathbb{R}}_{+};L^{\infty}({\mathbb{R}}^{2}))$.

Next, we state our result in the three-dimensional case.

######  Theorem B (existence and uniqueness in 3D).

Consider any initial data $(\rho_{0},u_{0})$ satisfying (1.2) such that
$u_{0}\in\mathcal{D}^{1,2}({\mathbb{R}}^{3})$, $\nabla\cdot u_{0}=0$ and
$\rho_{0}$ satisfies (1.5). Then system (INS) supplemented with initial data
$(\rho_{0},u_{0})$ admits a _unique_ solution $(\rho,u)$ on the time interval
$(0,T_{0})$, in the sense of Definition 1.1, satisfying in addition the
following regularity properties:555The $L^{1}_{t}Lip_{x}$ regularity below is
essential for Theorem C.

$\displaystyle 0\leq\rho\leq\rho_{*},~{}u\in
L^{\infty}(0,T_{0};\mathcal{D}^{1,2}({\mathbb{R}}^{3})),~{}\sqrt{\rho}u_{t},~{}\Delta
u\in L^{2}((0,T_{0})\times{\mathbb{R}}^{3}),$ $\displaystyle P\in
L^{2}(0,T_{0};\mathcal{D}^{1,2}({\mathbb{R}}^{3})),\ \nabla u\in
L^{1}(0,T_{0};L^{\infty}({\mathbb{R}}^{3})),$ $\displaystyle\sqrt{\rho
t}u_{t}\in L^{\infty}(0,T_{0};L^{2}({\mathbb{R}}^{3})),~{}\sqrt{t}u_{t}\in
L^{2}(0,T_{0};\mathcal{D}^{1,2}({\mathbb{R}}^{3})),$
$\displaystyle\nabla^{2}(\sqrt{t}u),~{}\nabla(\sqrt{t}P)\in
L^{q}(0,T_{0};L^{r}({\mathbb{R}}^{3})),~{}\mbox{for
all}~{}r\in[2,6],~{}q\in[2,{4r}/{(3r-6)}],$

where $T_{0}:=\frac{c\nu^{3}}{\rho_{*}^{3}\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{3})}^{4}}$ for some universal constant $c$.
Finally, there exists a universal constant $c_{0}>0$ such that if

(1.6)
$\displaystyle\rho_{*}^{\frac{3}{2}}\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{3})}\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{3})}\leq c_{0}\nu^{2},$

then the local-in-time solution can be extended globally-in-time.

######  Remark 1.2.

The initial condition $\sqrt{\rho_{0}}u_{0},\nabla u_{0}\in
L^{2}({\mathbb{R}}^{d})$ together with assumption (aNV) or (VB) imply that
$u_{0}\in L^{2}({\mathbb{R}}^{d})$, $d=2$ or $3$. Under (aNV) or (VB), there
exists a positive constant $C_{*}$ depending only on the factors in condition
(aNV) or (VB) such that on the time interval $(0,\infty)$ ($d=2$) and on the
life-span $[0,T_{*})$ of the solution in Theorem B ($d=3$), inequality (A.6)
implies

(1.7) $\displaystyle\|u(t,\cdot)\|_{L^{2}({\mathbb{R}}^{d})}\leq
C_{*}\big{(}\|\sqrt{\rho}u(t,\cdot)\|_{L^{2}({\mathbb{R}}^{d})}+\|\nabla
u(t,\cdot)\|_{L^{2}({\mathbb{R}}^{d})}\big{)}.$

In the two-dimensional far-field vacuum case, i.e. under (FFV-2) and (Compa),
we also have $u_{0}\in L^{2}({\mathbb{R}}^{2})$ by assumption. However, we
cannot show that $u(t,\cdot)\in L^{2}({\mathbb{R}}^{2})$. Indeed, instead of
the interpolation inequality, we have the weighted estimate (A.3).

As a by-product, we obtain the following result, which give a positive answer
to Lions’ question ([42, page 34], see above in the preamble) in the whole-
space case. As mentioned above the case of bounded domains or the Torus was
solved in [16].

######  Theorem C (solution to Lions’ problem).

Assume $\gamma\in(0,1)$ if $d=2$, and $\gamma\in(0,\frac{1}{2})$ if $d=3.$ Let
$\Omega_{0}\subset{\mathbb{R}}^{d}$ be a bounded simply connected domain with
boundary $\partial\Omega_{0}\in\mathcal{C}^{1,\gamma}$. Suppose that the
initial velocity satisfies all the conditions in Theorem A or Theorem B, and
initial density

$\rho_{0}(x)=1-\mathbf{1}_{\Omega_{0}}(x)\quad{\rm
or}\quad\rho_{0}(x)=\mathbf{1}_{\Omega_{0}}(x),\quad x\in{\mathbb{R}}^{d}.$

Then for each case the unique global solution $(\rho,u)$ of system (INS)
provided by previous theorems satisfies, respectively,

$\displaystyle\rho(t,x)=1-\mathbf{1}_{\Omega_{t}}(x)\quad{\rm
or}\quad\rho(t,x)=\mathbf{1}_{\Omega_{t}}(x)\quad{\rm
with}\quad\partial\Omega_{t}\in\mathcal{C}^{1,\gamma},$

where $\Omega_{t}:=X(t,\Omega_{0})$ and $X$ is the flow associated to the
velocity $u,$ that is, the unique solution to

$\displaystyle X(t,y)=y+\int_{0}^{t}u(s,X(s,y))\,ds,\quad
y\in{\mathbb{R}}^{d}.$

######  Remark 1.3.

Here, contrary to Theorem A and Theorem B, only one bubble is allowed. As a
consequence, the case of two bubbles is open.

###  1.3. A few remarks on the main results

##### Main novelties

First, our results settle the uniqueness question for the difficult two-
dimensional system in the presence of far-field vacuum. Notice that this is a
novelty of the whole-space case because in the bounded domain and Torus case
treated previously by Danchin and Mucha [16] there is no far-field vacuum.
Notice that the far-field vacuum case is not accessible by the methods of
[16].

Second, let us stress that in Theorem A and Theorem B, the density is bounded
but does not satisfy any smoothness assumption. Notice that this is not the
case of other works concerned with the whole-space. For example, in [43] the
initial density satisfies the condition

$\bar{x}^{\alpha}\rho_{0}\in L^{1}({\mathbb{R}}^{2})\cap
H^{1}({\mathbb{R}}^{2})\cap W^{1,p}({\mathbb{R}}^{2}),\quad p>2,$

while in [9] the authors assume that $\rho_{0}\in W^{1,p}({\mathbb{R}}^{d})$,
with $p>d$, $d=2$ or $3$. These conditions do not allow for non smooth
densities with jump discontinuities like patches or vacuum bubbles.

Third, we emphasize that in both theorems, Theorem A and Theorem B, we
construct finite-energy weak solutions in the sense of Definition 1.1. These
solutions have additional regularity properties that follow from the
assumption that the initial velocity belongs to the spaces
$\widetilde{\mathcal{D}}^{1,2}({\mathbb{R}}^{2})$ or
$\mathcal{D}^{1,2}({\mathbb{R}}^{3})$. In this framework, we can prove the
uniqueness of the solutions that we construct. Notice that we do not rely on
Lions’ theorem of existence of finite-energy weak solutions (Theorem Theorem
above), but prove the existence of the solutions with the properties stated
above, see Section 2 for the existence part of Theorem B in the three-
dimensional case, and see Section 3 for the existence part of Theorem A in the
two-dimensional case. Finally, let us also remark that condition (FFV-1) is
not needed neither in 2D nor 3D. In 2D, the far-field vacuum condition (FFV-1)
is replaced by the condition (FFV-2) and (Compa).

##### Some further estimates

We state here certain further boundedness properties for the velocity that can
be obtained from the estimates of the paper.

In 3D under (aNV) or (VB), we already know, see Theorem B and Remark 1.2 that
$u\in L^{\infty}_{t}L^{2}_{x}$ on the life-span of the solution. This in
combination with the techniques of the paper and the following compatibility
assumption

(Compa-3D) $\displaystyle-\Delta u_{0}+\nabla P_{0}=\sqrt{\rho_{0}}g,\quad{\rm
for}~{}g\in L^{2}({\mathbb{R}}^{2}).$

gives that $u\in L^{\infty}_{t,x}$ on the life-span of the solution.

In 2D under (aNV) or (VB), we already know, see Theorem A and Remark 1.2 that
$u\in L^{\infty}(0,\infty;L^{2}({\mathbb{R}}^{2}))$. Combining this with the
assumption (Compa) gives that $u\in
L^{\infty}({\mathbb{R}}_{+}\times{\mathbb{R}}^{2})$. In the case when neither
(aNV) nor (VB), we get at best the weighted-boundedness of the velocity

(1.8)
$\sup_{t\in[0,T]}\|\bar{x}^{-b}{u}(t,\cdot)\|_{L^{\infty}({\mathbb{R}}^{2})}<\infty,$

see Proposition 3.8 under (FFV-2) and (Compa).

###  1.4. Main difficulties and strategy for the proofs

First of all, let us point to the main difficulties. Those emerge from the
facts that: (i) the density is rough (our analysis includes density patches
and vacuum bubbles) and (ii) the density may vanish on some part of
${\mathbb{R}}^{d}$.

##### Dealing with the vacuum

If there is vacuum at initial time, the velocity equation is degenerate and it
becomes particularly difficult to gain control of the velocity itself. It is
important to note here that there is an essential difference between the two-
and three-dimensional cases. In the three-dimensional case, see Theorem B, we
can propagate the assumption that
$u_{0}\in\mathcal{D}^{1,2}({\mathbb{R}}^{3})$ and hence show that $u\in
L^{\infty}((0,T_{0});\mathcal{D}^{1,2}({\mathbb{R}}^{3}))$. Hence $u\in
L^{\infty}((0,T_{0});L^{6}({\mathbb{R}}^{3}))$ by Sobolev’s embedding. In the
two-dimensional case in the presence of vacuum, the quantity that is naturally
controlled by the energy estimate (1.3) is $\sqrt{\rho}u$, which generally
does not ensure that the velocity $u\in L^{p}({\mathbb{R}}^{2})$ for some
$p\in[1,\infty]$. Hence, one needs to find a way to control $u$ from
$\|\sqrt{\rho}u(t,\cdot)\|_{L^{2}({\mathbb{R}}^{2})}$ and $\|\nabla
u(t,\cdot)\|_{L^{2}({\mathbb{R}}^{2})}$. Such estimates can be proved, see
Appendix A, under specific assumptions that control the size of the vacuum
region and the behavior of the density near the vacuum region. In the most
favorable case, i.e. condition (aNV) or (VB), one can show a direct
interpolation estimate for the $L^{2}$ norm of the velocity in terms of
$\|\sqrt{\rho}u(t,\cdot)\|_{L^{2}({\mathbb{R}}^{2})}$ and $\|\nabla
u(t,\cdot)\|_{L^{2}({\mathbb{R}}^{2})}$, see Proposition A.6. In the case of
the far-field vacuum assumption, which is the most difficult case handled in
our work, one can only show an estimate for weighted $L^{m}$ norms of the
velocity, $m\geq 2$, see Proposition A.4.

##### The compatibility condition for the far-field vacuum case

A compatibility condition in the spirit of (Compa) first appeared in the works
[9, 10]. Roughly speaking, such a condition boils down to assuming that
$\sqrt{\rho}\partial_{t}u$ belongs to $L^{2}({\mathbb{R}}^{2})$ at initial
time. The compatibility condition was removed in the recent papers [36, 43].
In these works the existence and uniqueness of strong solutions is proved.
However, without a compatibility condition, one is unable to infer certain
information, such as continuity in time, on the velocity near initial time.
One barely has time weighted estimates on the velocity.

To handle the two-dimensional far-field vacuum (Theorem A (Case 2)), which is
the most difficult case, we show persistence of the initial condition (FFV-2).
Doing so, we can only obtain bounds for $\rho u$ and time-weighted estimates
of $u$. Condition (Compa) is then crucial to get the space-weighted
boundedness of the velocity near initial time, see (1.8) and Proposition 3.8,
which in turn is key for the uniqueness in Subsection 4.2.

The compatibility condition (Compa) creates important difficulties when trying
to approximate the initial data to construct an approximate sequence of smooth
solutions. Our construction in Subsubsection 3.2.1 is inspired by [10].
However, since we apply existence results for smooth data, we need to further
regularize the right-hand-side $\sqrt{\rho_{0}}g$ of (Compa). We then require
that $u_{0}\in L^{2}({\mathbb{R}}^{2})$ and $\nabla u_{0}\in
L^{1}({\mathbb{R}}^{2})$ so as to be able to show the appropriate convergence
and boundedness properties of the constructed sequence of approximate data.

##### Outline of the proofs

The general strategy is that of proofs à la Hoff [30].

We first get lower-order estimates on the velocity. We give propagate the
$\widetilde{\mathcal{D}}^{1,2}({\mathbb{R}}^{2})$ or the
$\mathcal{D}^{1,2}({\mathbb{R}}^{3})$ regularity of the initial data for the
velocity. In the three-dimensional case, this estimate directly yields a
$L^{\infty}_{t}L^{6}_{x}$ bound on $u$. In the two-dimensional case, these
estimates, in combination with the interpolation estimate of Proposition A.6
(Case 1 of Theorem A) or the weighted interpolation estimate of Proposition
3.8 (Case 2 of Theorem A), lead to estimates of the velocity itself.

The next step is to get higher-order estimates on the velocity. We carry out
time-weighted estimates of $\sqrt{\rho}\dot{u}$. Finally we transfer (shift)
the integrability of $\sqrt{\rho}\dot{u}$ to second-order derivatives of the
velocity via maximal regularity estimates for the stationary Stokes system.
This enables us to get the crucial Lipschitz estimate for the velocity.

The Lipschitz estimate for the velocity is key to the uniqueness proofs. To
show the uniqueness, we carry out a duality proof, which is a new approach in
this context. In particular our approach differs from the Lagrangian approach
used in [16].

For more details concerning the relationship between the results in the paper,
see Figure 1.

Figure 1. Relationships between the results in the paper

###  1.5. Some notations and assumptions

Throughout, we use $B_{n}$ to denote the ball
$\\{x\in{\mathbb{R}}^{d};~{}|x|\leq n\\}.$ The notation $(f)_{+}=\max(f,0)$
stands for the positive part of a function $f$. The japanese bracket is
defined as follows $\langle x\rangle:=(e+|x|^{2})^{1/2}$ and we recall that
the notation $\bar{x}$ stands for $\bar{x}:=\langle x\rangle\,(\ln\langle
x\rangle)^{2}$. We define material derivative
$\dot{v}:=\partial_{t}v+u\cdot\nabla v$ for functions
$v:{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}.$ Sometimes, we
denote $\partial_{t}v$ by $v_{t}.$ For simplicity, we will fix the viscosity
to be $1$ thanks to a standard rescaling argument.

###  1.6. Outline of the paper

Section 2 is concerned with the three-dimensional case. It is devoted to the
proof of the existence part of Theorem B. Section 3 handles the two-
dimensional case. It is devoted to the existence part of Theorem A. The proof
is more involved, especially in the far-field vacuum case, than in 3D. The
proof of uniqueness partis of Theorem A and Theorem B is carried out in
Section 4. Theorem C is proved in Section 5. Some frequently used functional
spaces and essential interpolation inequalities are recalled in Appendix A.

##  2\. Proof of the existence results in the three-dimensional case, Theorem
B

This section mainly concerns the proof of the existence part in Theorem B (3D
case). The idea is to take advantage of classical results to construct smooth
approximate solutions without vacuum. After that, we show persistence of
$\mathcal{D}^{1,2}({\mathbb{R}}^{3})$ regularity and some time-weighted
estimates for time derivatives of the velocity independent of the lower bounds
of the density. Those estimates will enable us to shift the regularity to
obtain $L^{1}_{t}Lip_{x}$ estimates for the velocity, which is important for
our proof of uniqueness. Finally, we pass to the limit via classical
compactness argument.

Recall that the initial data $(\rho_{0},u_{0})$ satisfies conditions (1.2) and
(1.5), and that $u_{0}\in\mathcal{D}^{1,2}({\mathbb{R}}^{3}),\nabla\cdot
u_{0}=0.$ Thus, we consider smoothed-out initial data

$\displaystyle\rho_{0}^{\varepsilon}\in\mathcal{C}^{\infty}({\mathbb{R}}^{3})~{}\,\,{\rm
with}~{}\,\,\varepsilon\leq\rho_{0}^{\varepsilon}\leq
2\rho_{*}u_{0}^{\varepsilon}\in\mathcal{C}_{0}^{\infty}({\mathbb{R}}^{3})~{}\,\,{\rm
with}~{}\,\,\nabla\cdot u_{0}^{\varepsilon}=0$

such that as $\varepsilon\to 0^{+}$

$\displaystyle u_{0}^{\varepsilon}\to u_{0}\quad{\rm
in}~{}~{}\mathcal{D}^{1,2}({\mathbb{R}}^{3}),\quad\sqrt{\rho_{0}^{\varepsilon}}u_{0}^{\varepsilon}\,\rightharpoonup\,\sqrt{\rho_{0}}u_{0}\quad{\rm
in}~{}~{}L^{2}({\mathbb{R}}^{3})$

and

$\displaystyle\rho_{0}^{\varepsilon}\rightharpoonup\rho_{0}\quad{\rm
weak*~{}in}~{}~{}L^{\infty}({\mathbb{R}}^{3}).$

Then, by the strong solution theory (e.g. [13, Theorem 0.2]), we know that
there exists a local-in-time strong solution
$(\rho^{\varepsilon},u^{\varepsilon})$ to (INS).

###  2.1. Uniform estimates

In what follows, we focus on uniform estimates for the approximate solutions
$(\rho^{\varepsilon},u^{\varepsilon})$. We often use the following Sobolev
embedding

(2.1) $\|z\|_{L^{6}({\mathbb{R}}^{3})}\leq C\|\nabla
z\|_{L^{2}({\mathbb{R}}^{3})}\quad{\rm
for}~{}~{}z\in\dot{H}^{1}({\mathbb{R}}^{3}),$

with a universal positive constant $C$. For notational simplicity, we omit the
superscript $\varepsilon$ and write the solution $(\rho,u)$.

