# Discontinuous solutions for the Navier-Stokes equations with density-
dependent viscosity

and  Sagbo Marcel ZODJI  Université Paris Cité and Sorbonne Université, CNRS,
IMJ-PRG, F-75013 Paris, France<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

We prove existence and uniqueness of global weak solutions of the Navier-
Stokes equations that govern the motion of a compressible viscous fluid with
density-dependent viscosity in two space dimensions. The initial velocity
belongs to the Sobolev space $H^{1}(\mathbb{R}^{2})$ and the initial fluid
density is Hölder continuous on both sides of a $\mathscr{C}^{1+\alpha}$
interface with some geometrical assumption. We show that this configuration
persists over time: the initial interface is transported by the flow to an
interface that is as regular as the initial one.

Our result generalizes previous known of Hoff [21], Hoff and Santos [22]
concerning the propagation of regularity for discontinuity surfaces by
allowing more general nonlinear pressure law and density-dependent viscosity.
Moreover, it supplements the work by Danchin, Fanelli and Paicu [7] with
global well-posedness, even for density-dependent viscosity and we obtain
uniqueness in a large space.

###### Key words and phrases:

Compressible Navier-Stokes equations, Density-dependent viscosity, Density
patch problem, Intermediate regularity, Free boundary problem

###### 2020 Mathematics Subject Classification:

35R35, 35A02, 35Q30, 76N10

##  1\. Introduction and main result

In this paper, we study the problem of existence and uniqueness of global weak
solutions, with intermediate regularity, for the Navier-Stokes equations
describing the motion of compressible fluid with density-dependent viscosity
in $\mathbb{R}^{2}$. Our main interest is to generalize the result by Hoff
[21], Hoff and Santos [22] about propagation of discontinuous surfaces by
allowing nonlinear pressure law and density-dependent viscosity. On the other
hand, this result supplements the work by Danchin, Fanelli, Paicu [7] with
global well-posedness, even for density-dependent viscosity. Indeed, we
consider the following system:

(1.1) $\displaystyle\begin{cases}\partial_{t}\rho+\operatorname{div}(\rho
u)=0,\\\ \partial_{t}(\rho u)+\operatorname{div}(\rho u\otimes u)+\nabla
P(\rho)=\operatorname{div}(2\mu(\rho)\mathbb{D}u)+\nabla(\lambda(\rho)\operatorname{div}u)\end{cases}$

describing the motion of a compressible viscous fluid at constant temperature.
Above, $\rho=\rho(t,x)\geqslant 0$ and $u=u(t,x)\in\mathbb{R}^{2}$ are
respectively the density and the velocity of the fluid and they are the
unknowns of the problem. Also, $P=P(\rho)$, $\mu=\mu(\rho)$,
$\lambda=\lambda(\rho)$ are respectively the pressure, dynamic and kinematic
viscosity law of the fluid and they are given $\mathscr{C}^{2}$ functions of
the density. The equations (1.1) is supplemented with initial data

(1.2) $\displaystyle\rho_{|t=0}=\rho_{0}\quad\text{ and }\quad u_{|t=0}=u_{0}$

with the following behavior at infinity:

$\lim_{\lvert
x\rvert\to\infty}\rho_{0}(x)=\widetilde{\rho}\quad\text{and}\quad\lim_{\lvert
x\rvert\to\infty}u_{0}(x)=0$

for some $\widetilde{\rho}>0$ and $\widetilde{P}=P(\widetilde{\rho})$. The
initial velocity $u_{0}$ belongs to the Sobolev space $H^{1}(\mathbb{R}^{2})$
and the initial density is $\alpha$-Hölder continuous on both sides of an
curve $\mathcal{C}(0)\in\mathscr{C}^{1+\alpha}$ defined by:

(1.3)
$\displaystyle\mathcal{C}(0):=\left\\{\gamma_{0}(s)\in\mathbb{R}^{2}\;\colon\;s\in[0,2\pi]\right\\}.$

In particular, $\rho_{0}$ can experiences discontinuity across the curve
$\mathcal{C}(0)$. The curve $\mathcal{C}(0)$ is assumed to be the boundary of
a bounded simply connected set $D(0)\subset\mathbb{R}^{2}$ and verifies the
following non-self interception condition:

(1.4)
$\displaystyle\exists\;c_{\gamma}(0)>0,\quad\forall\;(s,s^{\prime})\in[0,2\pi]\times[0,2\pi],\quad\lvert\gamma_{0}(s)-\gamma_{0}(s^{\prime})\rvert\geqslant
c_{\gamma}(0)^{-1}\lvert s-s^{\prime}\rvert.$

_The purpose of this paper is to establish existence and uniqueness of global
weak solution of the system ( 1.1) with initial data $(\rho_{0},u_{0})$ in the
spirit of the works by Hoff [21], Hoff and Santos [22]. The regularity of the
solution helps to propagate the $\mathscr{C}^{1+\alpha}$ and the non-self
interception (1.4) properties of the initial curve at all over time. We obtain
that discontinuities in the initial density persist all over the time. The
extension to density-dependent viscosity is not trivial and the analysis of
the model is more subtle. We obtain uniqueness in a large space. On the other
hand this result supplements the work by Danchin, Fanelli and Paicu [7] with
global well-posedness even with density-dependent viscosity._

We will now proceed with the review of known results on the propagation of
discontinuity surfaces in the mathematical analysis of the Navier-Stokes
equations for compressible fluids.

###  1.1. Review of known results

Since its definitive formulation in the mid-19 th century, the Navier-Stokes
equations have consistently captivated the attention of numerous
mathematicians. The inaugural achievement in this realm is attributed to [32]
who proved a local well-posedness of strong solution in the whole
$\mathbb{R}^{3}$. The density belongs to $\mathscr{C}^{1+\alpha}$ while the
velocity belongs to $\mathscr{C}^{2+\alpha}$ for some $\alpha\in(0,1)$. We
also refer to the work by Solonnikov [36] which provided local well-posedness
of system (1.1) in $\mathscr{C}^{2}$ bounded subset
$\Omega\subset\mathbb{R}^{d},\;d\in\\{2,3\\}$, with Sobolev regularity. In
fact the initial density is bounded away from vacuum and belongs to
$W^{1,p}(\Omega)$ for some $p>d$ whereas the initial velocity belongs to the
Sobolev-Slobodetskii space $W^{2,1}_{q}(\Omega)$. Nash’s work takes into
account heat conducting fluids with viscosity laws which may depend on density
or temperature. On the other hand, Solonnikov did not take into account
temperature and the viscosities are constant. The first global result for
system (1.1) is obtained by Matsumura and Nishida [30] for small initial data
in the whole space $\mathbb{R}^{3}$. In the case of constant viscosity, the
initial data is required to exhibit smallness in
$H^{3}(\mathbb{R}^{3})$,_whereas for non-constant viscosity, smallness in the
space $H^{4}(\mathbb{R}^{3})$ is necessary for the initial data_. Later,
regularity of the initial data was weaken in [3, 6, 18] with small initial
data in the critical Besov space, _for constant viscosity setting_. But still,
density remains a continuous function in space.

_In the constant viscosity setting_ , the classical solutions constructed in
the referenced papers, come with the following energy balance:

(1.5) $\displaystyle E(t)+\mu\int_{0}^{t}\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+(\mu+\lambda)\int_{0}^{t}\lVert\operatorname{div}u\rVert_{L^{2}(\mathbb{R}^{d})}=E(0)=:E_{0}.$

Above $E(t)$ is the total energy functional defined by

(1.6) $\displaystyle E(t):=\int_{\mathbb{R}^{d}}\rho(t,x)\left[\dfrac{\lvert
u(t,x)\rvert^{2}}{2}+\int_{\widetilde{\rho}}^{\rho(t,x)}s^{-2}\left(P(s)-\widetilde{P}\right)ds\right]dx.$

In the particular case when the pressure law is of the form
$P(\rho)=a\rho^{\gamma}$, global weak solution of the system (1.1) is obtained
for the first time by P-L Lions [29] and Feireisl, Novotný, Petzeltová [13]
with some restriction on the adiabatic constant $\gamma$. The initial data is
assumed to have finite initial energy, that is $E(0)<\infty$ and the solution
verify (1.5) with inequality. In [2], Bresch and Desjardins established
existence of a global weak solution for the Navier-Stokes equation with
density-dependent viscosity. Their result requires some Sobolev regularity
assumption on the initial density, which goes beyond our framework since the
density is assumed to be discontinuous across a hyper-surface such that its
weak gradient contains Dirac masses.

In the last three decades, people have been interested in studying the
propagation of discontinuity surfaces for models coming from fluid mechanics
as the Euler or Navier-Stokes equations. These discontinuity surfaces are the
set of singularity points of certain quantities such as vorticity for Euler
equations or density for Navier-Stokes equations. For instance, we refer to
the so-called density-patch problem proposed by P-L Lions [28] for the
incompressible Navier-Stokes equations: _assuming
$\rho_{0}=\mathbb{1}_{D_{0}}$ for some domain $D_{0}\subset\mathbb{R}^{2}$,
the question is whether or not for any time $t>0$, the density is
$\rho(t)=\mathbb{1}_{D(t)}$, with $D(t)$ a domain with the same regularity as
the initial one_. This problem has been considered in [8, 9, 10, 12, 14, 16,
25, 26, 27, 34] where a satisfactory answer was given for various regularities
of $D_{0}$ including density-dependent viscosity.

As far as we know, there are not enough results in the literature regarding
the similar density-patch problem for the Navier-Stokes equations for
compressible fluids. In fact, on the one hand, classical solutions are too
regular and they do not take into account discontinuous initial density. On
the other hand, the weak solutions constructed by P-L Lions [29] or Feireisl,
Novotný, Petzeltvá [13] allow discontinuous initial density, but the velocity
is too weak in order to track down discontinuity in the density. As explained,
for instance in [16], a good approach to tracking discontinuous surfaces is to
construct weak solutions for the full model in a class that allows the study
of its dynamics. The first result regarding the propagation of discontinuous
surfaces for the compressible model is credited to Hoff in his work of 2002
[21], which is in continuation of his works [19, 20]. Indeed, in his pioneer
work [19], Hoff provided bounds for the following functionals

(1.7) $\displaystyle\mathcal{A}_{1}(t)=\sup_{[0,t]}\sigma\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{t}\sigma\lVert\sqrt{\rho}\dot{u}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\quad\text{and}\quad\mathcal{A}_{2}(t)=\sup_{[0,t]}\sigma^{d}\lVert\sqrt{\rho}\dot{u}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{t}\sigma^{d}\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$

owing to some smallness condition on the initial data. Above, the material
derivative of the velocity $\dot{u}$, and the time weight $\sigma$ are defined
as follows:

$\dot{u}:=\partial_{t}u+(u\cdot\nabla)u\quad\text{and}\quad\sigma(t):=\min\\{1,t\\}.$

He observed that the so-called effective flux defined by

(1.8) $\displaystyle
F:=(2\mu+\lambda)\operatorname{div}u-P(\rho)+\widetilde{P}$

and the vorticity $\operatorname{curl}u$ solve the following elliptic
equations:

(1.9) $\displaystyle\Delta
F=\operatorname{div}(\rho\dot{u})\quad\text{and}\quad\mu\Delta\operatorname{curl}u=\operatorname{curl}(\rho\dot{u}).$

Let us notice that the effective flux was discovered by Hoff and Smoller in
[23] and it plays a crucial role in the mathematical analysis of the
compressible model, for instance, in the propagation of oscillations [35] or
the construction of weak solutions [20, 29, 13]. By using the regularity of
$\dot{u}$ provided by the functionals $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$
in (1.7), he obtained the fact that the effective flux and vorticity belong,
at least, to $L^{8/3}((1,\infty),L^{\infty}(\mathbb{R}^{d}))$, and this helps
him to propagate the $L^{\infty}(\mathbb{R}^{d})$ norm of the density.
Thereby, he obtained existence of global weak solution for the Navier-Stokes
equations with linear pressure law in a first paper [19] before considering
nonlinear pressure law (gamma-law) in a second paper [20]. The initial density
is assumed to be small around a positive state in $L^{2}(\mathbb{R}^{d})\cap
L^{\infty}(\mathbb{R}^{d})$, far from zero and the initial velocity is small
in $L^{2}(\mathbb{R}^{d})$ but can be large in $L^{2^{d}}(\mathbb{R}^{d})$.
These weak solutions are in between the unique, global, classical one
constructed by Matsumura and Nishida in [31] and the one with finite initial
energy constructed by P-L Lions [29] or Feireisl, Novotný and Petzeltová [13].
In particular, discontinuous initial density are allowed and the regularity of
the velocity at positive time helps to track down discontinuity in the
density. As an example, in 2008, Hoff and Santos [22] studied the Lagrangian
structure for such weak solutions of the Navier-Stokes equations. Basically,
they write the velocity as sum of two terms: the first one is associated with
the effective flux and the vorticity and is, at least, Lipschitz at positive
time while the second one is associated with the pressure and is less regular
in space than the first one. In fact its gradient belongs to
$L^{\infty}((0,\infty),BMO(\mathbb{R}^{d}))$ space. They need the initial
velocity to be a little more regular ($u_{0}\in H^{s}(\mathbb{R}^{d})$ for
$s>0$ in $d=2$ and $s>1/2$ in $d=3$) in order to lower the singularity at
initial time of the first term of the velocity. Hence, the velocity gradient
belongs to $L^{1}_{\text{loc}}([0,\infty),BMO(\mathbb{R}^{d}))$ and this is
sufficient to define a continuous flow map for the velocity field $u$ and it
turns out that continuous manifold maintains its regularity over time.
However, a bit more regularity is lost, for instance, an initial
$\mathscr{C}^{\alpha}$-manifold is transported to some
$\mathscr{C}^{\alpha(t)}$-manifold with $\alpha(t)$ decaying exponentially to
zero. Let us notice that they constructed solution of the heat equation with
specific initial velocity $u_{0}\in H^{s}(\mathbb{R}^{3})$ for $s<1/2$, that
has infinitely many integral curves approaching $x=0$ as $t$ goes to $0$.

This decomposition of the velocity was used some years before by Hoff [21] to
propagate regularity of discontinuity surfaces in $\mathbb{R}^{2}$. In fact,
for initial velocity in $H^{\beta}(\mathbb{R}^{2})$, both the effective flux
and vorticity belong to
$L^{1}_{\text{loc}}([0,\infty),\mathscr{C}^{\alpha}(\mathbb{R}^{2}))$, and as
a result, the gradient of the regular part of the velocity does as well, for
all $0<\alpha<\beta<1$. Assuming that initially the density is Hölder
continuous on both sides of a $\mathscr{C}^{1+\alpha}$ interface (with
geometrical assumption (1.4)) across which it is discontinuous, then he showed
that the second part of the velocity is at least Lipschitz. In fact, its
regularity allows to show that the initial interface is transported to an
interface that is as regular as the initial one. In particular, the gradient
of the second term of the velocity (associated with the pressure) is Hölder
continuous in the tangential direction of the transported interface.

It is important to notice that the approximate sequence of the density is
constructed in a large space that precludes any non-linear pressure law. More
recently, we have established in [38], the existence of local weak solutions
for the two-fluid model with density-dependent viscosity and with
discontinuous initial data. In particular, the regularity of the local
solution is close to that needed in order to propagate the regularity of the
interface. These local solutions allow general nonlinear pressure law and will
be used as building blocks for global solutions in this paper. On the other
hand, in [21], the analysis requires the kinematic viscosity $\lambda$ to be
small (see [21, Section 2.3]). These constraints will be removed in this
paper.

In 2020, Danchin, Fanelli and Paicu [7] have shown that, under assumptions of
tangential regularity of the initial density, the less regular part of the
velocity is Lipschitz. They achieved this by investigating tangential
regularity estimates in the spirit of the works by Chemin [4, 5] for ideal
flows. In particular they propagate the $W^{1,p}$ regularity for hyper-
surfaces up to a finite time: global propagation of the interface’s regularity
does not hold even for small initial data.

Finally, we may notice that in [24], the same question was investigated in a
two-dimensional ball. The authors considered the case of an initial density
and initial velocity that are small in $H^{4}$ on both sides of an interface,
which is nearly circular and $H^{9/2}$ regular.

_All of the above results concern the case of constant viscosity. However,
when the viscosity becomes density-dependent, the effective flux and the
vorticity lose regularity, and the analysis becomes more subtle._

In this part, we will make different observations to show that for the Navier-
Stokes equations with density-dependent viscosity, the effective flux and the
vorticity are not smooth as in the constant viscosity setting. In particular,
we will show that at the interface, these quantities are continuous exactly
where the dynamic viscosity is continuous.

The first observation consists in applying the divergence and the rotational
operators to the momentum equation in order to express the effective flux and
the vorticity as follows:

(1.10) $\displaystyle
F:=(2\mu(\rho)+\lambda(\rho))\operatorname{div}u-P(\rho)+\widetilde{P}=-(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})+K\\{(\mu(\rho)-\widetilde{\mu})\mathbb{D}u\\},$
(1.11)
$\displaystyle\mu(\rho)\operatorname{curl}u=-(-\Delta)^{-1}\operatorname{curl}(\rho\dot{u})+K^{\prime}\\{(\mu(\rho)-\widetilde{\mu})\mathbb{D}u\\}.$

We refer to section 3.1 for technical details leading to these expressions.
Above, $K$ and $K^{\prime}$ are second-order Riesz operators well-known to map
linearly $L^{p}(\mathbb{R}^{d})$ into itself for $1<p<\infty$. At the end
point they map $L^{\infty}(\mathbb{R}^{d})$ into the $BMO(\mathbb{R}^{d})$
space. The first terms of the effective flux and vorticity are regular as in
constant viscosity setting and the second terms are the same order as the
gradient of $u$. Even if the gradient of $u$ belongs to
$L^{\infty}(\mathbb{R}^{d})$, the effective flux will just belong to
$BMO(\mathbb{R}^{d})$ and this is not sufficient to obtain a
$L^{\infty}(\mathbb{R}^{d})$ bound for the density. This observation is also
done by Bresch and Burtea in [1] in the study of the an-isotropic Navier-
Stokes equations.