####  2.1.1. Persistence of $\mathcal{D}^{1,2}{({\mathbb{R}}^{3})}$ regularity
for the velocity

######  Proposition 2.1 (gradient estimate).

Let $(\rho,u)$ be a smooth enough solution to system (INS) on
$[0,T_{*})\times{\mathbb{R}}^{3}.$ There exist positive universal constants
$c_{0},C$ such that if

(2.2)
$\rho_{*}^{\frac{3}{2}}\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{3})}\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{3})}\leq c_{0}$

then it holds for all $t\in[0,T_{*})$

(2.3)
$\|u(t,\cdot)\|_{\mathcal{D}^{1,2}({\mathbb{R}}^{3})}^{2}+2\int_{0}^{t}\|(\sqrt{\rho}{u}_{t},\Delta
u)(s,\cdot)\|_{L^{2}({\mathbb{R}}^{3})}^{2}\,ds+2\int_{0}^{t}\|P(s,\cdot)\|_{\mathcal{D}^{1,2}({\mathbb{R}}^{3})}^{2}\,ds\\\
\leq\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{3})}^{2}\exp(C\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{3})}^{2}\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{3})}^{2}).$

If (2.2) is not satisfied, then (2.3) holds true on $[0,T]$ provided that

(2.4) $T\leq\frac{c}{\rho_{*}^{3}\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{3})}^{4}}\quad{\rm
for~{}some~{}universal~{}constant}~{}c.$

###### Proof.

In order to estimate the second derivative of $u$ and the gradient of the
pressure, we rewrite the momentum equation of (INS) in the form

(2.5) $\left\\{\begin{aligned} &-\Delta u+\nabla P=-\rho\dot{u}\hskip
33.28966pt{\rm in}~{}(0,T)\times{\mathbb{R}}^{3},\\\ &\nabla\cdot u=0\hskip
85.07388pt{\rm in}~{}(0,T)\times{\mathbb{R}}^{3},\\\ &u\to 0,\quad{\rm
as}~{}|x|\to\infty.\end{aligned}\right.$

Applying the standard $L^{2}$-estimate to (2.5) yields that

(2.6) $\displaystyle\|\Delta u\|_{L^{2}({\mathbb{R}}^{3})^{2}}^{2}+\|\nabla
P\|_{L^{2}({\mathbb{R}}^{3})}^{2}\leq
C\Big{(}{\rho_{*}}\|\sqrt{\rho}u_{t}\|_{L^{2}({\mathbb{R}}^{3})}^{2}+\rho_{*}^{2}\|u\cdot\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}^{2}\Big{)}.$

Testing the momentum equation of (INS) against ${u_{t}}$ yields

$\displaystyle\frac{1}{2}\frac{d}{dt}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}^{2}+\|\sqrt{\rho}{u}_{t}\|_{L^{2}({\mathbb{R}}^{3})}^{2}$
$\displaystyle=-\langle\rho u\cdot\nabla u,u_{t}\rangle.$

Hence, by Hölder’s inequality

$\displaystyle\frac{1}{2}\frac{d}{dt}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}^{2}+\|\sqrt{\rho}{u}_{t}\|_{L^{2}({\mathbb{R}}^{3})}^{2}\leq$
$\displaystyle\|\sqrt{\rho}u\cdot\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}\|\sqrt{\rho}u_{t}\|_{L^{2}({\mathbb{R}}^{3})}$
$\displaystyle\leq$ $\displaystyle C\rho_{*}\|u\cdot\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}^{2}+\frac{1}{4}\|\sqrt{\rho}u_{t}\|_{L^{2}({\mathbb{R}}^{3})}^{2}$

Putting the above inequality together with (2.6) implies

(2.7) $\displaystyle\frac{d}{dt}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}^{2}+\frac{1}{\rho_{*}}\|(\sqrt{\rho}{u}_{t},\Delta
u,\nabla P)\|_{L^{2}({\mathbb{R}}^{3})}^{2}\leq$ $\displaystyle
C\rho_{*}\|u\|_{L^{6}({\mathbb{R}}^{3})}^{2}\|\nabla
u\|_{L^{3}({\mathbb{R}}^{3})}^{2}$ $\displaystyle\leq$ $\displaystyle
C\rho_{*}\|\nabla u\|_{L^{2}({\mathbb{R}}^{3})}^{3}\|\Delta
u\|_{L^{2}({\mathbb{R}}^{3})}$ $\displaystyle\leq$ $\displaystyle
C\rho_{*}^{3}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}^{6}+\frac{1}{2\rho_{*}}\|\Delta
u\|_{L^{2}({\mathbb{R}}^{3})}^{2},$

in which inequality (2.1) and the Gagliardo-Nirenberg inequality (A.1) were
used. Therefore, we see that whenever $T$ satisfies $C\rho_{*}^{3}\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{3})}^{2}\int_{0}^{T}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}^{2}\,ds<1,$ we have

$\displaystyle\|\nabla
u(t,\cdot)\|_{L^{2}({\mathbb{R}}^{3})}^{2}\leq\frac{\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{3})}^{2}}{1-C\rho_{*}^{3}\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{3})}^{2}\int_{0}^{T}\|\nabla
u(s,\cdot)\|_{L^{2}({\mathbb{R}}^{3})}^{2}\,ds}\quad{\rm
for~{}all}~{}t\in[0,T].$

Now, the energy inequality (1.3) implies that

$\displaystyle\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{3})}^{2}\int_{0}^{t}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}^{2}\,ds\leq\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{3})}^{2}\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{3})}^{2}\leq\frac{1}{2C\rho_{*}^{3}}\quad{\rm
for~{}any}~{}t\geq 0$

provided that $(\sqrt{\rho_{*}})^{3}\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{3})}\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{3})}\leq
c_{0}$ for small enough $c_{0}.$ Using inequality (2.1) again, we get (2.3).

In the case when the smallness condition is not satisfied, we get from (2.7)
that

$\displaystyle\|\nabla
u(t,\cdot)\|_{L^{2}({\mathbb{R}}^{3})}^{4}\leq\frac{\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{3})}^{4}}{1-2C\rho_{*}^{3}t\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{3})}^{4}},$

which eventually enables us to finish the proof of the second statement. ∎

######  Remark 2.2 (boundedness in space of the velocity).

Notice that thanks to inequality (2.3) we also have

$\displaystyle\int_{0}^{T}\|u(t,\cdot)\|_{L^{\infty}({\mathbb{R}}^{3})}^{4}\,dt$
$\displaystyle\leq\int_{0}^{T}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}^{2}\|\nabla^{2}u\|_{L^{2}({\mathbb{R}}^{3})}^{2}\,dt$
$\displaystyle\leq\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{3})}^{4}\exp(C\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{3})}^{2}\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{3})}^{2}).$

####  2.1.2. Estimates of the time derivative

Here, we want to bound time derivative $\sqrt{\rho t}{u}_{t}$ in the space
$L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))$ and $\sqrt{t}\nabla{u}_{t}$ in the
space $L^{2}((0,T)\times{\mathbb{R}}^{3}),$ since it is an important step
towards the proof of existence and higher-order spatial estimates for the
velocity.

######  Proposition 2.3 (time derivative estimates).

Let $(\rho,u)$ be a smooth enough solution to system (INS) on
$[0,T_{*})\times{\mathbb{R}}^{3}.$ Then for all $T\in[0,T_{*})$ it holds

(2.8) $\sup_{t\in[0,T]}\|\sqrt{\rho
t}{u}_{t}\|_{L^{2}({\mathbb{R}}^{3})}^{2}+\int_{0}^{T}\|\sqrt{t}{u}_{t}\|_{\mathcal{D}^{1,2}({\mathbb{R}}^{3})}^{2}\,ds\leq
C_{0,T},$

where $C_{0,T}$ is a constant depending only on $T,\rho_{*}$ and norms
$\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{3})},\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{3})}.$

###### Proof.

At first, applying the time derivative $\partial_{t}$ to the momentum equation
in system (INS) and multiplying the resulting equation by $\sqrt{t}$ yields

(2.9)
$\rho\big{(}\partial_{t}(\sqrt{t}{u}_{t})+u\cdot\nabla(\sqrt{t}{u}_{t})\big{)}-\Delta(\sqrt{t}{u}_{t})+\nabla(\sqrt{t}{P}_{t})\\\
=\frac{1}{2\sqrt{t}}\rho
u_{t}-\sqrt{t}\rho_{t}u_{t}-\sqrt{t}\rho_{t}u\cdot\nabla u-\sqrt{t}\rho
u_{t}\cdot\nabla u.$

Taking the $L^{2}$ scalar product with $\sqrt{t}{u}_{t}$, we get

(2.10) $\displaystyle\frac{1}{2}\frac{d}{dt}\|\sqrt{\rho
t}{u}_{t}\|_{L^{2}}^{2}+\|\nabla(\sqrt{t}{u}_{t})\|_{L^{2}}^{2}=A_{1}+\dots+A_{4}$

where

$\displaystyle
A_{1}:=\frac{1}{2}\|\sqrt{\rho}{u}_{t}\|_{L^{2}({\mathbb{R}}^{3})}^{2},$
$\displaystyle A_{2}:=-\langle t\rho_{t},|u_{t}|^{2}\rangle,$ $\displaystyle
A_{3}:=-\langle\sqrt{t}\rho_{t}u\cdot\nabla u,\sqrt{t}{u}_{t}\rangle,$
$\displaystyle A_{4}:=-\langle\sqrt{t}\rho u_{t}\cdot\nabla
u,\sqrt{t}{u}_{t}\rangle.$

In order to estimate $A_{1},A_{2},A_{3},A_{4},$ we proceed as follows.
Noticing that

$\displaystyle A_{1}$
$\displaystyle=\frac{1}{2}\|\sqrt{\rho}(\dot{u}-u\cdot\nabla
u)\|_{L^{2}({\mathbb{R}}^{3})}^{2}$
$\displaystyle\leq\|\sqrt{\rho}\dot{u}\|_{L^{2}({\mathbb{R}}^{d})}^{2}+\rho_{*}\|u\|_{L^{\infty}({\mathbb{R}}^{3})}^{2}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}^{2}.$

Using the equation
$\dot{\rho}=\partial_{t}\rho+u\cdot\nabla\rho=\partial_{t}\rho+\nabla\cdot(\rho
u)=0,$ we write

$\displaystyle A_{2}=-\langle t\rho u,\nabla(|u_{t}|^{2})\rangle$

and

$\displaystyle|A_{2}|\leq$ $\displaystyle
2\sqrt{\rho_{*}}\|u\|_{L^{\infty}({\mathbb{R}}^{3})}\|\sqrt{\rho
t}u_{t}\|_{L^{2}({\mathbb{R}}^{3})}\|\nabla(\sqrt{t}u_{t})\|_{L^{2}({\mathbb{R}}^{3})}$
$\displaystyle\leq$
$\displaystyle\frac{1}{8}\|\nabla(\sqrt{t}u_{t})\|_{L^{2}({\mathbb{R}}^{3})}^{2}+C\rho_{*}\|\sqrt{\rho
t}u_{t}\|_{L^{2}({\mathbb{R}}^{3})}^{2}\|u\|_{L^{\infty}({\mathbb{R}}^{3})}^{2}.$

Similarly, we write

$\displaystyle A_{3}=-\langle t\rho u,\nabla[(u\cdot\nabla u)\cdot
u_{t}]\rangle.$

and decompose

$\displaystyle|A_{3}|$ $\displaystyle\leq\langle|t\rho u|,|\nabla
u|^{2}\,|u_{t}|\rangle+\langle|t\rho
u|,|u|\,|\nabla^{2}u|\,|u_{t}|\rangle+\langle|t\rho u|,|u|\,|\nabla
u|\,|\nabla u_{t}|\rangle$ $\displaystyle=:A_{31}+A_{32}+A_{33}.$

Thanks to inequality (2.1) and Young’s inequality

$\displaystyle A_{31}$
$\displaystyle\leq\rho_{*}\sqrt{T}\|\sqrt{t}u_{t}\|_{L^{6}({\mathbb{R}}^{3})}\|u\|_{L^{6}({\mathbb{R}}^{3})}\|\nabla
u\|_{L^{3}({\mathbb{R}}^{3})}^{2}$ $\displaystyle\leq
C\rho_{*}\sqrt{T}\|\nabla(\sqrt{t}u_{t})\|_{L^{2}({\mathbb{R}}^{3})}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}^{2}\|\nabla^{2}u\|_{L^{2}({\mathbb{R}}^{3})}$
$\displaystyle\leq\frac{1}{8}\|\nabla(\sqrt{t}u_{t})\|_{L^{2}({\mathbb{R}}^{3})}^{2}+C\rho_{*}^{2}{T}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}^{4}\|\nabla^{2}u\|_{L^{2}({\mathbb{R}}^{3})}^{2}.$

As for $A_{32},A_{33}$, we have

$\displaystyle A_{32}$ $\displaystyle\leq\sqrt{\rho_{*}T}\|\sqrt{\rho
t}u_{t}\|_{L^{2}({\mathbb{R}}^{3})}\|u\|_{L^{\infty}({\mathbb{R}}^{3})}^{2}\|\nabla^{2}u\|_{L^{2}({\mathbb{R}}^{3})}$
$\displaystyle\leq\|\sqrt{\rho
t}u_{t}\|_{L^{2}({\mathbb{R}}^{3})}^{2}\|u\|_{L^{\infty}({\mathbb{R}}^{3})}^{4}+\rho_{*}T\|\nabla^{2}u\|_{L^{2}({\mathbb{R}}^{3})}^{2}$

and

$\displaystyle A_{33}$
$\displaystyle\leq\rho_{*}\sqrt{T}\|u\|_{L^{\infty}({\mathbb{R}}^{3})}^{2}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}\|\nabla(\sqrt{t}u_{t})\|_{L^{2}({\mathbb{R}}^{3})}$
$\displaystyle\leq\frac{1}{8}\|\nabla(\sqrt{t}u_{t})\|_{L^{2}({\mathbb{R}}^{3})}^{2}+C\rho_{*}^{2}{T}\|u\|_{L^{\infty}({\mathbb{R}}^{3})}^{4}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}^{2}.$

To handle $A_{4},$ we use inequality (2.1) and the Gagliardo-Nirenberg
inequality (A.1) to get

$\displaystyle|A_{4}|\leq$ $\displaystyle\sqrt{\rho_{*}}\|\sqrt{\rho
t}u_{t}\|_{L^{2}({\mathbb{R}}^{3})}\|\nabla
u\|_{L^{3}({\mathbb{R}}^{3})}\|\sqrt{t}u_{t}\|_{L^{6}({\mathbb{R}}^{3})}$
$\displaystyle\leq$ $\displaystyle\sqrt{\rho_{*}}\|\sqrt{\rho
t}u_{t}\|_{L^{2}({\mathbb{R}}^{3})}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}^{\frac{1}{2}}\|\nabla^{2}u\|_{L^{2}({\mathbb{R}}^{3})}^{\frac{1}{2}}\|\nabla(\sqrt{t}u_{t})\|_{L^{2}({\mathbb{R}}^{3})}$
$\displaystyle\leq$
$\displaystyle\frac{1}{8}\|\nabla(\sqrt{t}u_{t})\|_{L^{2}({\mathbb{R}}^{3})}^{2}+C{\rho_{*}}\|\sqrt{\rho
t}u_{t}\|_{L^{2}({\mathbb{R}}^{3})}^{2}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}\|\nabla^{2}u\|_{L^{2}({\mathbb{R}}^{3})}.$

By putting all the above estimates into (2.10), we find that

(2.11) $\frac{d}{dt}\|\sqrt{\rho
t}{u}_{t}\|_{L^{2}({\mathbb{R}}^{3})}^{2}+\|\nabla(\sqrt{t}{u}_{t})\|_{L^{2}({\mathbb{R}}^{3})}^{2}\leq\|\sqrt{\rho
t}u_{t}\|_{L^{2}({\mathbb{R}}^{3})}^{2}\left(1+\|u\|_{L^{\infty}({\mathbb{R}}^{3})}^{4}+\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}\|\nabla^{2}u\|_{L^{2}({\mathbb{R}}^{3})}\right)\\\
+C_{T}\left(\|\sqrt{\rho}\dot{u}\|_{L^{2}({\mathbb{R}}^{3})}^{2}+(\|u\|_{L^{\infty}({\mathbb{R}}^{d})}^{4}+1)\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}^{2}+(\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}^{4}+1)\|\nabla^{2}u\|_{L^{2}({\mathbb{R}}^{3})}^{2}\right).$

Using (2.3) and Remark 2.2 again, it is not difficult to conclude that
inequality (2.11) can be rewritten as

$\displaystyle\frac{d}{dt}\|\sqrt{\rho
t}{u}_{t}\|_{L^{2}({\mathbb{R}}^{3})}^{2}+\|\nabla(\sqrt{t}{u}_{t})\|_{L^{2}({\mathbb{R}}^{3})}^{2}\leq
B_{3d}(t)\|\sqrt{\rho t}u_{t}\|_{L^{2}({\mathbb{R}}^{3})}^{2}+B_{3d}(t)$

for some function $B_{3d}\in L^{1}(0,T),$ the norm of which bounded only in
terms of time $T,$ $\rho_{*}$ and norms
$\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{3})},\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{3})}.$

Finally, Gronwall’s lemma implies that for all $t\in[0,T]$

(2.12) $\displaystyle\|\sqrt{\rho
t}{u}_{t}\|_{L^{2}({\mathbb{R}}^{3})}^{2}+\int_{0}^{T}\|\nabla(\sqrt{t}{u}_{t})\|_{L^{2}({\mathbb{R}}^{3})}^{2}\,ds\leq\int_{0}^{T}B_{3d}(t)\,dt\exp\Big{(}\int_{0}^{T}B_{3d}(t)\,dt\Big{)}.$

This completes the proof of proposition 2.3. ∎

####  2.1.3. Shift of regularity

As a consequence of Proposition 2.3, we will get higher-order estimates for
the velocity, via considering the following multi-dimensional stationary
Stokes problem

(2.13) $\left\\{\begin{aligned}
&-\Delta(\sqrt{t}u)+\nabla(\sqrt{t}P)=-\rho\sqrt{t}(u_{t}+u\cdot\nabla
u)\hskip 33.28966pt{\rm in}~{}(0,T)\times{\mathbb{R}}^{d},\\\
&\nabla\cdot(\sqrt{t}u)=0\hskip 176.12262pt{\rm
in}~{}(0,T)\times{\mathbb{R}}^{d},\\\ &\sqrt{t}u\to
0,\quad|x|\to\infty.\end{aligned}\right.$

We have

######  Proposition 2.4 (higher-order estimates).

Let $(\rho,u)$ be a smooth enough solution to system (INS) on
$[0,T_{*})\times{\mathbb{R}}^{3}$. Then for all $T\in[0,T_{*})$ it holds that
for all $r\in[2,6]$ and $q\in[2,\frac{4r}{3r-6}]$

$\displaystyle\|\nabla^{2}(\sqrt{t}u)\|_{L^{q}(0,T;L^{r}({\mathbb{R}}^{3}))}+\|\nabla(\sqrt{t}P)\|_{L^{q}(0,T;L^{r}({\mathbb{R}}^{3}))}\leq
C_{0,T}.$

Moreover, we have the key Lipschitz estimate

$\displaystyle\int_{0}^{T}\|\nabla
u(t,\cdot)\|_{L^{\infty}({\mathbb{R}}^{3})}\,dt\leq C_{0,T}\,T^{\frac{1}{4}}$

and $C_{0,T}\,T^{\frac{1}{4}}\to 0$ as $T\to 0.$

###### Proof.

Applying the standard $L^{p}$ estimates for system (2.13) yields that for all
finite $p,$

(2.14)
$\displaystyle\|\nabla^{2}(\sqrt{t}u)\|_{L^{p}({\mathbb{R}}^{3})}+\|\nabla(\sqrt{t}P)\|_{L^{p}({\mathbb{R}}^{3})}\leq
C(p)\sqrt{\rho_{*}}\|\sqrt{\rho t}\dot{u}\|_{L^{p}({\mathbb{R}}^{3})}.$

Now, we estimate the material derivative in the right-hand side of above
inequality. On the one hand, we have by inequality (2.1) and Proposition 2.3

$\displaystyle\int_{0}^{T}\|\sqrt{\rho
t}u_{t}\|_{L^{6}({\mathbb{R}}^{3})}^{2}\,dt$
$\displaystyle\leq\rho_{*}\int_{0}^{T}\|\nabla(\sqrt{t}u_{t})\|_{L^{2}({\mathbb{R}}^{3})}^{2}\,dt\leq
C_{0,T}.$

Interpolating this with the estimate in Proposition 2.3 gives that for all
$r\in[2,6]$ and $q\in[2,\frac{4r}{3r-6}]$

(2.15) $\displaystyle\|\sqrt{\rho
t}u_{t}\|_{L^{q}(0,T;L^{r}({\mathbb{R}}^{3})}\leq C_{0,T}.$

In the other hand, taking $p=2$ in inequality (2.14) and using Young’s
inequality yield that

(2.16)
$\displaystyle\|\nabla^{2}(\sqrt{t}u)\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))}$
$\displaystyle\leq$ $\displaystyle C(\sqrt{\rho_{*}}\|\sqrt{\rho
t}u_{t}\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))}+\rho_{*}\|\sqrt{t}u\cdot\nabla
u\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))})$ $\displaystyle\leq$
$\displaystyle C(\sqrt{\rho_{*}}\|\sqrt{\rho
t}u_{t}\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))}+\rho_{*}\|\sqrt{t}u\|_{L^{\infty}(0,T;L^{\infty}({\mathbb{R}}^{3}))}\|\nabla
u\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))})$ $\displaystyle\leq$
$\displaystyle C(\rho_{*})\left(\|\sqrt{\rho
t}u_{t}\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))}+\|(\sqrt{t}\nabla
u,\sqrt{t}\nabla^{2}u)\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))}^{\frac{1}{2}}\|\nabla
u\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))}\right)$ $\displaystyle\leq$
$\displaystyle C_{0,T}.$

Then by the Gagliardo-Nirenberg inequality (A.1), Proposition 2.1 and estimate
(2.16)

$\displaystyle\|\sqrt{\rho t}u\cdot\nabla
u\|_{L^{2}(0,T;L^{6}({\mathbb{R}}^{3}))}\leq$
$\displaystyle\sqrt{\rho_{*}}\left(\int_{0}^{T}\|u\|_{L^{\infty}({\mathbb{R}}^{3})}^{2}\|\sqrt{t}\nabla
u\|_{L^{6}({\mathbb{R}}^{3})}^{2}\,dt)\right)^{{1}/{2}}$ $\displaystyle\leq$
$\displaystyle\sqrt{\rho_{*}}\left(\int_{0}^{T}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}\|\nabla^{2}u\|_{L^{2}({\mathbb{R}}^{3})}\|\sqrt{t}\nabla
u\|_{L^{6}({\mathbb{R}}^{3})}^{2}\,dt\right)^{1/2}$ $\displaystyle\leq$
$\displaystyle\sqrt{\rho_{*}}\left(\int_{0}^{T}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}\big{(}\|\nabla^{2}u\|_{L^{2}({\mathbb{R}}^{3})}^{2}+\|\sqrt{t}\nabla
u\|_{L^{6}({\mathbb{R}}^{3})}^{4}\big{)}\,dt\right)^{1/2}$ $\displaystyle\leq$
$\displaystyle C_{0,T}.$

Notice that the Gagliardo-Nirenberg inequality [26, page 54] and estimate
(2.16) give that

$\displaystyle\|\sqrt{\rho t}u\cdot\nabla
u\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))}\leq$ $\displaystyle\|\sqrt{\rho
t}u\|_{L^{\infty}(0,T;L^{\infty}({\mathbb{R}}^{3}))}\|\nabla
u\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))}$ $\displaystyle\leq$
$\displaystyle\sqrt{\rho_{*}}\|\sqrt{t}u\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))}^{\frac{1}{2}}\|\sqrt{t}\nabla^{2}u\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))}^{\frac{1}{2}}\|\nabla
u\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))}$ $\displaystyle\leq$
$\displaystyle C_{0,T}.$

Hence, by interpolation, for all $r\in[2,6]$ and $q\in[2,\frac{4r}{3r-6}]$

$\displaystyle\|\sqrt{\rho t}u\cdot\nabla
u\|_{L^{q}(0,T;L^{r}({\mathbb{R}}^{2}))}\leq C_{0,T},$

which together with (2.15) and (2.14) imply the desired estimate.

Finally, we have by inequality (A.1) again

$\displaystyle\int_{0}^{T}\|\nabla u\|_{L^{\infty}({\mathbb{R}}^{3})}\,dt\leq$
$\displaystyle\int_{0}^{T}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}^{\frac{1}{4}}\|\nabla^{2}(\sqrt{t}u)\|_{L^{6}({\mathbb{R}}^{3})}^{\frac{3}{4}}\,t^{-\frac{3}{8}}\,dt$
$\displaystyle\leq$ $\displaystyle\|\nabla
u\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))}^{\frac{1}{4}}\|\nabla^{2}(\sqrt{t}u)\|_{L^{2}(0,T;L^{6}({\mathbb{R}}^{3}))}^{\frac{3}{4}}\|t^{-\frac{3}{8}}\|_{L^{\frac{8}{5}}(0,T)}$
$\displaystyle\leq$ $\displaystyle C_{0,T}\,T^{\frac{1}{4}}.$

It remains to check in the proof of Proposition 2.3 that $C_{0,T}$ is
uniformly bounded as $T\to 0.$ This completes the proof of Proposition 2.4. ∎

###  2.2. Proof of existence

One can now turn to the proof of the existence of a solution stated in Theorem
B. Denoting by $T^{\varepsilon}$ the maximal time of existence for approximate
solutions $(\rho^{\varepsilon},u^{\varepsilon})$, then [49, Theorem 1.4] shows
that $T^{\varepsilon}\geq T_{0}$. On the other hand, in the case the smallness
condition (1.6) is satisfied, one can continue
$(\rho^{\varepsilon},u^{\varepsilon})$ globally-in-time thanks to the
$L^{1}_{t}Lip_{x}$ estimate in Proposition 2.4 and the blow-up criterion in
[13, Theorem 0.4] or [10, Theorem 4]. It remains to prove the convergence of
approximate solutions. At this stage, with all the estimates established in
the previous subsection, the standard compactness argument yields that, up to
a subsequence,

$\rho^{\varepsilon}\rightharpoonup\rho\quad{\rm
weak*~{}in}~{}L^{\infty}({\mathbb{R}}_{+}\times{\mathbb{R}}^{3})0\leq\rho\leq\rho_{*},$
$\nabla u^{\varepsilon}\rightharpoonup\nabla u\quad{\rm
weak~{}in}~{}L^{2}({\mathbb{R}}_{+};L^{2}({\mathbb{R}}^{3}),$
$u^{\varepsilon}\rightharpoonup u\quad{\rm
weak~{}in}~{}L^{2}({\mathbb{R}}_{+};L^{6}({\mathbb{R}}^{3}),$

for some $(\rho,u)$ satisfies all the regularity results stated in Theorem B,
except for the regularities of non-linear term $\sqrt{\rho}u_{t}.$ The
compactness result obtained in [22] then shows that

$\displaystyle{\rho^{\varepsilon}}\to{\rho}\quad{\rm in}~{}~{}\mathcal{C}_{\rm
loc}({\mathbb{R}}_{+};L^{q}_{\rm
loc}({\mathbb{R}}^{3})),\quad\forall\,q\in[0,\infty)$

and

$\displaystyle{\sqrt{\rho^{\varepsilon}}u^{\varepsilon}}\to{\sqrt{\rho}u}\quad{\rm
in}~{}~{}L^{2}_{\rm loc}({\mathbb{R}}_{+}\times{\mathbb{R}}^{3}).$

From which, we know from Proposition 2.3 that
$\sqrt{\rho^{\varepsilon}}u^{\varepsilon}_{t}\rightharpoonup\sqrt{\rho}u_{t}$
weak in $\mathcal{D}^{\prime}((0,T_{0})\times{\mathbb{R}}^{3}))$ and thus
$\sqrt{\rho}u_{t}\in L^{2}(0,T_{0};L^{2}({\mathbb{R}}^{3}))$ thanks to the
uniform bounds (2.3). Similarly, one has $\sqrt{\rho t}u_{t}\in
L^{\infty}(0,T_{0};L^{2}({\mathbb{R}}^{3})).$

Finally, all the compactness information above is enough to justify that the
couple $(\rho,u)$ is a weak solution to (INS) in the sense of distributions.
This finishes the proof of the existence part of Theorem B. ∎

##  3\. Proof of the existence results in the two-dimensional case, Theorem A

This section is concerned with the proof of the existence part in Theorem A
(2D case). The initial data $(\rho_{0},u_{0})$ satisfies condition (1.2) and
(1.5), and $\nabla u_{0}\in L^{2}({\mathbb{R}}^{2}),\nabla\cdot u_{0}=0$. We
analyse separately the case when: (i) (aNV) or (VB) is satisfied, see
Subsection 3.1 below, (ii) (FFV-2) is satisfied, see Subsection 3.2. The idea
is similar to the three-dimensional case. We first approximate the data or
mollify the equation666We use two different approximation procedures depending
on which case (i) or (ii) is considered and rely on known existence results
for smooth approximate solutions without vacuum. Then, we show the persistence
of the regularity $u\in\widetilde{\mathcal{D}}^{1,2}({\mathbb{R}}^{2})$ and
other higher-order estimates uniformly for the approximate solutions. The
actual existence proof is similar to the one in the three-dimensional case,
see Subsection 2.2.

Before going into the details of each case, let us recall an estimate obtained
in [43] for which the proof is valid for our approximate solutions. We mention
that the following lemma is motivated by the work [37] of Li and Xin on the
two-dimensional compressible Navier-Stokes equations with vacuum.

######  Proposition 3.1 (gradient estimate; [43, Lemma 3.2]).

There exists a universal constant $C>0$ such that for all $t>0$,

(3.1) $\|\nabla
u(t,\cdot)\|_{L^{2}({\mathbb{R}}^{2})}^{2}+2\int_{0}^{t}\|(\sqrt{\rho}\dot{u},\Delta
u,\nabla P)(s,\cdot)\|_{L^{2}({\mathbb{R}}^{2})}^{2}\,ds\\\ \leq\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{2})}^{2}\exp(C\rho_{*}\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{2})}^{2}).$

Notice that this result does not require any of the assumptions (aNV), (VB)
nor (FFV-2).

###  3.1. The case when $\eqref{cond1}$ or $\eqref{cond2}$ is satisfied

####  3.1.1. Approximation procedure

As in the three-dimensional case, we regularize the initial data. We require
that the approximate initial density $\rho_{0}^{\varepsilon}$ satisfies (aNV)
or (VB) whenever $\rho_{0}$ satisfies (aNV) or (VB). Therefore, we construct a
sequence $\rho_{0}^{\varepsilon}$ such that
$\rho_{0}^{\varepsilon}\geq\rho_{0}$. To do this we build the approximation on
the sequence of balls $B_{n}$, $n\rightarrow\infty$.

Let $\varepsilon>0$ and $n\in{\mathbb{N}}$, $n\geq 1$. Let $K_{\varepsilon}$
denote a non-negative mollifier. We approximate the initial velocity $u_{0}$
by $u_{0}^{\varepsilon,n}\in\mathcal{C}^{\infty}_{0}(B_{n})$ in the following
way

$u_{0}^{\varepsilon,n}=(u_{0}\varphi_{n})\star K_{\varepsilon},$

where $\varphi_{n}=\varphi(\cdot/n)$ with
$\varphi\in\mathcal{C}^{\infty}_{0}(B_{1})$ a cut-off function such that
$\varphi=1$ on $B_{1/2}$. As for the density, we approximate it in the
following way on $B_{n}$

$\rho_{0}^{\varepsilon,n}:=\rho_{0}\star K_{\varepsilon}+\|\rho_{0}\star
K_{\varepsilon}-\rho_{0}\|_{L^{\infty}(B_{n})}+\frac{1}{n}e^{-|x|^{2}}.$

Notice that

$\displaystyle\rho_{0}^{\varepsilon,n}\in\mathcal{C}^{\infty}({\mathbb{R}}^{2})~{}\,\,{\rm
with}~{}\,\,\frac{1}{n}e^{-n^{2}}\leq\rho_{0}^{\varepsilon}\leq
2\rho_{*}u_{0}^{\varepsilon,n}\in\mathcal{C}_{0}^{\infty}({\mathbb{R}}^{2})~{}\,\,{\rm
with}~{}\,\,\nabla\cdot u_{0}^{\varepsilon,n}=0.$

Moreover, $\|\rho_{0}\star
K_{\varepsilon}-\rho_{0}\|_{L^{\infty}(B_{n})}\longrightarrow 0$ when
$\varepsilon\rightarrow 0^{+}$ and $\rho_{0}^{\varepsilon,n}\geq\rho_{0}$.
Therefore if $\eqref{cond1}$ (resp. (VB)) is satisfied for $\rho_{0}$ it is
also satisfied for $\rho_{0}^{\varepsilon,n}$.

We consider the solutions $(\rho^{\varepsilon,n},u^{\varepsilon,n})$ to the
following mollified version of (INS)

(INS-$\varepsilon$) $\left\\{\begin{aligned}
&\partial_{t}\rho^{\varepsilon,n}+\nabla\cdot\big{(}\rho^{\varepsilon,n}(K_{\varepsilon}\star
u^{\varepsilon,n})\big{)}=0,\\\ &\partial_{t}(\rho
u^{\varepsilon,n})+\nabla\cdot\big{(}\rho^{\varepsilon,n}(K_{\varepsilon}\star
u^{\varepsilon,n})\otimes u^{\varepsilon,n}\big{)}+\nabla
P^{\varepsilon,n}=\nu\Delta u^{\varepsilon,n},\\\ &\nabla\cdot
u^{\varepsilon,n}=0.\end{aligned}\right.$

in the domain $B_{n}$ with no-slip boundary condition $u^{\varepsilon,n}=0$ on
$\partial B_{n}$ and initial data
$(\rho^{\varepsilon,n}_{0},u^{\varepsilon,n}_{0})$. According to [42, Theorem
2.6], there exists a smooth global-in-time solution
$(\rho^{\varepsilon,n},u^{\varepsilon,n})\in\mathcal{C}^{\infty}([0,\infty)\times\overline{B}_{n})$
to (INS-$\varepsilon$).

Our objective is now to derive uniform estimates in $\varepsilon$ and $n$ for
the approximate solutions $(\rho^{\varepsilon,n},u^{\varepsilon,n})$. For
notational simplicity, we drop the superscripts $\varepsilon$ and $n$ below
and simply write the solution $(\rho,u)$. Notice that all the estimates below
are on $B_{n}$.777Notice that as in the previous subsection, the estimates
that do not involve second-order space derivatibves or first-order time
derivatives of the velocity can be extended to ${\mathbb{R}}^{2}$ thanks to
the fact that $u^{\varepsilon,n}$ satisfies no-slip boundary conditions on
$\partial B_{n}$.

####  3.1.2. Lower-order estimates

Here, we establish some lower-order bounds following from condition (aNV) or
(VB) for $\rho_{0}.$

######  Proposition 3.2 ($L^{\infty}_{t}L^{2}_{x}$ bound for the velocity).

We have

(3.2) $\displaystyle\|u\|_{L^{\infty}({\mathbb{R}}_{+};L^{2}(B_{n}))}\leq
C_{*}(\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{2})}+\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{2})})$

and

(3.3) $\displaystyle\|u\|_{L^{4}({\mathbb{R}}_{+};L^{\infty}(B_{n}))}\leq
C_{*}\big{(}\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}(B_{n})}+\|\nabla
u_{0}\|_{L^{2}(B_{n})}\big{)}^{2}\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{2})}^{2}\exp(\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{2})}^{2}),$

with positive constant $C_{*}$ depending only on the factors in condition
(aNV) or (VB).

###### Proof.

Using the Lagrangian flow maps and the fact that they are measure preserving
due to incompressibility, $\rho$ satisfies (aNV) or (VB) along the evolution
with the same constants $d$, $\delta_{0}$ and
$\|({1}/{{\eta}})\,\mathbf{1}_{\eta<\delta_{0}}\|_{L^{1}}$ in the case when
(aNV) is satisfied, $p$, $d$, $\bar{\eta}$ and
$\|(\bar{\eta}-\eta)_{+}\|_{L^{p}}$ in the case when (VB). Then by Proposition
A.6 and (3.1), we get

$\displaystyle\|u\|_{L^{\infty}({\mathbb{R}}_{+};L^{2}(B_{n}))}$
$\displaystyle\leq C_{*}(\|\sqrt{\rho}u\|_{L^{2}(B_{n})}+\|\nabla
u\|_{L^{2}(B_{n})})$ $\displaystyle\leq
C_{*}(\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{2})}+\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{2})}).$

Using (3.2) and (3.1), we further get

$\displaystyle\int_{0}^{\infty}\|u(t,\cdot)\|_{L^{\infty}(B_{n})}^{4}\,dt$
$\displaystyle\leq\int_{0}^{\infty}\|u\|_{L^{2}(B_{n})}^{2}\|\nabla^{2}u\|_{L^{2}(B_{n})}^{2}\,dt$
$\displaystyle\leq
C_{*}\big{(}\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{2})}+\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{2})}\big{)}^{2}\|\nabla^{2}u\|_{L^{2}({\mathbb{R}}_{+};L^{2}(R_{n}))}^{2}$
$\displaystyle\leq
C_{*}\big{(}\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{2})}+\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{2})}\big{)}^{2}\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{2})}^{2}\exp(\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{2})}^{2}).\qed$

####  3.1.3. Estimates of the time derivative

In this step, we want to bound time derivatives $\sqrt{\rho t}{u}_{t}$ in
$L^{\infty}_{\rm loc}({\mathbb{R}}_{+};L^{2}(B_{n}))$ and
$\sqrt{t}\nabla{u}_{t}$ in $L^{2}_{\rm loc}({\mathbb{R}}_{+};L^{2}(B_{n})),$
since it is an important step towards higher-order spatial estimates for the
velocity.

######  Proposition 3.3 (time derivative estimates).

For any $T>0,$ there exits a positive constant $C_{0,*}(T)$ depending only on
$C_{*},T,\rho_{*}$ and $\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}},\|\nabla
u_{0}\|_{L^{2}}$ such that

(3.4) $\|\sqrt{\rho
t}{u}_{t}\|_{L^{\infty}(0,T;L^{2}(B_{n}))}^{2}+\|\nabla(\sqrt{t}{u}_{t})\|_{L^{2}(0,T;L^{2}(B_{n}))}^{2}\leq
C_{0,*}(T).$

###### Proof.

The proof is similar to the three-dimensional case. Recall that by testing the
equation (2.9) with $\sqrt{t}{u}_{t}$, we are led to

(3.5) $\displaystyle\frac{1}{2}\frac{d}{dt}\|\sqrt{\rho
t}{u}_{t}\|_{L^{2}}^{2}+\|\nabla(\sqrt{t}{u}_{t})\|_{L^{2}}^{2}=A_{1}+\dots+A_{4}$

with

$\displaystyle
A_{1}=\frac{1}{2}\|\sqrt{\rho}{u}_{t}\|_{L^{2}({\mathbb{R}}^{d})}^{2},$
$\displaystyle A_{2}=-\langle t\rho_{t},|u_{t}|^{2}\rangle,$ $\displaystyle
A_{3}=-\langle\sqrt{t}\rho_{t}u\cdot\nabla u,\sqrt{t}{u}_{t}\rangle,$
$\displaystyle A_{4}=-\langle\sqrt{t}\rho u_{t}\cdot\nabla
u,\sqrt{t}{u}_{t}\rangle.$

In order to estimate $A_{1},A_{2},A_{3},A_{4},$ we proceed as follows.
Noticing that

$\displaystyle A_{1}$
$\displaystyle=\frac{1}{2}\|\sqrt{\rho}(\dot{u}-u\cdot\nabla
u)\|_{L^{2}(B_{n})}^{2}$
$\displaystyle\leq\|\sqrt{\rho}\dot{u}\|_{L^{2}(B_{n})}^{2}+\rho_{*}\|u\|_{L^{\infty}(B_{n})}^{2}\|\nabla
u\|_{L^{2}(B_{n})}^{2}.$

Using the equation $\partial_{t}\rho+\nabla\cdot(\rho u)=0,$ we write

$\displaystyle A_{2}=-\langle t\rho u,\nabla(|u_{t}|^{2})\rangle$

and

$\displaystyle|A_{2}|\leq$ $\displaystyle
2\sqrt{\rho_{*}}\|u\|_{L^{\infty}(B_{n})}\|\sqrt{\rho
t}u_{t}\|_{L^{2}(B_{n})}\|\nabla(\sqrt{t}u_{t})\|_{L^{2}(B_{n})}$
$\displaystyle\leq$
$\displaystyle\frac{1}{8}\|\nabla(\sqrt{t}u_{t})\|_{L^{2}(B_{n})}^{2}+C\rho_{*}\|\sqrt{\rho
t}u_{t}\|_{L^{2}(B_{n})}^{2}\|u\|_{L^{\infty}(B_{n})}^{2}.$

Similarly, we write

$\displaystyle A_{3}=-\langle t\rho u,\nabla[(u\cdot\nabla u)\cdot
u_{t}]\rangle,$

and decompose

$\displaystyle|A_{3}|$ $\displaystyle\leq\langle|t\rho u|,|\nabla
u|^{2}\,|u_{t}|\rangle+\langle|t\rho
u|,|u|\,|\nabla^{2}u|\,|u_{t}|\rangle+\langle|t\rho u|,|u|\,|\nabla
u|\,|\nabla u_{t}|\rangle$ $\displaystyle=:A_{31}+A_{32}+A_{33}.$

To bound $A_{31}$, we write by the Gagliardo-Nirenberg inequality (A.1)

$\displaystyle A_{31}$ $\displaystyle\leq\sqrt{\rho_{*}T}\|\sqrt{\rho
t}u_{t}\|_{L^{2}(B_{n})}\|\nabla
u\|_{L^{4}(B_{n})}^{2}\|u\|_{L^{\infty}(B_{n})}$
$\displaystyle\leq\|\sqrt{\rho
t}u_{t}\|_{L^{2}(B_{n})}^{2}\|u\|_{L^{\infty}(B_{n})}^{2}+\rho_{*}T\|\nabla
u\|_{L^{4}(B_{n})}^{4}$ $\displaystyle\leq\|\sqrt{\rho
t}u_{t}\|_{L^{2}(B_{n})}^{2}\|u\|_{L^{\infty}(B_{n})}^{2}+\rho_{*}T\|\nabla
u\|_{L^{2}(B_{n})}^{2}\|\nabla^{2}u\|_{L^{2}(B_{n})}^{2}.$