The second observation will involve computing the exact jump of the effective
flux, vorticity, and velocity at the interface. The discussion follows the
same lines as in [33] where the case of constant viscosity is analyzed. First,
we observe that there is a balance of forces applied to the interface, which
suggests the continuity of the stress tensor in the normal direction of the
interface, that is:

(1.12) $\displaystyle\llbracket\Pi^{j}\rrbracket\cdot n_{x}=0,\quad\text{
where
}\quad\Pi^{jk}=2\mu(\rho)\mathbb{D}^{jk}u+\left(\lambda(\rho)\operatorname{div}u-P(\rho)+\widetilde{P}\right)\delta^{jk},$

is the stress tensor, $n_{x}$ is the outward normal vector field of the
interface and $\llbracket g\rrbracket$ denotes the jump of $g$ at the
interface:

$\llbracket g\rrbracket(\sigma)=\underbrace{\lim_{r\to
0}g(\sigma+rn_{x}(\sigma))}_{g_{+}(\sigma)}-\underbrace{\lim_{r\to 0}g(\sigma-
rn_{x}(\sigma))}_{g_{-}(\sigma)},\quad\forall\sigma\in\mathcal{C}.$

Next, since the velocity $u$ is continuous in the whole space, and its
gradient is assumed to be continuous on both sides of the interface, then the
velocity gradient is also continuous in the tangential direction of the
interface. Basically, discontinuities in the gradient of $u$ will only occur
in the normal direction $n_{x}$ of the interface. In other words, there exists
a vector field $\mathrm{a}=\mathrm{a}(t,\sigma)\in\mathbb{R}^{2}$ such that

(1.13) $\displaystyle\llbracket\nabla u\rrbracket=\mathrm{a}\otimes
n^{t}_{x}.$

From this, one easily deduces that such vector field is given by:

(1.14)
$\displaystyle\mathrm{a}=(\mathrm{a}\cdot\tau_{x})\tau_{x}+(\mathrm{a}\cdot
n_{x})n_{x}=\llbracket\operatorname{curl}u\rrbracket\tau_{x}+\llbracket\operatorname{div}u\rrbracket
n_{x},$

where $\tau_{x}$ is the tangential vector field of the interface. By
considering (1.13), one can rewrite (1.12) as follows111
$<a>:=\dfrac{a_{+}+a_{-}}{2}$ is the average of $a$ at the interface.:

(1.15) $\displaystyle<\mu(\rho)>\left(\mathrm{a}^{j}+\mathrm{a}\cdot
n_{x}n^{j}_{x}\right)+2\llbracket\mu(\rho)\rrbracket<\mathbb{D}^{jk}u>n^{k}_{x}+\llbracket\lambda(\rho)\operatorname{div}u-P(\rho)\rrbracket
n^{j}_{x}=0.$

Next, one multiplies the above by $n^{j}_{x}$ before summing over $j$ in order
to obtain the following:

$2<\mu(\rho)>\mathrm{a}\cdot
n_{x}+2\llbracket\mu(\rho)\rrbracket<\mathbb{D}^{jk}u>n^{j}_{x}n^{k}_{x}+\llbracket\lambda(\rho)\operatorname{div}u-P(\rho)\rrbracket=0$

and, since $\mathrm{a}\cdot n_{x}=\llbracket\operatorname{div}u\rrbracket$,
then the jump of the effective flux at the interface is given by the
following:

(1.16)
$\displaystyle\llbracket(2\mu(\rho)+\lambda(\rho))\operatorname{div}u-P(\rho)\rrbracket=2\llbracket\mu(\rho)\rrbracket\left(<\operatorname{div}u>-<\mathbb{D}^{jk}u>n^{j}_{x}n^{k}_{x}\right).$

As above, we take the scalar product of (1.15) with the tangential vector
$\tau_{x}$ and use the fact that
$\llbracket\operatorname{curl}u\rrbracket=\mathrm{a}\cdot\tau_{x}$ in order to
obtain the following:

(1.17)
$\displaystyle\llbracket\mu(\rho)\operatorname{curl}u\rrbracket=\llbracket\mu(\rho)\rrbracket\left(<\operatorname{curl}u>-2<\mathbb{D}^{jk}u>n^{k}_{x}\tau^{j}_{x}\right).$

It then turns out that when the dynamic viscosity is continuous at the
interface, for instance, when it is constant, the effective flux and vorticity
are also continuous at the interface. Another condition for these quantities
to be continuous at the interface is to show that the terms in brackets
vanish, but the latter seems not true in general.

_In view of all the above observations, it is less clear whether the effective
flux and the vorticity are continuous at the interface. However, their jumps
are expressed in terms of the jump of the viscosity at the interface. As we
will see in section 3.2, the viscous damping of the density will cause the
jump of the viscosity to decrease exponentially in time. Consequently, the
jumps of the effective flux, the vorticity, and the gradient of the velocity
will also decay exponentially in time, as observed in [21, 22, 33] in case of
constant viscosity_. We are in position to state our main result.

###  1.2. Statement of the main result

We consider the classical Navier-Stokes equations

(1.18) $\displaystyle\begin{cases}\partial_{t}\rho+\operatorname{div}(\rho
u)=0,\\\ \partial_{t}(\rho u)+\operatorname{div}(\rho u\otimes u)+\nabla
P(\rho)=\operatorname{div}(2\mu(\rho)\mathbb{D}u)+\nabla(\lambda(\rho)\operatorname{div}u),\end{cases}$

in $d=2$, with initial data $(\rho_{0},u_{0})$ that fulfills,

(1.19) $\displaystyle u_{0}\in H^{1}(\mathbb{R}^{2})\quad\text{ and
}\quad\rho_{0}-\widetilde{\rho}\in
L^{2}(\mathbb{R}^{2})\cap\mathscr{C}^{\alpha}_{pw,\gamma_{0}}(\mathbb{R}^{2}).$

Above, $\widetilde{\rho}>0$ is the density reference and we set
$P(\widetilde{\rho})=\widetilde{P}$ and
$\mu(\widetilde{\rho})=\widetilde{\mu}$. We assume that the initial density is
far away from vacuum that is, there exist constants $0<\rho_{*}<\rho^{*}$ such
that:

(1.20) $\displaystyle 0<\rho_{*}<\rho_{0}<\rho^{*}.$

The pressure law $P=P(\rho)$ and viscosity laws $\mu=\mu(\rho)$,
$\lambda=\lambda(\rho)$ are assumed to be regular functions of the density,
that is:

(1.21) $\displaystyle
P,\;\mu,\;\lambda\in\mathscr{C}^{2}([\rho_{*}/4,4\rho^{*}]).$

Additionally, the pressure is assumed to be an increasing function of the
density and the viscosity are far away from vacuum, that is, there exist
constants $0<\mu_{*}<\mu^{*}$ such that

(1.22) $\displaystyle
P^{\prime}_{[\rho_{*}/4,4\rho^{*}]}>0\quad\text{and}\quad\mu_{*}\leqslant\mu_{[\rho_{*}/4,4\rho^{*}]},\;\lambda_{[\rho_{*}/4,4\rho^{*}]}\leqslant\mu^{*}.$

On the other hand the initial interface
$\mathcal{C}(0)\in\mathscr{C}^{1+\alpha}$ is:

(1.23) $\displaystyle\mathcal{C}(0)=\\{\gamma_{0}(s)\in\mathbb{R}^{2}\colon
s\in[0,2\pi]\\}$

where $\mathcal{C}(0)$ is the boundary of an open, bounded, simply connected
set $D(0)$ and fulfills the condition (1.4). The initial data will be measured
by the following constant:

(1.24) $\displaystyle c_{0}:=\lVert
u_{0}\rVert_{H^{1}(\mathbb{R}^{2})}^{2}+\lVert\rho_{0}-\widetilde{\rho}\rVert_{L^{2}(\mathbb{R}^{2})\cap\mathscr{C}^{\alpha}_{pw,\gamma_{0}}(\mathbb{R}^{2})}^{2}+\lVert\llbracket\rho_{0}\rrbracket\rVert_{L^{2}(\mathcal{C}(0))\cap
L^{\infty}(\mathcal{C}(0))}^{2}.$

######  Theorem 1.1.

Let $(\rho_{0},u_{0})$ be an initial data associated with the Navier-Stokes
equations (1.18) and fulfills (1.19) and (1.20). Assume that the pressure law
$P=P(\rho)$ and the viscosity laws $\mu=\mu(\rho)$ and $\lambda=\lambda(\rho)$
satisfy the assumptions as described above.

There are constants $c>0$ and $[\mu]_{0}>0$ such that if:

$c_{0}\leqslant c\quad\text{ and
}\quad\lVert\mu(\rho_{0})-\widetilde{\mu}\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma_{0}}(\mathbb{R}^{2})}\leqslant[\mu]_{0}$

then, there exists an unique solution $(\rho,u)$ to the Cauchy problem
associated to (1.18) with initial data $(\rho_{0},u_{0})$ verifying:

(1.25)
$\displaystyle\forall\;t>0,\quad\mathcal{A}_{1}(t)+\mathcal{A}_{2}(t)+\mathcal{A}_{3}(t)+\sqrt{\text{\textbeta}(t)}+\sqrt{\text{\texttheta}(t)}+\sup_{[0,t]}\lVert
u,\;\rho-\widetilde{\rho}\rVert^{2}_{L^{2}(\mathbb{R}^{2})}+\int_{0}^{t}\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\leqslant Cc_{0},$

where

(1.26)
$\displaystyle\begin{cases}\displaystyle\mathcal{A}_{1}(t)=\sup_{[0,t]}\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\int_{0}^{t}\lVert\sqrt{\rho}\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2},&\displaystyle\mathcal{A}_{2}(t)=\sup_{[0,t]}\sigma\lVert\sqrt{\rho}\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\int_{0}^{t}\sigma\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2},\\\
\displaystyle\mathcal{A}_{3}(t)=\sup_{[0,t]}\sigma^{2}\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\int_{0}^{t}\sigma^{2}\lVert\sqrt{\rho}\ddot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2},&\displaystyle\text{\texttheta}(t)=\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau,\\\
\displaystyle\text{\textbeta}(t)=\sup_{[0,t]}\lVert
f(\rho)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2})}^{4}+\int_{0}^{t}\lVert
f(\rho(\tau))\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau,&\displaystyle
f(\rho):=\int_{\widetilde{\rho}}^{\rho}\dfrac{2\mu(s)+\lambda(s)}{s}ds\end{cases}$

and $r_{\alpha}:=\max\\{7/5;\;1+2\alpha\\}<3$. The constant $C$ in (1.25)
depends on $\alpha,\rho_{*},\,\rho^{*},\,\mu_{*},\,\mu^{*}$ and
$c_{\gamma}(0)$ and the regularity of the initial interface.

######  Remark 1.2.

On the one hand, 1.1 generalises the works by Hoff [21], Hoff and Santos [22]
by allowing nonlinear pressure law and density-dependent viscosity. By
propagating interface regularity over time, we complete Danchin, Fanelli, and
Paicu’s work [7].

On the other hand, our result takes into account the viscosity of the form
$1+\mu(\rho)$, which fall out of the framework of the entropy of Bresch-
Desjardins and which is relevant for the models of suspensions, see for
example [17].

######  Remark 1.3.

The fact that $r_{\alpha}<3$ makes the velocity gradient belongs to
$L^{1}_{\text{loc}}([0,\infty),\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2}))$
and this is sufficient in order to propagate the regularity of the initial
curve. We have not been able to establish that $\nabla u$ belongs to
$L^{1}([0,\infty),\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2}))$222global
in time. Consequently, the characteristic $c_{\gamma}(t)$ and the
$\mathscr{C}^{1+\alpha}$ norm of the interface $\gamma(t)$ grow exponentially
in time as:

$c_{\gamma}(t)+\lVert\nabla\gamma(t)\rVert_{\mathscr{C}^{\alpha}}\leqslant
Ce^{Ct^{3/4}}$

but less than the growth obtained in [21].

######  Remark 1.4.

The function $f(\rho)$ defined in (1.26) verifies the following:

(1.27) $\llbracket f(\rho(t,\gamma(t,s)))\rrbracket=\llbracket
f(\rho_{0},\gamma_{0}(s))\rrbracket\exp\left[\int_{0}^{t}-\left(g(\tau,s)+2h(\tau,s)<\operatorname{div}u(\tau,\gamma(\tau,s))>\right.\right.\\\
\left.\left.-<\mathbb{D}^{jk}u(\tau,\gamma(\tau,s))>n^{j}_{x}n^{k}_{x}(\tau,s)\right)d\tau\vphantom{\int_{0}^{t}}\right]$

where the functions $g$ and $h$ are defined as:

$g(t,s)=\dfrac{\llbracket P(\rho(t,\gamma(t,s)))\rrbracket}{\llbracket
f(\rho(t,\gamma(t,s)))\rrbracket}\quad\text{ and }\quad
h(t,s)=\dfrac{\llbracket\mu(\rho(t,\gamma(t,s)))\rrbracket}{\llbracket
f(\rho(t,\gamma(t,s)))\rrbracket}.$

For constant viscosity, $f$ is the logarithm function, $h=0$, and the
exponential decay rate is immediate as soon as the pressure is an increasing
function of the density. This observation is made by Hoff in [21] and Hoff and
Santos [22].

The assumption of increasing pressure ensures that the function $g$ is lower
bounded away from zero by a constant that depends on the lower and upper
bounds of the density and viscosity. On the one hand, the function $h$ is
upper bounded by a constant depending of the lower and upper bounds on the
density and viscosity. Thus, from the estimate of the
$L^{4}((1,\infty),L^{\infty}(\mathbb{R}^{2}))$ norm of the velocity gradient,
given by the functional , we deduce the exponential decay rate of the jump of
$\llbracket f(\rho)\rrbracket$. On the other hand, the upper bounds for $g$
and $h$ allow us to deduce the exponential decay rate of the jump of the
pressure and the viscosity. Furthermore, the expressions for the jumps of the
effective flux in (1.16) and the vorticity in (1.17) at the interface lead to
the exponential decay of the jump of these quantities. In consequence, the
vector field $\mathrm{a}$ given by (1.14) decays exponentially in time so as
for the jump of the gradient of velocity since (1.13) holds.

It is important to notice that if initially the density is continuous at a
point $\gamma_{0}(s)$ of the interface, then the density and the velocity
gradient at time $t$ are continuous at $\gamma(t,s)$.

######  Remark 1.5.

As we will see in (3.45), we write from the momentum equation (1.18) the
following:

$\displaystyle\widetilde{\mu}\Delta
u^{j}=\rho\dot{u}-\partial_{j}\left(\dfrac{\widetilde{\mu}+\lambda(\rho)}{2\mu(\rho)+\lambda(\rho)}F\right)+\partial_{j}\left(\dfrac{2\mu(\rho)-\widetilde{\mu}}{2\mu(\rho)+\lambda(\rho)}(P(\rho)-\widetilde{P})\right)-\partial_{k}\\{2(\mu(\rho)-\widetilde{\mu})\mathbb{D}^{jk}u\\}$

from which we deduce that the velocity gradient can be expressed as the sum of
four terms:

(1.28) $\displaystyle\nabla u=\nabla u_{*}+\nabla u_{F}+\nabla u_{P}+\nabla
u_{\delta}.$

The first term of the above expression is regular in the whole space, for
instance, $\nabla u_{*}\in
L^{1}_{\text{loc}}([0,\infty),\mathscr{C}^{\alpha}(\mathbb{R}^{2}))$. This is
due to the regularity of $\dot{u}$ provided by functionals $\mathcal{A}_{1}$
and $\mathcal{A}_{2}$. The other terms are less regular: there are
discontinuous across the interface $\mathcal{C}$ and we need the quantitative
Hölder estimate from A.1 and the extension result of piecewise Hölder
functions A.3 in order to estimate their norms. But it is less clear, whether
$\nabla u_{P}\in
L^{2}((1,\infty),\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2}))$, even
though this holds locally in time. Of course, a uniform estimate is required,
as our goal is to close all estimates using a bootstrap argument. So the idea
is to achieve more integrability at positive times for $\nabla u_{*}$. This is
the key role of the functional $\mathcal{A}_{3}$. A similar functional was
derived in [11] in the context of the incompressible Navier-Stokes model.

#### Outline of the paper

The rest of this paper is structured as follows. In the following section,
section 2, we present the main ideas of the proof of 1.1, while technical
details are postponed in section 3.

##  2\. Sketch of the proof of the main result

In this section, we present the main ideas of the proof of 1.1. In fact, we
derive some a priori estimates for weak solutions for the following system.

(2.1) $\displaystyle\begin{cases}\partial_{t}\rho+\operatorname{div}(\rho
u)=0,\\\ \partial_{t}(\rho u)+\operatorname{div}(\rho u\otimes u)+\nabla
P(\rho)=\operatorname{div}(2\mu(\rho)\mathbb{D}u)+\nabla(\lambda(\rho)\operatorname{div}u),\end{cases}$

The existence and uniqueness of such a local solution is the purpose of our
recent contribution [38], and our result can be summarized in the following
theorem.

######  Theorem 2.1.

Let $(\rho_{0},u_{0})$ be initial data associated with the system (2.1) that
satisfies the conditions (1.19),

$0<\rho_{*}\leqslant\rho_{0}\leqslant\rho^{*},\quad
0<\mu_{*}\leqslant\mu(\rho_{0}),\,\lambda(\rho_{0})\leqslant\mu^{*}$

and the compatibility condition

(2.2) $\displaystyle(\rho\dot{u})_{|t=0}=\operatorname{div}(\Pi)_{|t=0}\in
L^{2}(\mathbb{R}^{2}).$

There exists $[\mu]>0$ such that if

(2.3)
$\displaystyle\dfrac{4}{\widetilde{\mu}}\lVert\mu(\rho_{0})-\widetilde{\mu}\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma_{0}}(\mathbb{R}^{2})}\leqslant[\mu],$

then, there exists a time $T>0$ and an unique solution $(\rho,u)$ of the
system (2.1) verifying the following:

  1. (1)

$\forall\;0\leqslant t\leqslant T$, with $\displaystyle\partial
D(t)=:\mathcal{C}(t)=\left\\{\gamma(t,s)\in\mathbb{R}^{2},\;\;s\in[0,2\pi]\right\\}$
and $\gamma(t)\in\mathscr{C}^{1+\alpha}$;

  2. (2)

$P(\rho)-\widetilde{P}\in\mathscr{C}([0,T],L^{2}(\mathbb{R}^{2})\cap\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2}))$;

  3. (3)

$u\in\mathscr{C}([0,T],H^{1}(\mathbb{R}^{2}))\cap
L^{\infty}((0,T),\dot{W}^{1,6}(\mathbb{R}^{2}))\cap
L^{16}((0,T),\dot{W}^{1,8}(\mathbb{R}^{2}))$, $\sigma^{r/4}\nabla u\in
L^{4}((0,T),\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2}))$ for $0<r<2$;

  4. (4)

$\dot{u}\in\mathscr{C}([0,T],L^{2}(\mathbb{R}^{2}))\cap
L^{2}((0,T),\dot{H}^{1}(\mathbb{R}^{2}))$, $\sqrt{\sigma}\nabla\dot{u}\in
L^{\infty}((0,T),L^{2}(\mathbb{R}^{2}))\cap
L^{4}((0,T),L^{4}(\mathbb{R}^{2}))$;

  5. (5)

$\sqrt{\sigma}\ddot{u}\in L^{2}((0,T)\times\mathbb{R}^{2})$,
$\sigma\ddot{u}\in L^{\infty}((0,T),L^{2}(\mathbb{R}^{2}))\cap
L^{2}((0,T),\dot{H}^{1}(\mathbb{R}^{2}))$,

where $\sigma(t)=\min\\{1,t\\}$.

######  Remark 2.2.

The velocity $u$ and its the material derivatives $\dot{u}$ are continuous in
the whole space and their gradients, $\nabla u$ and $\nabla\dot{u}$ are Hölder
continuous on both sides of the interface $\gamma$. Moreover, the second
material derivative of the velocity, $\ddot{u}$ is continuous at least across
the interface $\gamma$.

Let us notice that 2.1 comes with the following blow-up criterion:

######  Theorem 2.3 (Blow-up criterion).

Let $(\rho,u)$ be the solution constructed in 2.1 defined up to a maximal time
$T^{*}$. Assume that, for all $0<t<T^{*}$ we have:

$\displaystyle\limsup_{t\to T^{*}}$
$\displaystyle\left\\{c_{\gamma}(t)+\lVert\nabla\gamma(t)\rVert_{\mathscr{C}^{\alpha}}+\bigg{\|}\dfrac{1}{\rho(t)},\;\;\dfrac{1}{\mu(\rho(t))},\;\;\dfrac{1}{\lambda(\rho(t))}\bigg{\|}_{L^{\infty}(\mathbb{R}^{2})}\right\\}$
$\displaystyle+\limsup_{t\to T^{*}}$ $\displaystyle\left\\{\lVert
u(t)\rVert_{H^{1}(\mathbb{R}^{2})}+\lVert(\rho\dot{u})(t)\rVert_{L^{2}(\mathbb{R}^{2})}+\lVert
P(\rho(t))-\widetilde{P}\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(t)}(\mathbb{R}^{2})}\right\\}<\infty$

and

$\dfrac{1}{\widetilde{\mu}}\sup_{[0,t]}\lVert\mu(\rho,c)-\widetilde{\mu}\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2})}\leqslant[\mu]$

then $T^{*}=\infty$.