As for $A_{32},A_{33}$, we have

$\displaystyle A_{32}$ $\displaystyle\leq\sqrt{\rho_{*}T}\|\sqrt{\rho
t}u_{t}\|_{L^{2}(B_{n})}\|u\|_{L^{\infty}(B_{n})}^{2}\|\nabla^{2}u\|_{L^{2}(B_{n})}$
$\displaystyle\leq\|\sqrt{\rho
t}u_{t}\|_{L^{2}(B_{n})}^{2}\|u\|_{L^{\infty}(B_{n})}^{4}+\rho_{*}T\|\nabla^{2}u\|_{L^{2}(B_{n})}^{2}$

and

$\displaystyle A_{33}$
$\displaystyle\leq\rho_{*}\sqrt{T}\|u\|_{L^{\infty}(B_{n})}^{2}\|\nabla
u\|_{L^{2}(B_{n})}\|\nabla(\sqrt{t}u_{t})\|_{L^{2}(B_{n})}$
$\displaystyle\leq\frac{1}{8}\|\nabla(\sqrt{t}u_{t})\|_{L^{2}(B_{n})}^{2}+C\rho_{*}^{2}{T}\|u\|_{L^{\infty}(B_{n})}^{4}\|\nabla
u\|_{L^{2}(B_{n})}^{2}.$

To handle $A_{4}$, we need to use Proposition A.6. More precisely, since
condition (aNV) or (VB) is also satisfied by $\rho$, so the functional
inequality (A.8) (taking $\eta=\sqrt{t}u_{t}$) yields that

(3.6) $\displaystyle\|\sqrt{t}u_{t}\|_{L^{2}(B_{n})}\leq
C_{*}\Big{(}\|\sqrt{\rho
t}u_{t}\|_{L^{2}(B_{n})}+\|\nabla(\sqrt{t}u_{t})\|_{L^{2}(B_{n})}\Big{)}.$

Then for all $r\in[2,\infty),$ we have by interpolation inequality

$\displaystyle\|\sqrt{t}u_{t}\|_{L^{r}(B_{n})}\leq$ $\displaystyle
C(\|\sqrt{t}u_{t}\|_{L^{2}(B_{n})}+\|\sqrt{t}\nabla u_{t}\|_{L^{2}(B_{n})})$
$\displaystyle\leq$ $\displaystyle C_{*}(\|\sqrt{\rho
t}u_{t}\|_{L^{2}(B_{n})}+\nabla(\sqrt{t}u_{t})\|_{L^{2}(B_{n})}),$

which implies that

$\displaystyle|A_{4}|\leq$ $\displaystyle\|\sqrt{\rho
t}u_{t}\|_{L^{4}(B_{n})}^{2}\|\nabla u\|_{L^{2}(B_{n})}$ $\displaystyle\leq$
$\displaystyle\|\sqrt{\rho t}u_{t}\|_{L^{2}(B_{n})}^{\frac{1}{2}}\|\sqrt{\rho
t}u_{t}\|_{L^{6}(B_{n})}^{\frac{3}{2}}\|\nabla u\|_{L^{2}(B_{n})}$
$\displaystyle\leq$ $\displaystyle C_{*}\rho_{*}^{\frac{3}{4}}\|\sqrt{\rho
t}u_{t}\|_{L^{2}(B_{n})}^{\frac{1}{2}}\big{(}\|\sqrt{\rho
t}u_{t}\|_{L^{2}(B_{n})}+\|\nabla(\sqrt{t}u_{t})\|_{L^{2}(B_{n})}\big{)}^{\frac{3}{2}}\|\nabla
u\|_{L^{2}(B_{n})}$ $\displaystyle\leq$ $\displaystyle
C_{*}\rho_{*}^{\frac{3}{4}}\|\sqrt{\rho t}u_{t}\|_{L^{2}(B_{n})}^{2}\|\nabla
u\|_{L^{2}(B_{n})}+C_{*}\rho_{*}^{3}\|\sqrt{\rho
t}u_{t}\|_{L^{2}(B_{n})}^{2}\|\nabla u\|_{L^{2}(B_{n})}^{4}.$

Finally, by plugging all the above estimates of $A_{i}~{}(i=1,\ldots\,4)$ into
(3.5), we get

(3.7) $\displaystyle\frac{d}{dt}\|\sqrt{\rho
t}{u}_{t}\|_{L^{2}(B_{n})}^{2}+\|\nabla(\sqrt{t}{u}_{t})\|_{L^{2}(B_{n})}^{2}$
$\displaystyle\leq$ $\displaystyle\|\sqrt{\rho
t}u_{t}\|_{L^{2}(B_{n})}^{2}\left(1+\|u\|_{L^{\infty}(B_{n})}^{4}+\|\nabla
u\|_{L^{2}(B_{n})}+\|\nabla u\|_{L^{2}(B_{n})}^{4}\right)$
$\displaystyle+C_{0,*}({T})\left(\|\sqrt{\rho}{u}_{t}\|_{L^{2}(B_{n})}^{2}+(\|u\|_{L^{\infty}(B_{n})}^{4}+1)\|\nabla
u\|_{L^{2}(B_{n})}^{2}+(\|\nabla
u\|_{L^{2}(B_{n})}^{2}+1)\|\nabla^{2}u\|_{L^{2}(B_{n})}^{2}\right).$

To conclude, thanks to inequalities (3.1) and (3.3), it is not difficult to
find that inequality (3.7) can be rewritten as

$\displaystyle\frac{d}{dt}\|\sqrt{\rho
t}{u}_{t}\|_{L^{2}(B_{n})}^{2}+\|\nabla(\sqrt{t}{u}_{t})\|_{L^{2}(B_{n})}^{2}\leq
B_{2d}(t)\|\sqrt{\rho t}u_{t}\|_{L^{2}(B_{n})}^{2}+B_{2d}(t)$

for some function $B_{2d}\in L^{1}(0,T),$ the norm of which bounded only in
terms of $C_{*},T,$ $\rho_{*}$ and norms
$\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}(B_{n})},\|\nabla u_{0}\|_{L^{2}(B_{n})}.$

Therefore, Gronwall’s lemma implies that for all $t\in[0,T]$

$\displaystyle\|\sqrt{\rho
t}{u}_{t}\|_{L^{2}}^{2}+\int_{0}^{T}\|\nabla(\sqrt{t}{u}_{t})\|_{L^{2}}^{2}\,dt\leq\int_{0}^{T}B_{2d}(t)\,dt\exp\left(\int_{0}^{T}B_{2d}(t)\,dt\right).$

This completes the proof of proposition 3.3. ∎

####  3.1.4. Shift of regularity

As a consequence of Proposition 3.3, we will get higher-order estimates for
the velocity, via considering the Stokes problem (2.13). We have

######  Proposition 3.4 (higher-order estimates).

Let $T>0.$ For all $r\in[2,\infty)$ and $q\in[2,\frac{2r}{r-2}),$ we have

(3.8)
$\displaystyle\|\nabla^{2}(\sqrt{t}u)\|_{L^{q}(0,T;L^{r}(B_{n}))}+\|\nabla(\sqrt{t}P)\|_{L^{q}(0,T;L^{r}(B_{n}))}\leq
C_{0,*}(T).$

Moreover, we have the following key Lipschitz estimate

(3.9) $\displaystyle\int_{0}^{T}\|\nabla
u(t,\cdot)\|_{L^{\infty}({\mathbb{R}}^{2})}\,dt\leq
C_{0,*}(T)T^{\frac{1}{3}}.$

In particular, $C_{0,*}(T)T^{\frac{1}{3}}\to 0$ as $T\to 0.$

###### Proof.

Applying the standard $L^{p}$ estimates for system (2.13), one gets for all
finite $p,$

(3.10)
$\displaystyle\|\nabla^{2}(\sqrt{t}u)\|_{L^{p}(B_{n})}+\|\nabla(\sqrt{t}P)\|_{L^{p}(B_{n})}\leq
C(p)\sqrt{\rho_{*}}\|\sqrt{\rho t}\dot{u}\|_{L^{p}(B_{n})}.$

To estimate the material derivative in the right-hand side of above
inequality, we use the Gagliardo-Nirenberg inequality (A.1) and (3.36) to
write that for all ${\bar{r}}\in[2,\infty)$

$\displaystyle\int_{0}^{T}\|\sqrt{\rho t}u_{t}\|_{L^{\bar{r}}(B_{n})}^{2}\,dt$
$\displaystyle\leq\int_{0}^{T}{\rho_{*}}(\|\sqrt{t}u_{t}\|_{L^{2}(B_{n})}+\|\nabla(\sqrt{t}u_{t})\|_{L^{2}(B_{n})})^{2}\,dt$
$\displaystyle\leq C_{*}{\rho_{*}}\int_{0}^{T}\left(\|\sqrt{\rho
t}u_{t}\|_{L^{2}(B_{n})}^{2}+\|\nabla(\sqrt{t}u_{t})\|_{L^{2}(B_{n})}^{2}\right)\,dt\leq
C_{0,*}(T),$

where in the last inequality we used bounds (3.4).

Then, interpolating above inequality with bounds (3.4) yields that for all
$r\in[2,\infty)$ and $q\in[2,r^{*})$ with $r^{*}:=\frac{2r}{r-2},$ one has

(3.11) $\displaystyle\|\sqrt{\rho t}u_{t}\|_{L^{q}(0,T;L^{r}(B_{n}))}\leq
C_{0,*}(T).$

For the convective term, we notice that the Gagliardo-Nirenberg inequality
(A.1), (3.1) and Proposition 3.2 imply that for all $\bar{r}\in[2,\infty)$

$\displaystyle\|\sqrt{\rho t}u\|_{L^{\infty}(0,T;L^{\bar{r}}(B_{n}))}\leq$
$\displaystyle\sqrt{\rho_{*}T}\big{(}\|u\|_{L^{\infty}(0,T;L^{2}(B_{n}))}+\|\nabla
u\|_{L^{\infty}(0,T;L^{2}(B_{n}))}\big{)}\leq C_{0,*}(T),$

and for all $\bar{r}\in[2,\infty)$ and
$\bar{m}\in[2,\frac{2\bar{r}}{\bar{r}-2}]$

$\displaystyle\|\nabla u\|_{L^{\bar{m}}(0,T;L^{\bar{r}}(B_{n}))}\leq$
$\displaystyle\|\nabla u\|_{L^{\infty}(0,T;L^{2}(B_{n}))}+\|\nabla
u\|_{L^{2}(0,T;\dot{H}^{1}(B_{n}))}\leq C_{0,*}(T).$

Thus, for all $r\in[2,\infty)$ and $q\in[2,r^{*})$, we write according to the
above two inequalities

(3.12) $\displaystyle\|\sqrt{\rho t}u\cdot\nabla
u\|_{L^{q}(0,T;L^{r}(B_{n}))}\leq C_{0,*}(T).$

Putting estimates (3.11) and (3.12) into (3.10) yields

$\displaystyle\|\nabla^{2}(\sqrt{t}u)\|_{L^{q}(0,T;L^{r}(B_{n}))}+\|\nabla(\sqrt{t}P)\|_{L^{q}(0,T;L^{r}(B_{n}))}\leq
C_{0,*}(T).$

To prove Lipschitz estimate, we write by inequality (A.1), (3.1) and (3.8)
that888Notice that the time-weighted estimate (3.8) enables us to gain
smallness in time in the final estimate (3.9).

$\displaystyle\int_{0}^{T}\|\nabla u\|_{L^{\infty}(B_{n})}\,dt\leq$
$\displaystyle\int_{0}^{T}\|\nabla
u\|_{L^{2}(B_{n})}^{\frac{1}{3}}\|\nabla^{2}(\sqrt{t}u)\|_{L^{4}(B_{n})}^{\frac{2}{3}}\,t^{-\frac{1}{3}}\,dt$
$\displaystyle\leq$ $\displaystyle\|\nabla
u\|_{L^{\infty}(0,T;L^{2}(B_{n}))}^{\frac{1}{3}}\|\nabla^{2}(\sqrt{t}u)\|_{L^{2}(0,T;L^{4}(B_{n}))}^{\frac{2}{3}}\|t^{-\frac{1}{3}}\|_{L^{\frac{3}{2}}(0,T)}$
$\displaystyle\leq$ $\displaystyle C_{0,*}(T)\,T^{\frac{1}{3}}.$

It is easy to notice that $C_{0,*}(T)$ stays bounded as $T\to 0,$ thus we
complete the proof of Proposition 3.4. ∎

###  3.2. The case when (FFV-2) is satisfied

This case corresponds to the far-field vacuum.

####  3.2.1. Approximation procedure

The construction is similar to the one in Section 3. Without loss of
generality, we assume that the initial density $\rho_{0}$ satisfies
$\|\rho_{0}\|_{L^{1}({\mathbb{R}}^{2})}=1,$ which implies that there exists a
positive constant $N_{0}$ such that

$\displaystyle\int_{B_{N_{0}}}\rho_{0}\,dx>\frac{1}{2}.$

We require that the approximate initial density satisfies (FFV-2) whenever the
initial density satisfies (FFV-2).

Let $\varepsilon>0$ and $n\in{\mathbb{N}}$, $n\geq 1$. Let $K_{\varepsilon}$
denote a non-negative mollifier. We approximate the density in the following
way on $B_{n}$

$\rho_{0}^{\varepsilon,n}:=\rho_{0}\star
K_{\varepsilon}+\frac{1}{n}e^{-|x|^{2}},$

Notice that

$\displaystyle\rho_{0}^{\varepsilon,n}\in\mathcal{C}^{\infty}({\mathbb{R}}^{2})~{}\,\,{\rm
with}~{}\,\,\frac{1}{n}e^{-n^{2}}\leq\rho_{0}^{\varepsilon}\leq
2\rho_{*}u_{0}^{\varepsilon,n}\in\mathcal{C}_{0}^{\infty}({\mathbb{R}}^{2})~{}\,\,{\rm
with}~{}\,\,\nabla\cdot u_{0}^{\varepsilon,n}=0.$

For $n\in{\mathbb{N}}$, $n\geq 1$ fixed and for $\varepsilon\rightarrow
0^{+}$,

(3.13) $\left\\{\begin{aligned}
&\int_{B_{N_{0}}}\rho^{\varepsilon,n}_{0}\,dx>\frac{1}{4},\\\
&\bar{x}^{\alpha}\rho_{0}^{\varepsilon,n}\rightharpoonup\bar{x}^{\alpha}\rho_{0}\quad{\rm
weak}*\quad{\rm in}~{}~{}L^{\infty}({\mathbb{R}}^{2}),\\\
&\bar{x}^{\alpha}\rho^{\varepsilon,n}_{0}\to\bar{x}^{\alpha}\rho_{0}\quad{\rm
in}~{}~{}L^{p}({\mathbb{R}}^{2}),\quad\forall~{}p\in[1,\infty).\end{aligned}\right.$

The construction of an approximate initial velocity
$u_{0}^{\varepsilon,n}\in\mathcal{C}^{\infty}_{0}(B_{n})$ is inspired by [10].
We let $u_{0}^{\varepsilon,n}\in\mathcal{C}^{\infty}(\overline{B}_{n})$ be the
solution to

(Approx-compa) $\displaystyle-\Delta u^{\varepsilon,n}_{0}+\nabla
P_{0}^{\varepsilon,n}=\sqrt{\rho^{\varepsilon,n}_{0}}g^{\varepsilon,n},\quad\nabla\cdot
u^{\varepsilon,n}_{0}=0\qquad\mbox{on}\quad B_{n},$

with no-slip boundary conditions $u^{\varepsilon,n}_{0}=0$ on $\partial
B_{n}$, with $g^{\varepsilon,n}:=(g\star K_{\varepsilon})\varphi_{n}$. Notice
that we have the following estimate

(3.14) $\|\nabla(u^{\varepsilon,n}_{0}-u_{0})\|_{L^{2}(B_{n})}\leq\
Cn\|\sqrt{\rho^{\varepsilon,n}_{0}}g^{\varepsilon,n}-\rho_{0}g\|_{L^{2}(B_{n})}+n^{-1}\|u_{0}\|_{L^{2}(B_{n}^{c})}+\|u_{0}\|_{H^{1}(B_{n}^{c})},$

where we used the rescaled trace inequality

$n^{-\frac{1}{2}}\|u_{0}\|_{L^{2}(\partial
B_{n})}+\|u_{0}\|_{H^{\frac{1}{2}}(\partial B_{n})}\leq
n^{-1}\|u_{0}\|_{L^{2}(B_{n}^{c})}+\|u_{0}\|_{H^{1}(B_{n}^{c})}.$

Furthermore, by the Poincaré-Sobolev inequality [26, estimate II.3.7], we have

(3.15)
$\displaystyle\begin{split}\|u^{\varepsilon,n}_{0}\|_{L^{2}({\mathbb{R}}^{2})}\leq\
&C\|\nabla u^{\varepsilon,n}_{0}\|_{L^{1}({\mathbb{R}}^{2})}=C\|\nabla
u^{\varepsilon,n}_{0}\|_{L^{1}(B_{n})}\\\ \leq\
&Cn\|\nabla(u^{\varepsilon,n}_{0}-u_{0})\|_{L^{2}(B_{n})}+C\|\nabla
u_{0}\|_{L^{1}({\mathbb{R}}^{2})}\\\ \leq\
&Cn^{2}\|\sqrt{\rho^{\varepsilon,n}_{0}}g^{\varepsilon,n}-\rho_{0}g\|_{L^{2}(B_{n})}+C\|\nabla
u_{0}\|_{L^{1}({\mathbb{R}}^{2})}.\end{split}$

We rely on (3.14), (3.15) and on the facts that999Notice that here we use the
assumption that $u_{0}$ in addition belongs to $L^{2}({\mathbb{R}}^{2})$ so
that $u_{0}\in H^{1}({\mathbb{R}}^{2})$.

$n^{-1}\|u_{0}\|_{L^{2}(B_{n}^{c})}+\|u_{0}\|_{H^{1}(B_{n}^{c})}\rightarrow
0,\qquad n\rightarrow\infty$

and

$Cn^{2}\|\sqrt{\rho^{\varepsilon,n}_{0}}g^{\varepsilon,n}-\rho_{0}g\|_{L^{2}(B_{n})}\rightarrow
0,\qquad\varepsilon\rightarrow 0$

when $n$ is fixed, to choose a sequence $(\varepsilon_{k},n_{k})$ on which
$(\rho^{\varepsilon_{k},n_{k}}_{0},u_{0}^{\varepsilon_{k},n_{k}})$ have the ad
hoc boundedness and convergence properties.

Let $k\in{\mathbb{N}}$. Assume $n_{1}<\ldots\ <n_{k-1}$ and
$\varepsilon_{1}>\ldots\ >\varepsilon_{k-1}>0$ are constructed. There exists
$n_{k}\in{\mathbb{N}}$, $n_{k}>n_{k-1}$

$\displaystyle\|u_{0}\|_{L^{2}(B_{n_{k}}^{c})}+\|u_{0}\|_{H^{1}(B_{n_{k}}^{c})}\leq\frac{1}{2k}.$

Then there exists $0<\varepsilon_{k}<\varepsilon_{k-1}$ such that

$Cn_{k}^{2}\|\sqrt{\rho^{\varepsilon_{k},n_{k}}_{0}}g^{\varepsilon_{k},n_{k}}-\rho_{0}g\|_{L^{2}(B_{n_{k}})}\leq\frac{1}{2k}.$

Hence, for all $k\in{\mathbb{N}}$,

$\|\nabla(u^{\varepsilon_{k},n_{k}}_{0}-u_{0})\|_{L^{2}(B_{n_{k}})}\leq\frac{1}{k},$

which implies

$\displaystyle\|\nabla(u^{\varepsilon_{k},n_{k}}_{0}-u_{0})\|_{L^{2}({\mathbb{R}}^{2})}=\
$
$\displaystyle\|\nabla(u^{\varepsilon_{k},n_{k}}_{0}-u_{0})\|_{L^{2}(B_{n_{k}})}+\|\nabla
u_{0}\|_{L^{2}(B_{n_{k}}^{c})}$ $\displaystyle\leq\ $
$\displaystyle\frac{1}{k}+\|\nabla u_{0}\|_{L^{2}(B_{n_{k}}^{c})}\rightarrow
0,\qquad n\rightarrow\infty,$

and101010Remark that here the assumption that $\nabla u_{0}\in
L^{1}({\mathbb{R}}^{2})$ plays a key role.

$\|u^{\varepsilon_{k},n_{k}}_{0}\|_{L^{2}({\mathbb{R}}^{2})}\leq C\|\nabla
u_{0}\|_{L^{1}({\mathbb{R}}^{2})}+\frac{1}{2}.$

Therefore

$\|u^{\varepsilon_{k},n_{k}}_{0}\|_{L^{2}({\mathbb{R}}^{2})},\quad\|\sqrt{\rho^{\varepsilon_{k},n_{k}}_{0}}u^{\varepsilon_{k},n_{k}}_{0}\|_{L^{2}({\mathbb{R}}^{2})}\quad\mbox{and}\quad\|\nabla
u^{\varepsilon_{k},n_{k}}_{0}\|_{L^{2}({\mathbb{R}}^{2})}$

are bounded uniformly in $k$ by

(3.16)
$\mathcal{A}_{0}:=\rho_{*}+\|\rho_{0}\|_{L^{1}({\mathbb{R}}^{2})}+\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{2})}+\|\nabla
u_{0}\|_{L^{1}({\mathbb{R}}^{2})}+\|\nabla u_{0}\|_{L^{2}({\mathbb{R}}^{2})}$

and we have the following convergence properties

$\displaystyle u_{0}^{\varepsilon_{k},n_{k}}\to u_{0}\quad{\rm
in}~{}~{}\widetilde{\mathcal{D}}^{1,2}({\mathbb{R}}^{2}),\quad\sqrt{\rho_{0}^{\varepsilon_{k},n_{k}}}u_{0}^{\varepsilon_{k},n_{k}}\,\rightharpoonup\,\sqrt{\rho_{0}}u_{0}\quad{\rm
in}~{}~{}L^{2}({\mathbb{R}}^{2}).$

Finally, elliptic estimates for the Neumann boundary value problem imply

(3.17) $\|\nabla P_{0}^{\varepsilon_{k},n_{k}}\|_{L^{2}(B_{n_{k}})}\leq
C\|\sqrt{\rho^{\varepsilon_{k},n_{k}}_{0}}g^{\varepsilon_{k},n_{k}}\|_{L^{2}(B_{n_{k}})}\leq
C\|\sqrt{\rho_{0}}g\|_{L^{2}({\mathbb{R}}^{2})}+\frac{1}{2},$

with $C$ uniform in $k$, and by (Approx-compa) and maximal regularity for the
stationary Stokes system, we have

(3.18) $\|\nabla^{2}u^{\varepsilon_{k},n_{k}}\|_{L^{2}(B_{n_{k}})}\leq
C\|\sqrt{\rho^{\varepsilon_{k},n_{k}}_{0}}g^{\varepsilon_{k},n_{k}}\|_{L^{2}(B_{n_{k}})}\leq
C\|\sqrt{\rho_{0}}g\|_{L^{2}({\mathbb{R}}^{2})}+\frac{1}{2}.$

We consider the solutions $(\rho^{k},u^{k})$ to the following mollified
version of (INS)

(INS-$\varepsilon_{k}$) $\left\\{\begin{aligned}
&\partial_{t}\rho^{k}+\nabla\cdot\big{(}\rho^{k}(K_{\varepsilon_{k}}\star
u^{k})\big{)}=0,\\\
&\partial_{t}(\rho^{k}u^{k})+\nabla\cdot\big{(}\rho^{k}(K_{\varepsilon_{k}}\star
u^{k})\otimes u^{k}\big{)}+\nabla P^{k}=\nu\Delta u^{k},\\\ &\nabla\cdot
u^{k}=0.\end{aligned}\right.$

in the domain $B_{n_{k}}$ with no-slip boundary condition $u^{k}=0$ on
$\partial B_{n_{k}}$ and initial data

$(\rho^{k}_{0},u^{k}_{0}):=(u_{0}^{\varepsilon_{k},n_{k}},\rho_{0}^{\varepsilon_{k},n_{k}}).$

According to [42, Theorem 2.6], there exists a smooth global-in-time solution
$(\rho^{k},u^{k})\in\mathcal{C}^{\infty}([0,\infty)\times\overline{B}_{n_{k}})$
to (INS-$\varepsilon$).

Our objective is now to derive uniform estimates in $k$ for the approximate
solutions $(\rho^{k},u^{k})$. Notice that all the estimates below are on
$B_{n_{k}}$.111111See comment in Footnote 7.

####  3.2.2. First uniform estimates

Since $(\rho^{k},u^{k})$ belongs to
$\mathcal{C}^{\infty}([0,\infty)\times\overline{B}_{n_{k}})$ and satisfies the
no-slip boundary condition $u^{k}=0$ on $\partial B_{n_{k}}$, we have all the
estimates from the proof of [43, Theorem 1.1]. More precisely, we have the
following strengthened version of Proposition 3.1 that in addition to gradient
estimates provides time-weighted estimates for the convective derivative.

######  Proposition 3.5 (gradient and convective-derivative estimates; [43,
Lemma 3.2, 3.2 and 3.4]).

There exists a positive $C_{0}$ depending only on $\mathcal{A}_{0}$ defined by
(3.16) such that $0\leq\rho^{k}\leq\rho_{*}$ and

(3.19) $\|(\sqrt{\rho^{k}}u^{k},\nabla
u^{k})\|_{L^{\infty}({\mathbb{R}}_{+};L^{2}(B_{n_{k}}))}^{2}+\|(\nabla
u^{k},\sqrt{\rho^{k}}\dot{u}^{k},\Delta u^{k},\nabla
P^{k})\|_{L^{2}({\mathbb{R}}_{+}\times B_{n_{k}})}^{2}\leq C_{0}$

and for all $T>0$ it holds that

(3.20)
$\sup_{t\in[0,T]}\|\sqrt{\rho^{k}t}\dot{u}^{k}\|_{L^{2}(B_{n_{k}})}^{2}+\int_{0}^{T}\|\sqrt{t}\nabla\dot{u}^{k}\|_{L^{2}(B_{n_{k}})}^{2}\,dt\leq
C_{0}.$

Moreover,

(3.21) $\inf_{t\in[0,T]}\int_{B_{N_{1}}}\rho^{k}(t,x)\,dx\geq\frac{1}{4},$

for some positive constant $N_{1}$ depending only on
$\|\rho_{0}\|_{L^{1}({\mathbb{R}}^{2})},\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{2})},N_{0}$
and $T.$

Remark that (3.19) is a combination of estimates [43, (3.4), Lemma 3.2 and
Lemma 3.3], (3.20) follows from [43, Lemma 3.3] and (3.21) follows from [43,
(3.24)].

At this stage, we want to shift the regularity in order to get
$L^{1}_{t}Lip_{x}$ estimate for the velocity. To achieve it, we first need to
show the following spatial-weighted estimates for the velocity and the
density.

######  Proposition 3.6 (weighted estimates for the velocity and the density).

For any $T>0,$ there exists a positive constant $C_{0,T}$ depending only on
$N_{0}$, $T$ and on $\mathcal{A}_{0}$ defined by (3.16) such that for all
$a\in(2,\infty)$ and $b\in(0,1]$

(3.22)
$\sup_{t\in[0,T]}\|\bar{x}^{-b}u^{k}(t,\cdot)\|_{L^{\frac{a}{b}}(B_{n_{k}})}\leq
C_{0,T},$ (3.23)
$\displaystyle\sup_{t\in[0,T]}\|\bar{x}^{-1}u^{k}(t,\cdot)\|_{L^{2}(B_{n_{k}})}$
$\displaystyle\leq C_{0,T},$ (3.24)
$\displaystyle\|\bar{x}^{-b}u^{k}\|_{L^{2+b}(0,T;L^{\infty}(B_{n_{k}}))}$
$\displaystyle\leq C_{0,T}.$

Moreover for $\alpha>1$, for all $p\in[1,\infty]$,

(3.25)
$\sup_{t\in[0,T]}\|\bar{x}^{\alpha}\rho^{k}(t,\cdot)\|_{L^{p}(B_{n_{k}})}\leq
C_{0,T}.$

###### Proof.

At first, we prove inequality (3.23). It can be derived directly from
classical weighted inequalities and the uniform estimates in Proposition 3.5.
Indeed, estimate (A.3) with inequality (3.21) and (3.19) implies that there
exists a positive constant $C$ depending only on $\rho_{*},m,\ell,N_{0},T$ and
norms
$\|\rho_{0}\|_{L^{1}({\mathbb{R}}^{2})},\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{2})}$
such that

(3.26) $\displaystyle\left(\int_{B_{n_{k}}}\frac{|u^{k}(t,x)|^{m}}{{\langle
x\rangle}^{2}}\,(\ln\langle x\rangle)^{-\ell}\,dx\right)^{1/m}$
$\displaystyle\leq C(\|\sqrt{\rho^{k}}u^{k}\|_{L^{2}(B_{n_{k}})}+\|\nabla
u^{k}\|_{L^{2}(B_{n_{k}})})$ $\displaystyle\leq
C(\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{2})}+\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{2})}),$

for all $m\in[2,\infty)$ and $\ell\in(1+\frac{m}{2},\infty).$ Recalling that
$\bar{x}=\langle x\rangle(\ln\langle x\rangle)^{2}$ and taking $m=2,\ell=4$ in
inequality (3.26), we easily get (3.23).

Taking $m=4,\ell=4$ in inequality (3.26), we find that

(3.27) $\displaystyle\|\langle
x\rangle^{-1}u^{k}(t,\cdot)\|_{L^{4}(B_{n_{k}})}$
$\displaystyle\leq\left(\int_{B_{n_{k}}}|u^{k}(t,x)|^{4}\langle
x\rangle^{-2}(\ln\langle x\rangle)^{-4}\,dx\right)^{\frac{1}{4}}$
$\displaystyle\leq
C(\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{2})}+\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{2})}),$

where we used the fact that

(3.28) $\displaystyle(\ln\langle x\rangle)^{\gamma}\leq C(\gamma)\langle
x\rangle,\quad{\rm
for~{}any}~{}~{}\gamma\in[1,\infty),~{}~{}x\in{\mathbb{R}}^{2}.$

The proof of (3.22) be follows similarly by using (3.26) and (3.28) again.

To prove (3.24), we use the Gagliardo-Nirenberg inequality (A.1) and Young’s
inequality to write that

$\displaystyle\|\bar{x}^{-b}{u}^{k}(t,\cdot)\|_{L^{\infty}(B_{n_{k}})}$
$\displaystyle\leq
C(\|\bar{x}^{-b}{u}^{k}(t,\cdot)\|_{L^{\frac{a}{b}}(B_{n_{k}})}+\|\nabla(\bar{x}^{-b}{u}^{k}(t,\cdot))\|_{L^{\frac{a}{b}}(B_{n_{k}})})$
$\displaystyle\leq
C(\|\bar{x}^{-b}{u}^{k}(t,\cdot)\|_{L^{\frac{a}{b}}(B_{n_{k}})}+\|\nabla{u}^{k}(t,\cdot)\|_{L^{\frac{a}{b}}(B_{n_{k}})}).$

Thus, taking $a=4$, one then has (3.24) thanks to inequality (3.22) and the
uniform estimate (3.19).

Now, we can prove (3.25). Noticing that $\bar{x}^{\alpha}\rho^{k}$ satisfies

$\partial_{t}(\bar{x}^{\alpha}\rho^{k})+u^{k}\cdot\nabla(\bar{x}^{\alpha}\rho^{k})=\alpha(\bar{x}^{\alpha}\rho^{k})u^{k}\cdot\nabla\ln\bar{x},$

then the standard $L^{p}$ estimate for the transport equation gives that

$\displaystyle\|\bar{x}^{\alpha}\rho^{k}\|_{L^{p}(B_{n_{k}})}^{p}\leq
C\|\bar{x}^{\alpha}\rho_{0}\|_{L^{p}({\mathbb{R}}^{2})}^{p}\exp(\|u^{k}\cdot\nabla\ln\bar{x}\|_{L^{1}(0,T;L^{\infty}(B_{n_{k}}))}).$

Using the fact that

(3.29) $\displaystyle|\nabla\bar{x}|\leq C(\ln\langle
x\rangle)^{2},\quad|\nabla\langle x\rangle|\leq C,\quad{\rm
for~{}any}~{}~{}x\in{\mathbb{R}}^{2},$

together with inequalities (A.1), (3.19), (3.27), one obtains

(3.30)
$\displaystyle\int_{0}^{T}\|u^{k}\cdot\nabla\ln\bar{x}\|_{L^{\infty}(B_{n_{k}})}\,dt$
$\displaystyle\leq C\int_{0}^{T}\|\langle
x\rangle^{-1}u^{k}\|_{L^{\infty}(B_{n_{k}})}\,dt$ $\displaystyle\leq
C\int_{0}^{T}\|\langle
x\rangle^{-1}u^{k}\|_{L^{4}(B_{n_{k}})}^{\frac{1}{2}}\|\nabla(\langle
x\rangle^{-1}u^{k})\|_{L^{4}(B_{n_{k}})}^{\frac{1}{2}}\,dt$ $\displaystyle\leq
C\int_{0}^{T}\|\langle x\rangle^{-1}u^{k}\|_{L^{4}(B_{n_{k}})}\|\nabla
u^{k}\|_{L^{4}(B_{n_{k}})}^{\frac{1}{2}}\,dt$ $\displaystyle\leq
C(\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{2})}+\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{2})})^{\frac{3}{2}}\,T^{\frac{3}{4}}.$

This finishes the proof of Proposition 3.6. ∎

####  3.2.3. Shift of regularity

######  Proposition 3.7 (improved higher-order estimates).

For any $T>0$ and for all $r\in[2,\infty)$ and $q\in[2,\frac{2r}{r-2}),$ we
have

(3.31)
$\displaystyle\|\nabla^{2}(\sqrt{t}u^{k})\|_{L^{q}(0,T;L^{r}(B_{n_{k}}))}+\|\nabla(\sqrt{t}P^{k})\|_{L^{q}(0,T;L^{r}(B_{n_{k}}))}\leq
C_{0,T}$

and

(3.32)
$\displaystyle\sup_{t\in[0,T]}\left(\|\nabla^{2}(\sqrt{t}u^{k})\|_{L^{2}(B_{n_{k}})}+\|\nabla(\sqrt{t}P^{k})\|_{L^{2}(B_{n_{k}})}\right)\leq
C_{0,T},$

where $C_{0,T}$ depends only on $N_{0}$, $T$ and on $\mathcal{A}_{0}$ defined
by (3.16). Moreover, we have the following key Lipschitz estimate

$\displaystyle\int_{0}^{T}\|\nabla
u^{k}(t,\cdot)\|_{L^{\infty}(B_{n_{k}})}\,dt\leq C_{0,T}T^{\frac{1}{3}}$

and $C_{0,T}T^{\frac{1}{3}}\to 0$ as $T\to 0.$

###### Proof.

Applying the standard $L^{p}$ estimates for Stokes problem (2.13), we have for
all finite $p,$

(3.33)
$\displaystyle\|\nabla^{2}(\sqrt{t}u^{k})\|_{L^{p}(B_{n_{k}})}+\|\nabla(\sqrt{t}P^{k})\|_{L^{p}(B_{n_{k}})}\leq
C(p)\|\sqrt{t}\rho\dot{u}^{k}\|_{L^{p}(B_{n_{k}})}.$

Obviously, inequality (3.20) yields (3.32) directly.

Similarly to the proof of inequality (3.26), applying Proposition A.4 to
$\dot{u}$ implies that for all $r\in[2,\infty)$ (taking $\ell=2r$)

$\displaystyle\left(\int_{B_{n_{k}}}\frac{|\sqrt{t}\dot{u}^{k}(t,x)|^{r}}{{\langle
x\rangle}^{2}}\,(\ln\langle x\rangle)^{-2r}\,dx\right)^{1/r}$
$\displaystyle\leq
C(\|\sqrt{\rho^{k}t}\dot{u}^{k}\|_{L^{2}(B_{n_{k}})}+\|\sqrt{t}\nabla\dot{u}^{k}\|_{L^{2}(B_{n_{k}})}).$

Thus by using $\alpha>1,$ one has

$\displaystyle\|\sqrt{t}\rho\dot{u}^{k}\|_{L^{r}(B_{n_{k}})}=\ $
$\displaystyle\left(\int_{B_{n_{k}}}(\bar{x}^{\alpha}\rho^{k})^{r}(\langle
x\rangle^{2-\alpha r}(\ln\langle x\rangle)^{2r-2\alpha
r}|\sqrt{t}\dot{u}^{k}|^{r}\langle x\rangle^{-2}(\ln\langle
x\rangle)^{-2r}\,dx\right)^{1/r}$ $\displaystyle\leq\ $ $\displaystyle
C\|\bar{x}^{\alpha}\rho^{k}\|_{L^{\infty}}(\|\sqrt{\rho^{k}t}\dot{u}^{k}\|_{L^{2}(B_{n_{k}})}+\|\sqrt{t}\nabla\dot{u}^{k}\|_{L^{2}(B_{n_{k}})}),$

which together with inequalities (3.25), (3.20) and (3.33) yields (3.31) in
the case $q=2$. Further interpolating with (3.20) this finally implies (3.31).

To prove the Lipschitz estimate, we write by inequalities (A.1), (3.19) and
(3.31) that

$\displaystyle\int_{0}^{T}\|\nabla u^{k}\|_{L^{\infty}(B_{n_{k}})}\,dt\leq$
$\displaystyle\int_{0}^{T}\|\nabla
u^{k}\|_{L^{2}(B_{n_{k}})}^{\frac{1}{3}}\|\nabla^{2}(\sqrt{t}u^{k})\|_{L^{4}(B_{n_{k}})}^{\frac{2}{3}}\,t^{-\frac{1}{3}}\,dt$
$\displaystyle\leq$ $\displaystyle\|\nabla
u^{k}\|_{L^{\infty}(0,T;L^{2}(B_{n_{k}}))}^{\frac{1}{3}}\|\nabla^{2}(\sqrt{t}u^{k})\|_{L^{2}(0,T;L^{4}(B_{n_{k}}))}^{\frac{2}{3}}\|t^{-\frac{1}{3}}\|_{L^{\frac{3}{2}}(0,T)}$
$\displaystyle\leq$ $\displaystyle C_{0,T}\,T^{\frac{1}{3}},$

where $C_{0,T}$ remains bounded when $T\rightarrow 0$. ∎

####  3.2.4. Weighted $L^{\infty}$ estimates

In order to prove uniqueness, we need the following improved weighted-estimate
for the velocity, with the help of initial condition (Compa).

######  Proposition 3.8 (almost boundedness of the velocity).

For any $T>0,$ there exits a constant $C_{\\#}$ depending only on the norms
$\|g\|_{L^{2}({\mathbb{R}}^{2})},\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{2})},\|\Delta u_{0}\|_{L^{2}({\mathbb{R}}^{2})}$
and on $C_{0,T}$ (defined in Proposition 3.7) such that for any $b\in(0,1]$,

(3.34)
$\sup_{t\in[0,T]}\|\bar{x}^{-b}{u}^{k}(t,\cdot)\|_{L^{\infty}(B_{n_{k}})}\leq
C_{\\#}.$

###### Proof.

At first, using the Gagliardo-Nirenberg inequality (A.1) and Young’s
inequality, one has for any $a\in(2,\infty)$,

(3.35) $\displaystyle\|\bar{x}^{-b}{u}^{k}(t,\cdot)\|_{L^{\infty}(B_{n_{k}})}$
$\displaystyle\leq
C(\|\bar{x}^{-b}{u}^{k}(t,\cdot)\|_{L^{\frac{a}{b}}(B_{n_{k}})}+\|\nabla
u^{k}(t,\cdot)\|_{L^{\frac{a}{b}}(B_{n_{k}})}).$

Let us bound the right-hand-side of (3.35) for $\frac{a}{b}=4$. The bound for
the first term follows simply from (3.22).