Let us notice that the solution constructed in 2.1 is not smooth: the density
can be discontinuous across a curve and the second weak derivative of the
velocity is not locally integrable in space; it could contains Dirac masses.
But the regularity of $u$ is sufficient in order to use $u$, $\dot{u}$ and
$\sigma^{2}\ddot{u}$ as a test function in the following various computations.
The a priori estimates we will derive will involve lower regularity on the
initial data; in particular, the compatibility condition (2.2) will not be
required.

The first one is the classical energy, which consists in taking the scalar
product of the momentum equation $\eqref{epq4}_{2}$ with the velocity $u$, and
then integrating it over time and space. Due to the continuity of the velocity
$u$ and the normal component of the stress tensor, one obtains the so-called
classical energy:

(2.4)
$\displaystyle\sup_{s\in[0,t]}E(s)+\int_{0}^{t}\int_{\mathbb{R}^{2}}\\{2\mu(\rho)\lvert\mathbb{D}u\rvert^{2}+\lambda(\rho)\lvert\operatorname{div}u\rvert^{2}\\}\leqslant
E(0),$

where $E(t)$ is the total energy at time $t$, defined by:

$E(t)=\int_{\mathbb{R}^{2}}\left\\{\rho\dfrac{\lvert
u\rvert^{2}}{2}+H_{1}(\rho)\right\\}(t,x)dx.$

Above, $H_{1}$ stands for the potential energy that solves the following ODE :

$\rho H_{1}^{\prime}(\rho)-H_{1}(\rho)=P(\rho)-\widetilde{P}.$

The potential energy $H_{1}$ is given by the formula:

$H_{1}(\rho)=\rho\int_{\widetilde{\rho}}^{\rho}s^{-2}(P(s)-\widetilde{P})ds.$

More generally, one can define, as in [1], the potential energy $H_{l}$ that
solves the ODE

$\rho H_{l}^{\prime}(\rho)-H_{l}(\rho)=\lvert
P(\rho)-\widetilde{P}\rvert^{l-1}(P(\rho)-\widetilde{P})$

and given by the formula:

$H_{l}(\rho)=\rho\int_{\widetilde{\rho}}^{\rho}s^{-2}\lvert
P(s)-\widetilde{P}\rvert^{l-1}(P(s)-\widetilde{P})ds.$

The next step is devoted to providing bounds for the Hoff functionals
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$, which, in Eulerian coordinates, are
analogous to the energies obtained in the proof of the local well-posedness
result in Lagrangian coordinates. These functionals yield estimates for the
material acceleration (see [19, 21, 1]) that is,

$\dot{u}=\partial_{t}u+(u\cdot\nabla)u,$

and consequently for the velocity. The first Hoff functional $\mathcal{A}_{1}$
is obtained while taking the scalar product of the momentum equation
$\eqref{epq4}$ written in the form

$\rho\dot{u}^{j}=\operatorname{div}(\Pi^{j}),$

with the material acceleration $\dot{u}$. The second Hoff functional
$\mathcal{A}_{2}$ is obtained by first, taking the material derivative

$\partial_{t}\cdot+\operatorname{div}(\,\cdot\;u)$

of the momentum equation, before taking the scalar product of the obtained
equation with the material acceleration $\sigma\dot{u}$. Under the assumption
that the density and viscosity are lower bounded away from vacuum and upper
bound, and that the dynamic viscosity is close to a constant state, we have
the following estimates for the functionals $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$.

######  Proposition 2.4.

Assume that the density and viscosity are uniformly lower and upper bounded:

(2.5) $\displaystyle
0<\underline{\rho}\leqslant\rho(t,x)\leqslant\overline{\rho},\quad
0<\underline{\mu}\leqslant\mu(\rho)\leqslant\overline{\mu}\quad\text{and}\quad
0<\underline{\lambda}\leqslant\lambda(\rho)\leqslant\overline{\lambda}$

and let

$\delta:=\dfrac{1}{\underline{\mu}}\sup_{[0,t]}\lVert\mu(\rho)-\widetilde{\mu}\rVert_{L^{\infty}(\mathbb{R}^{2})}.$

There exists a positive function $\text{\textkappa}=\text{\textkappa}(l)$ for
$l\in(1,\infty)$, such that, if

$\delta\left(\dfrac{2\underline{\mu}+\underline{\lambda}}{2\overline{\mu}+\overline{\lambda}}\right)^{-\tfrac{1}{l+1}}<\text{\textkappa}(l),$

for $l\in\\{2,3\\}$ then, the following bounds hold true for the Hoff
functionals:

(2.6) $\displaystyle\mathcal{A}_{1}(t)\leqslant
M\left[C_{0}+\mathcal{A}_{1}(t)^{1/2}\left(C_{0}+\mathcal{A}_{1}(t)\right)+\mathcal{A}_{1}(t)\left(C_{0}+\mathcal{A}_{1}(t)\right)\right],$
(2.7) $\displaystyle\mathcal{A}_{2}(t)\leqslant
M\left[C_{0}+\mathcal{A}_{1}(\sigma(t))+\mathcal{A}_{1}(t)\left(C_{0}+\mathcal{A}_{1}(t)\right)\right],$

for some constant $M$ which depends on the lower and upper bounds of the
density and viscosity as given in (2.5), as well as on $\delta$. Hereafter,
$C_{0}$ denotes a sum of powers of $c_{0}$, which is defined in (1.24) as the
norm of the initial data.

The proof of 2.4 is given in section 3.1. In the following proposition we
estimate the $L^{p}$ norm of the density and velocity gradient jump at the
interface.

######  Proposition 2.5.

Assume that the density is bounded far away from vacuum and upper bounded as:

(2.8) $\displaystyle
0<\underline{\rho}\leqslant\rho(t,x)\leqslant\overline{\rho}.$

Then, there are constants
$0<\underline{\text{\textnu}}<\overline{\text{\textnu}}$ and $M>0$ depending
on $\underline{\rho}$ and $\overline{\rho}$ such that the following
inequalities hold true:

(2.9) $\displaystyle\lVert\llbracket
f(\rho(t))\rrbracket\rVert_{L^{p}(\mathcal{C}(t))}\leqslant\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{p}(\mathcal{C}(0))}\exp\left(-\underline{\text{\textnu}}\;t+(4\overline{\text{\textnu}}+1/p)\int_{0}^{t}\lVert\nabla
u(\tau)\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau\right),$ (2.10)
$\displaystyle\lVert\llbracket\nabla
u(t)\rrbracket\rVert_{L^{p}(\mathcal{C}(t))}\leqslant M\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{p}(\mathcal{C}(0))}\left(1+\lVert\nabla
u(t)\rVert_{L^{\infty}(\mathbb{R}^{2})}\right)\exp\left(-\underline{\text{\textnu}}\;t+(4\overline{\text{\textnu}}+1/p)\int_{0}^{t}\lVert\nabla
u(\tau)\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau\right),$

for all $1\leqslant p\leqslant\infty$ where $f$ is defined in (1.26).

In the ideal setting where we have a priori estimate for

$\int_{0}^{t}\lVert\nabla u(\tau)\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau$

that is uniform with respect to time, the exponential decay of the jump of the
velocity gradient and $f(\rho)$ follows immediately. Of course, in that case,
there is no need to use the decay of jumps since the characteristics of the
interface will be uniformly bounded in time. Unfortunately, we are not able to
obtain such an estimate, and the exponential decay of the density and velocity
gradient jump at the interface turns out to be crucial in order to compensate
for the growth of the characteristics of the interface. Indeed, as we will see
in (3.45), we have from the momentum equation (1.18) the following expression
for the velocity gradient:

(2.11) $\displaystyle\nabla u=\nabla u_{*}+\nabla u_{F}+\nabla u_{P}+\nabla
u_{\delta}$

and it is not clear whether we can obtain an
$L^{2}((\sigma(t),t),\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2}))$
estimate for $\nabla u_{P}$ uniformly with respect to time. However, as soon
as we obtain a uniform estimate for $\text{\texttheta}(t)$ with respect to
time (see 2.8), then, thanks to Hölder and Young inequalities:

$\int_{0}^{t}\lVert\nabla
u(\tau)\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau\leqslant\dfrac{\underline{\text{\textnu}}}{4(4\overline{\text{\textnu}}+1/p)}t+M\text{\texttheta}(t),$

we can deduce from the proposition above the exponential decay in time of the
density and velocity gradient jump. The proof of 2.5 is given in section 3.2.

We notice that the functionals $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$
obtained in 2.4 do not yield significant time integrability for the material
derivative $\dot{u}$ such that we can control the
$L^{4}((\sigma(t),t),\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2}))$ norm
of $\nabla u_{*}$. Thus, this requires us to perform another energy estimate.
On the other hand, to achieve exponential decay in the case of constant
viscosity [21], the velocity gradient was required to be small in
$L^{\infty}_{t,x}$ relative to $\underline{\text{\textnu}}$. To obtain the
$L^{\infty}((\tau,\infty)\times\mathbb{R}^{2})$ ($\tau>0$) norm estimate for
the velocity gradient, the author performs an energy estimate that consists in
applying the material derivative
$\partial_{t}\,\cdot\,+\operatorname{div}(\,\cdot\;u)$ to the momentum
equation $\eqref{epq4}_{2}$ before multiplying the obtained equation by
$\dot{u}\lvert\dot{u}\rvert^{s-2}$. This energy estimate yields the following
bound

(2.12)
$\displaystyle\sup_{t\geqslant\tau}\lVert\dot{u}(t)\rVert_{L^{s}(\mathbb{R}^{2})}^{s}\leqslant
C(\tau)C_{0},$

for all $\tau>0$, from which he deduces the
$L^{\infty}((\tau,T)\times\mathbb{R}^{2})$ norm of the gradient of the regular
part of the velocity. However, a smallness condition on the kinematic
viscosity [21, Section 2.3] has been required. Alternatively, we will derive
functional $\mathcal{A}_{3}$, (see (1.26) for the definition), that controls
the following norm of $\dot{u}$

$\sup_{[0,T]}\sigma\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}.$

This implies in particular (2.12) and the bound of the energy
$\mathcal{A}_{3}$ does not require any smallness condition on the kinematic
viscosity. A similar functional was considered in [11] in the context of the
Navier-Stokes equations for incompressible fluids. To obtain this functional,
we first apply the material derivative
$\partial_{t}\cdot+\operatorname{div}(\,\cdot\,u)$ to the momentum equation
$\eqref{epq4}_{2}$ and obtain the fact that $\dot{u}$ verifies:

(2.13)
$\displaystyle\partial_{t}(\rho\dot{u}^{j})+\operatorname{div}(\rho\dot{u}^{j}u)=\partial_{k}(\dot{\Pi}^{jk})+\partial_{k}(\Pi^{jk}\operatorname{div}u)-\operatorname{div}(\partial_{k}u\Pi^{jk}).$

Since the left hand side above can be reduced to $\rho\ddot{u}$, we can test
the above equation with $\sigma^{2}\ddot{u}$ and we obtain the functional
$\mathcal{A}_{3}$. Moreover, the following bound holds for $\mathcal{A}_{3}$.

######  Proposition 2.6.

Assume that the hypotheses of 2.4 hold true for $l\in\\{2,3,5\\}$. Then, the
following bound holds true with a constant $M$ that depends on the lower and
upper bounds of the density and the viscosity, as well as $\delta$:

(2.14) $\displaystyle\mathcal{A}_{3}(t)\leqslant
M\left[C_{0}+\mathcal{A}_{1}(t)+\mathcal{A}_{2}(t)+\mathcal{L}(C_{0},\mathcal{A}_{1}(t),\mathcal{A}_{2}(t),\mathcal{A}_{3}(t))\right].$

Above, the last term is super-linear with respect to $C_{0}$,
$\mathcal{A}_{1}(t)$, $\mathcal{A}_{2}(t)$ and $\mathcal{A}_{3}(t)$.

The proof is given is section 3.3. Let us notice that, the functionals
$\mathcal{A}_{1}$, $\mathcal{A}_{2}$ and $\mathcal{A}_{3}$, are under control
as long as the density and the viscosity are upper bounded and bounded away
from vacuum as well, the dynamic viscosity is close to $\widetilde{\mu}$. The
final step of our analysis consists in translating (2.6), (2.7), (2.14), (2.9)
and (2.10) into piecewise Hölder estimates for the density $\rho$ and the
velocity gradient $\nabla u$.

In general, the $L^{\infty}((0,T)\times\mathbb{R}^{2})$ estimate for the
density follows as soon as the effective flux belongs to
$L^{1}((0,T),L^{\infty}(\mathbb{R}^{2}))$. As it is well-explained in [1], the
algebraic structure of the Navier-Stokes equations with density-dependent
viscosity does not allow to propagate the $L^{\infty}$ norm of the density as
it has been done by Hoff [19, 20] for the isotropic case. The issue arises
from the fact that the effective flux, which plays the role of connecting the
momentum equation and the mass equation, does not belong to
$L^{1}_{\text{loc}}((0,\infty),L^{\infty}(\mathbb{R}^{2}))$ when the density
is only bounded. In particular, the effective flux and the velocity gradient
are of the same order. In fact, the effective flux $F$ is given by:

$F=-(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})+K((\mu(\rho)-\widetilde{\mu})\mathbb{D}u)$

where $K$ is a singular integral operator, well-known to map
$L^{\infty}(\mathbb{R}^{2})$ into $BMO(\mathbb{R}^{2})$ space. Thus, even if
the velocity gradient is bounded, it is not obvious that the effective flux
will also be bounded. It is straightforward to show the following quantitative
Hölder estimate by combining A.2 and A.3.

######  Theorem 2.7.

Let $0<\alpha<1$, $1\leqslant p<\infty$ and $K$ an even higher-order Riesz
operator. We consider a curve $\mathcal{C}\in\mathscr{C}^{1+\alpha}$ that
satisfies (1.4), and for $g\in
L^{p}(\mathbb{R}^{2})\cap\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2})$ we
define the following function:

$w(x)=\int_{\mathbb{R}^{2}}K(x-y)g(y)dy.$

There exists a constant $C$ depending on $\alpha$ and $p$ such that:

(2.15) $\displaystyle\lVert
w\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2})}\leqslant
C\left[\lVert g\rVert_{L^{p}(\mathbb{R}^{2})}+\lVert\llbracket
g\rrbracket\rVert_{L^{p}(\mathcal{C})}+\lVert
g\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2})}+\lVert\llbracket
g\rrbracket\rVert_{L^{\infty}(\mathcal{C})}\mathcal{P}(\mathcal{C})\right]$

where

$\mathcal{P}(\mathcal{C}):=c_{\gamma}^{2}\lVert\nabla\gamma\rVert_{\mathscr{C}^{\alpha}}^{1/\alpha}+\left(1+\lvert\partial
D\rvert\right)\mathrm{P}(c_{\gamma}+\lVert\nabla\gamma\rVert_{L^{\infty}})\left(1+\lVert\nabla\gamma\rVert_{\dot{\mathscr{C}}^{\alpha}}\right)$

and $\mathrm{P}$ a polynomial that depends on $K$.

2.7 appears as an interpolation of the fact that singular integral operators
of Calderón-Zygmung type map linearly $L^{p}(\mathbb{R}^{2})$ into itself for
$1<p<\infty$ and $\dot{\mathscr{C}}^{\alpha}(\mathbb{R}^{2})$ into itself. By
the help of this quantitative Hölder estimate, one has the following estimates
for functionals  and .

######  Proposition 2.8.

Assume that the density is bounded away from vacuum and upper bounded (2.8).
There are constants $M$ and $M^{\prime}$ such that the following estimates
hold true:

(2.16) $\displaystyle\text{\textbeta}(t)\leqslant
M^{\prime}e^{M^{\prime}\text{\texttheta}(t)}\left(\lVert
f(\rho_{0})\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma_{0}}(\mathbb{R}^{2})}^{4}+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\right),$
$\displaystyle\text{\texttheta}(t)$ $\displaystyle\leqslant
M\left[C_{0}^{2}+\mathcal{A}_{1}^{2}(t)+\mathcal{A}_{2}^{2}(t)+\mathcal{A}_{3}(t)^{2}+\text{\textbeta}(t)(1+\text{\texttheta}(t))+(1+\text{\textbeta}(t))\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\right.$
(2.17)
$\displaystyle\left.+(1+K_{0})e^{M^{\prime}\text{\texttheta}(t)}\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{\infty}\cap
L^{2}(\mathcal{C}(0))}^{4}\left(1+\text{\texttheta}(t)+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\right)\right],$

with

$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau$
$\displaystyle\leqslant
M^{\prime}\left[C_{0}^{2}+\mathcal{A}_{1}(t)^{2}+\mathcal{A}_{2}(t)^{2}+\mathcal{A}_{3}(t)^{2}+\text{\textbeta}(t)\text{\texttheta}(t)\right.$
(2.18) $\displaystyle\left.+(1+K_{0})\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{\infty}\cap
L^{2}(\mathcal{C}(0))}^{4}\left(1+\text{\texttheta}(t)\right)e^{M^{\prime}\text{\texttheta}(t)}\right].$

The constant $M$ depends on $\delta$, $\alpha$ and the upper and lower bounds
of the density and viscosity whereas the constants $M^{\prime}$ depends on the
same quantities except $\delta$.

The proof of this proposition is given in section 3.4. It is obvious that the
estimate for  also holds for the dynamic viscosity $\mu(\rho)$. We finally
close the estimates in 2.4, 2.6 and 2.8 with the help of the bootstrap
argument similar to the one in [1], and we do not present the proof.

######  Lemma 2.9.

Let $(\rho,u)$ be a local solution of the equations (2.1) with initial data
$(\rho_{0},u_{0})$ that verifies (1.19) and (1.20) and the compatibility
condition:

$\operatorname{div}\\{2\mu(\rho_{0})\mathbb{D}u_{0}+(\lambda(\rho_{0})\operatorname{div}u_{0}-P(\rho_{0})+\widetilde{P})I_{d}\\}\in
L^{2}(\mathbb{R}^{2}).$

Assume that the solution $(\rho,u)$ is defined up to a maximal time $T^{*}$.
There exist constants $c>0$ and $[\mu]_{0}>0$, such that if

$c_{0}:=\lVert
u_{0}\rVert_{H^{1}(\mathbb{R}^{2})}^{2}+\lVert\rho_{0}-\widetilde{\rho}\rVert_{L^{2}(\mathbb{R}^{2})\cap\mathscr{C}^{\alpha}_{pw,\gamma_{0}}(\mathbb{R}^{2})}^{2}+\lVert\llbracket\rho_{0}\rrbracket\rVert_{L^{2}(\mathcal{C}(0))\cap
L^{\infty}(\mathcal{C}(0))}^{2}\leqslant c$

and

$\lVert\mu(\rho_{0})-\widetilde{\mu}\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma_{0}}(\mathbb{R}^{2})}\leqslant[\mu]_{0},$

then, there exists a constant $C$ depending on the lower and upper bounds of
the density and viscosity, as well on the initial interface $\mathcal{C}(0)$
such that for all $0<t<T^{*}$, the followings hold:

$0<\rho^{*}/4\leqslant\rho(t,x)\leqslant
4\rho^{*},\quad\quad\mu_{*}\leqslant\mu(\rho),\;\lambda(\rho)\leqslant\mu^{*},\quad\quad\sup_{[0,t]}\lVert\mu(\rho)-\widetilde{\mu}\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2})}\leqslant
C[\mu]_{0},$

and

$E(t)+\mathcal{A}_{1}(t)+\mathcal{A}_{2}(t)+\mathcal{A}_{3}(t)+\sqrt{\text{\textbeta}(t)}+\sqrt{\text{\texttheta}(t)}+\int_{0}^{t}\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\leqslant Cc_{0}.$

To complete the proof of 1.1, we have two remaining steps. First, we need to
construct a sequence $(\rho^{\delta},u^{\delta})$ that is globally defined in
time, and converges to $(\rho,u)$, which is a solution of the Navier-Stokes
equations (2.1). Usually, this sequence $(\rho^{\delta},u^{\delta})$
corresponds to the solution of the Cauchy problem (2.1) with initial data
$(\rho_{0}^{\delta},u_{0}^{\delta})$ where
$(\rho_{0}^{\delta},u_{0}^{\delta})$ is obtained by smoothing
$(\rho_{0},u_{0})$. In our case, this does not seem to be possible. The
problem arises because smoothing the initial data would lead to a loss of
density discontinuity. The approach we employed is more subtle and it is
presented in section 3.5 along with the proof of the stability of the
approximate sequence. The second step will be devoted to showing that the
regularity of the limit $(\rho,u)$ is sufficient to establish it uniqueness.
As in the proof of the local well-posedness result, the proof of uniqueness
follows the same lines as in [7] since the velocity limit $u$ comes with the
following regularity:

(2.19) $\displaystyle\nabla u\in
L^{1}_{\text{loc}}([0,\infty),L^{\infty}(\mathbb{R}^{2}))\quad\text{and}\quad\sqrt{\sigma}\nabla
u\in L^{2}_{\text{loc}}([0,\infty),L^{\infty}(\mathbb{R}^{2})).$

##  3\. Proofs

###  3.1. Proof of 2.4

###### Proof.

Let us begin the proof of this proposition by recalling that the first
functional is obtained while multiplying the momentum equation, written in the
form

(3.1) $\displaystyle\rho\dot{u}^{j}=\operatorname{div}\Pi^{j}$

by $\dot{u}^{j}$, then summing over $j$ and integrating on $\mathbb{R}^{2}$ in
order to obtain:

$\displaystyle\int_{\mathbb{R}^{2}}\rho\lvert\dot{u}^{j}\rvert^{2}$
$\displaystyle+\dfrac{d}{dt}\int_{\mathbb{R}^{2}}\left\\{\mu(\rho)\lvert\mathbb{D}u\rvert^{2}+\dfrac{\lambda(\rho)}{2}\lvert\operatorname{div}u\rvert^{2}\right\\}=-\int_{\mathbb{R}^{2}}2\mu(\rho)\mathbb{D}^{jk}u\partial_{k}u^{l}\partial_{l}u^{j}+\int_{\mathbb{R}^{2}}(\rho\mu^{\prime}(\rho)-\mu(\rho))\lvert\mathbb{D}u\rvert^{2}\operatorname{div}u$
$\displaystyle+\dfrac{1}{2}\int_{\mathbb{R}^{2}}(\rho\lambda^{\prime}(\rho)-\lambda(\rho))(\operatorname{div}u)^{3}-\int_{\mathbb{R}^{2}}\lambda(\rho)\operatorname{div}u\nabla
u^{l}\partial_{l}u+\dfrac{d}{dt}\int_{\mathbb{R}^{2}}\left(\operatorname{div}u\\{P(\rho)-\widetilde{P}\\}\right)+\int_{\mathbb{R}^{2}}\nabla
u^{l}\partial_{l}u\left(P(\rho)-\widetilde{P}\right)$
$\displaystyle+\int_{\mathbb{R}^{2}}(\operatorname{div}u)^{2}\left(\rho
P^{\prime}(\rho)-P(\rho)+\widetilde{P}\right).$

The above equality follows from the continuity of $\dot{u}$ and of the normal
component of the stress tensor at the interface. Next, we integrate the above
in time and we obtain:

(3.2) $\displaystyle\mathcal{A}_{1}(t)\lesssim C_{0}+\sup_{[0,t]}\lVert
P(\rho)-\widetilde{P}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\int_{0}^{t}\lVert\nabla
u\rVert_{L^{3}(\mathbb{R}^{2})}^{3}+\int_{0}^{t}\lVert
P(\rho)-\widetilde{P}\rVert_{L^{p}(\mathbb{R}^{2})}\lVert\nabla
u\rVert_{L^{2p^{\prime}}(\mathbb{R}^{2})}^{2}$

where we have used the classical energy (2.4) and where $p\geqslant 3$. On the
other hand, we take the material derivative of the momentum equation, written
in the form (3.1), before multiplying by $\dot{u}$ and space integration
yields:

$\displaystyle\dfrac{1}{2}\dfrac{d}{dt}\int_{\mathbb{R}^{2}}\rho\lvert\dot{u}^{j}\rvert^{2}+$
$\displaystyle\int_{\mathbb{R}^{2}}\left\\{2\mu(\rho)\lvert\mathbb{D}^{jk}\dot{u}\rvert^{2}+\lambda(\rho)\lvert\operatorname{div}\dot{u}\rvert^{2}\right\\}=\int_{\mathbb{R}^{2}}\partial_{k}\dot{u}^{j}\left\\{\mu(\rho)\partial_{j}u^{l}\partial_{l}u^{k}+\mu(\rho)\partial_{k}u^{l}\partial_{l}u^{j}+2\rho\mu^{\prime}(\rho)\mathbb{D}^{jk}u\operatorname{div}u\right\\}$
$\displaystyle+\int_{\mathbb{R}^{2}}\operatorname{div}\dot{u}\left\\{\lambda(\rho)\nabla
u^{l}\partial_{l}u+\rho\lambda^{\prime}(\rho)(\operatorname{div}u)^{2}-\rho
P^{\prime}(\rho)\operatorname{div}u\right\\}-\int_{\mathbb{R}^{2}}\partial_{k}\dot{u}^{j}\Pi^{jk}\operatorname{div}u+\int_{\mathbb{R}^{2}}\partial_{l}\dot{u}^{j}\partial_{k}u^{l}\Pi^{jk}.$

The computations leading to the above equality can be found in section B.1.
Next, we multiply the above by $\sigma$ before integrating in space, then
Hölder and Young inequalities yield:

(3.3) $\displaystyle\mathcal{A}_{2}(t)\lesssim
C_{0}+\mathcal{A}_{1}(\sigma(t))+\int_{0}^{t}\sigma\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{2})}^{4}+\int_{0}^{t}\sigma\lVert
P(\rho)-\widetilde{P}\rVert_{L^{4}(\mathbb{R}^{2})}^{4}.$

We turn to the estimation of the norm of the gradient of the velocity and the
pressure that appear in the estimate (3.2) and (3.3). First, we apply the
divergence operator to the momentum equation (3.1) in order to obtain an
elliptic equation on the so called effective flux:

$\operatorname{div}(\rho\dot{u})=\operatorname{div}\operatorname{div}(2\mu(\rho)\mathbb{D}u)+\Delta\\{\lambda(\rho)\operatorname{div}u-P(\rho)+\widetilde{P}\\}$

from which we deduce that

(3.4)
$\displaystyle\lambda(\rho)\operatorname{div}u-P(\rho)+\widetilde{P}=(-\Delta)^{-1}\operatorname{div}\operatorname{div}(2\mu(\rho)\mathbb{D}u)-(-\Delta)^{-1}\operatorname{div}(\rho\dot{u}).$

By the help of the fundamental solution of the Laplacian, we express the first
term of the right hand side above by the following,

$\displaystyle(-\Delta)^{-1}\operatorname{div}\operatorname{div}(2\mu(\rho)\mathbb{D}u)$
$\displaystyle=-p.v.\int_{\mathbb{R}^{2}}\Gamma(x-y)\operatorname{div}\operatorname{div}(2\mu(\rho)\mathbb{D}u)(y)dy$
$\displaystyle=-2\mu(\rho)\operatorname{div}u-p.v.\int_{\mathbb{R}^{2}}\partial_{ij}^{2}\Gamma(x-y)(2\mu(\rho)\mathbb{D}^{ij}u)(y)dy$

from which, we deduce the following representation of the effective flux:

(3.5) $\displaystyle
F=(2\mu(\rho)+\lambda(\rho))\operatorname{div}u-P(\rho)+\widetilde{P}=K\\{(\mu(\rho)-\widetilde{\mu})\mathbb{D}u\\}-(-\Delta)^{-1}\operatorname{div}(\rho\dot{u}).$

Above, we have used the fact that

$p.v.\int_{\mathbb{R}^{2}}\partial_{ij}^{2}\Gamma(x-y)\mathbb{D}^{ij}u(y)dy=0.$

On the other hand, we take the rotational operator to the momentum equation
(3.1) in order to obtain:

$\operatorname{curl}_{jk}(\rho\dot{u})=\operatorname{curl}_{jk}\operatorname{div}(2\mu(\rho)\mathbb{D}u)=\operatorname{curl}_{jk}\operatorname{div}(2(\mu(\rho)-\widetilde{\mu})\mathbb{D}u)+\widetilde{\mu}\Delta\operatorname{curl}_{jk}u.$

From which, we deduce

$\displaystyle\widetilde{\mu}\operatorname{curl}_{jk}u$
$\displaystyle=(-\Delta)^{-1}\operatorname{curl}_{jk}\operatorname{div}(2(\mu(\rho)-\widetilde{\mu})\mathbb{D}u)-(-\Delta)^{-1}\operatorname{curl}_{jk}(\rho\dot{u})$
$\displaystyle=-(\mu(\rho)-\widetilde{\mu})\operatorname{curl}_{jk}+K^{\prime}_{jklm}\\{(\mu(\rho)-\widetilde{\mu})\mathbb{D}^{lm}u\\}-(-\Delta)^{-1}\operatorname{curl}_{jk}(\rho\dot{u}).$

It then holds that the vorticity takes the following form:

$\mu(\rho)\operatorname{curl}_{jk}u=K^{\prime}_{jklm}\\{(\mu(\rho)-\widetilde{\mu})\mathbb{D}^{lm}u\\}-(-\Delta)^{-1}\operatorname{curl}_{jk}(\rho\dot{u}).$

Let us notice that the singular integral operators $K$ and $K^{\prime}$ are
homogeneous with symbols of degree zero, such that, in view of Calderón
Zygmung theory, see [37], they map the Lebesgue space $L^{p}(\mathbb{R}^{d})$
into itself for $1<p<\infty$, with norm $\kappa(p)$. It turns out that the
following $L^{p}$ estimates hold true for the vorticity and the effective
flux:

(3.6) $\displaystyle\lVert
F\rVert_{L^{p}(\mathbb{R}^{2})}\leqslant\kappa(p)\lVert\mu(\rho)-\widetilde{\mu}\rVert_{L^{\infty}(\mathbb{R}^{2})}\lVert\mathbb{D}u\rVert_{L^{p}(\mathbb{R}^{2})}+\lVert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rVert_{L^{p}(\mathbb{R}^{2})},$
(3.7)
$\displaystyle\lVert\mu(\rho)\operatorname{curl}u\rVert_{L^{p}(\mathbb{R}^{2})}\leqslant\kappa(p)\lVert\mu(\rho)-\widetilde{\mu}\rVert_{L^{\infty}(\mathbb{R}^{2})}\lVert\mathbb{D}u\rVert_{L^{p}(\mathbb{R}^{2})}+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\dot{u})\rVert_{L^{p}(\mathbb{R}^{2})}$

for all $1<p<\infty$. Also, theory of singular integrals operators, together
with the above estimates imply:

$\displaystyle\lVert\nabla u\rVert_{L^{p}(\mathbb{R}^{2})}$
$\displaystyle\leqslant\kappa(p)\left(\lVert\operatorname{div}u\rVert_{L^{p}(\mathbb{R}^{2})}+\lVert\operatorname{curl}u\rVert_{L^{p}(\mathbb{R}^{2})}\right)$
$\displaystyle\leqslant\kappa(p)\left(\dfrac{1}{2\underline{\mu}+\underline{\lambda}}\lVert
F+P(\rho)-\widetilde{P}\rVert_{L^{p}(\mathbb{R}^{2})}+\dfrac{1}{\underline{\mu}}\lVert\mu(\rho)\operatorname{curl}u\rVert_{L^{p}(\mathbb{R}^{2})}\right)$
$\displaystyle\leqslant\dfrac{\kappa(p)^{2}}{\underline{\mu}}\lVert\mu(\rho)-\widetilde{\mu}\rVert_{L^{\infty}(\mathbb{R}^{2})}\lVert\mathbb{D}u\rVert_{L^{p}(\mathbb{R}^{2})}+\dfrac{\kappa(p)}{2\underline{\mu}+\underline{\lambda}}\lVert
P(\rho)-\widetilde{P}\rVert_{L^{p}(\mathbb{R}^{2})}$
$\displaystyle+\dfrac{\kappa(p)}{\underline{\mu}}\left(\lVert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rVert_{L^{p}(\mathbb{R}^{2})}+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\dot{u})\rVert_{L^{p}(\mathbb{R}^{2})}\right).$

Assume that

(3.8)
$\displaystyle\delta:=\dfrac{\lVert\mu(\rho)-\widetilde{\mu}\rVert_{L^{\infty}(\mathbb{R}^{2})}}{\underline{\mu}}<\dfrac{1}{\kappa(p)^{2}}$

then, the first term of the right hand side of the above inequality can be
absorbed in the left hand side and it yields:

$\displaystyle\lVert\nabla u\rVert_{L^{p}(\mathbb{R}^{2})}$
$\displaystyle\leqslant\dfrac{\kappa(p)}{\underline{\mu}(1-\delta\kappa(p)^{2})}\left(\lVert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rVert_{L^{p}(\mathbb{R}^{2})}+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\dot{u})\rVert_{L^{p}(\mathbb{R}^{2})}\right)$
(3.9)
$\displaystyle+\dfrac{\kappa(p)}{(2\underline{\mu}+\underline{\lambda})(1-\delta\kappa(p)^{2})}\lVert
P(\rho)-\widetilde{P}\rVert_{L^{p}(\mathbb{R}^{2})}.$

We turn to the estimate of the $L^{p}$ norm of the pressure that appears in
the above inequality with the help of the potential energies. First, we recall
that the potential energy $H_{l}$ satisfies:

$\dfrac{d}{dt}\int_{\mathbb{R}^{2}}H_{l}(\rho)+\int_{\mathbb{R}^{2}}\lvert
P(\rho)-\widetilde{P}\rvert^{l-1}(P(\rho)-\widetilde{P})\operatorname{div}u=0.$

One can substitute the divergence of the velocity in the expression of the
effective flux (3.5) as:

$\operatorname{div}u=(2\mu(\rho)+\lambda(\rho))^{-1}\left(F+P(\rho)-\widetilde{P}\right)$

and then obtains, after Hölder inequality:

$\displaystyle\dfrac{d}{dt}\int_{\mathbb{R}^{2}}H_{l}(\rho)$
$\displaystyle+\int_{\mathbb{R}^{2}}(2\mu(\rho)+\lambda(\rho))^{-1}\lvert
P(\rho)-\widetilde{P}\rvert^{l+1}$
$\displaystyle=-\int_{\mathbb{R}^{2}}(2\mu(\rho)+\lambda(\rho))^{-1}\lvert
P(\rho)-\widetilde{P}\rvert^{l-1}(P(\rho)-\widetilde{P})F$
$\displaystyle\leqslant\dfrac{l}{l+1}\int_{\mathbb{R}^{2}}(2\mu(\rho)+\lambda(\rho))^{-1}\lvert
P(\rho)-\widetilde{P}\rvert^{l+1}+\dfrac{1}{l+1}\int_{\mathbb{R}^{2}}(2\mu(\rho)+\lambda(\rho))^{-1}\lvert
F\rvert^{l+1}.$

The first term in the right hand side can be absorbed in the left hand side
and it follows that:

(3.10)
$\displaystyle\dfrac{d}{dt}\int_{\mathbb{R}^{2}}H_{l}(\rho)+\dfrac{1}{l+1}\int_{\mathbb{R}^{2}}(2\mu(\rho)+\lambda(\rho))^{-1}\lvert
P(\rho)-\widetilde{P}\rvert^{l+1}\leqslant\dfrac{1}{l+1}\int_{\mathbb{R}^{2}}(2\mu(\rho)+\lambda(\rho))^{-1}\lvert
F\rvert^{l+1}.$

In order to express the $L^{l+1}$ norm of the effective flux, in the above
inequality, one goes back to (3.6) and then uses (3.9) and obtains the
following:

$\displaystyle\int_{\mathbb{R}^{2}}(2\mu(\rho)$
$\displaystyle+\lambda(\rho))^{-1}\lvert
F\rvert^{l+1}\leqslant\dfrac{1}{2\underline{\mu}+\underline{\lambda}}\lVert
K\\{(\mu(\rho)-\widetilde{\mu})\mathbb{D}u\\}\rVert_{L^{l+1}(\mathbb{R}^{2})}^{l+1}+\dfrac{1}{2\underline{\mu}+\underline{\lambda}}\int_{\mathbb{R}^{2}}\lvert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rvert^{l+1}$
$\displaystyle\leqslant\dfrac{\kappa(l+1)^{l+1}}{2\underline{\mu}+\underline{\lambda}}\lVert\mu(\rho)-\widetilde{\mu}\rVert_{L^{\infty}(\mathbb{R}^{2})}^{l+1}\lVert\mathbb{D}u\rVert_{L^{l+1}(\mathbb{R}^{2})}^{l+1}+\dfrac{1}{2\underline{\mu}+\underline{\lambda}}\int_{\mathbb{R}^{2}}\lvert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rvert^{l+1}$
$\displaystyle\leqslant\dfrac{\kappa(l+1)^{l+1}}{2\underline{\mu}+\underline{\lambda}}\left(\dfrac{\kappa(l+1)}{2\underline{\mu}+\underline{\lambda}}\dfrac{\lVert\mu(\rho)-\widetilde{\mu}\rVert_{L^{\infty}}}{1-\delta\kappa(l+1)^{2}}\right)^{l+1}\lVert
P(\rho)-\widetilde{P}\rVert_{L^{l+1}(\mathbb{R}^{2})}^{l+1}$
$\displaystyle+\dfrac{\kappa(l+1)^{l+1}}{2\underline{\mu}+\underline{\lambda}}\left(\dfrac{\kappa(l+1)}{\underline{\mu}}\dfrac{\lVert\mu(\rho)-\widetilde{\mu}\rVert_{L^{\infty}(\mathbb{R}^{2})}}{1-\delta\kappa(l+1)^{2}}\right)^{l+1}\left(\lVert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rVert_{L^{l+1}(\mathbb{R}^{2})}^{l+1}\right.$
$\displaystyle\left.+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\dot{u})\rVert_{L^{l+1}(\mathbb{R}^{2})}^{l+1}\right)+\dfrac{1}{2\underline{\mu}+\underline{\lambda}}\lVert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rVert^{l+1}_{L^{l+1}(\mathbb{R}^{2})}.$

In the following lines, we will replace the above estimate for the effective
flux in (3.10) and absorb the norm of the pressure that is the first term of
the right hand side of the above estimate in the left side of (3.10). And so
that, we need for example, for $l>1$, the following:

$\delta\left(\dfrac{2\underline{\mu}+\underline{\lambda}}{2\overline{\mu}+\overline{\lambda}}\right)^{-\tfrac{1}{l+1}}<\dfrac{1}{3\sqrt{(l+1)}\kappa(l+1)^{2}}.$

In conclusion, for all $l>1$ there exists a constant
$\text{\textkappa}=\text{\textkappa}(l)$ such that if