It remains to obtain estimate for $\nabla u^{k}$ in
$L^{\infty}(0,T;L^{4}(B_{n_{k}}))$. Estimate (3.32) can be used to obtain an
estimate away from initial time. The difficulty is that this estimate
degenerates near initial time. Our goal is now to get an estimate near $t=0$.
In order to achieve this, we need the compatibility condition (Compa). From
[43, Lemma 3.3], we know that there exists a positive constant $C$ depending
only on on $\mathcal{A}_{0}$ defined by (3.16) such that

(3.36)
$\displaystyle\frac{d}{dt}\Big{(}\|\sqrt{\rho}\dot{u}^{k}\|_{L^{2}(B_{n_{k}})}^{2}-2\sum_{j=1,2}\langle
P,\partial_{j}u^{k}\cdot\nabla(u^{k})^{j}\rangle\Big{)}+\|\nabla\dot{u}^{k}\|_{L^{2}(B_{n_{k}})}^{2}\leq
C\|\sqrt{\rho}\dot{u}^{k}\|_{L^{2}(B_{n_{k}})}^{4}.$

First, one has for all $0\leq\tau\leq t\leq T$

(3.37)
$\displaystyle\|\sqrt{\rho}\dot{u}^{k}(\tau,\cdot)\|_{L^{2}(B_{n_{k}})}^{2}=\langle\rho\dot{u}^{k},\dot{u}^{k}\rangle=\langle\Delta
u^{k},\dot{u}^{k}\rangle-\langle\nabla P^{k},\dot{u}^{k}\rangle$

and

$\displaystyle\langle\Delta u^{k},\dot{u}^{k}\rangle$
$\displaystyle=\langle(\Delta u^{k}-\nabla
P_{0}^{k})/{\sqrt{\rho}},\sqrt{\rho^{k}}\dot{u}^{k}\rangle+\langle\nabla
P_{0}^{k},u^{k}\cdot\nabla u^{k}\rangle$ $\displaystyle=\langle(\Delta
u^{k}-\nabla
P_{0}^{k})/{\sqrt{\rho^{k}}},\sqrt{\rho^{k}}\dot{u}^{k}\rangle-\sum_{j=1,2}\langle
P_{0}^{k},\partial_{j}u^{k}\cdot\nabla(u^{k})^{j}\rangle,$

and

$\displaystyle\langle\nabla P^{k},\dot{u}^{k}\rangle=\langle\nabla
P^{k},u^{k}\cdot\nabla u^{k}\rangle=-\sum_{j=1,2}\langle
P^{k},\partial_{j}u^{k}\cdot\nabla(u^{k})^{j}\rangle.$

Second, by using the well-known duality between ${\rm BMO}({\mathbb{R}}^{2})$
and the Hardy space $\mathcal{H}^{1}({\mathbb{R}}^{2})$ and the following
inequality (see for instance [42]), there exists a universal positive constant
$C$ such that for all vectors $w,z\in
L^{2}({\mathbb{R}}^{2};{\mathbb{R}}^{2})$ satisfying $\nabla\cdot
w=\nabla^{\perp}z=0,$

(3.38) $\|w\cdot z\|_{\mathcal{H}^{1}({\mathbb{R}}^{2})}\leq
C\|w\|_{L^{2}({\mathbb{R}}^{2})}\|z\|_{L^{2}({\mathbb{R}}^{2})},$

we have

(3.39) $\displaystyle|\langle
P^{k}-P_{0}^{k},\partial_{j}u^{k}\cdot\nabla(u^{k})^{j}\rangle|$
$\displaystyle\leq\|P^{k}-P_{0}^{k}\|_{{\rm
BMO}({\mathbb{R}}^{2})}\|\partial_{j}u^{k}\|_{L^{2}({\mathbb{R}}^{2})}\|\nabla(u^{k})^{j}\|_{L^{2}({\mathbb{R}}^{2})}$
$\displaystyle\leq
C(\rho_{*})\big{(}\|\sqrt{\rho_{0}}g\|_{L^{2}({\mathbb{R}}^{2})}+\|\sqrt{\rho^{k}}\dot{u}^{k}\|_{L^{2}({\mathbb{R}}^{2})}\big{)}\|\nabla
u^{k}\|_{L^{2}({\mathbb{R}}^{2})}^{2},$

where in the last inequality we used the embedding
$\dot{H}^{1}({\mathbb{R}}^{2})\hookrightarrow{\rm BMO}({\mathbb{R}}^{2})$, the
standard maximal regularity estimates for the stationary Stokes system applied
to (INS-$\varepsilon_{k}$) and estimate (3.17) for $P_{0}^{k}$.

Thus by Hölder’s and Young’s inequalities and (3.39), the identity (3.37)
leads to the following estimate

$\|\sqrt{\rho^{k}}\dot{u}^{k}(\tau,\cdot)\|_{L^{2}(B_{n_{k}})}^{2}\leq
C\Big{(}\|(\Delta u^{k}-\nabla
P_{0}^{k})/{\sqrt{\rho^{k}}}\|_{L^{2}(B_{n_{k}})}^{2}\\\
+\|{\sqrt{\rho^{k}}}u^{k}\cdot\nabla u^{k}\|_{L^{2}(B_{n_{k}})}^{2}+\|\nabla
u^{k}\|_{L^{2}(B_{n_{k}})}^{4}\Big{)}.$

Moreover, using Hölder’s inequality again, (A.1) and the estimate

$\displaystyle\sup_{t\in[0,T]}\|\sqrt{\rho}u^{k}(t,\cdot)\|_{L^{4}({\mathbb{R}}^{2})}$
$\displaystyle=\left(\int_{{\mathbb{R}}^{2}}(\bar{x}^{\alpha}\rho)^{2}\langle
x\rangle^{2-2\alpha}(\ln\langle x\rangle)^{4-4\alpha}|{u}^{k}|^{4}\langle
x\rangle^{-2}(\ln\langle x\rangle)^{-4}\,dx\right)^{1/4}$ $\displaystyle\leq
C\|\bar{x}^{\alpha}\rho\|_{L^{\infty}({\mathbb{R}}^{2})}\big{(}\|\sqrt{\rho^{k}}{u}^{k}\|_{L^{2}({\mathbb{R}}^{2})}+\|\nabla{u}^{k}\|_{L^{2}({\mathbb{R}}^{2})}\big{)}\leq
C_{0,T},$

the above inequality can be further rewritten as

$\|\sqrt{\rho^{k}}\dot{u}^{k}(\tau,\cdot)\|_{L^{2}(B_{n_{k}})}^{2}\leq
C\|(\Delta u^{k}-\nabla
P_{0}^{k})/{\sqrt{\rho^{k}}}\|_{L^{2}(B_{n_{k}})}^{2}\\\ +C_{0,T}\|\nabla
u^{k}\|_{L^{2}(B_{n_{k}})}\|\nabla^{2}u^{k}\|_{L^{2}(B_{n_{k}})}+\|\nabla
u^{k}\|_{L^{2}(B_{n_{k}})}^{4}.$

Therefore, by the fact that $u^{k}\in C_{t}H^{2}$ and by the approximate
compatibility condition (Approx-compa), we have

(3.40) $\limsup_{\tau\to
0^{+}}\|\sqrt{\rho}\dot{u}^{k}(\tau,\cdot)\|_{L^{2}(B_{n_{k}})}^{2}\\\ \leq
C(\|g\|_{L^{2}({\mathbb{R}}^{2})}^{2}+C_{0,T}\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{2})}\|\nabla^{2}u_{0}\|_{L^{2}({\mathbb{R}}^{2})}+\|\nabla
u_{0}\|_{L^{2}({\mathbb{R}}^{2})}^{4}):=C_{\\#}.$

Notice that we used (3.18) to bound
$\|\nabla^{2}u^{k}\|_{L^{2}({\mathbb{R}}^{2})}$ uniformly in $k$. By a variant
of inequality (3.39), we see that

$\frac{1}{2}\|\sqrt{\rho}\dot{u}^{k}(t,\cdot)\|_{L^{2}(B_{n_{k}})}^{2}-C\|\nabla
u^{k}(t,\cdot)\|_{L^{2}(B_{n_{k}})}^{4}\leq\
\|\sqrt{\rho^{k}}\dot{u}^{k}(t,\cdot)\|_{L^{2}(B_{n_{k}})}^{2}\\\
-2\sum_{j=1,2}\langle
P^{k},\partial_{j}u^{k}\cdot\nabla(u^{k})^{j}\rangle|(t).$

Then, integrating (3.36) on the time interval $[\tau,t]$ and letting $\tau\to
0^{+}$ in the resulting inequality, we obtain that there exits a positive time
$T_{\\#}$ depending only on $C_{\\#}$ such that

(3.41)
$\displaystyle\|\sqrt{\rho^{k}}\dot{u}^{k}(t,\cdot)\|_{L^{2}(B_{n_{k}})}^{2}\leq
CC_{\\#},\quad{\rm for~{}all~{}time}~{}t\in(0,T_{\\#}].$

Using the standard estimates for the Stokes system, we then get for all
$t\in(0,T_{\\#}]$,

$\displaystyle\|\Delta u^{k}(t,\cdot)\|_{L^{2}(B_{n_{k}})}^{2}+\|\nabla
P^{k}(t,\cdot)\|_{L^{2}(B_{n_{k}})}^{2}\leq
C\rho_{*}\|\sqrt{\rho^{k}}\dot{u}^{k}(t,\cdot)\|_{L^{2}(B_{n_{k}})}^{2}\leq
CC_{\\#}.$

This together with estimate (3.32) implies that

(3.42)
$\displaystyle\sup_{t\in[0,T]}\|\nabla^{2}u^{k}(t,\cdot)\|_{L^{2}(B_{n_{k}})}^{2}\leq
C(C_{\\#}+C_{0,T}).$

Finally, from the Gagliardo-Nirenberg inequality (A.1) and (3.19),

$\displaystyle\sup_{t\in[0,T]}\|\nabla
u^{k}(t,\cdot)\|_{L^{4}(B_{n_{k}})}\leq\ $
$\displaystyle\sup_{t\in[0,T]}(\|\nabla
u^{k}(t,\cdot)\|_{L^{2}(B_{n_{k}})}+\|\nabla^{2}u^{k}(t,\cdot)\|_{L^{2}(B_{n_{k}})})$
$\displaystyle\leq\ $ $\displaystyle C(C_{\\#}+C_{0,T}).$

This completes the proof of proposition 3.8. ∎

##  4\. Proof of the uniqueness results

In this section, we show the uniqueness of the solutions that we constructed
in the paper, both in the two- and three-dimensional case. The main difficulty
we have to face is that having only bounded solutions can not ensure $L^{2}$
stability for the transport equation. Of course, one may reformulate system
(INS) in Lagrangian coordinates to prove uniqueness. The advantage of doing so
is obvious: the density is constant along the flow. However, motivated by
Hoff’s paper [30] on the compressible Navier-Stokes equations, it is possible
to directly estimate the difference of the densities in $\dot{H}^{-1}$ (see
for example [17, 21]).

###  4.1. The case when $u$ is in $L^{2}({\mathbb{R}}^{2})$ or
$L^{6}({\mathbb{R}}^{3})$

######  Proposition 4.1 (uniqueness in 2D and 3D under control of a Lebesgue
norm of the velocity).

Consider two finite-energy weak solutions $(\rho,u)$ and
$(\bar{\rho},\bar{u})$ (in the sense of Definition 1.1) to system (INS)
corresponding to the same initial data $(\rho_{0},u_{0})$ satisfying (1.2).
Assume that

(4.1) $\left\\{\begin{aligned} &\nabla\bar{u}\in
L^{\infty}(0,T;L^{2}({\mathbb{R}}^{d}))\cap
L^{1}(0,T;L^{\infty}({\mathbb{R}}^{d})),\\\ &\sqrt{t}\dot{\bar{u}}\in
L^{2}(0,T;L^{6}({\mathbb{R}}^{d})),~{}\nabla(\sqrt{t}\dot{\bar{u}})\in
L^{2}((0,T)\times{\mathbb{R}}^{d}),\end{aligned}\right.$

and

  * •

In the two-dimensional case, assume in addition that $\rho_{0}$ satisfies
condition (aNV) or (VB),

  * •

In the three-dimensional case, assume in addition that,

(4.2) $\displaystyle\nabla u\in L^{4}(0,T;L^{2}({\mathbb{R}}^{3})),$

then $(\rho,u)\equiv(\bar{\rho},\bar{u})$ on
$[0,T]\times{\mathbb{R}}^{d}$.121212Let us stress that the uniqueness is in
the class of finite-energy weak solutions. Moreover, notice that the result in
2D is of weak-strong uniqueness type.

###### Proof.

We focus on the case when $T$ is small, say $T\leq\frac{1}{2}$. We remark that
a standard connectivity argument enables us to prove the the case of arbitrary
large $T$.

Step 1: control of the difference of the densities.
Define the difference $\delta\\!\rho:=\rho-\bar{\rho}$ and
$\delta\\!u:=u-\bar{u},$ then the system for $(\delta\\!\rho,\delta\\!u)$
reads

(4.3) $\left\\{\begin{aligned}
&\partial_{t}\delta\\!\rho+\bar{u}\cdot\nabla\delta\\!\rho+\delta\\!u\cdot\nabla\rho=0,\\\
&\rho(\partial_{t}\delta\\!u+u\cdot\nabla\delta\\!u)+\nabla\delta
P-\Delta\delta\\!u=-\delta\\!\rho\dot{\bar{u}}-\rho\delta\\!u\cdot\nabla\bar{u},\\\
&\nabla\cdot\delta\\!u=0,\\\
&(\delta\\!\rho,\delta\\!u)|_{t=0}=0.\end{aligned}\right.$

We perform estimates for $\delta\\!\rho$ in
$L^{\infty}(0,T;\dot{H}^{-1}({\mathbb{R}}^{d})$ and for $\delta\\!u$ in
$L^{2}((0,T)\times{\mathbb{R}}^{d}).$ We remark that the general strategy is
the same for dimensions $d=2,3.$ To achieve these estimates, we set

(4.4) $\displaystyle\phi:=-(-\Delta)^{-1}\delta\\!\rho\quad{\rm
so~{}that}\quad\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{d})}=\|\delta\\!\rho\|_{\dot{H}^{-1}({\mathbb{R}}^{d})}.$

Now, testing the first equation of (4.3) against $\phi$ yields

(4.5)
$\displaystyle\frac{1}{2}\frac{d}{dt}\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{d})}^{2}\leq|\langle\bar{u}\cdot\nabla\Delta\phi,\phi\rangle|+|\langle\delta\\!u\cdot\nabla\rho,\phi\rangle|.$

For the first term in the right-hand-side of (4.5), using that
$\nabla\cdot\bar{u}=0$ one has

$\displaystyle\langle\bar{u}\cdot\nabla\Delta\phi,\phi\rangle=\sum_{1\leq
i,j\leq d}\langle\bar{u}^{i}\partial_{ijj}\phi,\phi\rangle=\sum_{1\leq i,j\leq
d}\langle\partial_{j}\bar{u}^{i}\partial_{j}\phi,\partial_{i}\phi\rangle$

so that

$\displaystyle|\langle\bar{u}\cdot\nabla\Delta\phi,\phi\rangle|\leq\|\nabla\bar{u}\|_{L^{\infty}({\mathbb{R}}^{d})}\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{d})}^{2}.$

As for the second term in the right-hand-side of (4.5), noticing that
$\nabla\cdot\delta\\!u=0,$ then

$\displaystyle\langle\delta\\!u\cdot\nabla\rho,\phi\rangle=-\langle\delta\\!u\cdot\nabla\phi,\rho\rangle,$

and thus

$\displaystyle|\langle\delta\\!u\cdot\nabla\rho,\phi\rangle|\leq\sqrt{\rho_{*}}\|\sqrt{\rho}\delta\\!u\|_{L^{2}({\mathbb{R}}^{d})}\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{d})}.$

Collecting the above two estimates into (4.5), we discover that

$\displaystyle\|\nabla\phi(t,\cdot)\|_{L^{2}({\mathbb{R}}^{d})}\leq\int_{0}^{t}\|\nabla\bar{u}(s,\cdot)\|_{L^{\infty}({\mathbb{R}}^{d})}\|\nabla\phi(s,\cdot)\|_{L^{2}({\mathbb{R}}^{d})}\,ds+\sqrt{\rho_{*}}\int_{0}^{t}\|\sqrt{\rho}\delta\\!u(s,\cdot)\|_{L^{2}({\mathbb{R}}^{d})}\,ds.$

Hence, using (4.4) and denoting

$D(t):=\sup_{0<s\leq
t}\,s^{-\frac{1}{2}}\|\delta\\!\rho(s,\cdot)\|_{\dot{H}^{-1}({\mathbb{R}}^{d})},$

we get after using Young’s inequality and Gronwall’s lemma, for all
$t\in[0,T],$

(4.6) $\displaystyle
D(t)\leq\sqrt{\rho_{*}}\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,t)\times{\mathbb{R}}^{d})}\exp(\|\nabla\bar{u}\|_{L^{1}(0,T;L^{\infty}({\mathbb{R}}^{d}))}).$

Step 2: duality argument.
At this stage, in order to control the difference $\sqrt{\rho}\delta\\!u$ in
$L^{2}((0,t)\times{\mathbb{R}}^{d})$, we introduce the solution $v$ to the
following linear backward parabolic system:

(4.7) $\left\\{\begin{aligned} &\rho(\partial_{t}v+u\cdot\nabla v)+\nabla
Q+\Delta v=\rho\delta\\!u,\\\ &\nabla\cdot v=0,\\\
&v|_{t=T}=0.\end{aligned}\right.$

Let us first claim an important a priori estimate for the above system:

(4.8) $\sup_{t\in[0,T]}\|\left(\sqrt{\rho}v,\nabla
v\right)(t,\cdot)\|_{L^{2}({\mathbb{R}}^{d})}^{2}+\int_{0}^{T}\left(\|\nabla
v(t,\cdot)\|_{L^{2}}^{2}+\|(\sqrt{\rho}\dot{v},\nabla^{2}v,\nabla
Q)(t,\cdot)\|_{L^{2}}^{2}\right)\,dt\\\ \leq
C\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{d})}^{2},$

that we prove below. Solving the above problem is not included in the
classical theory for the linear evolutionary Stokes system, since the
coefficients are rough and may vanish. It is not hard to find that if
$(\rho,u)$ are regular with $\rho$ bounded from below away from zero then the
existence issue can be solved thanks to the a priori estimate (4.8). In our
setting, the existence may be established by a regularizing procedure of
$(\rho,u),$ after using estimate (4.8), as in the proof of the existence part
of Theorem A and B.

For the moment, we assume that estimate (4.8) is satisfied. Testing the
equation of $\delta\\!u$ in system (4.3) by $v$ yields that

(4.9)
$\displaystyle\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{d})}^{2}\leq\int_{0}^{T}|\langle\delta\\!\rho\dot{\bar{u}},v\rangle|\,dt+\int_{0}^{T}|\langle\rho\delta\\!u\cdot\nabla\bar{u},v\rangle|\,dt.$

We have, by Hölder’s inequality

$\int_{0}^{T}|\langle\delta\\!\rho\dot{\bar{u}},v\rangle|\,dt\leq\|t^{-1/2}\delta\\!\rho\|_{L^{\infty}(0,T;\dot{H}^{-1}({\mathbb{R}}^{d}))}\|\sqrt{t}\nabla(\dot{\bar{u}}\cdot
v)\|_{L^{1}(0,T;L^{2}({\mathbb{R}}^{d}))}\\\ \qquad\leq
D(T)(\|\nabla(\sqrt{t}\dot{\bar{u}})\|_{L^{2}((0,T)\times{\mathbb{R}}^{d})}\|v\|_{L^{2}(0,T;L^{\infty}({\mathbb{R}}^{d}))}+\|\sqrt{t}\dot{\bar{u}}\|_{L^{2}(0,T;L^{6}({\mathbb{R}}^{d}))}\|\nabla
v\|_{L^{2}(0,T;L^{3}({\mathbb{R}}^{d}))}),$

and

$\displaystyle\int_{0}^{T}|\langle\rho\delta\\!u\cdot\nabla\bar{u},v\rangle|\,dt\leq\sqrt{\rho_{*}}\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{d})}\|\nabla\bar{u}\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{d}))}\|v\|_{L^{2}(0,T;L^{\infty}({\mathbb{R}}^{d}))}.$

By estimate (4.8) and Gagliardo-Nirenberg’s inequality [26, Lemma II.3.3], we
see that in 2D

$\displaystyle\|v\|_{L^{4}(0,T;L^{\infty}({\mathbb{R}}^{2}))}$
$\displaystyle\leq\left(\int_{0}^{T}\|v\|_{L^{2}({\mathbb{R}}^{2})}^{2}\|\nabla^{2}v\|_{L^{2}({\mathbb{R}}^{2})}^{2}\,dt\right)^{1/4}$
$\displaystyle\leq C\|(\sqrt{\rho}v,\nabla
v)\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{2}))}^{1/2}\|\nabla^{2}v\|_{L^{2}((0,T)\times{\mathbb{R}}^{2})}^{1/2}$
$\displaystyle\leq
C\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{2})},$

and in 3D

$\displaystyle\|v\|_{L^{4}(0,T;L^{\infty}({\mathbb{R}}^{3}))}$
$\displaystyle\leq\left(\int_{0}^{T}\|\nabla
v\|_{L^{2}({\mathbb{R}}^{3})}^{2}\|\nabla^{2}v\|_{L^{2}({\mathbb{R}}^{3})}^{2}\,dt\right)^{1/4}$
$\displaystyle\leq C\|\nabla
v\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))}^{1/2}\|\nabla^{2}v\|_{L^{2}((0,T)\times{\mathbb{R}}^{3})}^{1/2}$
$\displaystyle\leq
C\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{3})}.$

Moreover, one has (recall that $T\leq 1/2$)

$\displaystyle\|\nabla v\|_{L^{2}(0,T;L^{3}({\mathbb{R}}^{d}))}$
$\displaystyle\leq C\|\nabla
v\|_{L^{2}((0,T)\times{\mathbb{R}}^{d})}^{1-\frac{d}{6}}\|\nabla^{2}v\|_{L^{2}((0,T)\times{\mathbb{R}}^{d})}^{\frac{d}{6}}$
$\displaystyle\leq CT^{\frac{1}{2}-\frac{d}{12}}\|\nabla
v\|_{L^{\infty}((0,T;L^{2}({\mathbb{R}}^{d}))}^{1-\frac{d}{6}}\|\nabla^{2}v\|_{L^{2}((0,T)\times{\mathbb{R}}^{d})}^{\frac{d}{6}}$
$\displaystyle\leq
CT^{\frac{1}{3}}\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{d})}.$

So that one can conclude that

$\displaystyle\|v\|_{L^{2}(0,T;L^{\infty}({\mathbb{R}}^{d}))}+\|\nabla
v\|_{L^{2}(0,T;L^{3}({\mathbb{R}}^{d}))}\leq
CT^{\frac{1}{4}}\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{d})}.$

Taking these estimates into (4.9) and using assumption (4.1), we get

$\displaystyle\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{d})}^{2}\leq
CT^{\frac{1}{3}}(D(T)\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{d})}+\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{d})}^{2}).$

This together with inequality (4.6) implies that if $T$ is small enough then

$\displaystyle\sqrt{\rho}\delta\\!u\equiv 0D(t)\equiv 0\quad{\rm on}~{}[0,T].$

Now, since we already know that $\sqrt{\rho}\delta\\!u$ and $\delta\\!\rho$
are zero on $[0,T],$ a direct $L^{2}$ estimate for the equation of
$\delta\\!u$ in system (4.3) gives directly that

(4.10)
$\displaystyle\int_{0}^{t}\|\nabla\delta\\!u(s,\cdot)\|_{L^{2}({\mathbb{R}}^{d})}^{2}\,ds=0.$

In the 2D case, using again the condition (aNV) or (VB) together with
Proposition A.6, one gets $\delta\\!u\equiv 0$ on $[0,T].$ In the 3D case, one
gets same conclusion by recalling that $\delta\\!u\in
L^{2}({\mathbb{R}}_{+};L^{6}({\mathbb{R}}^{3})).$

Step 3: proof of estimate (4.8).
It remains to prove estimate (4.8). Actually, the proof is similar to
Proposition 3.1. To achieve it, we proceed as follows. At first, testing the
equation by $v,$ one finds that

(4.11)
$\sup_{t\in(0,T)}\|\sqrt{\rho}v(t,\cdot)\|_{L^{2}({\mathbb{R}}^{d})}^{2}+\int_{0}^{T}\|\nabla
v(t,\cdot)\|_{L^{2}({\mathbb{R}}^{d})}^{2}\,dt\\\
\leq\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{d})}^{2}+T\sup_{t\in(0,T)}\|\sqrt{\rho}v(t,\cdot)\|_{L^{2}({\mathbb{R}}^{d})}^{2}.$

Secondly, in order to obtain the higher-order estimates of $v$, we apply the
standard maximal regularity estimates for the stationary Stokes system to
(4.7) to get that for any $p\in(1,\infty),$

(4.12) $\|\Delta v\|_{L^{p}({\mathbb{R}}^{d})}+\|\nabla
Q\|_{L^{p}({\mathbb{R}}^{d})}\leq
C(p)\|\rho\dot{v}\|_{L^{p}({\mathbb{R}}^{d})}\leq
C(p)\sqrt{\rho_{*}}\|\sqrt{\rho}\dot{v}\|_{L^{p}({\mathbb{R}}^{d})}.$

Testing the first equation of (4.7) by $\dot{v}$ gives that

$\displaystyle\|\sqrt{\rho}\dot{v}(t,\cdot)\|_{L^{2}({\mathbb{R}}^{d})}^{2}+\langle\Delta
v,\dot{v}\rangle+\langle\nabla
Q,\dot{v}\rangle=\langle\rho\delta\\!u,\dot{v}\rangle.$

Since

$\displaystyle\langle\Delta v,\dot{v}\rangle=-\frac{d}{dt}\|\nabla
v\|_{L^{2}}^{2}+\sum_{1\leq i,j,k\leq
d}\langle\partial_{kk}v^{i},u^{j}\partial_{j}v^{i}\rangle$

and

$\displaystyle\langle\partial_{kk}v^{i},u^{j}\partial_{j}v^{i}\rangle=-\langle\partial_{k}v^{i},\partial_{k}u^{j}\partial_{j}v^{i}\rangle.$

thus

(4.13) $\displaystyle-\frac{d}{dt}\|\nabla
v\|_{L^{2}}^{2}+\|\sqrt{\rho}\dot{v}(t,\cdot)\|_{L^{2}({\mathbb{R}}^{d})}^{2}=\langle\rho\delta\\!u,\dot{v}\rangle-\langle\nabla
Q,\dot{v}\rangle+\sum_{1\leq i,j,k\leq
d}\langle\partial_{k}v^{i},\partial_{k}u^{j}\partial_{j}v^{i}\rangle.$

Note that in the 2D case, we have by the Gagliardo-Nirenberg inequality (A.1)

(4.14) $\displaystyle\sum_{1\leq i,j,k\leq
d}|\langle\partial_{k}v^{i},\partial_{k}u^{j}\partial_{j}v^{i}\rangle|\leq$
$\displaystyle C\|\nabla v\|_{L^{4}({\mathbb{R}}^{2})}^{2}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{2})}$ $\displaystyle\leq$ $\displaystyle C\|\nabla
v\|_{L^{2}({\mathbb{R}}^{2})}\|\nabla^{2}v\|_{L^{2}({\mathbb{R}}^{2})}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{2})}$ $\displaystyle\leq$
$\displaystyle\frac{1}{C{\rho_{*}}}\|\nabla^{2}v\|_{L^{2}({\mathbb{R}}^{2})}^{2}+C\rho_{*}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{2})}^{2}\|\nabla v\|_{L^{2}({\mathbb{R}}^{2})}^{2},$

while in the 3D case,

(4.15) $\displaystyle\sum_{1\leq i,j,k\leq
d}|\langle\partial_{k}v^{i},\partial_{k}u^{j}\partial_{j}v^{i}\rangle|\leq$
$\displaystyle C\|\nabla v\|_{L^{4}({\mathbb{R}}^{3})}^{2}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}$ $\displaystyle\leq$ $\displaystyle C\|\nabla
v\|_{L^{2}({\mathbb{R}}^{3})}^{\frac{1}{2}}\|\nabla^{2}v\|_{L^{2}({\mathbb{R}}^{3})}^{\frac{3}{2}}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}$ $\displaystyle\leq$
$\displaystyle\frac{1}{C{\rho_{*}}}\|\nabla^{2}v\|_{L^{2}({\mathbb{R}}^{3})}^{2}+C\rho_{*}^{3}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}^{4}\|\nabla v\|_{L^{2}({\mathbb{R}}^{3})}^{2}.$

Meanwhile, using that $\nabla\cdot v=0$, we write

$\displaystyle\langle\nabla Q,\dot{v}\rangle=-\langle
Q,\nabla\cdot(u\cdot\nabla v)\rangle=-\sum_{1\leq i,j\leq d}\langle
Q,\partial_{i}u^{j}\partial_{j}v^{i}\rangle=-\sum_{1\leq i\leq d}\langle
Q,\partial_{i}u\cdot\nabla v^{i}\rangle.$

Therefore in the 2D case, inequality (3.38) implies

(4.16) $\displaystyle|\langle\nabla Q,\dot{v}\rangle|$ $\displaystyle\leq
C\|Q\|_{{\rm BMO}({\mathbb{R}}^{2})}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{2})}\|\nabla v\|_{L^{2}({\mathbb{R}}^{2})}$
$\displaystyle\leq C\|\nabla Q\|_{L^{2}({\mathbb{R}}^{2})}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{2})}\|\nabla v\|_{L^{2}({\mathbb{R}}^{2})}$
$\displaystyle\leq\frac{1}{C{\rho_{*}}}\|\nabla
Q\|_{L^{2}({\mathbb{R}}^{2})}^{2}+C\rho_{*}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{2})}^{2}\|\nabla v\|_{L^{2}({\mathbb{R}}^{2})}^{2}.$

As for the 3D case, one has

(4.17) $\displaystyle|\langle\nabla Q,\dot{v}\rangle|$ $\displaystyle\leq
C\|Q\|_{L^{6}({\mathbb{R}}^{3})}\|\nabla u\|_{L^{2}({\mathbb{R}}^{3})}\|\nabla
v\|_{L^{3}({\mathbb{R}}^{3})}$ $\displaystyle\leq
C\|Q\|_{\mathcal{D}^{1,2}({\mathbb{R}}^{3})}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}\|\nabla
v\|_{L^{2}({\mathbb{R}}^{3})}^{\frac{1}{2}}\|\nabla^{2}v\|_{L^{2}({\mathbb{R}}^{3})}^{\frac{1}{2}}$
$\displaystyle\leq\frac{1}{C{\rho_{*}}}(\|\nabla^{2}v\|_{L^{2}}^{2}+\|Q\|_{\mathcal{D}^{1,2}({\mathbb{R}}^{3})}^{2})+C\rho_{*}^{3}\|\nabla
u\|_{L^{2}({\mathbb{R}}^{3})}^{4}\|\nabla v\|_{L^{2}({\mathbb{R}}^{3})}^{2}.$

It is easy to find that

$\displaystyle|\langle\rho\delta\\!u,\dot{v}\rangle|\leq\|\sqrt{\rho}\delta\\!u\|_{L^{2}({\mathbb{R}}^{d})}\|\sqrt{\rho}\dot{v}\|_{L^{2}({\mathbb{R}}^{d})}.$

Putting the above estimate and (4.12), (4.14)-(4.17) into (4.13), we conclude
after applying Gronwall’s lemma that, in the 2D case

$\sup_{t\in(0,T)}\|\nabla
v(t,\cdot)\|_{L^{2}({\mathbb{R}}^{2})}^{2}+\int_{0}^{T}\|(\sqrt{\rho}\dot{v},\nabla^{2}v,\nabla
Q)(t,\cdot)\|_{L^{2}({\mathbb{R}}^{2})}^{2}\,dt\\\
\leq\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{2})}^{2}\exp(C\rho_{*}\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}({\mathbb{R}}^{2})}^{2})$

and in the 3D case

$\sup_{t\in(0,T)}\|\nabla
v(t,\cdot)\|_{L^{2}({\mathbb{R}}^{3})}^{2}+\int_{0}^{T}\|(\sqrt{\rho}\dot{v},\nabla^{2}v,\nabla
Q)(t,\cdot)\|_{L^{2}({\mathbb{R}}^{3})}^{2}\,dt\\\
\leq\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{3})}^{2}\exp(C\rho_{*}^{3}\|\nabla
u\|_{L^{4}(0,T;L^{2}({\mathbb{R}}^{3}))}^{4}).$

Finally, these estimates together with estimate (4.11) prove (4.8). This
completes the proof of Proposition 4.1. ∎

###  4.2. The case of two-dimensional far-field vacuum

######  Proposition 4.2 (uniqueness in 2D in the case of far-field vacuum).

Let $d=2$. Consider two finite-energy weak solutions $(\rho,u)$ and
$(\bar{\rho},\bar{u})$ (in the sense of Definition 1.1) to system (INS)
corresponding to the same initial data $(\rho_{0},u_{0})$ satisfying (1.2).
Assume in addition that131313Here as in the whole paper, see (FFV-2).
$\rho\bar{x}^{\alpha}\in L^{1}({\mathbb{R}}^{2})\cap
L^{\infty}({\mathbb{R}}^{2})$ and

(4.18) $\left\\{\begin{aligned} &\bar{x}^{-1}\bar{u}\in
L^{\infty}(0,T;L^{2}({\mathbb{R}}^{2})),~{}~{}\bar{x}^{-1}\bar{u},~{}\bar{x}^{-1}u\in
L^{\infty}((0,T)\times{\mathbb{R}}^{2}),~{}~{}\langle x\rangle^{-1}\bar{u}\in
L^{1}(0,T;L^{\infty}({\mathbb{R}}^{2})),\\\ &\nabla\bar{u}\in
L^{\infty}(0,T;L^{2}({\mathbb{R}}^{2}))\cap
L^{1}(0,T;L^{\infty}({\mathbb{R}}^{2})),\\\ &\nabla(\sqrt{t}\dot{\bar{u}})\in
L^{2}((0,T)\times{\mathbb{R}}^{2}),\sqrt{t}\dot{\bar{u}}\bar{x}^{-\beta}\in
L^{2}(0,T;L^{6}({\mathbb{R}}^{2}))~{}~{}{\rm
for~{}some}~{}\beta\in\Big{(}\frac{1}{3},\frac{1}{2}\Big{)}.\end{aligned}\right.$

Then $(\rho,u)\equiv(\bar{\rho},\bar{u})$ on
$[0,T]\times{\mathbb{R}}^{2}$.141414Notice that the uniqueness is for finite-
energy weak solutions. However, this result is not of weak-strong uniqueness
type.

###### Proof.

Step 1: control of the difference of the densities.
Recall the system (4.3) satisfied by the difference. We define151515Notice
that in contrast with Subsection 4.1, we work here, in the far-field case,
with a weighted version of the difference of the densities.
$\delta\\!\varrho:=\bar{x}^{\beta}\delta\\!\rho$ (note that
$\beta<\frac{\alpha}{2}$) and thus get the following equation

(4.19)
$\displaystyle\partial_{t}\delta\\!\varrho+\bar{u}\cdot\nabla\delta\\!\varrho+\bar{x}^{\beta}\delta\\!u\cdot\nabla\rho=\alpha\delta\\!\varrho\bar{u}\cdot\nabla\ln\bar{x}.$

Similarly to the previous subsection, we define

(4.20) $\displaystyle\phi:=-(-\Delta)^{-1}\delta\\!\varrho\quad{\rm
so~{}that}\quad\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{2})}=\|\delta\\!\varrho\|_{\dot{H}^{-1}({\mathbb{R}}^{2})}.$

Now, testing the first equation of (4.19) against $\phi$ yields that

(4.21)
$\displaystyle\frac{1}{2}\frac{d}{dt}\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{2})}^{2}\leq|\langle\bar{u}\cdot\nabla\Delta\phi,\phi\rangle|+|\langle\bar{x}^{\beta}\delta\\!u\cdot\nabla\rho,\phi\rangle|+\beta|\langle\Delta\phi\bar{u}\cdot\nabla\ln\bar{x},\phi\rangle|.$

#### First term in the right-hand-side of (4.21)

One has

(4.22)
$\displaystyle|\langle\bar{u}\cdot\nabla\Delta\phi,\phi\rangle|\leq\|\nabla\bar{u}\|_{L^{\infty}({\mathbb{R}}^{2})}\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{2})}^{2}.$

#### Second term in the right-hand-side of (4.21)

Noticing that $\nabla\cdot\delta\\!u=0$, we get

$\displaystyle\langle\bar{x}^{\beta}\delta\\!u\cdot\nabla\rho,\phi\rangle=-\langle\delta\\!u\cdot\nabla\phi,\bar{x}^{\beta}\rho\rangle-\langle\delta\\!u\cdot\nabla\bar{x}^{\beta},\rho\phi\rangle.$

By Hölder’s inequality

(4.23)
$\displaystyle|\langle\delta\\!u\cdot\nabla\phi,\bar{x}^{\beta}\rho\rangle|$
$\displaystyle\leq\|\sqrt{\rho}\delta\\!u\|_{L^{2}({\mathbb{R}}^{2})}\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{2})}\|\sqrt{\rho}\bar{x}^{\beta}\|_{L^{\infty}({\mathbb{R}}^{2})}$
$\displaystyle\leq\|\sqrt{\rho}\delta\\!u\|_{L^{2}({\mathbb{R}}^{2})}\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{2})}(1+\|{\rho}\bar{x}^{2\beta}\|_{L^{\infty}({\mathbb{R}}^{2})}).$

One also has

(4.24)
$\displaystyle|\langle\delta\\!u\cdot\nabla\bar{x}^{\beta},\rho\phi\rangle|\leq\|\phi\|_{{\rm
BMO}({\mathbb{R}}^{2})}\|\rho\delta\\!u\cdot\nabla\bar{x}^{\beta}\|_{\mathcal{H}^{1}({\mathbb{R}}^{2})}\leq\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{2})}\|\rho\delta\\!u\cdot\nabla\bar{x}^{\beta}\|_{\mathcal{H}^{1}({\mathbb{R}}^{2})}.$

Using the following inequality from [44, Theorem D],

(4.25) $\displaystyle\|f\|_{\mathcal{H}^{1}({\mathbb{R}}^{2})}\leq
C\|f\|_{L^{1}({\mathbb{R}}^{2})}\left(|\ln\|f\|_{L^{1}({\mathbb{R}}^{2})}|+\ln(e+\|f\|_{L^{\infty}({\mathbb{R}}^{2})})\right),$

with $f=\rho\delta\\!u\cdot\nabla\bar{x}^{\beta}$ noticing

$\displaystyle\|\rho\delta\\!u\cdot\nabla\bar{x}^{\beta}\|_{L^{1}({\mathbb{R}}^{2})}$
$\displaystyle\leq
C\|\sqrt{\rho}\delta\\!u\|_{L^{2}({\mathbb{R}}^{2})}\|\sqrt{\rho}\bar{x}^{\beta-1}(\ln\langle
x\rangle)^{2}\|_{L^{2}({\mathbb{R}}^{2})}$ $\displaystyle\leq
C\|\sqrt{\rho}\delta\\!u\|_{L^{2}({\mathbb{R}}^{2})}\|{\rho}\bar{x}^{2\beta}\|_{L^{1}}^{\frac{1}{2}},$

and that

$\displaystyle\|{\rho}\delta\\!u\cdot\nabla\bar{x}^{\beta}\|_{L^{\infty}({\mathbb{R}}^{2})}$
$\displaystyle\leq\|{\rho}\bar{x}^{\beta}(\ln\langle
x\rangle)^{2}\|_{L^{\infty}({\mathbb{R}}^{2})}\|\bar{x}^{-1}\delta\\!u\|_{L^{\infty}({\mathbb{R}}^{2})}$
$\displaystyle\leq
C\|{\rho}\bar{x}^{2\beta}\|_{L^{\infty}({\mathbb{R}}^{2})}(\|\bar{x}^{-1}u\|_{L^{\infty}({\mathbb{R}}^{2})}+\|\bar{x}^{-1}\bar{u}\|_{L^{\infty}({\mathbb{R}}^{2})}),$

we see that estimate (4.24) yields

$\displaystyle|\langle\delta\\!u\cdot\nabla\bar{x}^{\beta},\rho\phi\rangle|$
$\displaystyle\leq
C\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{2})}\|\sqrt{\rho}\delta\\!u\|_{L^{2}({\mathbb{R}}^{2})}(|\ln\|\rho\delta\\!u\cdot\nabla\bar{x}^{\beta}\|_{L^{1}({\mathbb{R}}^{2})}|+C)$
$\displaystyle\leq
C\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{2})}\|\sqrt{\rho}\delta\\!u\|_{L^{2}({\mathbb{R}}^{2})}.$

Notice that to get the last inequality, we used that

$\displaystyle\|\rho\delta\\!u\cdot\nabla\bar{x}^{\beta}\|_{L^{1}({\mathbb{R}}^{2})}$
$\displaystyle\leq\|{\rho}\bar{x}^{\beta}(\ln\langle
x\rangle)^{2}\|_{L^{1}({\mathbb{R}}^{2})}\|\bar{x}^{-1}\delta\\!u\|_{L^{\infty}({\mathbb{R}}^{2})}$
$\displaystyle\leq
C\|{\rho}\bar{x}^{2\beta}\|_{L^{1}({\mathbb{R}}^{2})}(\|\bar{x}^{-1}u\|_{L^{\infty}({\mathbb{R}}^{2})}+\|\bar{x}^{-1}\bar{u}\|_{L^{\infty}({\mathbb{R}}^{2})})\leq
C.$

These results combined with inequality (4.23) imply that

(4.26)
$\displaystyle|\langle\bar{x}^{\beta}\delta\\!u\cdot\nabla\rho,\phi\rangle|\leq
C\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{2})}\|\sqrt{\rho}\delta\\!u\|_{L^{2}({\mathbb{R}}^{2})}.$

#### Third term in the right-hand-side of (4.21)

We rewrite it into

(4.27)
$\displaystyle|\langle\Delta\phi\bar{u}\cdot\nabla\ln\bar{x},\phi\rangle|\leq|\langle\nabla\phi\cdot\nabla(\bar{u}\cdot\nabla\ln\bar{x}),\phi\rangle|+|\langle|\nabla\phi|^{2},\bar{u}\cdot\nabla\ln\bar{x}\rangle|.$

Similarly, as above one has

$\displaystyle|\langle\nabla\phi\cdot\nabla(\bar{u}\cdot\nabla\ln\bar{x}),\phi\rangle|\leq$
$\displaystyle\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{2})}\|\nabla\phi\cdot\nabla(\bar{u}\cdot\nabla\ln\bar{x})\|_{\mathcal{H}^{1}({\mathbb{R}}^{2})}.$

We now rely on (4.25) with
$f:=\nabla\phi\cdot\nabla(\bar{u}\cdot\nabla\ln\bar{x})$ to estimate the
right-hand-side in the previous estimate. Noticing from inequalities (3.28)
and (3.29) that

$\displaystyle\|\nabla\phi\cdot\nabla(\bar{u}\cdot\nabla\ln\bar{x})\|_{L^{1}({\mathbb{R}}^{2})}$
$\displaystyle\leq\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{2})}(\|\nabla\bar{u}\nabla\ln\bar{x}\|_{L^{2}({\mathbb{R}}^{2})}+\|\bar{u}\langle
x\rangle^{-2}\|_{L^{2}({\mathbb{R}}^{2})})$ $\displaystyle\leq
C\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{2})}(\|\nabla\bar{u}\|_{L^{2}({\mathbb{R}}^{2})}+\|\bar{u}\bar{x}^{-1}\|_{L^{2}({\mathbb{R}}^{2})})$

and additionally using the Gagliardo-Nirenberg inequality (A.1) we have

$\displaystyle\|\nabla\phi\cdot\nabla(\bar{u}\cdot\nabla\ln\bar{x})\|_{L^{\infty}({\mathbb{R}}^{2})}$
$\displaystyle\leq
C\|\nabla\phi\|_{L^{\infty}({\mathbb{R}}^{2})}(\|\nabla\bar{u}\|_{L^{\infty}({\mathbb{R}}^{2})}+\|\bar{u}\bar{x}^{-1}\|_{L^{\infty}({\mathbb{R}}^{2})})$
$\displaystyle\leq
C(\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{2})}+\|\bar{x}^{\beta}\delta\\!\rho\|_{L^{4}({\mathbb{R}}^{2})})(\|\nabla\bar{u}\|_{L^{\infty}({\mathbb{R}}^{2})}+\|\bar{u}\bar{x}^{-1}\|_{L^{\infty}({\mathbb{R}}^{2})}).$

Thus one has

(4.28)
$\displaystyle|\langle\nabla\phi\cdot\nabla(\bar{u}\cdot\nabla\ln\bar{x}),\phi\rangle|\leq
C\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{2})}^{2}(|\ln\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{2})}|+\|\nabla\bar{u}\|_{L^{\infty}({\mathbb{R}}^{2})}+\|\bar{u}\bar{x}^{-1}\|_{L^{\infty}({\mathbb{R}}^{2})}+1).$

For the remaining term in (4.27), notice that

$\displaystyle|\langle|\nabla\phi|^{2},\bar{u}\cdot\nabla\ln\bar{x}\rangle|\leq\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{2})}^{2}\|\bar{u}\langle
x\rangle^{-1}\|_{L^{\infty}({\mathbb{R}}^{2})}.$

#### Final estimate of (4.21)

Putting the above estimates (4.22), (4.26), (4.28) together into (4.21), we
get

$\displaystyle\frac{d}{dt}\|\nabla\phi(t,\cdot)\|_{L^{2}({\mathbb{R}}^{2})}^{2}\leq$
$\displaystyle
C\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{2})}^{2}(|\ln\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{2})}|+\|\nabla\bar{u}\|_{L^{\infty}({\mathbb{R}}^{2})}+\|\bar{u}\langle
x\rangle^{-1}\|_{L^{\infty}({\mathbb{R}}^{2})}+1)$
$\displaystyle\quad+\|\nabla\phi\|_{L^{2}({\mathbb{R}}^{2})}\|\sqrt{\rho}\delta\\!u\|_{L^{2}({\mathbb{R}}^{2})}.$

Hence, denoting

$\bar{D}(t):=\sup_{0<s\leq
t}\,s^{-\frac{1}{2}}\|\delta\\!\varrho(s,\cdot)\|_{\dot{H}^{-1}({\mathbb{R}}^{2})},$

we further get that for all $t\in[0,T],$

(4.29) $\displaystyle\bar{D}(t)\leq$ $\displaystyle
C\int_{0}^{t}\bar{D}(s)\left(|\ln\bar{D}(s)|+|\ln
s|+\|\nabla\bar{u}\|_{L^{\infty}({\mathbb{R}}^{2})}+\|\bar{u}\langle
x\rangle^{-1}\|_{L^{\infty}({\mathbb{R}}^{2})}+1\right)\,ds$
$\displaystyle\quad+C\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,t)\times{\mathbb{R}}^{2})}.$

Step 2: duality argument.
At this stage, in order to control the difference $\sqrt{\rho}\delta\\!u$ in
$L^{2}((0,t)\times{\mathbb{R}}^{2})$, we will estimate the solution $v$ to the
linear backward parabolic system (4.7) as in Subsection 4.1. Indeed, in our
current setting, the solvability of problem (4.7) can be achieved by following
the steps in Subsection 3.2. Moreover, similarly to estimate (4.8) and also to
(3.22), (3.24), we have

(4.30) $\sup_{t\in(0,T)}\|\left(\sqrt{\rho}v,\nabla
v\right)(t,\cdot)\|_{L^{2}({\mathbb{R}}^{2})}+\|(\nabla
v,\nabla^{2}v)\|_{L^{2}(0,T;L^{2}({\mathbb{R}}^{2}))}+\|\bar{x}^{-\beta}v\|_{L^{2+\beta}(0,T;L^{\infty}({\mathbb{R}}^{2}))}\\\
+\|\bar{x}^{-\frac{1}{2}-\beta}v\|_{L^{\infty}(0,T;L^{3}({\mathbb{R}}^{2}))}\leq
C\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{2})}.$

Now, recall that

(4.31)
$\displaystyle\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{2})}^{2}\leq\int_{0}^{T}|\langle\delta\\!\rho\dot{\bar{u}},v\rangle|\,dt+\int_{0}^{T}|\langle\rho\delta\\!u\cdot\nabla\bar{u},v\rangle|\,dt.$

By Hölder’s inequality

$\displaystyle\int_{0}^{T}|\langle\rho\delta\\!u\cdot\nabla\bar{u},v\rangle|\,dt$
$\displaystyle\leq$
$\displaystyle\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{2})}\|\nabla\bar{u}\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{2}))}\|\sqrt{\rho}v\|_{L^{2}(0,T;L^{\infty}({\mathbb{R}}^{2}))}$
$\displaystyle\leq$ $\displaystyle
C\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{2})}\|\nabla\bar{u}\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{2}))}\|\bar{x}^{-\beta}v\|_{L^{2}(0,T;L^{\infty}({\mathbb{R}}^{2}))}(1+\|\rho\bar{x}^{2\beta}\|_{L^{\infty}((0,T\times{\mathbb{R}}^{2})}),$

and

$\displaystyle\int_{0}^{T}|\langle\delta\\!\rho\dot{\bar{u}},v\rangle|\,dt$
$\displaystyle\leq\|t^{-1/2}\delta\\!\rho\bar{x}^{\beta}\|_{L^{\infty}(0,T;\dot{H}^{-1}({\mathbb{R}}^{2}))}\|\sqrt{t}\nabla(\bar{x}^{-\beta}\dot{\bar{u}}\cdot
v)\|_{L^{1}(0,T;L^{2}({\mathbb{R}}^{2}))}$
$\displaystyle\leq\bar{D}(T)\|\sqrt{t}\nabla(\bar{x}^{-\beta}\dot{\bar{u}}\cdot
v)\|_{L^{1}(0,T;L^{2}({\mathbb{R}}^{2}))}.$

Using inequalities (3.28), (3.29) and $\beta<\frac{1}{2}$, we have

$\displaystyle|\nabla(\bar{x}^{-\beta}\dot{\bar{u}}\cdot v)|\leq
C(|\nabla\dot{\bar{u}}||\bar{x}^{-\beta}v|+|\dot{\bar{u}}\bar{x}^{-\beta}||\nabla
v|+|\dot{\bar{u}}\bar{x}^{-\beta}||v\bar{x}^{-\frac{1}{2}-\beta}|),$

which implies that

$\displaystyle\|\sqrt{t}\nabla(\bar{x}^{-\beta}\dot{\bar{u}}\cdot
v)\|_{L^{1}(0,T;L^{2}({\mathbb{R}}^{2}))}$ $\displaystyle\leq$
$\displaystyle\|\sqrt{t}\nabla\dot{\bar{u}}\|_{L^{2}((0,T)\times{\mathbb{R}}^{2})}\|\bar{x}^{-\beta}v\|_{L^{2}(0,T;L^{\infty}({\mathbb{R}}^{2}))}+\|\sqrt{t}\dot{\bar{u}}\bar{x}^{-\beta}\|_{L^{2}(0,T;L^{6}({\mathbb{R}}^{2}))}\|\nabla
v\|_{L^{2}(0,T;L^{3}({\mathbb{R}}^{2}))}$
$\displaystyle\quad+T^{\frac{1}{2}}\|\sqrt{t}\dot{\bar{u}}\bar{x}^{-\beta}\|_{L^{2}(0,T;L^{6}({\mathbb{R}}^{2}))}\|v\bar{x}^{-\frac{1}{2}-\beta}\|_{L^{\infty}(0,T;L^{3}({\mathbb{R}}^{2}))}$

So by estimate (4.30) we can bound the right-hand-side of (4.31) and get

$\displaystyle\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{2})}^{2}\leq
CT^{\frac{\beta}{6}}\big{(}\bar{D}(T)\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{2})}+\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{2})}^{2}\big{)}.$

Clearly, the above inequality implies that, if $T$ is small enough then

(4.32)
$\displaystyle\|\sqrt{\rho}\delta\\!u\|_{L^{2}((0,T)\times{\mathbb{R}}^{2})}\leq
C\bar{D}(T)T^{\frac{\beta}{6}}.$

Putting the above inequality into (4.29), we get for $T$ small enough

$\displaystyle\bar{D}(T)\leq
C\left(\int_{0}^{T}\bar{D}(t)\left(\|\nabla\bar{u}\|_{L^{\infty}({\mathbb{R}}^{2})}+\|\bar{u}\langle
x\rangle^{-1}\|_{L^{\infty}({\mathbb{R}}^{2})}+|\ln\bar{D}(t)|-\ln
t\right)\,dt\right).$

From Osgood’s lemma [6, Lemma 3.4], we then infer that $\bar{D}(t)\equiv 0$ on
$[0,T],$ and thus $\sqrt{\rho}\delta\\!u\equiv 0,~{}\delta\\!\rho\equiv 0$ on
$[0,T]$ thanks to inequality (4.32) and equation (4.19), respectively.
Finally, $\delta\\!u\equiv 0$ and $\delta\\!\rho\equiv 0$ on $[0,\infty)$ can
be concluded similarly as previous subsection. In particular, one needs to use
the functional inequality (3.26). ∎

###  4.3. Proof of uniqueness

To complete the proof of uniqueness part of Theorem A and Theorem B, it
suffices to observe that all the assumptions in Proposition 4.1 and
Proposition 4.2 are satisfied by those solutions constructed in Theorem A and
Theorem B.∎

##  5\. Proof of Theorem C

We remark that the general strategy is the same in dimensions $d=2$ and $3$.
Assume that $\partial\Omega_{0}$ corresponds to the level set $\\{f_{0}=0\\}$
of some $\mathcal{C}^{1,\gamma}$ function $f_{0}:\Omega_{0}\to{\mathbb{R}}.$
Then we know that $\partial\Omega_{t}=X(t,f^{-1}_{0}(\\{0\\}))$ corresponds to
the level set $\\{f_{t}=0\\}$ with $f_{t}:=f_{0}\circ(X(t,\cdot))^{-1},$ where
$(X(t,\cdot))^{-1}$ is the inverse function of $X(t,\cdot)$. Indeed, as the
flow is incompressible, the Jacobian of $X$ is identically equal to $1$. Hence
the classical inverse function theorem ensures the existence and regularity of
$X^{-1}.$

Now, fix some $T>0.$ In the 2D case, according to Theorem A, for all
$r\in[2,\infty)$ we can find $q>2$ such that

$\displaystyle\|\nabla^{2}u\|_{L^{1}(0,T;L^{r}({\mathbb{R}}^{2})}\leq\|\nabla^{2}(\sqrt{t}u)\|_{L^{q}(0,T;L^{r}({\mathbb{R}}^{2}))}\|{t}^{-\frac{1}{2}}\|_{L^{q^{\prime}}(0,T)}\leq
C_{0,T}$

and by the Gagliardo-Nirenberg inequality (A.1) and Young’s inequality

$\displaystyle\|\nabla u\|_{L^{1}(0,T;L^{r}({\mathbb{R}}^{2})}\leq C(\|\nabla
u\|_{L^{1}(0,T;L^{2}({\mathbb{R}}^{2}))}+\|\Delta
u\|_{L^{1}(0,T;L^{2}({\mathbb{R}}^{2}))})\leq C_{0,T}.$

By Sobolev’s embedding, one has $\nabla u\in
L^{1}(0,T;\mathcal{C}^{0,\gamma})$ for all $\gamma\in(0,1).$ Consequently, the
flow $X(t,\cdot)$ is $\mathcal{C}^{0,\gamma}$ and so is $f_{t}.$

For the 3D case, Theorem B ensures that for all $r\in[2,6)$, $\nabla u\in
L^{1}(0,T;L^{r}({\mathbb{R}}^{3}))$ and we can find $q>2$ such that
$\nabla^{2}(\sqrt{t}u)\in{L^{q}(0,T;L^{r}({\mathbb{R}}^{3}))}.$ Thus $\nabla
u\in L^{1}(0,T;W^{1,r}({\mathbb{R}}^{3})).$ This finally implies that $f_{t}$
is a $\mathcal{C}^{0,\gamma}$ function, if $\gamma<\frac{1}{2}.$∎

##  Appendix A Functional spaces and inequalities

For the reader’s convenience, we here recall a few results that are used
repeatedly in the paper. Let us first recall the definitions of homogeneous
Sobolev spaces.

######  Definition A.1 (homogeneous Sobolev space).

Let $s$ be in $\mathbb{R}.$ The homogeneous Sobolev space
$\dot{H}^{s}(\mathbb{R}^{d})$ is the set of tempered distributions $u$ on
$\mathbb{R}^{d},$ with Fourier transform in $L^{1}_{loc}(\mathbb{R}^{d}),$
satisfying

$\|u\|_{\dot{H}^{s}({\mathbb{R}}^{d})}:=\||\xi|^{s}\mathcal{F}(u)(\xi)\|_{L^{2}({\mathbb{R}}^{d})}<\infty.$

We often use the following Gagliardo-Nirenberg inequalities.

######  Proposition A.2 (Gagliardo-Nirenberg inequalities; [26, Lemma
II.3.3]).

If $(q,r)\in(1,\infty)^{2}$, there exists a constant $C$ depending on $q$ and
$r$ such that

(A.1) $\|z\|_{L^{p}({\mathbb{R}}^{d})}\leq C\|\nabla
z\|_{L^{r}({\mathbb{R}}^{d})}^{\theta}\|z\|_{L^{q}({\mathbb{R}}^{d})}^{1-\theta}$

with
$\frac{1}{p}=\theta\left(\frac{1}{r}-\frac{1}{d}\right)+\frac{1-\theta}{q},~{}0\leq\theta\leq
1.$

Let us now recall some properties of the spaces
$\widetilde{D}^{1,2}({\mathbb{R}}^{2})$ and
$\mathcal{D}^{1,2}({\mathbb{R}}^{3})$ that are used in the paper. For more
details, see Appendix A and B of [42]. The first one is the following weighted
estimate for elements of the space $\widetilde{D}^{1,2}({\mathbb{R}}^{2}).$

######  Lemma A.3 ([42, Theorem B.1]).

For $m\in[2,\infty)$ and $\ell\in(1+\frac{m}{2},\infty),$ there exists a
positive constant $C$ depending on $m$ and $l$ such that for all
$z\in\widetilde{D}^{1,2}({\mathbb{R}}^{2}),$

(A.2) $\displaystyle\left(\int_{{\mathbb{R}}^{2}}\frac{|z|^{m}}{{\langle
x\rangle}^{2}}\,(\log\langle x\rangle)^{-\ell}\,dx\right)^{1/m}\leq
C\left(\|z\|_{L^{2}(B_{1})}+\|\nabla z\|_{L^{2}({\mathbb{R}}^{2})}\right).$

Then one has the following Proposition due to Li and Xin [37], which is a
combination of Lemma A.3 with the Poincaré inequality; see [37], (2.6) and
(2.8) in the proof of Lemma 2.4 therein.

######  Proposition A.4 ([37, Lemma 2.4]).

Let $m\in[2,\infty)$ and $\ell\in(1+\frac{m}{2},\infty)$. Let
$z\in\widetilde{D}^{1,2}({\mathbb{R}}^{2})$ and $\eta\in
L^{\infty}({\mathbb{R}}^{2})$. Assume that $\sqrt{\eta}z\in
L^{2}({\mathbb{R}}^{2})$ and satisfies

$\displaystyle 0\leq\eta\leq\eta^{*},\quad M\leq\int_{B_{R}}\eta\,dx,$

for positive constants $\eta^{*},M,R.$ Then there exists a positive constant
$C$ depending only on $m,\ell,\eta^{*},M,R$ such that

(A.3) $\left(\int_{{\mathbb{R}}^{2}}\frac{|z|^{m}}{{\langle
x\rangle}^{2}}\,(\log\langle x\rangle)^{-\ell}\,dx\right)^{1/m}\leq
C(\|\sqrt{\eta}z\|_{L^{2}({\mathbb{R}}^{2})}+\|\nabla
z\|_{L^{2}({\mathbb{R}}^{2})}).$

We also needed the following conditional $L^{2}$ bound for elements of the
space $\widetilde{D}^{1,2}({\mathbb{R}}^{2}).$

######  Lemma A.5 ([42, Lemma B.1 and Remark B.1]).

$\widetilde{D}^{1,2}({\mathbb{R}}^{2})$ is a Hilbert space for the scalar
product $\langle\nabla z,\nabla
w\rangle+\langle\mathbf{1}_{B_{1}}\,z,\,w\rangle$ and an equivalent norm is
given by $\|\nabla z\|_{L^{2}({\mathbb{R}}^{2})}+|\int_{B_{1}}z\,dx|.$
Moreover, we have
$\widetilde{D}^{1,2}({\mathbb{R}}^{2})\cap(L^{1}({\mathbb{R}}^{2})+L^{2}({\mathbb{R}}^{2}))\hookrightarrow
H^{1}({\mathbb{R}}^{2}),$ more precisely, there exists a constant $C>0$ such
that for all $z\in\widetilde{D}^{1,2}({\mathbb{R}}^{2})$ satisfying
$z=z_{1}+z_{2}$ with $z_{1}\in L^{1}({\mathbb{R}}^{2}),\,z_{2}\in
L^{2}({\mathbb{R}}^{2})$

(A.4) $\|z\|_{L^{2}({\mathbb{R}}^{2})}\leq
C\left(\|z_{1}\|_{L^{1}({\mathbb{R}}^{2})}^{\frac{1}{2}}\|\nabla
z\|_{L^{2}({\mathbb{R}}^{2})}^{\frac{1}{2}}+\|z_{2}\|_{L^{2}({\mathbb{R}}^{2})}\right).$

A similar result holds for all $z\in\mathcal{D}^{1,2}({{\mathbb{R}}^{3}}).$
There exists a constant $C>0$ such that for all
$z\in\mathcal{D}^{1,2}({\mathbb{R}}^{3})$ satisfying $z=z_{1}+z_{2}$ with