(3.11)
$\displaystyle\delta\left(\dfrac{2\underline{\mu}+\underline{\lambda}}{2\overline{\mu}+\overline{\lambda}}\right)^{-\tfrac{1}{l+1}}<\text{\textkappa}(l)$

then, (3.9) holds for $p=l+1$ and the following inequality also:

(3.12) $\displaystyle\dfrac{d}{dt}\int_{\mathbb{R}^{2}}H_{l}(\rho)+\lVert
P(\rho)-\widetilde{P}\rVert^{l+1}_{L^{l+1}(\mathbb{R}^{2})}\lesssim\lVert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rVert_{L^{l+1}(\mathbb{R}^{2})}^{l+1}+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\dot{u})\rVert_{L^{l+1}(\mathbb{R}^{2})}^{l+1}.$

The objective of the above computations is to know the appropriate smallness
condition on the viscosity (3.11). Let us notice that, here, in order to close
the estimates for functional $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$, we will
only need the smallness condition on the viscosity (3.11) for $l\in\\{2,3\\}$
however, latter, one will need the case $l=5$. The constants involved in (3.9)
and (3.11) and the latter estimates depend not only on the lower and upper
bounds of the density, viscosity but also on $\delta$. In particular, when
$\delta$ approaches

$\text{\textkappa}(l)\left(\dfrac{2\underline{\mu}+\underline{\lambda}}{2\overline{\mu}+\overline{\lambda}}\right)^{\tfrac{1}{l+1}},$

the underlying constants become large. Assuming the smallness condition (3.11)
for $l=2$, the inequality (3.12) holds true for $l=2$ and one can integrate it
in time in order to obtain:

(3.13)
$\displaystyle\sup_{[0,t]}\int_{\mathbb{R}^{2}}H_{2}(\rho)+\int_{0}^{t}\lVert
P(\rho)-\widetilde{P}\rVert^{3}_{L^{3}(\mathbb{R}^{2})}\lesssim\int_{\mathbb{R}^{2}}H_{2}(\rho_{0})+\int_{0}^{t}\left(\lVert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rVert_{L^{3}(\mathbb{R}^{2})}^{3}+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\dot{u})\rVert_{L^{3}(\mathbb{R}^{2})}^{3}\right).$

Next, we assume again that the smallness condition (3.11) holds for $l=3$,
then, one can multiply (3.12) by $\sigma$ before integrating in time and
obtains:

$\sup_{[0,t]}\sigma\int_{\mathbb{R}^{2}}H_{3}(\rho)+\int_{0}^{t}\sigma\lVert
P(\rho)-\widetilde{P}\rVert^{4}_{L^{4}(\mathbb{R}^{2})}\lesssim\int_{0}^{\sigma(t)}\int_{\mathbb{R}^{2}}H_{3}(\rho)\\\
+\int_{0}^{t}\sigma\left(\lVert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rVert_{L^{4}(\mathbb{R}^{2})}^{4}+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\dot{u})\rVert_{L^{4}(\mathbb{R}^{2})}^{4}\right).$

By combining the above estimate, the inequality (3.9), Gagliardo-Nirenberg
inequality and the fact that
$H_{3}(\rho)\lesssim_{\overline{\rho}}H_{1}(\rho)$, one obtains the following:

(3.14) $\displaystyle\int_{0}^{t}\sigma\left(\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{2})}^{4}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{4}(\mathbb{R}^{2})}^{4}\right)\lesssim
C_{0}+\mathcal{A}_{1}(t)\left(C_{0}+\mathcal{A}_{1}(t)\right).$

Above, we also use the fact that

$\sup_{[0,t]}\lVert
P(\rho)-\widetilde{P}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\lesssim_{\overline{\rho}}\sup_{[0,t]}\int_{\mathbb{R}^{2}}H_{1}(\rho).$

We now go back to (3.3) from which we deduce the following bound for the
second Hoff functional:

$\mathcal{A}_{2}(t)\lesssim
C_{0}+\mathcal{A}_{1}(\sigma(t))+\mathcal{A}_{1}(t)\left(C_{0}+\mathcal{A}_{1}(t)\right)$

and we turn to the bound for the first Hoff functional. We may first recall
that from (3.2), one has the following:

$\mathcal{A}_{1}(t)\lesssim C_{0}+\int_{0}^{t}\lVert\nabla
u\rVert_{L^{3}(\mathbb{R}^{2})}^{3}+\int_{0}^{t}\lVert
P(\rho)-\widetilde{P}\rVert_{L^{p}(\mathbb{R}^{2})}\lVert\nabla
u\rVert_{L^{2p^{\prime}}(\mathbb{R}^{2})}^{2}$

for some $p>3$ and all integral above will be splitted into two

$\int_{0}^{t}=\int_{0}^{\sigma(t)}+\int_{\sigma(t)}^{t}.$

For the lower time, we take $p=3$ and by Hölder and Young inequalities, we
have from (3.9) and (3.13) the following:

(3.15) $\displaystyle\int_{0}^{\sigma(t)}\left(\lVert\nabla
u\rVert_{L^{3}(\mathbb{R}^{2})}^{3}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{3}(\mathbb{R}^{2})}^{3}\right)\lesssim
C_{0}+\mathcal{A}_{1}(t)^{1/2}\left(C_{0}+\mathcal{A}_{1}(t)\right).$

For large time, we will take $p=4$, and by Hölder, Young, Gagliardo-Nirenberg
and interpolation inequalities, we have:

(3.16) $\displaystyle\int_{\sigma(t)}^{t}\lVert\nabla
u\rVert_{L^{3}(\mathbb{R}^{2})}^{3}+\int_{\sigma(t)}^{t}\lVert
P(\rho)-\widetilde{P}\rVert_{L^{4}(\mathbb{R}^{2})}\lVert\nabla
u\rVert_{L^{8/3}(\mathbb{R}^{2})}^{2}\lesssim
C_{0}+\mathcal{A}_{1}(t)\left(C_{0}+\mathcal{A}_{1}(t)\right).$

Finally, we have the following estimate for the first Hoff functional:

$\mathcal{A}_{1}(t)\lesssim
C_{0}+\mathcal{A}_{1}(t)^{1/2}\left(C_{0}+\mathcal{A}_{1}(t)\right)+\mathcal{A}_{1}(t)\left(C_{0}+\mathcal{A}_{1}(t)\right).$

The functional $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ are under control as
long as the density and the viscosity are bounded away from vacuum and upper
bounded. Additionally, the dynamic viscosity should be a small perturbation of
the constant state $\widetilde{\mu}$. This ends the proof of 2.4. ∎

###  3.2. Proof of 2.5

###### Proof.

Let us begin the proof of 2.5 by rewriting the mass equation
$\eqref{epq4}_{1}$. We recall that it can be written as follows:

$\partial_{t}\log\rho+u\cdot\nabla\log\rho+\operatorname{div}u=0.$

We multiply the above by $2\mu(\rho)+\lambda(\rho)$ and substitute the last
term by the help of the effective flux in order to obtain:

$\partial_{t}f(\rho)+u\cdot\nabla f(\rho)+P(\rho)-\widetilde{P}=-F$

where $f(\rho)$ is:

(3.17) $\displaystyle
f(\rho)=\int_{\widetilde{\rho}}^{\rho}\dfrac{2\mu(s)+\lambda(s)}{s}ds.$

In particular, if $X$ is the flow of the velocity $u$, then for all
$x\in\mathbb{R}^{2}$, one has:

(3.18)
$\displaystyle\dfrac{d}{dt}f(\rho(t,X(t,x)))+P(\rho(t,X(t,x)))-\widetilde{P}=-F(t,X(t,x)).$

In particular along the interface $\mathcal{C}(t)$ parameterized by
$\gamma(t,s)=X(t,\gamma_{0}(s))$, one has:

(3.19)
$\displaystyle\dfrac{d}{dt}f(\rho(t,\gamma(t,s)))+P(\rho(t,\gamma(t,s)))-\widetilde{P}=-F(t,\gamma(t,s))$

and then, by taking the jump at $\gamma(t,s)$, one has:

$\displaystyle\dfrac{d}{dt}\llbracket f(\rho(t,\gamma(t,s)))$
$\displaystyle\rrbracket+g(t,s)\llbracket
f(\rho(t,\gamma(t,s)))\rrbracket=-\llbracket F(t,\gamma(t,s))\rrbracket$
$\displaystyle=-2h(t,s)\llbracket
f(\rho(t,\gamma(t,s)))\rrbracket\left(<\operatorname{div}u>-<\mathbb{D}^{jk}u>n^{j}_{x}n^{k}_{x}\right)(t,\gamma(t,s))$

where we have used the expression of the jump of the effective flux given in
(1.16). Above, the function $g$ and $h$ are defined as:

$g(t,s):=\dfrac{\llbracket P(\rho(t,\gamma(t,s)))\rrbracket}{\llbracket
f(\rho(t,\gamma(t,s)))\rrbracket}\quad\text{ and }\quad
h(t,s):=\dfrac{\llbracket\mu(\rho(t,\gamma(t,s)))\rrbracket}{\llbracket
f(\rho(t,\gamma(t,s)))\rrbracket}.$

To achieve an exponential time decay of the jump of $f(\rho)$ and subsequently
deduce the exponential time decay of the pressure jump, we require that $g$ is
lower and upper bounded, and $h$ is just bounded, that is, for all
$0<\underline{\rho}\leqslant\rho,\,\rho^{\prime}\leqslant\overline{\rho}$:

(3.20) $\displaystyle
0<\underline{\text{\textnu}}\leqslant\dfrac{P(\rho)-P(\rho^{\prime})}{f(\rho)-f(\rho^{\prime})}\leqslant\overline{\text{\textnu}}\quad\text{and}\quad\bigg{|}\dfrac{\mu(\rho)-\mu(\rho^{\prime})}{f(\rho)-f(\rho^{\prime})}\bigg{|}\leqslant\overline{\text{\textnu}}$

for some constant $0<\underline{\text{\textnu}}<\overline{\text{\textnu}}$
that depend on the lower and upper bound of the density. We observe that when
the pressure law and viscosity laws are proportional, the condition (3.20)
obviously holds. In fact the above condition states somehow that the
difference of the pressure, of the dynamic viscosity $\mu$ and of $f(\rho)$
vanish at the same state to which the density converges exponentially. The
first condition in (3.20) naturally holds due to the fact that the pressure
and $f(\rho)$ are increasing functions of $\rho$ whereas the second inequality
(3.20) holds as soon as the viscosity is an increasing function of $\rho$. The
same condition holds also for the kinematic viscosity $\lambda$. It then
follows that

$\displaystyle\dfrac{d}{dt}\left\\{e^{\int_{0}^{t}g(\tau,s)d\tau}\llbracket
f(\rho(t,\gamma(t,s)))\rrbracket\right\\}=-2h(t,s)e^{\int_{0}^{t}g(\tau,s)d\tau}\llbracket
f(\rho(t,\gamma(t,s)))\rrbracket\left(<\operatorname{div}u>-<\mathbb{D}^{jk}u>n^{j}_{x}n^{k}_{x}\right)(t,\gamma(t,s))$

and consequently:

$\lvert\llbracket
f(\rho(t,\gamma(t,s)))\rrbracket\rvert\leqslant\lvert\llbracket
f(\rho_{0}(\gamma_{0}(s)))\rrbracket\rvert\exp\left(-\underline{\text{\textnu}}t+4\overline{\text{\textnu}}\int_{0}^{t}\lVert\nabla
u(\tau)\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau\right).$

Thus, the $L^{\infty}(\mathcal{C}(t))$ norm of the jump of $f(\rho)$ is
bounded as follows:

$\lVert\llbracket
f(\rho(t))\rrbracket\rVert_{L^{\infty}(\mathcal{C}(t))}\leqslant\lVert\llbracket
f(\rho_{0},)\rrbracket\rVert_{L^{\infty}(\mathcal{C}(0))}\exp\left(-\underline{\text{\textnu}}t+4\overline{\text{\textnu}}\int_{0}^{t}\lVert\nabla
u(\tau)\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau\right).$

Moreover, for all $1\leqslant p\leqslant\infty$, one has the following
$L^{p}(\mathcal{C}(t))$ norm of $\llbracket f(\rho(t))\rrbracket$:

(3.21) $\displaystyle\lVert\llbracket
f(\rho(t))\rrbracket\rVert_{L^{p}(\mathcal{C}(t))}\leqslant\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{p}(\mathcal{C}(0))}\exp\left(-\underline{\text{\textnu}}t+(4\overline{\text{\textnu}}+1/p)\int_{0}^{t}\lVert\nabla
u(\tau)\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau\right).$

From the condition (3.20), a similar estimate holds for the jump of the
viscosity $\mu(\rho)$ and the pressure $P(\rho)$. Since the kinematic
viscosity $\lambda$ fulfills the same condition as the dynamic viscosity $\mu$
in (3.20) a similar estimate also holds for the jump of $\lambda(\rho)$. Next,
we go back to (1.16) and (1.17) in order to express the jump of
$\operatorname{div}u$ and $\operatorname{curl}u$ as follows:

$<2\mu(\rho)+\lambda(\rho)>\llbracket\operatorname{div}u\rrbracket=\llbracket
P(\rho)\rrbracket-\llbracket\lambda(\rho)\rrbracket<\operatorname{div}u>-2\llbracket\mu(\rho)\rrbracket<\mathbb{D}^{jk}u>n_{x}^{j}n_{x}^{k},$
$<\mu(\rho)>\llbracket\operatorname{curl}u\rrbracket=-2\llbracket\mu(\rho)\rrbracket<\mathbb{D}^{jk}u>n_{x}^{k}\tau_{x}^{j}.$

From these expressions and the $L^{p}(\mathcal{C}(t))$ estimate (3.21), we
deduce the $L^{p}(\mathcal{C}(t))$ norm of the jump of $\operatorname{div}u$
and $\operatorname{curl}u$ before deriving the following estimate for the jump
of the velocity gradient using (1.13) and (1.14):

(3.22) $\lVert\llbracket\nabla
u(t)\rrbracket\rVert_{L^{p}(\mathcal{C}(t))}\leqslant M\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{p}(\mathcal{C}(0))}\left(1+\lVert\nabla
u(t)\rVert_{L^{\infty}(\mathbb{R}^{2})}\right)\\\
\exp\left(-\underline{\text{\textnu}}t+(4\overline{\text{\textnu}}+1/p)\int_{0}^{t}\lVert\nabla
u(\tau)\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau\right),\;p\in[1,\infty].$

Above the constant $M$ depends only on the lower and upper bounds of the
density and viscosity. This ends the proof of 2.5. ∎

###  3.3. Proof of 2.6

###### .

Let us recall that the functional $\mathcal{A}_{3}$ naturally comes while
testing the equation (2.13) with $\sigma^{2}\ddot{u}$. By doing so, we obtain
the following:

$\displaystyle\int_{0}^{t}\sigma^{2}\lVert\sqrt{\rho}\ddot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}$
$\displaystyle+\sigma^{2}(t)\int_{\mathbb{R}^{2}}\left\\{\mu(\rho)\lvert\mathbb{D}\dot{u}\rvert^{2}+\dfrac{\lambda(\rho)}{2}\lvert\operatorname{div}\dot{u}\rvert^{2}\right\\}=2\int_{0}^{\sigma(t)}\sigma\int_{\mathbb{R}^{2}}\left\\{\mu(\rho)\lvert\mathbb{D}\dot{u}\rvert^{2}+\dfrac{\lambda(\rho)}{2}\lvert\operatorname{div}\dot{u}\rvert^{2}\right\\}$
$\displaystyle-\sigma^{2}\int_{\mathbb{R}^{2}}\rho
P^{\prime}(\rho)\operatorname{div}\dot{u}\operatorname{div}u+\int_{0}^{\sigma(t)}\sigma\int_{\mathbb{R}^{2}}\rho
P^{\prime}(\rho)\operatorname{div}\dot{u}\operatorname{div}u+\int_{0}^{t}\sigma^{2}\int_{\mathbb{R}^{2}}\rho
P^{\prime}(\rho)(\operatorname{div}\dot{u})^{2}$
$\displaystyle+\int_{0}^{t}\sigma^{2}\int_{\mathbb{R}^{2}}\operatorname{div}\dot{u}(\operatorname{div}u)^{2}(\rho
P^{\prime}(\rho)-P(\rho)+\widetilde{P})-\int_{0}^{t}\sigma^{2}\int_{\mathbb{R}^{2}}\partial_{l}\dot{u}^{j}\operatorname{div}u\partial_{j}u^{l}(\rho
P^{\prime}(\rho)-P(\rho)+\widetilde{P})$ (3.23)
$\displaystyle+\sigma^{2}(t)I_{1}(t)-2\int_{0}^{\sigma(t)}\sigma
I_{1}(s)ds+\int_{0}^{t}\sigma^{2}I_{2}(s)ds+\int_{0}^{t}\sigma^{2}I_{3}(s)ds$

where terms $I_{1},\;I_{2},\;I_{3}$ are of the type

(3.24) $\displaystyle
I_{1}=\int_{\mathbb{R}^{2}}\varphi(\rho)\partial_{j_{1}}\dot{u}^{j_{2}}\partial_{j_{3}}u^{j_{4}}\partial_{j_{5}}u^{j_{6}}+\int_{\mathbb{R}^{2}}\partial_{j_{1}}\dot{u}^{j_{2}}\partial_{j_{3}}u^{j_{4}}(P(\rho)-\widetilde{P}),$
(3.25) $\displaystyle
I_{2}=\int_{\mathbb{R}^{2}}\varphi(\rho)\partial_{j_{1}}\dot{u}^{j_{2}}\partial_{j_{3}}\dot{u}^{j_{4}}\partial_{j_{5}}u^{j_{6}}+\int_{\mathbb{R}^{2}}\partial_{j_{1}}\dot{u}^{j_{2}}\partial_{j_{3}}\dot{u}^{j_{4}}(P(\rho)-\widetilde{P})+\int_{\mathbb{R}^{2}}\psi(\rho)\partial_{j_{1}}\dot{u}^{j_{2}}\partial_{j_{3}}u^{j_{4}}\partial_{j_{5}}\dot{u}^{j_{6}},$
(3.26) $\displaystyle
I_{3}=\int_{\mathbb{R}^{2}}\varphi(\rho)\partial_{j_{1}}\dot{u}^{j_{2}}\partial_{j_{3}}u^{j_{4}}\partial_{j_{5}}u^{j_{6}}\partial_{j_{7}}u^{j_{8}}+\int_{\mathbb{R}^{2}}\partial_{j_{1}}\dot{u}^{j_{2}}\partial_{j_{3}}u^{j_{4}}\partial_{j_{5}}u^{j_{6}}(P(\rho)-\widetilde{P}),$

and where $\varphi$ is either the viscosity $\mu$ or $\lambda$ or
$\rho\mu^{\prime}$ , $\rho\lambda^{\prime}$, $\rho^{2}\mu^{\prime\prime}$ or
$\rho^{2}\lambda^{\prime\prime}$ whereas $\psi$ is either $\rho P^{\prime}$ or
$\rho^{2}P^{\prime\prime}$. The computations leading to (3.23) can be found in
section B.2 and use the fact that the local solution $(\rho,u)$ given by 2.1
enjoys the following Rankine Hugoniot condition at the interface (see section
B.1):

$\llbracket\dot{\Pi}^{jk}\rrbracket
n^{k}_{x}+\llbracket\Pi^{jk}\operatorname{div}u\rrbracket
n^{k}_{x}-\llbracket\partial_{k}u^{l}\Pi^{jk}\rrbracket n_{x}^{l}=0$

and the fact that the second derivative of the velocity, $\ddot{u}$, is
continuous at the interface, see 2.2. In the following, we will estimate the
terms appearing in the left hand side of (3.23) in term of $C_{0}$ and
functional $\mathcal{A}_{1}$, $\mathcal{A}_{2}$ and $\mathcal{A}_{3}$. It is
clear that the first term on the right hand side of (3.23) is controlled by
the second Hoff functional:

$2\int_{0}^{\sigma(t)}\sigma\int_{\mathbb{R}^{2}}\left\\{\mu(\rho)\lvert\mathbb{D}\dot{u}\rvert^{2}+\dfrac{\lambda(\rho)}{2}\lvert\operatorname{div}\dot{u}\rvert^{2}\right\\}\leqslant\mathcal{A}_{2}(\sigma(t)),$

whereas the second term on the right hand side can be estimated as follows:

$\sigma^{2}(t)\bigg{|}\int_{\mathbb{R}^{2}}\rho
P^{\prime}(\rho)\operatorname{div}\dot{u}\operatorname{div}u\bigg{|}\leqslant\eta\mathcal{A}_{3}(t)+C_{\eta}\mathcal{A}_{1}(t)$

for some positive $\eta$ small in order to absorb the first term in the right
hand side of (3.23). Next, the third term of the right hand side of (3.23) can
be bounded by the help of the initial energy and the second Hoff functional:

$\bigg{|}\int_{0}^{\sigma(t)}\sigma\int_{\mathbb{R}^{2}}\rho
P^{\prime}(\rho)\operatorname{div}\dot{u}\operatorname{div}u\bigg{|}\lesssim
E_{0}^{1/2}\mathcal{A}_{2}(t)^{1/2},$

and it is obvious that the fifth term is controlled by the second functional:

$\bigg{|}\int_{0}^{t}\sigma^{2}\int_{\mathbb{R}^{2}}\rho
P^{\prime}(\rho)(\operatorname{div}\dot{u})^{2}\bigg{|}\lesssim\mathcal{A}_{2}(t).$

We observe the following bound for the sixth and seventh term of the right
hand side of (3.23)

$\displaystyle\bigg{|}\int_{0}^{t}\sigma^{2}\int_{\mathbb{R}^{2}}\operatorname{div}\dot{u}(\operatorname{div}u)^{2}(\rho
P^{\prime}(\rho)-P(\rho)+\widetilde{P})\bigg{|}$
$\displaystyle+\bigg{|}\int_{0}^{t}\sigma^{2}\int_{\mathbb{R}^{2}}\partial_{l}\dot{u}^{j}\operatorname{div}u\partial_{j}u^{l}(\rho
P^{\prime}(\rho)-P(\rho)+\widetilde{P})\bigg{|}$
$\displaystyle\lesssim\left[\int_{0}^{t}\sigma\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\right]^{1/2}\left[\int_{0}^{t}\sigma\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{2})}^{4}\right]^{1/2}$ (3.27)
$\displaystyle\lesssim\mathcal{A}_{2}(t)^{1/2}\left(C_{0}+\mathcal{A}_{1}(t)\left(C_{0}+\mathcal{A}_{1}(t)\right)\right)^{1/2},$

where we have used the estimate (3.14) to control the
$L^{4}((0,T)\times\mathbb{R}^{2})$ norm of the gradient of the velocity. We
now turn to the last four terms of (3.23). The first term among the underlying
terms can be bounded, thanks to Hölder and Young’s inequalities, by:

$\bigg{|}\sigma^{2}(t)I_{1}(t)\bigg{|}\leqslant\eta\sigma^{2}(t)\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+C_{\eta}\sigma^{2}(t)\left(\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{2})}^{4}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{4}(\mathbb{R}^{2})}^{4}\right)$

for some $\eta$ small in order to absorb the first term on the right hand side
in the left hand side of (3.23). On the other hand, owing to the fact that the
$L^{\infty}((0,T),L^{4}(\mathbb{R}^{2}))$ norm of the pressure is controlled
by the classical energy, we deduce from (3.9) the following:

$\displaystyle\sup_{[0,t]}\sigma\left(\lVert\nabla
u\rVert^{4}_{L^{4}(\mathbb{R}^{2})}+\lVert
P(\rho)-\widetilde{P}\rVert^{4}_{L^{4}(\mathbb{R}^{2})}\right)$
$\displaystyle\lesssim
C_{0}+\sup_{[0,t]}\sigma\left(\lVert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rVert_{L^{4}(\mathbb{R}^{2})}^{4}+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\dot{u})\rVert_{L^{4}(\mathbb{R}^{2})}^{4}\right)$
$\displaystyle\lesssim
C_{0}+\sup_{[0,t]}\sigma\lVert\rho\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\left(\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\right)$ (3.28)
$\displaystyle\lesssim
C_{0}+\mathcal{A}_{2}(t)\left(C_{0}+\mathcal{A}_{1}(t)\right).$

Thus, we have

(3.29) $\displaystyle\sup_{[0,t]}\lvert\sigma
I_{1}\rvert\leqslant\eta\mathcal{A}_{3}(t)+C_{\eta}\left(C_{0}+\mathcal{A}_{2}(t)\left(C_{0}+\mathcal{A}_{1}(t)\right)\right).$

Following (3.27), the second term can be bounded by:

$\bigg{|}\int_{0}^{\sigma(t)}sI_{1}(s)ds\bigg{|}\lesssim\mathcal{A}_{2}(t)^{1/2}\left(C_{0}+\mathcal{A}_{1}(t)\left(C_{0}+\mathcal{A}_{1}(t)\right)\right)^{1/2}.$

Next, the penultimate term of (3.23) is, thanks to Hölder and interpolation
inequalities:

$\displaystyle\bigg{|}\int_{0}^{t}\sigma^{2}I_{2}(s)ds\bigg{|}$
$\displaystyle\lesssim\int_{0}^{t}\sigma^{2}\lVert\nabla\dot{u}\rVert_{L^{8/3}(\mathbb{R}^{2})}^{2}\left(\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{2})}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{4}(\mathbb{R}^{2})}\right)$
$\displaystyle\lesssim\int_{0}^{t}\sigma^{2}\lVert\nabla\dot{u}\rVert_{L^{3}(\mathbb{R}^{2})}^{3/2}\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{1/2}\left(\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{2})}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{4}(\mathbb{R}^{2})}\right)$
$\displaystyle\lesssim\left[\int_{0}^{t}\sigma\left(\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{2})}^{4}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{4}(\mathbb{R}^{2})}^{4}\right)\right]^{1/4}\left[\int_{0}^{t}\sigma\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\right]^{1/4}\left[\int_{0}^{t}\sigma^{3}\lVert\nabla\dot{u}\rVert_{L^{3}(\mathbb{R}^{2})}^{3}\right]^{1/2}$
$\displaystyle\lesssim\mathcal{A}_{2}(t)^{1/4}\left(C_{0}+\mathcal{A}_{1}(t)\left(C_{0}+\mathcal{A}_{1}(t)\right)\right)^{1/4}\left[\int_{0}^{t}\sigma^{3}\lVert\nabla\dot{u}\rVert_{L^{3}(\mathbb{R}^{2})}^{3}\right]^{1/2}$

where we have used the inequality (3.9) to control the
$L^{4}((0,T)\times\mathbb{R}^{2})$ norm of the pressure and of the velocity
gradient. Finally, with the help of the Hölder inequality, the last term of
equation (3.23) can be bounded as follows:

$\displaystyle\bigg{|}\int_{0}^{t}\sigma^{2}I_{3}(s)ds\bigg{|}$
$\displaystyle\lesssim\int_{0}^{t}\sigma^{2}\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}\left(\lVert\nabla
u\rVert_{L^{6}(\mathbb{R}^{2})}^{3}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{6}(\mathbb{R}^{2})}^{3}\right)$
$\displaystyle\lesssim\left[\int_{0}^{t}\sigma\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\right]^{1/2}\left[\int_{0}^{t}\sigma^{3}\left(\lVert\nabla
u\rVert_{L^{6}(\mathbb{R}^{2})}^{6}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{6}(\mathbb{R}^{2})}^{6}\right)\right]^{1/2}$
$\displaystyle\lesssim\mathcal{A}_{2}(t)^{1/2}\left[\int_{0}^{t}\sigma^{3}\left(\lVert\nabla
u\rVert_{L^{6}(\mathbb{R}^{2})}^{6}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{6}(\mathbb{R}^{2})}^{6}\right)\right]^{1/2}.$

In order to control the $L^{6}((0,T)\times\mathbb{R}^{2})$ norm of the
pressure and of the velocity gradient, we will assume that the smallness
condition for the viscosity (3.11) holds for $l=5$, multiply (3.12) by
$\sigma$ and integrate in time and finally obtain, after considering (3.9):

$\displaystyle\int_{0}^{t}\sigma\left(\lVert\nabla
u\rVert_{L^{6}(\mathbb{R}^{2})}^{6}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{6}(\mathbb{R}^{2})}^{6}\right)$
$\displaystyle\lesssim
C_{0}+\int_{0}^{t}\sigma\left(\lVert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rVert_{L^{6}(\mathbb{R}^{2})}^{6}+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\dot{u})\rVert_{L^{6}(\mathbb{R}^{2})}^{6}\right)$
$\displaystyle\lesssim
C_{0}+\int_{0}^{t}\sigma\lVert\rho\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{4}\left(\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\right)$ (3.30)
$\displaystyle\lesssim
C_{0}+\mathcal{A}_{1}(t)\mathcal{A}_{2}(t)\left(\mathcal{A}_{1}(t)+C_{0}\right).$

In sum, we estimate all terms in the right hand side of (3.23) in term of
$C_{0}$, functional $\mathcal{A}_{1}$, $\mathcal{A}_{2}$, $\mathcal{A}_{3}$
and the following term

$\int_{0}^{t}\sigma^{3}\lVert\nabla\dot{u}\rVert_{L^{3}(\mathbb{R}^{2})}^{3},$

that we will be analyzed in the following lines. The idea is similar to what
has be done previously to estimate the $L^{3}((0,T)\times\mathbb{R}^{2})$ norm
of $\nabla u$ and $L^{4}((0,T)\times\mathbb{R}^{2}))$ norm of
$\sigma^{1/4}\nabla u$. We begin by recalling that the equation (2.13) can be
written as follows:

(3.31)
$\displaystyle\rho\ddot{u}^{j}=\partial_{k}(\dot{\Pi}^{jk})+\partial_{k}(\Pi^{jk}\operatorname{div}u)-\operatorname{div}(\partial_{k}u\Pi^{jk})$

and we apply the divergence operator in order to observe that $F_{*}$ defined
by

$F_{*}:=(2\mu(\rho)+\lambda(\rho))\operatorname{div}\dot{u}-\lambda(\rho)\nabla
u^{l}\partial_{l}u-\rho\lambda^{\prime}(\rho)(\operatorname{div}u)^{2}+\rho
P^{\prime}(\rho)\operatorname{div}u$

can be expressed as follows:

(3.32)
$F_{*}=-(-\Delta)^{-1}\operatorname{div}(\rho\ddot{u})-(-\Delta)^{-1}\partial_{jk}\\{\mu(\rho)\partial_{j}u^{l}\partial_{l}u^{k}+\mu(\rho)\partial_{k}u^{l}\partial_{l}u^{j}+2\rho\mu^{\prime}(\rho)\mathbb{D}^{jk}u\operatorname{div}u\\}\\\
+(-\Delta)^{-1}\partial_{jk}(\Pi^{jk}\operatorname{div}u)-(-\Delta)^{-1}\partial_{j}\operatorname{div}(\partial_{k}u\Pi^{jk})+K\\{(\mu(\rho)-\widetilde{\mu})\mathbb{D}\dot{u}\\}.$

On the other hand, we can apply the curl to the equation (3.31) and obtain
that the curl of the material derivative $\dot{u}$ can be expressed as:

(3.33)
$\mu(\rho)\operatorname{curl}\dot{u}=-(-\Delta)^{-1}\operatorname{curl}(\rho\ddot{u})+K^{\prime}\\{(\mu(\rho)-\widetilde{\mu})\mathbb{D}\dot{u}\\}+(-\Delta)^{-1}\operatorname{curl}\partial_{k}(\Pi^{jk}\operatorname{div}u)\\\
-(-\Delta)^{-1}\operatorname{curl}\operatorname{div}(\partial_{k}u\Pi^{jk})-(-\Delta)^{-1}\operatorname{curl}\partial_{k}\\{\mu(\rho)\partial_{j}u^{l}\partial_{l}u^{k}+\mu(\rho)\partial_{k}u^{l}\partial_{l}u^{j}+2\rho\mu^{\prime}(\rho)\mathbb{D}^{jk}u\operatorname{div}u\\}.$

Since the smallness condition (3.9) holds true for $p=3$, one deduces from
singular operators theory that:

$\lVert\nabla\dot{u}\rVert_{L^{3}(\mathbb{R}^{2})}\lesssim\lVert(-\Delta)^{-1}\operatorname{div}(\rho\ddot{u})\rVert_{L^{3}(\mathbb{R}^{2})}+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\ddot{u})\rVert_{L^{3}(\mathbb{R}^{2})}+\lVert\nabla
u\rVert_{L^{6}(\mathbb{R}^{2})}^{2}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{6}(\mathbb{R}^{2})}^{2}+\lVert\nabla
u\rVert_{L^{3}(\mathbb{R}^{2})}.$

It then turns out that the aforementioned term is bounded as follows:

(3.34)
$\int_{0}^{t}\sigma^{5/2}\lVert\nabla\dot{u}\rVert_{L^{3}(\mathbb{R}^{2})}^{3}\lesssim\int_{0}^{t}\sigma^{5/2}\left(\lVert(-\Delta)^{-1}\operatorname{div}(\rho\ddot{u})\rVert_{L^{3}(\mathbb{R}^{2})}^{3}+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\ddot{u})\rVert_{L^{3}(\mathbb{R}^{2})}^{3}\right)\\\
+\int_{0}^{t}\sigma^{5/2}\lVert\nabla
u\rVert_{L^{3}(\mathbb{R}^{2})}^{3}+\int_{0}^{t}\sigma^{5/2}\left(\lVert\nabla
u\rVert_{L^{6}(\mathbb{R}^{2})}^{6}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{6}(\mathbb{R}^{2})}^{6}\right).$

Let us notice that the last two terms of the above inequality are controlled
by (3.30) for the last term, and (3.15), (3.16) for the previous one. The
first two terms on the right hand side of (3.34) can be bounded using the
Gagliardo-Nirenberg inequality:

$\displaystyle\int_{0}^{t}\sigma^{5/2}\left(\right.$
$\displaystyle\left.\lVert(-\Delta)^{-1}\operatorname{div}(\rho\ddot{u})\rVert_{L^{3}(\mathbb{R}^{2})}^{3}+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\ddot{u})\rVert_{L^{3}(\mathbb{R}^{2})}^{3}\right)$
$\displaystyle\lesssim\int_{0}^{t}\sigma^{5/2}\lVert\rho\ddot{u}\rVert_{L^{2}(\mathbb{R}^{2})}\left(\lVert(-\Delta)^{-1}\operatorname{div}(\rho\ddot{u})\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\lVert(-\Delta)^{-1}\operatorname{div}(\rho\ddot{u})\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\right)$
$\displaystyle\lesssim\int_{0}^{t}\sigma^{5/2}\lVert\rho\ddot{u}\rVert_{L^{2}(\mathbb{R}^{2})}\left(\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{2})}^{4}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{4}(\mathbb{R}^{2})}^{4}+\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\right).$

Above we have used (3.32) and (3.33) in order to express

$\lVert(-\Delta)^{-1}\operatorname{div}(\rho\ddot{u})\rVert_{L^{2}(\mathbb{R}^{2})}\quad\text{
and
}\quad\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\ddot{u})\rVert_{L^{2}(\mathbb{R}^{2})}.$

Finally, we have:

$\displaystyle\int_{0}^{t}\sigma^{5/2}$
$\displaystyle\left(\lVert(-\Delta)^{-1}\operatorname{div}(\rho\ddot{u})\rVert_{L^{3}(\mathbb{R}^{2})}^{3}+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\ddot{u})\rVert_{L^{3}(\mathbb{R}^{2})}^{3}\right)$
$\displaystyle\lesssim\left[\int_{0}^{t}\sigma^{2}\lVert\sqrt{\rho}\ddot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\right]^{1/2}\left[\int_{0}^{t}\sigma^{3}\left(\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{4}+\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{2})}^{8}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{4}(\mathbb{R}^{2})}^{8}+\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{2})}^{4}\right)\right]^{1/2}$
$\displaystyle\lesssim\mathcal{A}_{3}(t)\mathcal{A}_{2}(t)^{1/2}+C_{0}^{1/2}\mathcal{A}_{1}(t)^{1/2}\mathcal{A}_{3}(t)^{1/2}$
$\displaystyle+\mathcal{A}_{3}(t)^{1/2}\sup_{[0,t]}\sigma^{1/2}\left(\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{2})}^{2}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{4}(\mathbb{R}^{2})}^{2}\right)\left[\int_{0}^{t}\sigma\left(\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{2})}^{4}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{4}(\mathbb{R}^{2})}^{4}\right)\right]^{1/2}.$

The norm of the pressure and the velocity gradient are controlled in (3.14)
and (3.28). The estimate of the functional $\mathcal{A}_{3}$ is then achieved.
To obtain regularity about higher-order material derivatives of velocity, one
can follow the procedure as before by applying the material derivative
$\partial_{t}\,\cdot+\operatorname{div}(\,\cdot\,u)$ to (3.31) and so on. It
is clear that the computations are quite lengthy, but they follow the same
principle. ∎

###  3.4. Proof of 2.8

###### .

The proof of 2.8 is divided in two parts: in the first one we obtain estimates
for the functional  and the second part is devoted to the estimates for  and
for the effective flux $F$. We begin the first part of the proof by noticing
that from the fact that the pressure and $f$ are increasing functions of the
density on $[\underline{\rho},\overline{\rho}]$ and the fact that the
viscosity $2\mu(\rho)+\lambda(\rho)$ is bounded away from vacuum, we have the
following:

(3.35)
$\displaystyle\forall\;\underline{\rho}\leqslant\rho\leqslant\overline{\rho},\quad\quad
0<\underline{\text{\textnu}}\leqslant\dfrac{P(\rho)-\widetilde{P}}{f(\rho)}\leqslant\overline{\text{\textnu}}.$

The constants $\underline{\text{\textnu}},\;\overline{\text{\textnu}}$ depend
on the lower and upper bounds of the density and viscosity. In fact, (3.35)
can be obtained, for example, with the help of the Cauchy mean value theorem.
We deduce from (3.18) the following:

$f(\rho(\tau,X(\tau,x))=f(\rho_{0}(x))e^{-\int_{0}^{\tau}g_{1}(\tau^{\prime},x)d\tau^{\prime}}-\int_{0}^{\tau}e^{-\int_{\tau^{\prime}}^{\tau}g_{1}(\tau^{\prime\prime},x)d\tau^{\prime\prime}}F(\tau^{\prime},X(\tau^{\prime},x))d\tau^{\prime}$

where the function $g_{1}$ is:

$g_{1}(t,x):=\dfrac{P(\rho(t,X(t,x)))-\widetilde{P}}{f(\rho(t,X(t,x)))}\in[\underline{\text{\textnu}},\,\overline{\text{\textnu}}].$

Thus, we have:

(3.36) $\displaystyle\lVert
f(\rho(\tau))\rVert_{L^{\infty}(\mathbb{R}^{2})}\leqslant
e^{-\underline{\text{\textnu}}\,\tau}\lVert
f(\rho_{0})\rVert_{L^{\infty}(\mathbb{R}^{2})}+\int_{0}^{\tau}e^{\underline{\text{\textnu}}\,(\tau^{\prime}-\tau)}\lVert
F(\tau^{\prime})\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau^{\prime},$

from which we deduce:

(3.37) $\displaystyle\sup_{[0,t]}\lVert
f(\rho)\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}\leqslant\lVert
f(\rho_{0})\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}+M\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}$

for all $0<r_{\alpha}<3$ with $M$ depending on $r_{\alpha}$ and
$\underline{\text{\textnu}}$. To obtain the above estimate, we express the
last term of (3.36) into two terms as follows:

$\int_{0}^{\tau}e^{\underline{\text{\textnu}}\,(\tau^{\prime}-\tau)}\lVert
F(\tau^{\prime})\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau^{\prime}=\int_{0}^{\tau}e^{\underline{\text{\textnu}}\,(\tau^{\prime}-\tau)}\lVert
F(\tau^{\prime})\rVert_{L^{\infty}(\mathbb{R}^{2})}\mathbb{1}_{\\{\tau^{\prime}\leqslant
1\\}}d\tau^{\prime}+\int_{0}^{\tau}e^{\underline{\text{\textnu}}\,(\tau^{\prime}-\tau)}\lVert
F(\tau^{\prime})\rVert_{L^{\infty}(\mathbb{R}^{2})}\mathbb{1}_{\\{\tau^{\prime}>1\\}}d\tau^{\prime}.$

Thus by the help of Hölder’s inequality we have the following:

$\int_{0}^{\tau}e^{\underline{\text{\textnu}}\,(\tau^{\prime}-\tau)}\lVert
F(\tau^{\prime})\rVert_{L^{\infty}(\mathbb{R}^{2})}\mathbb{1}_{\\{\tau^{\prime}\leqslant
1\\}}d\tau^{\prime}\leqslant\left(\dfrac{3}{3-r_{\alpha}}\right)^{3/4}\lVert\sigma^{r_{\alpha}/4}F\rVert_{L^{4}((0,\sigma(t)),L^{\infty}(\mathbb{R}^{2}))},$

and

$\int_{0}^{\tau}e^{\underline{\text{\textnu}}\,(\tau^{\prime}-\tau)}\lVert
F(\tau^{\prime})\rVert_{L^{\infty}(\mathbb{R}^{2})}\mathbb{1}_{\\{\tau^{\prime}>1\\}}d\tau^{\prime}\leqslant\left(\dfrac{3}{4\underline{\text{\textnu}}}\right)^{3/4}\lVert
F\rVert_{L^{4}((\sigma(t),t),L^{\infty}(\mathbb{R}^{2}))}.$

Moreover, by Young’s inequality for convolution, one has:

$\bigg{\|}\int_{0}^{\tau}e^{\underline{\text{\textnu}}\,(\tau^{\prime}-\tau)}\lVert
F(\tau^{\prime})\rVert_{L^{\infty}(\mathbb{R}^{2})}\mathbb{1}_{\\{\tau^{\prime}\leqslant
1\\}}d\tau^{\prime}\bigg{\|}_{L^{4}(0,t)}\leqslant\left(\dfrac{1}{4\underline{\text{\textnu}}}\right)^{1/4}\left(\dfrac{3}{3-r_{\alpha}}\right)^{3/4}\lVert\sigma^{r_{\alpha}/4}F\rVert_{L^{4}((0,\sigma(t)),L^{\infty}(\mathbb{R}^{2}))},$

and

$\bigg{\|}\int_{0}^{\tau}e^{\underline{\text{\textnu}}\,(\tau^{\prime}-\tau)}\lVert
F(\tau^{\prime})\rVert_{L^{\infty}(\mathbb{R}^{2})}\mathbb{1}_{\\{\tau^{\prime}>1\\}}d\tau^{\prime}\bigg{\|}_{L^{4}(0,t)}\leqslant\dfrac{1}{\underline{\text{\textnu}}}\lVert
F\rVert_{L^{4}((\sigma(t),t),L^{\infty}(\mathbb{R}^{2}))}.$

Thus, we have the following:

(3.38) $\displaystyle\int_{0}^{t}\lVert
f(\rho(\tau))\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}d\tau\lesssim\lVert
f(\rho_{0})\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F(\tau)\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}d\tau.$

We turn to the estimate of the
$\dot{\mathscr{C}}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2})$ norm for the
$f(\rho)$. Let $x_{i}^{0}$, $i\in\\{1,2\\}$ two points lying on the same side
of the interface and let $x_{i}(t):=X(t,x_{i})$. From the mass equation
(3.19), one has the following:

(3.39)
$\displaystyle\dfrac{d}{dt}f(\rho(t,x_{i}(t)))\Bigg{|}_{i=1}^{i=2}+P(\rho(t,x_{i}(t)))\Bigg{|}_{i=1}^{i=2}=-F(t,x_{i}(t))\Bigg{|}_{i=1}^{i=2}.$

Using the fact that

(3.40) $\displaystyle
g_{2}(t,x,y):=\dfrac{P(\rho(t,x))-P(\rho(t,y))}{f(\rho(t,x))-f(\rho(t,y))}\in[\underline{\text{\textnu}},\,\overline{\text{\textnu}}]$

we have from (3.39):

(3.41) $\displaystyle
f(\rho(\tau,x_{i}(\tau)))\bigg{|}_{i=1}^{i=2}=f(\rho_{0}(x_{i}^{0}))\bigg{|}_{i=1}^{i=2}e^{-\int_{0}^{\tau}g_{2}(\tau^{\prime},x_{2}(\tau^{\prime}),x_{1}(\tau^{\prime}))d\tau^{\prime}}-\int_{0}^{\tau}e^{-\int_{\tau^{\prime}}^{\tau}g_{2}(\tau^{\prime\prime},x_{2}(\tau^{\prime\prime}),x_{1}(\tau^{\prime\prime}))d\tau^{\prime\prime}}F(\tau^{\prime},x_{i}(\tau^{\prime}))\bigg{|}_{i=1}^{i=2}d\tau^{\prime}.$

On the other hand, we have for all $0\leqslant\tau^{\prime}\leqslant\tau$, the
following:

$\lvert x_{2}(\tau^{\prime})-x_{1}(\tau^{\prime})\rvert\leqslant
e^{\int_{\tau^{\prime}}^{\tau}\lVert\nabla
u(\tau^{\prime\prime})\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau^{\prime\prime}}\lvert
x_{2}(\tau)-x_{1}(\tau)\rvert$

and thus (3.41) implies:

$\displaystyle\lVert
f(\rho(\tau))\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}$
$\displaystyle\leqslant
e^{\int_{0}^{\tau}\left[-g_{2}(\tau^{\prime},x_{2}(\tau^{\prime}),x_{1}(\tau^{\prime}))+\lVert\nabla
u(\tau^{\prime})\rVert_{L^{\infty}(\mathbb{R}^{2})}\right]d\tau^{\prime}}\lVert
f(\rho_{0})\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma_{0}}(\mathbb{R}^{2})}$
(3.42)
$\displaystyle+\int_{0}^{\tau}e^{\int_{\tau^{\prime}}^{\tau}\left[-g_{2}(\tau^{\prime\prime},x_{2}(\tau^{\prime\prime}),x_{1}(\tau^{\prime\prime}))+\lVert\nabla
u(\tau^{\prime\prime})\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau^{\prime\prime}\right]}\lVert
F(\tau^{\prime})\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}d\tau^{\prime}.$

Also, we have for all $0<r_{\alpha}<3$, the following:

(3.43) $\displaystyle\int_{\tau^{\prime}}^{\tau}\lVert\nabla
u(\tau^{\prime\prime})\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau^{\prime\prime})}(\mathbb{R}^{2})}d\tau^{\prime\prime}\leqslant\varepsilon(\tau-\tau^{\prime})+C_{\varepsilon}\text{\texttheta}(t)$

for $\varepsilon>0$ small and a constant $C_{\varepsilon}$ depending on
$\varepsilon$ and $r_{\alpha}$. Here we choose
$\varepsilon=\dfrac{\underline{\text{\textnu}}}{4}$ and we deduce from the
lower bound of $g_{2}$ in (3.40), (3.42) and (3.43) the following estimates
for $f(\rho)$:

(3.44) $\displaystyle\sup_{[0,t]}\lVert
f(\rho)\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2})}^{4}+\int_{0}^{t}\lVert
f(\rho(\tau))\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\leqslant
M^{\prime}e^{M^{\prime}\text{\texttheta}(t)}\left(\lVert
f(\rho_{0})\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma_{0}}(\mathbb{R}^{2})}^{4}+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F(\tau)\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\right).$

Finally, by gathering (3.37), (3.38) and (3.44) we obtain the estimates for
the functional  and this closes the first part of the proof of 2.8. The second
part concerns the estimates for the functional  and the Hölder norm of the
effective flux. For this purpose, we recall that from the momentum equation,
we have the following expression for the velocity gradient:

$\displaystyle\widetilde{\mu}\Delta u^{j}$
$\displaystyle=\rho\dot{u}-\partial_{j}\left((\widetilde{\mu}+\lambda(\rho))\operatorname{div}u-P(\rho)+\widetilde{P}\right)-\partial_{k}\\{(2\mu(\rho)-\widetilde{\mu})\mathbb{D}^{jk}u\\}$
(3.45)
$\displaystyle=\rho\dot{u}-\partial_{j}\left(\dfrac{\widetilde{\mu}+\lambda(\rho)}{2\mu(\rho)+\lambda(\rho)}F\right)+\partial_{j}\left(\dfrac{2\mu(\rho)-\widetilde{\mu}}{2\mu(\rho)+\lambda(\rho)}(P(\rho)-\widetilde{P})\right)-\partial_{k}\\{2(\mu(\rho)-\widetilde{\mu})\mathbb{D}^{jk}u\\}.$

Thus, the velocity gradient can be expressed as the sum of four terms:

$\nabla u=\nabla u_{*}+\nabla u_{F}+\nabla u_{P}+\nabla u_{\delta}$

and consequently, we have:

$\displaystyle\text{\texttheta}(t)$
$\displaystyle\leqslant\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u_{*}(\tau)\rVert_{\mathscr{C}^{\alpha}(\mathbb{R}^{2})}^{4}d\tau+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u_{P}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau$
(3.46) $\displaystyle+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u_{F}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u_{\delta}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau.$

Notice that $u_{*}$ is the regular part of the velocity associated to the
first term of the right hand side of (3.45). The gradient of this regular part
is Hölder continuous in the whole $\mathbb{R}^{2}$ with a norm that can be
estimated as follows:

$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u_{*}\rVert_{\mathscr{C}^{\alpha}(\mathbb{R}^{2})}^{4}$
$\displaystyle\lesssim\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\rho\dot{u}\rVert_{L^{6}(\mathbb{R}^{2})}^{12/5}\left(\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{2})}^{8/5}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{2}(\mathbb{R}^{2})}^{8/5}\right)+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{4\alpha}\lVert\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{4(1-\alpha)}$
$\displaystyle\lesssim\left(\mathcal{A}_{1}(t)^{4/5}+E_{0}^{4/5}\right)\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{8/5}\lVert\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{4/5}$
(3.47)
$\displaystyle+\left(\int_{0}^{t}\sigma^{1+(r_{\alpha}-1)/\alpha}\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{4}\right)^{\alpha}\left(\int_{0}^{t}\sigma\lVert\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{4}\right)^{1-\alpha}.$

We then choose $r_{\alpha}\geqslant 7/5$ in order to control the integral in
the first term above by

$\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{8/5}\lVert\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{4/5}\lesssim\mathcal{A}_{2}(t)\mathcal{A}_{1}(t)^{1/5}$

and, in order to control the singularity at initial time for the second term
of (3.47), we choose $r_{\alpha}\geqslant 2\alpha+1$ and then, we obtain:

$\left(\int_{0}^{t}\sigma^{1+(r_{\alpha}-1)/\alpha}\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{4}\right)^{\alpha}\left(\int_{0}^{t}\sigma\lVert\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{4}\right)^{1-\alpha}\lesssim\mathcal{A}_{3}(t)^{\alpha}\mathcal{A}_{2}(t)^{\alpha}\mathcal{A}_{2}(t)^{1-\alpha}\mathcal{A}_{1}(t)^{1-\alpha}.$

Finally, we choose

$r_{\alpha}=\max\\{7/5;\,1+2\alpha\\}<3$

and then, we obtain the following estimate for the regular part of the
velocity:

(3.48) $\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u_{*}(\tau)\rVert_{\mathscr{C}^{\alpha}(\mathbb{R}^{2})}^{4}d\tau\leqslant
M^{\prime}\left[C_{0}^{2}+\mathcal{A}_{1}(t)^{2}+\mathcal{A}_{2}(t)^{2}+\mathcal{A}_{3}(t)^{2}\right].$

The others terms of the expression of the velocity gradient (3.45) are less
regular than $\nabla u_{*}$, in particular, they are discontinuous across the
interface $\gamma$ and we need the quantitative Hölder estimate 2.7 to bound
their piecewise Hölder norms. For this purpose, we will first estimate
$\mathcal{P}(\mathcal{C}(\tau))$. Obviously, we have:

$\begin{cases}\displaystyle c_{\gamma}(\tau)\leqslant
c_{\gamma}(0)\exp\left(\int_{0}^{\tau}\lVert\nabla
u(\tau^{\prime})\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau^{\prime}\right),\;\quad\quad\lvert\mathcal{C}(\tau)\rvert\leqslant\lvert\mathcal{C}(0)\rvert\exp\left(\int_{0}^{\tau}\lVert\nabla
u(\tau^{\prime})\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau^{\prime}\right),\\\
\displaystyle\lVert\nabla\gamma(\tau)\rVert_{\mathscr{C}^{\alpha}}\leqslant\left(\lVert\nabla\gamma_{0}\rVert_{\mathscr{C}^{\alpha}}+\lVert\nabla\gamma_{0}\rVert_{L^{\infty}}^{1+\alpha}\int_{0}^{\tau}\lVert\nabla
u(\tau^{\prime})\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma(\tau^{\prime})}(\mathbb{R}^{2})}d\tau^{\prime}\right)\exp\left((2+\alpha)\int_{0}^{\tau}\lVert\nabla
u(\tau^{\prime})\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau^{\prime}\right).\end{cases}$

Thus, the following estimate holds:

(3.49) $\displaystyle\mathcal{P}(\mathcal{C}(\tau))\leqslant
K_{0}\left(1+\int_{0}^{\tau}\lVert\nabla
u(\tau^{\prime})\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma(\tau^{\prime})}(\mathbb{R}^{2})}d\tau^{\prime}\right)\exp\left(m\int_{0}^{\tau}\lVert\nabla
u(\tau^{\prime})\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau^{\prime}\right)$

with the constant $K_{0}$ depending on the regularity of the initial interface
and $m$ a positive constant that depends on polynomial in the definition of
$\mathcal{P}(\mathcal{C})$ in 2.7.

By applying the quantitative Hölder estimate, the last term of (3.46) can be
bounded as follows:

$\displaystyle\lVert\nabla
u_{\delta}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}$
$\displaystyle\lesssim\lVert\nabla
u(\tau)\rVert_{L^{2}(\mathbb{R}^{2})}+\lVert\llbracket(\mu(\rho(\tau))-\widetilde{\mu})\mathbb{D}u(\tau)\rrbracket\rVert_{L^{2}(\mathcal{C}(\tau))}+\lVert
2(\mu(\rho(\tau))-\widetilde{\mu})\mathbb{D}u\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}$
(3.50)
$\displaystyle+\lVert\llbracket(\mu(\rho(\tau))-\widetilde{\mu})\mathbb{D}u(\tau)\rrbracket\rVert_{L^{\infty}(\mathcal{C}(\tau))}\mathcal{P(\mathcal{C}}(\tau)).$

Obviously, we have

$\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{2})}^{4}\leqslant
M^{\prime}\mathcal{A}_{1}(t)E_{0}$

and on the other hand we consider (3.43) with
$\varepsilon=\dfrac{\underline{\text{\textnu}}}{4(4\overline{\text{\textnu}}+1/2)}$
in order to obtain a decay rate for the $L^{2}$ norm of the jump of $f(\rho)$
(3.21), and for the $L^{2}$ norm of the jump of the velocity gradient (3.22)
from which we deduce the following:

(3.51)
$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\llbracket(\mu(\rho(\tau))-\widetilde{\mu})\mathbb{D}u(\tau)\rrbracket\rVert_{L^{2}(\mathcal{C}(\tau))}^{4}d\tau\leqslant
M^{\prime}\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{2}(\mathcal{C}(0))}^{4}(1+\text{\texttheta}(t))e^{M^{\prime}\text{\texttheta}(t)}.$

In a similar way, the following estimate holds by gathering (3.21), (3.22),
(3.43) and (3.49):

(3.52)
$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\llbracket(\mu(\rho(\tau))-\widetilde{\mu})\mathbb{D}u(\tau)\rrbracket\rVert_{L^{\infty}(\mathcal{C}(\tau))}^{4}\mathcal{P(\mathcal{C}}(\tau))^{4}d\tau\leqslant
M^{\prime}K_{0}\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{\infty}(\mathcal{C}(0))}^{4}\left(1+\text{\texttheta}(t)\right)e^{M^{\prime}\text{\texttheta}(t)}.$

Finally, the penultimate term of (3.50) is bounded by:

(3.53) $\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
2(\mu(\rho(\tau))-\widetilde{\mu})\mathbb{D}u(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\leqslant
M^{\prime}\text{\textbeta}(t)\text{\texttheta}(t).$

All the previous computations give the following estimate:

(3.54) $\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u_{\delta}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\\\
\leqslant
M^{\prime}\left[C_{0}+\mathcal{A}_{1}(t)^{2}+\text{\textbeta}(t)\text{\texttheta}(t)+(1+K_{0})\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{\infty}(\mathcal{C}(0))\cap
L^{2}(\mathcal{C}(0))}^{4}\left(1+\text{\texttheta}(t)\right)e^{M^{\prime}\text{\texttheta}(t)}\right].$

To estimate the third term of the right-hand side of (3.46), we make use of
the quantitative Hölder estimate 2.7 and we obtain:

(3.55) $\displaystyle\lVert\nabla
u_{F}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}\lesssim\lVert\mathscr{F}(\tau)\rVert_{L^{4}(\mathbb{R}^{2})}+\lVert\llbracket\mathscr{F}(\tau)\rrbracket\rVert_{L^{4}(\mathcal{C}(\tau))}+\lVert\mathscr{F}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}+\lVert\llbracket\mathscr{F}(\tau)\rrbracket\rVert_{L^{\infty}(\mathcal{C}(\tau))}\mathcal{P}(\mathcal{C}(\tau))$

with

$\mathscr{F}=\dfrac{\widetilde{\mu}+\lambda(\rho)}{2\mu(\rho)+\lambda(\rho)}F$

and we recall that the effective flux is:

(3.56) $\displaystyle
F=-(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})+K\\{(\mu(\rho)-\widetilde{\mu})\mathbb{D}u\\}.$

Obviously from (3.14) and Gagliardo-Nirenberg inequality, we have:

(3.57)
$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\mathscr{F}\rVert_{L^{4}(\mathbb{R}^{2})}^{4}\lesssim\int_{0}^{t}\sigma^{r_{\alpha}}\left(\lVert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rVert_{L^{4}(\mathbb{R}^{2})}^{4}+\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{2})}^{4}\right)\leqslant
M\left(C_{0}+\left(C_{0}+\mathcal{A}_{1}(t)\right)\mathcal{A}_{1}(t)\right).$

On the other hand, we notice that the jump of $\mathscr{F}$ verifies:

(3.58)
$\displaystyle\lvert\llbracket\mathscr{F}\rrbracket\rvert\lesssim\lvert\llbracket
f(\rho)\rrbracket\rvert\lvert<F>\rvert+\lvert\llbracket
F\rrbracket\rvert\quad\text{ and }\quad\llbracket
F\rrbracket=2\llbracket\mu(\rho)\rrbracket\left(<\operatorname{div}u>-<\mathbb{D}^{jk}u>n^{j}_{x}n^{k}_{x}\right).$

Thus, we have:

(3.59)
$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\llbracket\mathscr{F}(\tau)\rrbracket\rVert_{L^{4}(\mathcal{C}(\tau))}^{4}d\tau\leqslant
M^{\prime}\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{4}(\mathcal{C}(0))}^{4}\left(\text{\texttheta}(t)+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}\right)e^{M^{\prime}\text{\texttheta}(t)}.$

Also, from the expression of $\mathscr{F}$ we have:

(3.60)
$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\mathscr{F}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\leqslant
M^{\prime}\int_{0}^{t}\sigma^{r_{\alpha}}\left(\lVert
F(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau+\lVert
f(\rho(\tau))\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}\lVert
F(\tau)\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}d\tau\right).$

Finally, as it has been done previously, we combine from (3.58), (3.49),
(3.43) and (3.21) in order to obtain the following:

(3.61)
$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\llbracket\mathscr{F}(\tau)\rrbracket\rVert_{L^{\infty}(\mathcal{C}(\tau))}^{4}\mathcal{P}(\mathcal{C}(\tau))^{4}d\tau\leqslant
M^{\prime}K_{0}\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{\infty}(\mathcal{C}(0))}^{4}e^{M^{\prime}\text{\texttheta}(t)}\left(\text{\texttheta}(t)+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}\right).$

We sum (3.57), (3.59), (3.60) and (3.61) and we obtain the following estimate
for the third term of (3.46):

(3.62) $\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u_{F}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\leqslant
M\left[C_{0}+\left(C_{0}+\mathcal{A}_{1}(t)\right)\mathcal{A}_{1}(t)+(1+\text{\textbeta}(t))\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\right.\\\
\left.+\lVert\llbracket f(\rho_{0})\rrbracket\rVert_{L^{2}\cap
L^{\infty}(\mathcal{C}(0))}^{4}(1+K_{0})e^{M^{\prime}\text{\texttheta}(t)}\left(\text{\texttheta}(t)+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}\right)\right].$

To obtain an estimate for the remainder term of (3.46), we apply again the
quantitative Hölder estimate 2.7:

$\displaystyle\lVert\nabla
u_{P}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}\lesssim\lVert\mathscr{P}(\tau)\rVert_{L^{4}(\mathbb{R}^{2})}+\lVert\llbracket\mathscr{P}(\tau)\rrbracket\rVert_{L^{4}(\mathcal{C}(\tau))}+\lVert\mathscr{P}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}+\lVert\llbracket\mathscr{P}(\tau)\rrbracket\rVert_{L^{\infty}(\mathcal{C}(\tau))}\mathcal{P}(\mathcal{C}(\tau))$

where

$\mathscr{P}=\dfrac{2\mu(\rho)-\widetilde{\mu}}{2\mu(\rho)+\lambda(\rho)}(P(\rho)-\widetilde{P}).$

On the one hand, from (3.14), the following estimate holds:

$\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\mathscr{P}\rVert_{L^{4}(\mathbb{R}^{2})}^{4}d\tau\leqslant
M\left(C_{0}+\mathcal{A}_{1}(t)(C_{0}+\mathcal{A}_{1}(t))\right)$

and on the other hand from (3.43) and (3.21) we have:

$\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\llbracket\mathscr{P}(\tau)\rrbracket\rVert_{L^{4}(\mathcal{C}(\tau))}^{4}d\tau\leqslant
M^{\prime}\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{4}(\mathcal{C}(0))}^{4}e^{M^{\prime}\text{\texttheta}(t)}.$

As it has been done previously, we have from (3.49), (3.43) and (3.21) the
following estimate:

$\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\llbracket\mathscr{P}(\rho(\tau))\rrbracket\rVert_{L^{\infty}(\mathcal{C}(\tau))}^{4}\mathcal{P}(\mathcal{C}(\tau))^{4}d\tau\leqslant
M^{\prime}K_{0}\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{\infty}(\mathcal{C}(0))}^{4}e^{M^{\prime}\text{\texttheta}(t)}.$

Finally, we have:

$\int_{0}^{t}\lVert\mathscr{P}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\leqslant
M^{\prime}\text{\textbeta}(t).$

Gathering all of these estimates, we have the following estimate for the
remainder term of (3.46):

(3.63) $\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u_{P}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\leqslant
M\left[C_{0}+\mathcal{A}_{1}(t)(C_{0}+\mathcal{A}_{1}(t))+(1+K_{0})e^{M^{\prime}\text{\texttheta}(t)}\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{2}\cap
L^{\infty}(\mathcal{C}(0))}^{4}+\text{\textbeta}(t)\right].$

The Hölder norm of the effective flux that appears in (3.62) and (3.63) can be
estimaed by summing (3.48) and (3.54) since the effective flux is given by
(3.56):

$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau$
$\displaystyle\leqslant
M^{\prime}\left[C_{0}^{2}+\mathcal{A}_{1}(t)^{2}+\mathcal{A}_{2}(t)^{2}+\mathcal{A}_{3}(t)^{2}+\text{\textbeta}(t)\text{\texttheta}(t)\right.$
$\displaystyle\left.+(1+K_{0})\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{\infty}\cap
L^{2}(\mathcal{C}(0))}^{4}\left(1+\text{\texttheta}(t)\right)e^{M^{\prime}\text{\texttheta}(t)}\right].$

Finally the proof of 2.8 follows by summing (3.48), (3.54), (3.62) and (3.63):

$\displaystyle\text{\texttheta}(t)$ $\displaystyle\leqslant
M\left[C_{0}^{2}+\mathcal{A}_{1}^{2}(t)+\mathcal{A}_{2}^{2}(t)+\mathcal{A}_{3}(t)^{2}+\text{\textbeta}(t)(1+\text{\texttheta}(t))+(1+\text{\textbeta}(t))\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\right.$
$\displaystyle\left.+(1+K_{0})e^{M^{\prime}\text{\texttheta}(t)}\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{\infty}\cap
L^{2}(\mathcal{C}(0))}^{4}\left(1+\text{\texttheta}(t)+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\right)\right].$

∎

###  3.5. Construction and stability of approximate sequence

We begin the last step of the proof of 1.1 by identifying which initial datum
to smooth and which not to smooth. First, the initial interface $\gamma_{0}$
and the initial density and velocity have the required regularity as stated in
the local well-posedness theorem, 2.1. Therefore, it is not necessary to
smooth them. However, the stress tensor at the initial time only fulfills

$\operatorname{div}(\Pi_{0})=\operatorname{div}(2\mu(\rho_{0})\mathbb{D}u_{0}+(\lambda(\rho_{0})\operatorname{div}u_{0}-P(\rho_{0})+\widetilde{P})I_{2})\in
H^{-1}(\mathbb{R}^{2}).$

In order to preserve the discontinuity in the initial density, we will
consider, for each $\delta>0$, $(\rho_{0},u_{0}^{\delta})$ as initial data for
the equations (2.1) where $u_{0}^{\delta}$ solves the following elliptic
equation:

(3.64)
$\displaystyle-\operatorname{div}(2\mu(\rho_{0})\mathbb{D}u_{0}^{\delta}+(\lambda(\rho_{0})\operatorname{div}u_{0}^{\delta}-P(\rho_{0})+\widetilde{P})I_{2})+c^{\delta}u_{0}^{\delta}=-\operatorname{div}(w_{\delta}*\Pi_{0}).$

Above, $w_{\delta}:=\delta^{-2}w(\cdot/\delta)$ with $w$ a smooth non negative
function supported in the unit ball centered at the origin and whose integral
equal $1$. Also the constant $c^{\delta}$ is defined by:

$c^{\delta}:=\lVert
w_{\delta}*\Pi_{0}-\Pi_{0}\rVert_{L^{2}(\mathbb{R}^{2})}\quad\text{and
verifies}\quad c^{\delta}\xrightarrow[]{\delta\to 0}0.$

Since the viscosity are bounded away from vacuum and upper bounded and the
pressure $P(\rho_{0})-\widetilde{P}$ belongs to $L^{2}(\mathbb{R}^{2})$, the
existence of unique solution $u_{0}^{\delta}\in H^{1}(\mathbb{R}^{2})$ for the
elliptic equation (3.64) follows from the Lax-Milgram theorem. Moreover the
sequence $(u_{0}^{\delta})_{\delta}$ comes with the following estimate:

(3.65) $\displaystyle M^{-1}\lVert\nabla
u_{0}^{\delta}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+c^{\delta}\lVert
u_{0}^{\delta}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\leqslant
M\left(\lVert\Pi_{0}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\lVert
P(\rho_{0})-\widetilde{P}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\right)$

with a constant $M>1$ depending on the lower bound of the initial viscosity.
To show that the sequence $(u^{\delta}_{0})_{\delta}$ converges strongly to
$u_{0}$ in $H^{1}(\mathbb{R}^{2})$, we add $\operatorname{div}(\Pi_{0})$ to
both sides of (3.64) and we obtain:

$-\operatorname{div}\\{2\mu(\rho_{0})\mathbb{D}(u^{\delta}_{0}-u_{0})+\lambda(\rho_{0})\operatorname{div}(u_{0}^{\delta}-u_{0})I_{2}\\}+c^{\delta}u_{0}^{\delta}=-\operatorname{div}(w_{\delta}*\Pi_{0}-\Pi_{0}).$

Next, we test the above equation with $u_{0}^{\delta}-u_{0}$ and we obtain:

$\displaystyle
2\int_{\mathbb{R}^{2}}\mu(\rho_{0})\lvert\mathbb{D}(u_{0}^{\delta}-u_{0})\rvert^{2}+\int_{\mathbb{R}^{2}}\lambda(\rho_{0})\lvert\operatorname{div}(u_{0}^{\delta}-u_{0})\rvert^{2}+c^{\delta}\int_{\mathbb{R}^{2}}u_{0}^{\delta}(u_{0}^{\delta}-u_{0})=\int_{\mathbb{R}^{2}}\nabla(u_{0}^{\delta}-u_{0})\colon(w_{\delta}*\Pi_{0}-\Pi_{0}).$

The following estimate holds from Hölder and Young inequalities:

(3.66) $\displaystyle
M^{-1}\lVert\nabla(u_{0}^{\delta}-u_{0})\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\dfrac{1}{2}c^{\delta}\lVert
u_{0}^{\delta}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\leqslant M\lVert
w_{\delta}*\Pi_{0}-\Pi_{0}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\dfrac{1}{2}c^{\delta}\lVert
u_{0}\rVert_{L^{2}(\mathbb{R}^{2})}^{2},$

from which we deduce, owing to the definition of $c^{\delta}$, see (3.65), the
following convergence:

$u_{0}^{\delta}\quad\xrightarrow[]{\delta\to 0}\quad u_{0}\quad\text{ in
}\quad\dot{H}^{1}(\mathbb{R}^{2}).$

As well, we have from (3.66) that

$\limsup_{\delta\to 0}\lVert
u_{0}^{\delta}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\leqslant\lVert
u_{0}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}$

from which we deduce:

$u_{0}^{\delta}\quad\xrightarrow[]{\delta\to 0}\quad u_{0}\quad\text{ in
}\quad L^{2}(\mathbb{R}^{2}).$

In conclusion, we recover the strong convergence of the sequence of initial
velocity $(u_{0}^{\delta})$ to $u_{0}$ in $H^{1}(\mathbb{R}^{2})$. Also from
(3.64), we have, for all $\delta>0$, the compatibility condition:

(3.67)
$\displaystyle\operatorname{div}(2\mu(\rho_{0})\mathbb{D}u_{0}^{\delta}+(\lambda(\rho_{0})\operatorname{div}u_{0}^{\delta}-P(\rho_{0})+\widetilde{P})I_{2})=-c^{\delta}u_{0}^{\delta}+\operatorname{div}(w_{\delta}*\Pi_{0})\in
L^{2}(\mathbb{R}^{2}).$

With regard to the regularity of the initial density and interface
$\gamma_{0}$, the fact that the initial viscosity $\mu(\rho_{0})$ is a small
perturbation around the constant state $\widetilde{\mu}$, along with the
regularity of $u_{0}^{\delta}$ and (3.67) for each $\delta>0$, 2.1 guarantee
the existence of a local solution $(\rho^{\delta},u^{\delta})$ for the
equation (2.1) with initial data:

$\rho^{\delta}_{|t=0}=\rho_{0}\quad\text{ and }\quad
u^{\delta}_{|t=0}=u^{\delta}_{0}.$

The solution is defined up to a time $T_{\delta}>0$ and enjoys the regularity
in 2.1. This regularity is sufficient for the computations carried out in the
previous sections to make sense. As a consequence, 2.9 holds true for solution
$(\rho^{\delta},u^{\delta})$. In particular, all the conditions outlined in
2.3 are fulfilled, resulting in $T_{\delta}=+\infty$. We turn to the final
step of this section, which consists of showing the stability of the sequence
$(\rho^{\delta},u^{\delta})$.

We begin by writing the equations solved by $(\rho^{\delta},u^{\delta})$:

(3.68)
$\displaystyle\begin{cases}\partial_{t}\rho^{\delta}+\operatorname{div}(\rho^{\delta}u^{\delta})=0,\\\
\partial_{t}(\rho^{\delta}u^{\delta})+\operatorname{div}(\rho^{\delta}u^{\delta}\otimes
u^{\delta})+\nabla
P(\rho^{\delta})=\operatorname{div}(2\mu(\rho^{\delta})\mathbb{D}u^{\delta})+\nabla(\lambda(\rho^{\delta})\operatorname{div}u^{\delta}).\end{cases}$

The velocity sequence $(u^{\delta})_{\delta}$ comes with the following
estimate: for all $T>0$ and $\delta>0$

(3.69)
$\lVert\rho^{\delta}-\widetilde{\rho}\rVert_{L^{\infty}((0,T),L^{2}(\mathbb{R}^{2}))}^{2}+\lVert\rho^{\delta}-\widetilde{\rho}\rVert_{L^{\infty}((0,T)\times\mathbb{R}^{2})}^{2}+\lVert
u^{\delta}\rVert_{L^{\infty}((0,T),H^{1}(\mathbb{R}^{2}))}^{2}\\\
+\lVert\nabla
u^{\delta}\rVert_{L^{3}((0,T)\times\mathbb{R}^{2})}^{2}+\lVert\dot{u}^{\delta}\rVert_{L^{2}((0,T)\times\mathbb{R}^{2})}^{2}\leqslant
MC_{0}.$

Hereafter, $M$ is a constant that depends on the lower and upper bounds of the
density and viscosity. Thus, there exist $\rho-\widetilde{\rho}\in
L^{\infty}((0,\infty),L^{2}(\mathbb{R}^{2}))\cap
L^{\infty}((0,\infty)\times\mathbb{R}^{2})$ and $u\in
L^{\infty}((0,\infty),H^{1}(\mathbb{R}^{2}))$ such that:

(3.70)
$\displaystyle\begin{cases}\rho^{\delta}-\widetilde{\rho}\;\;&\rightharpoonup^{*}\;\;\rho-\widetilde{\rho}\quad\text{
in }\quad L^{\infty}((0,T)\times\mathbb{R}^{2}),\\\
\rho^{\delta}-\widetilde{\rho}\;\;&\xrightarrow{}\quad\rho-\widetilde{\rho}\quad\text{
strongly in }\quad\mathscr{C}([0,T],L^{2}_{w}(\mathbb{R}^{2})),\\\
u^{\delta}\;\;&\rightharpoonup^{*}\;\;u\quad\text{ in
}L^{\infty}((0,T),H^{1}(\mathbb{R}^{2})),\\\ u^{\delta}&\xrightarrow{}\quad
u\quad\text{ strongly in
}\mathscr{C}([0,T],L^{2}_{\text{loc}}(\mathbb{R}^{2})).\end{cases}$

Moreover, by interpolation, we have, for all $2\leqslant q<\infty$,

$u^{\delta}\xrightarrow{}u\quad\text{ strongly in }\quad
L^{\infty}_{\text{loc}}((0,\infty),L^{q}_{\text{loc}}(\mathbb{R}^{2})).$

The density $\rho^{\delta}$ is discontinuous across the interface
$\mathcal{C}^{\delta}$ transported by the velocity $u^{\delta}$ and given by:

$\mathcal{C}^{\delta}(t)=\left\\{\gamma^{\delta}(t,s)\in\mathbb{R}^{2}\colon
s\in[0,2\pi]\right\\}$

with

(3.71)
$\displaystyle\gamma^{\delta}(t,s)=\gamma_{0}(s)+\int_{0}^{t}u^{\delta}(\tau,\gamma^{\delta}(\tau,s))d\tau.$

As well, the velocity $u^{\delta}$ comes with the following estimate: for all
$T>0$, $\delta>0$

(3.72) $\displaystyle\sup_{[0,T]}\sigma\lVert\nabla
u^{\delta}\rVert_{L^{4}(\mathbb{R}^{2})}^{4}+\int_{0}^{T}\sigma^{r_{\alpha}}\lVert\nabla
u^{\delta}\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma^{\delta}}(\mathbb{R}^{2})}^{4}\leqslant
MC_{0}$

from which we deduce, together with (3.69), that the sequence
$(\gamma^{\delta})_{\delta}$ verifies: for all $T>0$ and $n\in\mathbb{N}^{*}$,

$\sup_{\delta}\lVert\nabla\gamma^{\delta}\rVert_{L^{\infty}((0,T)\times(0,2\pi))}\leqslant
M(T)C_{0}\quad\text{ and
}\quad\sup_{\delta}\lVert\partial_{t}\gamma^{\delta}\rVert_{L^{\infty}((1/n,T)\times(0,2\pi))}\leqslant
M(n)C_{0}.$

The constant $M(T)$ and $M(n)$ depend not only on the lower and upper bounds
of the density and viscosity but also on time $T$ and $n$ respectively. So, up
to a subsequence, $(\gamma^{\delta})_{\delta}$ convergences uniformly in
$[0,T]\times[0,2\pi]$ to some $\gamma\in W^{1,\infty}((0,T)\times(0,2\pi))$.
Moreover, from (3.69) and embedding, the velocity sequence verifies:

$\sup_{\delta}\lVert
u^{\delta}\rVert_{L^{3}((0,\infty),\mathscr{C}^{1/3}(\mathbb{R}^{2}))}\leqslant
C_{0}$

such that we can let $\delta$ goes to zero in (3.71) and obtain the fact that
the interface limit $\gamma$ verifies:

(3.73)
$\displaystyle\gamma(t,s)=\gamma_{0}(s)+\int_{0}^{t}u(\tau,\gamma(\tau,s))d\tau.$

On the other hand, we have:

$\sup_{\delta}\int_{0}^{\infty}\sigma^{r_{\alpha}}\lVert\nabla
u^{\delta}\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma^{\delta}}(\mathbb{R}^{2})}^{4}\leqslant
C_{0}$

in such way that, by lower semi-continuity of the norm and the fact that
$r_{\alpha}<3$ we have $\nabla u\in
L^{1}_{\text{loc}}([0,\infty),\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2}))$.
Thus, the regularity of the velocity ensures the uniqueness of $\gamma$ that
satisfies (3.73). Moreover, the interface has the following regularity,

$\gamma\in\mathscr{C}([0,\infty),\mathscr{C}^{1+\alpha}([0,2\pi]))$

and verifies the non-intersection condition with a constant $c_{\gamma}(t)$
that satisfies: