# A well-posedness result for the compressible two-fluid model with density-
dependent viscosity

Sagbo Marcel ZODJI  Université Paris Cité and Sorbonne Université, CNRS, IMJ-
PRG, F-75013 Paris, France<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study a system of PDE’s describing the motion of two
compressible viscous fluids occupying the whole space
$\mathbb{R}^{d}\;(d\in\\{2,3\\}$). First, we prove a local well-posedness
result for discontinuous initial data. We require the dynamic viscosity to be
a small perturbation of a constant state and the divergence of the initial
stress tensor to belong to $L^{2}(\mathbb{R}^{d})$.

In the case where the pressure and viscosity depend only on the total density,
the two-fluid model coincides with the classical Navier-Stokes equations. In
$d=2$, we achieve global well-posedness with small initial data and there is
no requirement for the compatibility condition on the stress tensor mentioned
above to hold. As opposed to the constant viscosity case, the vorticity and
effective flux are no longer continuous at the interface and the delicate part
of the proof is to show that the jumps of those quantities decay exponentially
in time.

On the one hand, this result generalizes previous known of D. Hoff [17]
concerning the propagation of regularity for discontinuity surfaces by
allowing more general nonlinear pressure law and density-dependent viscosity.
On the other hand, this result supplements the work by Danchin, Fanelli and
Paicu [8] with global well-posedness, even for density-dependent viscosity and
we obtain uniqueness in a large space.

###### keywords:

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

###### 1991 Mathematics Subject Classification:

76T17, 35R35 (primary); 35A02, 35Q30, 76N10 (secondary)

##  1\. Introduction and main results

###  1.1. Presentation of the model

In this paper, we are concerned with the following system of PDE’s describing
the evolution of two compressible fluids with density-dependent viscosity in
the whole space $\mathbb{R}^{d},\;d\in\\{2,3\\}$,

$\displaystyle\begin{cases}\partial_{t}\rho_{\pm}+\operatorname{div}(\rho_{\pm}u)=0,\\\
\partial_{t}(\rho u^{j})+\operatorname{div}(\rho
u^{j}u)=\operatorname{div}\Pi^{j}.\end{cases}$ (1.1)

Above, $\rho_{\pm}=\rho_{\pm}(t,x)\geqslant 0$ is the density of the phase
fluid labelled $\pm$, $\rho:=\rho_{+}+\rho_{-}$ is the total density of the
mixture and $u=u(t,x)\in\mathbb{R}^{d}$ is the single velocity of the mixture.
The stress tensor $\Pi$ is given by

$\displaystyle\Pi^{jk}=2\mu\mathbb{D}^{jk}u+(\lambda\operatorname{div}u-P+\widetilde{P})\delta^{jk}\quad\text{with}\quad
2\mathbb{D}^{jk}u=\partial_{j}u^{k}+\partial_{k}u^{j},$ (1.2)

where $P=P(\rho_{+},\rho_{-})$, $\mu=\mu(\rho_{+},\rho_{-})$,
$\lambda=\lambda(\rho_{+},\rho_{-})$ are respectively pressure, dynamic and
kinematic viscosity of the mixture and are given $\mathscr{C}^{2}$ function of
$\rho_{\pm}$ and finally $\widetilde{P}$ is a positive equilibrium pressure.
We refer to [20] for the derivation of the system (1.1) for the more general
case of a mixture of two heat conducting fluids. It is assumed that the two
pure components of the mixture are immiscible, so that each fluid is
implicitly assumed to occupy its own domain and that $\rho_{+}\rho_{-}=0$. In
particular they are separated by a sharp interface $\mathcal{C}$. The relevant
physical situation for multi-phase flows is exactly where the total density
$\rho$ experiences discontinuity across the interface $\mathcal{C}$. Since the
two densities $\rho_{\pm}$ of the mixture are assumed to be different, we see
that the class of _piecewise regular_ functions is suitable for the study of
this system. This motivates the following notation. Notation Let
$\mathcal{C}=\\{\gamma(s)\in\mathbb{R}^{d},\;s\in\mathbb{T}^{d-1}\\}$ be a
hyper-surface that is the boundary of an open, simply connected set $D$ such
that

$\mathbb{R}^{d}=D\cup\mathcal{C}\cup(\mathbb{R}^{d}\setminus\overline{D}).$

We define the set of piecewise $\alpha$-Hölder space with respect to $\gamma$
as follows:

$\dot{\mathscr{C}}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d}):=\dot{\mathscr{C}}^{\alpha}(\overline{D})\cap\dot{\mathscr{C}}^{\alpha}(\mathbb{R}^{d}\setminus
D)$

and the non-homogeneous Hölder space is:

$\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d}):=L^{\infty}(\mathbb{R}^{d})\cap\dot{\mathscr{C}}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d}).$

This space strictly contains the Hölder space
$\mathscr{C}^{\alpha}(\mathbb{R}^{d})$. In particular, there exists functions
in $\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})$ that have
discontinuities across the interface $\mathcal{C}$. Moreover, we define the
following space: for $I\subset\mathbb{R}$,

$L^{p}(I,\;\;\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})):=\left\\{g=g(t,x)\colon\begin{cases}\displaystyle\int_{I}\lVert
g(t)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(t)}(\mathbb{R}^{d})}^{p}dt<\infty\quad\text{
if }\quad 1\leqslant p<\infty,\\\ \displaystyle\sup_{t\in I}\text{ess}\lVert
g(t)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(t)}(\mathbb{R}^{d})}<\infty\quad\text{
if }\quad p=\infty,\end{cases}\right\\}$

for a time dependent hyper-surface $\mathcal{C}=\mathcal{C}(t)$. For
$I=(0,T)$, the corresponding space will sometimes be shortly denoted by
$L^{p}_{T}\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})$ for finite $T$ and
$L^{p}\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})$ for $T=\infty$.
Alternatively, the local-in-time space is defined as follows:

$L_{\text{loc}}^{p}([0,\infty),\;\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})):=\left\\{g\in
L^{p}(I,\;\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})),\quad\forall\,I\subset[0,\infty)\;\;\text{bounded
}\right\\}.$

A suitable way to rewrite the system (1.1) is to introduce the so-called
fractional density or concentration $c$ defined by

$c:=\dfrac{\rho_{+}}{\rho}.$

From the first two equations of (1.1), one checks easily, assuming some
regularity, that the concentration solves the transport equation

$\displaystyle\partial_{t}c+u\cdot\nabla c=0.$ (1.3)

Let us notice that the domain occupied by the phase $\pm$ can be defined by:

$D_{\pm}(t):=\\{x\in\mathbb{R}^{d}\colon\rho_{\pm}(t,x)>0\\}.$

As well, one can write the domain occupied by the one phase fluid with the
help of the concentration as follows:

$\displaystyle D_{+}(t)=\\{x\in\mathbb{R}^{d}\colon c(t,x)=1\\},\;\quad
D_{-}(t)=\\{x\in\mathbb{R}^{2}\colon c(t,x)=0\\}$ (1.4)

and consequently the interface is:

$\mathcal{C}(t):=\partial D_{+}(t)=\partial D_{-}(t).$

Assume that the velocity is regular, for instance Lipschitz, then it admits a
continuous flow map $X(t)$ that helps together with (1.3) and (1.4) to obtain
that the initial interface is just transported by the fluid

$\mathcal{C}(t)=X(t)\mathcal{C}(0).$

On the other hand, by summing the mass equation on the density of the one
phase $\rho_{\pm}$ $\eqref{epNS}_{1}$, one obtains that the total density
solves a mass equation with velocity $u$. Besides, the density $\rho_{\pm}$ of
the one phase fluid can be written in terms of the total density and the
concentration as follows:

$\rho_{+}=c\rho\quad\text{ and }\quad\rho_{-}=(1-c)\rho,$

thus, the pressure can be written as function of the total density and the
concentration $P=P(\rho,c)$ such that $P_{+}(\rho)=P(\rho,1)$ and
$P_{-}(\rho)=P(\rho,0)$ as well as the viscosities. Hence, formally
$(c,\rho,u)$ solves:

$\displaystyle\begin{cases}\partial_{t}c+u\cdot\nabla c=0,\\\
\partial_{t}\rho+\operatorname{div}(\rho u)=0,\\\ \partial_{t}(\rho
u)+\operatorname{div}(\rho u\otimes u)+\nabla
P(\rho,c)=\operatorname{div}(2\mu(\rho,c)\mathbb{D}u)+\nabla(\lambda(\rho,c)\operatorname{div}u).\end{cases}$
(1.5)

Conversely, if $(c,\rho,u)$ solves (1.5), one can set

$\rho_{+}=c\rho\quad\text{and}\quad\rho_{-}=(1-c)\rho$

and checks easily that $\rho_{\pm}$ solve the equations $\eqref{epNS}_{1}$.
Just by replacing

$\rho=\rho_{+}+\rho_{-}\quad\text{and}\quad
c=\dfrac{\rho_{+}}{\rho_{+}+\rho_{-}}$

in the pressure and the viscosities, one notices that $(\rho_{+},\rho_{-},u)$
solves (1.1). In fact, we obtain that $(\rho_{+},\rho_{-},u)$ solves (1.1) if
and only if $(c,\rho,u)$ solves (1.5) owing to some change of pressure and
viscosity law, at least for classical solutions. The equivalence also holds
true for weak solutions _à la Lions_ due to the result in [38, Lemma 2.5],
which is a version of DiPerna-Lions theory for renormalized solutions. Thus,
we will only work with the system (1.5). The equations (1.5) are supplemented
with initial data

$\displaystyle
c_{|t=0}=c_{0},\quad\;{\rho}_{|t=0}=\rho_{0}\quad\text{and}\quad
u_{|t=0}=u_{0},$ (1.6)

with the following behavior at infinity

$\displaystyle\lim_{\lvert
x\rvert\to\infty}\rho_{0}(x)=\widetilde{\rho},\quad\lim_{\lvert
x\rvert\to\infty}P(\rho_{0}(x),c_{0}(x))=\widetilde{P},\quad\lim_{\lvert
x\rvert\to\infty}\mu(\rho_{0}(x),c_{0}(x))=\widetilde{\mu}\quad\text{and}\quad\lim_{\lvert
x\rvert\to\infty}\lambda(\rho_{0}(x),c_{0}(x))=\widetilde{\lambda},$ (1.7)

where $\widetilde{\rho},\,\widetilde{P},\,\widetilde{\mu}$ and
$\widetilde{\lambda}$ are positive constants.

The purpose of this paper is two-folded.

  * •

On the one hand, we construct a local-in-time solution $(c,\rho,u)$ of the
equations (1.5) with discontinuous initial data and with density-dependent
viscosity: the initial density $\rho_{0}$ is assumed to be Hölder continuous
on both sides of a suitable interface $\mathcal{C}(0)$ through which it is
discontinuous. The initial velocity belongs to the Sobolev space
$H^{1}(\mathbb{R}^{d})$ with the following compatibility condition:

$\displaystyle\operatorname{div}(\Pi)_{|t=0}\in L^{2}(\mathbb{R}^{d})$ (1.8)

which expresses somehow the continuity of the stress tensor. This result is
more general than the work by Tani [37] and it can be used to serve as
building block for global solutions.

  * •

On the second hand, we observe that when the pressure and the viscosity laws
depend only on the density, that is, they do not depend on the concentration
$c$, then the system (1.5) becomes the well-known Navier-Stokes equations for
compressible fluids. We establish existence and uniqueness of global weak
solution with intermediary regularity for the Navier-Stokes equations with
density-dependent viscosity in $\mathbb{R}^{2}$, in the spirit of the work by
Hoff [15, 17].

Unfortunately, we do not have enough information in order to prove existence
of global weak solution for the two-fluid model, even in the case of small
initial data.

###  1.2. Review of known results and statement of the main results

The study of the two-phase model (1.1) goes back to Tani’s work [37] where the
author showed existence of a local-in-time solution for the system (1.1)
including temperature. Like the pioneering work of John Nash [30], his proof
relies on a change in Lagrangian coordinates, followed by the construction of
an approximate solution using the kernel associated to the linearized
equations. The solution is constructed in a piecewise Hölder spaces : the
density is $\mathscr{C}^{1+\alpha}$ and the velocity is
$\mathscr{C}^{2+\alpha}$, on both sides of an interface $\mathcal{C}$ that is
$\mathscr{C}^{2+\alpha}$. To our knowledge, this result is the only one
concerning interfaces with Hölder regularity.

In the literature, other results mostly focus on interfaces with Sobolev
regularity. As an example, Denisova in [10] has proved the local well-
posedness of the system (1.1) including surface tension with different
constant viscosities for each component of the mixture and with Sobolev-
Slobodetskii interface $\mathcal{C}\in W^{3/2}_{2}$. Her proof also relies on
a change in Lagrangian coordinates and the density and the velocity are
piecewise $W_{2}^{1+l}$ (for $l\in(1/2,1)$) regular on both sides of the
interface $\mathcal{C}$.

In contrast to the aforementioned results based on a change in Lagrangian
coordinates, there are other works [21, 23, 22] by Jang _et al._ that rely on
a change into flattened coordinates. They considered the two-phase problem in
a cylinder, taking into account scenarios with and without surface tension.
This problem involves two fluids, with one fluid on top of another. In this
context, there are two interfaces to consider: the interface between the lower
and upper fluids, and the interface between the upper fluid and the
atmosphere. Both interfaces are assumed to be represented by graphs.
Additionally, the viscosities of the two fluids are assumed to be constant and
different from each other. In [21], they successfully constructed local-in-
time solution in Sobolev framework, in [22], they studied the Rayleigh-Taylor
instability which occurs when the heavier fluid is on top of a lighter fluid.
They successfully proved that when the surface tension is below a critical
value, then the system is nonlinearly unstable. Finally in [23], they proved
that if the heavier fluid is on top of the lighter one and the surface tension
is sufficiently large, then, the Rayleigh-Taylor instability is prevented and
in the converse configuration, the system is stable around the equilibrium
state with non-negative surface tension. Moreover, they studied the zero
surface tension limit problem.

On the other hand, there are also results [35, 40] that address the two-phase
problem involving a compressible fluid and an incompressible fluid. In [35],
Shibata has proved the local well-posedness of this two-phase model, including
temperature and surface tension, with an almost flat interface. He worked on
an exterior domain and with a viscosity that depends on both density and
temperature. The initial density of the compressible fluid has a Sobolev
$W^{1,p}$ regularity and the initial velocity and temperature have the Besov
regularity $B^{2(1-1/p)}_{q,p}$, on both sides of the interface, for some
$1<p,q<\infty$ that verify:

$\dfrac{2}{p}+\dfrac{d}{q}<1.$

The same Compressible-Incompressible model is considered in [40] by Yan and
Zhao with surface tension in horizontal periodic domain. The viscosities of
the fluids involved are different from each other and assumed to be constant.
They successfully achieved a global well-posedness result by making the
assumption that the initial density of the compressible fluid is close to a
constant state in $H^{4}$ with respect to the domain occupied by the
compressible fluid, and the initial velocity is small in $H^{4}$ on both sides
of the interface. To our knowledge, this result stands as the only one to
address the existence of strong global solution of the two-phase problem.

Finally, we may notice that Kubo _et al._ in [26] have addressed the local
well-posedness of the Compressible-Compressible model with different approach.
Their methods rely on proving the existence of a $\mathcal{R}$-bounded
operator and maximal regularity to the linearised system. The interface is
uniformly $W^{2-1/r}_{r}$ for $r>d$, the density has a $W^{1,r}$ regularity on
both sides of the interface and the initial velocity belongs to the Besov
space $B^{2(1-1/p)}_{r,p}$ for some $p>2$ on both sides of the initial
interface.

As explained, for instance in [13], a different approach to tracking the
interface is to construct weak solutions for the full model in a class that
allows the study of its dynamics. To the best of our knowledge, the only
available results for weak solutions pertain to fluid mixture with constant
viscosities that are the same for both components. For instance, we refer to
the work by Vasseur, Wen, and Yu [38] in a three-dimensional bounded domain
where they obtained a weak solution with finite initial energy. The main
difficulty in their proof lies in the compactness of an approximate sequence
of the pressure that depends on two variables, in contrast to the single-
variable case [28, 11]. They successfully overcome the difficulty by
considering the following pressure law

$P(\rho_{+},\rho_{-})=A_{+}{\rho_{+}}^{\gamma_{+}}+A_{-}{\rho_{-}}^{\gamma_{-}}$

with the following closeness constraints on $\gamma_{\pm}$:

$\gamma_{\pm}>9/5\quad\text{and}\quad\max\left\\{\dfrac{3\gamma_{+}}{4};\gamma_{+}-1;\dfrac{3(\gamma_{+}+1)}{5}\right\\}<\gamma_{-}<\min\left\\{\dfrac{4\gamma_{+}}{3};\gamma_{+}+1;\dfrac{5\gamma_{+}}{3}-1\right\\}.$

This result has been improved by Novotný [31], Wen [39] by allowing more
general pressure law, Novotný and Pokortný [33] and Novotný _et al._ [25] for
multi-components fluid models. It is important to notice that, although the
pressure law of the components of the mixture are different, the viscosity of
the components are supposed to be the same constant. The solution that they
constructed is too weak in order to track down discontinuity in the density.
In one dimension, Bresch, Burtea and Lagoutière in [2, Section 4] successfully
solved the global well-posedness for non common constant viscosity, with
numerical considerations in [3]. In our first result, we consider density-
dependent viscosity for both components of the mixture, and we achieve the
construction of unique weak solution in $\mathbb{R}^{d},\,(d\in\\{2,3\\})$
that allows the study of the dynamics of the interface.

We consider the Cauchy problem associated with (1.5) with initial data (1.6).
We assume that the initial concentration $c_{0}$ is the characteristic
function of $D(0)^{c}$ where $D(0)$ is an open, bounded, simply connected
domain in $\mathbb{R}^{d}$ such that its boundary

$\partial
D(0)=:\mathcal{C}(0)=\\{\gamma_{0}(s)\in\mathbb{R}^{d},\;s\in\mathbb{T}^{d-1}\\}$

with $\gamma_{0}\in\mathscr{C}^{1+\alpha}$ verifying the non-self intersection
condition:

$\displaystyle\exists\;c_{\gamma}(0)>0,\quad\forall\;(s,s^{\prime})\in(\mathbb{T}^{d-1})^{2},\;\quad\lvert\gamma_{0}(s)-\gamma_{0}(s^{\prime})\rvert\geqslant
c_{\gamma}(0)^{-1}\lvert s-s^{\prime}\rvert.$ (1.9)

Also the initial density $\rho_{0}$ and velocity $u_{0}$ have the following
regularity:

$\displaystyle u_{0}\in H^{1}(\mathbb{R}^{d}),\quad
P(\rho_{0},c_{0})-\widetilde{P}\in
L^{2}(\mathbb{R}^{d})\cap\mathscr{C}^{\alpha}_{pw,\gamma_{0}}(\mathbb{R}^{d}).$
(1.10)

Finally, we assume that the initial data is such that initially the material
derivative that is

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

belongs to $L^{2}(\mathbb{R}^{d})$, in other words,

$\displaystyle(\rho\dot{u})_{|t=0}=\operatorname{div}(\Pi)_{|t=0}\in
L^{2}(\mathbb{R}^{d}).$ (1.11)

A very technical step requires us to obtain estimate for the
$L^{8/d}((0,T),L^{4}(\mathbb{R}^{d}))$ norm of $\nabla\dot{u}$ and in order to
do so, we need to look at the second material derivative of the velocity
$\ddot{u}$, which is

$\ddot{u}:=\partial_{t}\dot{u}+(u\cdot\nabla)\dot{u}.$

This variable appears while applying the material derivative
$\partial_{t}\,\cdot+\operatorname{div}(\;\cdot\;u)$ to the momentum equation
$\eqref{ep1.1}_{3}$. Our first result reads as follows.

Let $(c_{0},\rho_{0},u_{0})$ be initial data associated with the system (1.5)
that satisfies the conditions (1.9), (1.10), (1.11) for some $0<\alpha<1$ and

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

There exists $[\mu]>0$ depending on the dimension $d\in\\{2,3\\}$ and the
initial interface $\mathcal{C}(0)$ such that if

$\displaystyle\dfrac{4}{\widetilde{\mu}}\lVert\mu(\rho_{0},c_{0})-\widetilde{\mu}\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma_{0}}(\mathbb{R}^{d})}\leqslant[\mu],$
(1.12)

then, there exists a time $T>0$ and an unique solution $(c,\rho,u)$ of the
system (1.5) verifying the following:

  1. (1)

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

  2. (2)

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

  3. (3)

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

  4. (4)

$\dot{u}\in\mathscr{C}([0,T],L^{2}(\mathbb{R}^{d}))\cap
L^{2}((0,T),\dot{H}^{1}(\mathbb{R}^{d}))$, $\sqrt{\sigma}\nabla\dot{u}\in
L^{\infty}((0,T),L^{2}(\mathbb{R}^{d}))\cap
L^{8/d}((0,T),L^{4}(\mathbb{R}^{d}))$;

  5. (5)

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

where $\sigma(t)=\min\\{1,t\\}$. Because of the roughness of the viscosity, we
use a perturbation argument to obtain $L^{p}$ and piece-wise Hölder estimates
for the velocity gradient. In fact, as we will see in (2.26), we have the
following expression for the velocity gradient:

$\displaystyle\nabla u$
$\displaystyle=\frac{1}{\widetilde{\mu}}(-\Delta)^{-1}\nabla\left[\mathcal{P}\operatorname{div}\\{2(\mu-\widetilde{\mu})\mathbb{D}u\\}-\mathcal{P}\left(\rho_{0}\partial_{t}u\right)+\mathcal{P}\operatorname{div}F\right]$
$\displaystyle+\dfrac{1}{2\widetilde{\mu}+\lambda}(-\Delta)^{-1}\nabla\left[\mathcal{Q}\operatorname{div}\\{2(\mu-\widetilde{\mu})\mathbb{D}u\\}-\mathcal{Q}\left(\rho_{0}\partial_{t}u\right)+\mathcal{Q}\left(\operatorname{div}F\right)\right]$
(1.13)

where $\mathcal{P}$ is the Leray projector onto the space of divergence-free
vector fields and where $\mathcal{Q}$ is the projector onto the space of curl-
free vector fields. Thus, we need the smallness condition (1.12) on the
dynamic viscosity in order to ensure that the first and third terms on the
right-hand side of (1.13) are small in comparison to its left-hand side. The
solution $(c,\rho,u)$ constructed in the above theorem exhibits the following
properties.

  1. (1)

There exists a constant $R$ depending on the time $T$, on the norms of the
initial data $(c_{0},\rho_{0},u_{0})$ and on the initial interface
$\mathcal{C}(0)$ such that:

$\sup_{[0,T]}\left\\{\lVert
u\rVert_{H^{1}(\mathbb{R}^{d})}+\lVert\dot{u}\rVert_{L^{2}(\mathbb{R}^{d})}\right\\}\leqslant
R.$

  2. (2)

From (2) and (4), one has
$\rho\dot{u}\in\mathscr{C}([0,T],L^{2}(\mathbb{R}^{d}))$ and one deduces from
the momentum equation (1.5) that
$\operatorname{div}(\Pi)\in\mathscr{C}([0,T],L^{2}(\mathbb{R}^{d}))$ so that
the compatibility condition (1.11) is propagated in time.

  3. (3)

From (3), one deduces by interpolation that the velocity $u$ is continuous in
time and space. On the other hand, from (4), its material derivative,
$\dot{u}$ is continuous in space for almost every positive time. Adding the
fact that its material derivative $\ddot{u}$ verifies (5), then $\ddot{u}$ is
at least continuous across the interface $\mathcal{C}(t)$. Roughly, it
follows, that for all $s\in\mathbb{T}^{d-1}$, one has:

$\llbracket\dot{u}(t,\gamma(t,s))\rrbracket=0\implies\dfrac{d}{dt}\llbracket\dot{u}(t,\gamma(t,s))\rrbracket=\llbracket\ddot{u}(t,\gamma(t,s)\rrbracket=0.$

  4. (4)

From the last part of (4), we deduce that, $\nabla u\in
L^{1}((0,T),\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d}))$ and, of course,
this is the key that helps the propagation the regularity of the initial
interface $\mathcal{C}(0)$. In particular $\mathcal{C}(t)$ belongs to
$\mathscr{C}^{1+\alpha}$ and verifies the non self intersection condition
(1.9) with constant $c_{\gamma}(t)$ verifying :

$\displaystyle\begin{cases}\displaystyle c_{\gamma}(t)&\displaystyle\leqslant
c_{\gamma}(0)\exp\left(\int_{0}^{t}\lVert\nabla
u(\tau)\rVert_{L^{\infty}(\mathbb{R}^{d})}d\tau\right),\\\
\displaystyle\lVert\nabla\gamma(t)\rVert_{\mathscr{C}^{\alpha}}&\displaystyle\leqslant\left(\lVert\nabla\gamma_{0}\rVert_{\mathscr{C}^{\alpha}}+\lVert\nabla\gamma_{0}\rVert_{L^{\infty}}^{1+\alpha}\int_{0}^{t}\lVert\nabla
u(\tau)\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{d})}d\tau\right)\exp\left((2+\alpha)\int_{0}^{t}\lVert\nabla
u(\tau)\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau\right).\end{cases}$ (1.14)

  5. (5)

The dynamic viscosity $\mu(\rho,c)$ remains close to $\widetilde{\mu}$:

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

  6. (6)

We notice that

$\displaystyle\rho\ddot{u}=\operatorname{div}(2\mu(\rho,c)\mathbb{D}\dot{u})+\nabla(\lambda(\rho,c)\operatorname{div}\dot{u})+\operatorname{div}\mathcal{F},$
(1.15)

where $\mathcal{F}$ is a lower order term that belongs, at least, to
$L^{2}((\tau,T),\;\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d}))$ for all
$0<\tau<T$. This equation is obtained by applying the material derivative
$\partial_{t}\cdot\,+\operatorname{div}(\,\cdot\;u)$ to the momentum equation
$\eqref{ep1.1}_{3}$. The computations follow the same line as in section B.1.
Using the regularity of $\ddot{u}$ in (5), the smallness of the dynamic
viscosity around $\widetilde{\mu}$, (1.15), the regularity of $\mathcal{F}$
and the quantitative Hölder estimate [12, 13], we have that

$\nabla\dot{u}\in
L^{2}((\tau,T),\;\mathscr{C}^{\beta}_{pw,\gamma}(\mathbb{R}^{d}))\quad\text{with}\quad\beta=\begin{cases}\alpha,\quad\text{in}\quad
d=2,\\\ \min\\{\alpha,\,1/2\\},\quad\text{in}\quad d=3.\end{cases}$

In particular, $\nabla\dot{u}$ is also continuous on both sides of the
interface $\gamma$.

Let us notice that section 1.2 comes with the following blow-up criterion:
[Blow-up criterion] Let $(c,\rho,u)$ be the solution constructed in section
1.2 defined up to a maximal time $T^{*}$. Assume that, for all $0<t<T^{*}$ we
have:

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

and

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

then $T^{*}=\infty$. Unfortunately, we do not have enough information in order
to prove global existence for the two-fluid model, even in the case of small
data. The difficulty lies in the fact that we are not able to estimate the
$L^{1}((0,T),\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d}))$ norm of
$\nabla u$ uniformly with respect to time. In particular, see (1.14), the
characteristic $c_{\gamma}(t)$ and the $\mathscr{C}^{1+\alpha}$ norm of the
interface $\gamma(t)$ are growing exponentially with respect to time.

In the study of the Navier-Stokes equations in $d=2$, David Hoff [17, 18] has
observed that the exponential growth of these quantities can be compensated by
the exponential decay of the pressure jump across the interface. _For the two-
fluid model, when the pressure laws of the two phases are different, this
exponential decay of the pressure jump does not hold, even when both phases
have the same constant viscosity_. The minimal requirements seem to be that:

$0<\text{\textnu}_{*}\leqslant\dfrac{P(\rho,1)-P_{-}(\rho^{\prime},0)}{f(\rho,1)-f(\rho^{\prime},0)}\leqslant\text{\textnu}^{*}\quad\text{
and }\quad\dfrac{\lvert\mu(\rho,1)-\mu(\rho^{\prime},0)\rvert}{\lvert
f(\rho,1)-f(\rho^{\prime},0)\rvert}+\dfrac{\lvert\lambda(\rho,1)-\lambda(\rho^{\prime},1)\rvert}{\lvert
f(\rho,1)-f(\rho^{\prime},0)\rvert}\leqslant\text{\textnu}^{*},$

at least, for all $\rho$ close to $\widetilde{\rho}(1)$ and $\rho^{\prime}$
close to $\widetilde{\rho}(0)$ where
$\widetilde{P}=P(\widetilde{\rho}(1),1)=P(\widetilde{\rho}(0),0)$ and where
$f$ is defined, up to a constant, by:

$f(\rho,c)=\int^{\rho}\dfrac{2\mu(s,c)+\lambda(s,c)}{s}ds.$

The only case in which this seems to occur is when the pressure and viscosity
laws are proportional. However, we are able to prove a global well-posedness
result in the spirit of [17] for a simplified situation. As announced above,
our second result address the global well-posedness for the Navier-Stokes
equations with density-dependent viscosity in $\mathbb{R}^{2}$. The initial
density is discontinuous and the initial velocity belongs to
$H^{1}(\mathbb{R}^{2})$. In particular, this global well-posedness result does
not require the compatibility condition (1.8). Thus, we generalize previous
known result of D. Hoff [17] concerning the propagation of regularity for
discontinuity surfaces, by allowing general nonlinear pressure law (more than
gamma-law) and density-dependent viscosity. 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 [8] with global well-posedness for
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.

As far as we know, there are not enough results in the literature regarding
the propagation of regularity for discontinuity surfaces for Navier-Stokes
equations for compressible fluids. The initial result is credited to D. Hoff
in his work of 2002 [17], which is part of a logical continuation of his works
[15, 16]. Indeed, in his pioneer work [15], D. Hoff provides bounds for the
following functionals

$\displaystyle\mathcal{A}_{1}(t):=\sup_{[0,t]}\sigma\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{t}\sigma\lVert\sqrt{\rho}\dot{u}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\quad\text{and}\quad\mathcal{A}_{2}(t):=\sup_{[0,t]}\sigma^{d}\lVert\sqrt{\rho}\dot{u}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{t}\sigma^{d}\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$
(1.16)

owing to some smallness condition on the initial data. We recall that 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

$\displaystyle F:=(2\mu+\lambda)\operatorname{div}u-P(\rho)+\widetilde{P}$
(1.17)

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

$\displaystyle\Delta
F=\operatorname{div}(\rho\dot{u})\quad\text{and}\quad\mu\Delta\operatorname{curl}u=\operatorname{curl}(\rho\dot{u}).$
(1.18)

Let us notice that the effective flux was discovered by Hoff and Smoller in
[19] and plays a crucial role in the mathematical analysis of the compressible
model, for instance, in the propagation of oscillations [34] or the
construction of weak solutions [16, 28, 11]. By using the regularity of
$\dot{u}$ provided by the functionals $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$
in (1.16), 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 [15] before considering
nonlinear pressure law (gamma-law) in a second paper [16]. 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 vacuum 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 [29] and the one with finite initial
energy constructed by Lions [28] or Feireisl, Novotný and Petzeltová [11]. 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 [18] 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 gradient of the
velocity 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 D. Hoff [17]
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 (1.9)) 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 weak space that requires the pressure to be linear. On the
other hand, the analysis requires the kinematic viscosity $\lambda$ to be
small for $\alpha$ near to $1$ (see [17, Section 2.3]). These constraints will
be removed in this paper.

In 2020, Danchin, Fanelli and Paicu [8] 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 J-Y Chemin [5, 6] for ideal
flows. In particular they propagate the $W^{1,p}$ regularity for hyper-
surfaces up to a finite time (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 vorticity
lose regularity, and the analysis becomes more subtle_. Let us notice that
there exists an interesting work on weak solutions for the compressible
Navier-Stokes equations with density-dependent viscosity by Bresch and
Desjardins [4]. On the one hand, they require 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. 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 [14].

In this part, we will make different observations to show that for Navier-
Stokes equations with density-dependent viscosity, the effective flux and the
vorticity are not as 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:

$\displaystyle
F:=(2\mu(\rho)+\lambda(\rho))\operatorname{div}u-P(\rho)+\widetilde{P}=-(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})+K\\{(\mu(\rho)-\widetilde{\mu})\mathbb{D}u\\},$
(1.19)
$\displaystyle\mu(\rho)\operatorname{curl}u=-(-\Delta)^{-1}\operatorname{curl}(\rho\dot{u})+K^{\prime}\\{(\mu(\rho)-\widetilde{\mu})\mathbb{D}u\\}.$
(1.20)

We refer to section 3.2.1 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 [32] 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:

$\displaystyle\llbracket\Pi^{j}\rrbracket\cdot n_{x}=0$ (1.21)

where $n_{x}$ 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, the discontinuity of 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

$\displaystyle\llbracket\nabla u\rrbracket=\mathrm{a}\otimes n^{t}_{x}.$
(1.22)

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

$\displaystyle\mathrm{a}=(\mathrm{a}\cdot\tau_{x})\tau_{x}+(\mathrm{a}\cdot
n_{x})n_{x}=\llbracket\operatorname{curl}u\rrbracket\tau_{x}+\llbracket\operatorname{div}u\rrbracket
n_{x},$ (1.23)

where $\tau_{x}$ is the tangential vector field of the interface. By
considering (1.22), one can rewrite (1.21) as follows111
$<a>:=\dfrac{a_{+}+a_{-}}{2}$ is the average of $a$ at the interface:

$\displaystyle<\mu(\rho)>\left(\mathrm{a}^{j}+\mathrm{a}\cdot
n_{x}n^{j}_{x}\right)+2\llbracket\mu(\rho)\rrbracket<\mathbb{D}^{jk}u>n^{k}_{x}+\llbracket\lambda(\rho)\operatorname{div}u-P(\rho)\rrbracket
n^{j}_{x}=0.$ (1.24)

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:

$\displaystyle\llbracket(2\mu(\rho)+\lambda(\rho))\operatorname{div}u-P(\rho)\rrbracket=2\llbracket\mu(\rho)\rrbracket\left(<\operatorname{div}u>-<\mathbb{D}^{jk}u>n^{j}_{x}n^{k}_{x}\right).$
(1.25)

As above, we take the scalar product of (1.24) 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:

$\displaystyle\llbracket\mu(\rho)\operatorname{curl}u\rrbracket=\llbracket\mu(\rho)\rrbracket\left(<\operatorname{curl}u>-2<\mathbb{D}^{jk}u>n^{k}_{x}\tau^{j}_{x}\right).$
(1.26)

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 3.2.2, 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 [17, 18, 32] in case of
constant viscosity_. Unfortunately, this exponential decay does not hold for
the two-fluid model, even when considering the same constant viscosity for
each component of the mixture. It is for this reason that we do not address
the global well-posedness of the two-fluid model in this paper. We are in
position to state our second result.

We consider in $d=2$ the classical Navier-Stokes equations

$\displaystyle\begin{cases}\partial_{t}\rho+\operatorname{div}(\rho u)=0,\\\
\partial_{t}(\rho u)+\operatorname{div}(\rho u\otimes u)+\nabla
P(\rho)=\operatorname{div}(2\mu(\rho)\mathbb{D}u)+\nabla(\lambda(\rho)\operatorname{div}u),\end{cases}$
(1.27)

with initial data $(\rho_{0},u_{0})$ that fulfills,

$\displaystyle u_{0}\in H^{1}(\mathbb{R}^{2})\quad\text{ and
}\quad\rho_{0}-\widetilde{\rho}\in
L^{2}(\mathbb{R}^{2})\cap\mathscr{C}^{\alpha}_{pw,\gamma_{0}}(\mathbb{R}^{2}).$
(1.28)

Above, $\widetilde{\rho}>0$ is the reference density such that
$P(\widetilde{\rho})=\widetilde{P}$ and
$\mu(\widetilde{\rho})=\widetilde{\mu}$ with constant
$\widetilde{P},\;\widetilde{\mu}$ defined in (1.7). We assume that the initial
density is far away from vacuum that is, there exist constants
$0<\rho_{*}<\rho^{*}$ such that:

$\displaystyle 0<\rho_{*}<\rho_{0}<\rho^{*}.$ (1.29)

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:

$\displaystyle P,\;\mu,\;\lambda\in\mathscr{C}^{2}([\rho_{*}/4,4\rho^{*}]).$
(1.30)

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

$\displaystyle
P^{\prime}_{[\rho_{*}/4,4\rho^{*}]}>0\quad\text{and}\quad\mu_{*}\leqslant\mu_{[\rho_{*}/4,4\rho^{*}]},\;\lambda_{[\rho_{*}/4,4\rho^{*}]}\leqslant\mu^{*}.$
(1.31)

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

$\displaystyle\mathcal{C}(0)=\\{\gamma_{0}(s)\in\mathbb{R}^{2}\colon
s\in[0,2\pi]\\}$ (1.32)

where $\mathcal{C}(0)$ is the boundary of an open, bounded, simply connected
set $D(0)$ and fulfills the condition (1.9). The initial data will be measured
by the following constant:

$\displaystyle c_{0}:=\lVert
u_{0}\rVert_{H^{1}(\mathbb{R}^{2})}^{2}+\lVert\rho_{0}-\widetilde{\rho}\rVert_{L^{2}(\mathbb{R}^{2})\cap\mathscr{C}^{\alpha}_{pw,\gamma_{0}}(\mathbb{R}^{2})}^{2}+\lVert\llbracket\rho_{0}\rrbracket\rVert_{L^{2}(\mathcal{C}(0))\cap
L^{\infty}(\mathcal{C}(0))}^{2}.$ (1.33)

Let $(\rho_{0},u_{0})$ be an initial data associated with the Navier-Stokes
equations (1.27) and fulfills (1.28) and (1.29). 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 (1.27) with initial data $(\rho_{0},u_{0})$ verifying:

$\displaystyle\forall\;t>0,\quad\mathcal{A}_{1}(t)+\mathcal{A}_{2}(t)+\mathcal{A}_{3}(t)+\sqrt{\text{\textbeta}(t)}+\sqrt{\text{\texttheta}(t)}+\sup_{[0,t]}\lVert
u,\;\rho-\widetilde{\rho}\rVert^{2}_{L^{2}(\mathbb{R}^{2})}+\int_{0}^{t}\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\leqslant Cc_{0},$ (1.34)

where

$\displaystyle\begin{cases}\displaystyle\mathcal{A}_{1}(t)=\sup_{[0,t]}\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\int_{0}^{t}\lVert\sqrt{\rho}\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2},&\displaystyle\mathcal{A}_{2}(t)=\sup_{[0,t]}\sigma\lVert\sqrt{\rho}\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\int_{0}^{t}\sigma\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2},\\\
\displaystyle\mathcal{A}_{3}(t)=\sup_{[0,t]}\sigma^{2}\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\int_{0}^{t}\sigma^{2}\lVert\sqrt{\rho}\ddot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2},&\displaystyle\text{\texttheta}(t)=\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau,\\\
\displaystyle\text{\textbeta}(t)=\sup_{[0,t]}\lVert
f(\rho)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2})}^{4}+\int_{0}^{t}\lVert
f(\rho(\tau))\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau,&\displaystyle
f(\rho):=\int_{\widetilde{\rho}}^{\rho}\dfrac{2\mu(s)+\lambda(s)}{s}ds\end{cases}$
(1.35)

and $r_{\alpha}:=\max\\{7/5;\;1+2\alpha\\}<3$. The constant $C$ in (1.34)
depends on $\alpha,\rho_{*},\,\rho^{*},\,\mu_{*},\,\mu^{*}$ and
$c_{\gamma}(0)$ and the regularity of the initial interface. The initial data
is less regular than in section 1.2, in particular, we do not require the
compatibility condition (1.11). 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 [17]. The function $f(\rho)$ defined in
(1.35) verifies the following:

$\displaystyle\llbracket f(\rho(t,\gamma(t,s)))\rrbracket=\llbracket
f(\rho_{0},\gamma_{0}(s))\rrbracket\exp\left[\int_{0}^{t}-\left(g(\tau,s)+2h(\tau,s)<\operatorname{div}u(\tau,\gamma(\tau,s))>-<\mathbb{D}^{jk}u(\tau,\gamma(\tau,s))>n^{j}_{x}n^{k}_{x}(\tau,s)\right)d\tau\right]$
(1.36)

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 viscosities, $f$ 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 D. Hoff in [17, 18].

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 (1.25) and the vorticity in (1.26) at the interface lead to
the exponential decay of the jump of these quantities. In consequence, the
vector field $\mathrm{a}$ given by (1.23) decays exponentially in time so as
for the jump of the gradient of velocity since (1.22) 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 $\gamma(t,s)$. As we will see in
(3.62), we write from the momentum equation (1.27) 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:

$\displaystyle\nabla u=\nabla u_{*}+\nabla u_{F}+\nabla u_{P}+\nabla
u_{\delta}.$ (1.37)

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 appendix A and the extension result of piecewise Hölder
functions appendix A 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 [9] in the context of the incompressible Navier-Stokes model.

##### Outline of the paper

The remainder of the paper is organized as follows: in the following section,
section 2, we present the proof of the local result section 1.2. It relies on
a change into Lagrangian coordinates followed by homotopy arguments. section 3
is devoted to the proof of the global result. It follows from routine a priori
estimates for the velocity $u$, the material derivative of the velocity
$\dot{u}$ and the Hölder estimate for $\nabla u$ and the density. We achieve
higher integrability of $\nabla\dot{u}$ compared to what is often provided in
[15, 17] by computing a third functional.

##  2\. Proof of the local wellposedness result

This part is devoted to the proof of section 1.2. It will be divided into
three sections: in the first one, section 2.1, we rewrite system (1.5) in
Lagrangian coordinates, and in the second section, section 2.2 we study the
linearized system associated to (1.5). Finally the study of the full nonlinear
system is presented in section 2.3.

###  2.1. Change of variables in Lagrangian coordinates

The aim of this section is to write the system (1.5) in Lagrangian
coordinates. We know that for a Lipschitz vector field $u$, one can define a
flow map $\mathscr{X}$ given by the Duhamel formula:

$\mathscr{X}(t,y)=y+\int_{0}^{t}u(s,\mathscr{X}(s,y))ds$

and adopting the following notation: for all function $g=g(t,x)$ we denote by
$\overline{g}$ the function:

$\overline{g}(t,y):=g(t,\mathscr{X}(t,y))$

then, the flow of the vector field $u$ can be written in term of
$\overline{u}$ by :

$\mathscr{X}(t,y)=y+\int_{0}^{t}\overline{u}(s,y)ds.$

And for any $g=g(t,x)$, one has that

$\displaystyle\partial_{t}\overline{g}(t,y)=(\partial_{t}g+u\cdot\nabla
g)(t,\mathscr{X}(t,y)).$ (2.1)

Let us notice that by chain rule, for any vector field $G$, one has

$D\overline{G}=DG\circ\mathscr{X}\cdot D\mathscr{X},$

but, since

$\displaystyle(D\mathscr{X}(t,y))_{ij}=\delta^{ij}+\int_{0}^{t}\partial_{j}\overline{u}^{i}(s,y)ds,$
(2.2)

it holds that, at least, for small time, the Jacobian matrix $D\mathscr{X}$ of
the change of variable is inversible and then

$\overline{DG}(t,y)=D\overline{G}(t,y)\cdot(D\mathscr{X}(t,y))^{-1}=\dfrac{1}{\det
D\mathscr{X}(t,y)}D\overline{G}(t,y)\cdot\operatorname{Adj}D\mathscr{X}(t,y).$

In particular, one has that:

$\displaystyle\overline{\partial_{j}G^{i}}=\dfrac{1}{\det
D\mathscr{X}}\partial_{k}\overline{G}^{i}\left(\operatorname{Adj}D\mathscr{X}\right)_{kj}.$
(2.3)

On the other hand, from the Piola formula: for any matrix
$A\in\mathcal{M}_{n\times n}(\mathbb{R})$,

$\partial_{k}(\operatorname{Adj}(A))_{kj}=0$

one deduces that:

$\displaystyle\overline{\partial_{j}G^{i}}=\dfrac{1}{\det
D\mathscr{X}}\partial_{k}\left[\overline{G}^{i}\left(\operatorname{Adj}D\mathscr{X}\right)_{kj}\right].$
(2.4)

The first form (2.3) will be referred as the non conservative form and the
second (2.4) is the conservative one and both will be used. From the non
conservative form, it turns out that

$\overline{\operatorname{div}G}=\dfrac{1}{\det
D\mathscr{X}}Tr(D\overline{G}\cdot\operatorname{Adj}D\mathscr{X}),$

and by setting

$A(t,y)=(D\mathscr{X}(t,y))^{-1}$

one has that

$\displaystyle\operatorname{div}_{A}G:=\overline{\operatorname{div}G}=D\overline{G}:A=A^{T}:\nabla\overline{G}.$
(2.5)

On the other hand, for all matrix
$B\in\mathscr{C}^{1}([0,T],\mathcal{M}_{n\times n}(\mathbb{R}))$, one has:

$\displaystyle\dfrac{d\det B}{ds}(s)$ $\displaystyle=\dfrac{\partial\det
B}{\partial b_{ij}}\dfrac{\partial b_{ij}}{\partial s}(s)$
$\displaystyle=(-1)^{i+j}M_{ij}(B(s))\dfrac{\partial b_{ij}}{\partial s}(s)$
$\displaystyle=\operatorname{Adj}(B(s))_{ji}\left(\dfrac{\partial B}{\partial
s}(s)\right)_{ij}$ $\displaystyle=\dfrac{\partial B(s)}{\partial
s}:\operatorname{Adj}(B(s)).$

In particular, for $B=D\mathscr{X}$, and remembering from (2.2) that

$\dfrac{\partial D\mathscr{X}}{\partial t}=D\overline{u}$

one deduces that

$\dfrac{\partial\det D\mathscr{X}}{\partial
t}(t,y)=D\overline{u}(t,y):\operatorname{Adj}D\mathscr{X}(t,y)).$

Thus, from the identity (2.5), one has that

$\displaystyle\dfrac{\partial\det D\mathscr{X}}{\partial t}(t,y)=\det
D\mathscr{X}(t,y)\operatorname{div}_{A}u(t,y).$ (2.6)

Finally, if we combine (2.1) and the previous identity, one notices that the
continuity equation on the density reads:

$\dfrac{\partial(\overline{\rho}\det D\mathscr{X})}{\partial t}(t,y)=0$

and consequently

$\overline{\rho}(t,y)=\dfrac{\rho_{0}(y)}{\det D\mathscr{X}(t,y)}.$

Let us turn now to the rewriting of the momentum equation in Lagrangian
coordinates. First, from what precedes, it is clear that

$\left[\partial_{t}(\rho u^{j})+\operatorname{div}(\rho
u^{j}u)\right](t,\mathscr{X}(t,y))=\dfrac{1}{\det
D\mathscr{X}(t,y)}\rho_{0}(y)\partial_{t}\overline{u}(t,y)$

so it only remains to compute the term of the right hand side of the momentum
equation. From the conservative form (2.4), one has that:

$\displaystyle\overline{\operatorname{div}\Pi^{j}}$
$\displaystyle=\dfrac{1}{\det
D\mathscr{X}(t,y)}\partial_{k}\left\\{(\operatorname{Adj}D\mathscr{X})_{kl}\overline{\Pi}^{j}_{l}\right\\}$
$\displaystyle=\dfrac{1}{\det
D\mathscr{X}(t,y)}\partial_{k}\left\\{(\operatorname{Adj}D\mathscr{X})_{kl}\left\\{2\mu(\overline{\rho},c_{0})\mathbb{D}_{A}^{jl}\overline{u}+\left(\lambda(\overline{\rho},c_{0})\operatorname{div}_{A}\overline{u}-P(\overline{\rho},c_{0})+\widetilde{P}\right)\delta_{jl}\right\\}\right\\}$

where, by the help of the non conservative form (2.3), one has

$2\mathbb{D}_{A}v=Dv\cdot A+A^{T}\cdot\nabla v.$

This ends the transformation of the momentum equation in Lagrangian
coordinates. Finally, if $(c,\rho,u)$ solves the system (1.5), then
$(\overline{c},\overline{\rho},\overline{u})$ solves:

$\displaystyle\begin{cases}\partial_{t}\overline{c}=0,\\\
\partial_{t}\left\\{\overline{\rho}J\right\\}=0,\\\
\rho_{0}\partial_{t}\overline{u}=\operatorname{div}\left\\{\operatorname{Adj}(D\mathscr{X})\left\\{2\mu(\overline{\rho},c_{0})\mathbb{D}_{A}\overline{u}+\left(\lambda(\overline{\rho},c_{0})\operatorname{div}_{A}\overline{u}-P(\overline{\rho},c_{0})+\widetilde{P}\right)Id\right\\}\right\\}\end{cases}$
(2.7)

where $J:=\lvert\det D\mathscr{X}\rvert$.

###  2.2. Study of the linearised system

####  2.2.1. Statement of the main result

The aim of this section is to prove the local-wellposedness of the linear non-
homogeneous parabolic equation

$\displaystyle\rho_{0}u_{t}-\operatorname{div}(2\mu\mathbb{D}u+\lambda\operatorname{div}uI_{d})=\operatorname{div}F$
(2.8)

associated to (2.7). The initial density $\rho_{0}=\rho_{0}(x)$ and the
viscosities $\mu=\mu(t,x),\;\lambda=\lambda(t,x)$, are supposed to verify:

$\displaystyle\begin{cases}0<\rho_{*}\leqslant\rho_{0}\leqslant\rho^{*},\;0<\mu_{*}\leqslant\mu,\,\lambda\leqslant\mu^{*},\\\
\mu,\;\lambda\in
L^{\infty}((0,T),\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})),\\\
\mu_{t},\;\lambda_{t}\in L^{\infty}((0,T),L^{4}(\mathbb{R}^{d}))\cap
L^{1}((0,T),L^{\infty}(\mathbb{R}^{d}))\cap
L^{16}((0,T),L^{8}(\mathbb{R}^{d})),\sqrt{\sigma}(\mu_{t},\,\lambda_{t})\in
L^{2}((0,T),L^{\infty}(\mathbb{R}^{d})),\\\ \mu_{tt},\,\lambda_{tt}\in
L^{1}((0,T),L^{2}(\mathbb{R}^{d})),\;\sqrt{\sigma}(\mu_{tt},\lambda_{tt})\in
L^{8/d}((0,T),L^{4}(\mathbb{R}^{d})),\end{cases}$ (2.9)

for positive constants $\rho_{*},\rho^{*},\mu_{*},\mu^{*}$ and time
$0<T<\infty$. Above, $\gamma$ is a parametrization of an hyper-surface
$\mathcal{C}\in\mathscr{C}^{1+\alpha}$ in $\mathbb{R}^{d}$, $d\in\\{2,3\\}$:

$\mathcal{C}:=\left\\{\gamma(s)\in\mathbb{R}^{d}\colon
s\in\mathbb{T}^{d-1}\right\\}$

that verifies the following non self-intersection condition:

$\exists\;c_{\gamma}>0,\quad\forall\;(s,s^{\prime})\in(\mathbb{T}^{d-1})^{2},\;\quad\lvert\gamma(s)-\gamma(s^{\prime})\rvert\geqslant
c_{\gamma}^{-1}\lvert s-s^{\prime}\rvert.$

We refer to section 1.1 for the definition of the space of piecewise Hölder
continuous functions. The source term $F$ in (2.8) and the initial datum
$u_{0}$ are such that $(F,u_{0})\in Y_{T}(\mu_{0},\lambda_{0})$ where:

$\displaystyle Y_{T}(\mu_{0},\lambda_{0}):=\left\\{(F,u_{0})\right.$
$\displaystyle\left.\in\mathscr{C}([0,T],L^{2}(\mathbb{R}^{d}))\times
H^{1}(\mathbb{R}^{d})\colon F\in
L^{\infty}((0,T),L^{6}(\mathbb{R}^{d})),\right.$ $\displaystyle\left.F\in
L^{16}((0,T),L^{8}(\mathbb{R}^{d})),\;\sigma^{r/4}F\in
L^{4}((0,T),\,\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})),\right.$
$\displaystyle\left.F_{t}\in
L^{2}((0,T)\times\mathbb{R}^{d}),\;\sqrt{\sigma}F_{t}\in
L^{8/d}((0,T),L^{4}(\mathbb{R}^{d})),\right.$ $\displaystyle\left.\sigma
F_{tt}\in
L^{2}((0,T)\times\mathbb{R}^{d}),\;\operatorname{div}(2\mu_{0}\mathbb{D}u_{0}+\lambda_{0}\operatorname{div}u_{0}I_{d}-F_{|t=0})\in
L^{2}(\mathbb{R}^{d})\right\\}.$

Above, $\mu_{0}:=\mu_{|t=0}$, $\lambda_{0}:=\lambda_{|t=0}$ and $0<r<2$
depends on $0<\alpha<1$ and the dimension $d\in\\{2,3\\}$. This dependence is
explicitly given in (2.39). The existence and uniqueness of solution to (2.8)
is established in the following space:

$\displaystyle X_{T}:=\left\\{u\in\mathscr{C}([0,T],\right.$
$\displaystyle\left.H^{1}(\mathbb{R}^{d}))\colon\right.\nabla u\in
L^{\infty}((0,T),L^{6}(\mathbb{R}^{d}))\cap
L^{16}((0,T),L^{8}(\mathbb{R}^{d})),\,\sigma^{r/4}\nabla u\in
L^{4}((0,T),\,\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})),$
$\displaystyle\left.u_{t}\in\mathscr{C}([0,T],L^{2}(\mathbb{R}^{d}))\cap
L^{2}((0,T),\dot{H}^{1}(\mathbb{R}^{d})),\;\sqrt{\sigma}\nabla u_{t}\in
L^{8/d}((0,T),L^{4}(\mathbb{R}^{d}))\cap
L^{\infty}((0,T),L^{2}(\mathbb{R}^{d})),\right.$
$\displaystyle\left.\sqrt{\sigma}u_{tt}\in
L^{2}((0,T)\times\mathbb{R}^{d}),\;\sigma u_{tt}\in
L^{\infty}((0,T),L^{2}(\mathbb{R}^{d}))\cap
L^{2}((0,T),\dot{H}^{1}(\mathbb{R}^{d}))\right\\}.$ (2.10)

The time weights are inspired by the pioneer work of Hoff [15]. One of the
main difficulties is to obtain estimates for the gradient of $u$, of the
equations (2.8), allowing to propagate the regularity of the interface
$\gamma$. The difficulty stems from the fact that the viscosities do not have
any Sobolev regularity.

The main result of this section is stated in the following theorem:

Let $0<T<\infty$, the viscosities $\mu,\,\lambda$ and the initial density
$\rho_{0}$ verify (2.9), $0<\alpha<1$. There exists a constant $[\mu]$
depending on the dimension $d$, on $\alpha$ and on the interface $\gamma$,
such that if:

$\displaystyle\dfrac{1}{\widetilde{\mu}}\lVert\mu-\widetilde{\mu}\rVert_{L^{\infty}((0,T),\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d}))}\leqslant[\mu],$
(2.11)

then, for all $(F,u_{0})\in Y_{T}(\mu_{0},\lambda_{0})$, there exists an
unique solution of the Cauchy problem:

$\displaystyle\begin{cases}\rho_{0}u_{t}-\operatorname{div}(\mu\mathbb{D}u)-\nabla(\lambda\operatorname{div}u)=\operatorname{div}F,\\\
u_{|t=0}=u_{0},\end{cases}$ (2.12)

in $X_{T}$. Moreover, the following estimates hold true for some constant $M$
that depends on the upper and lower bounds of the density and viscosities (see
first line of (2.9)):

  1. (1)

Estimates for $u$ and $u_{t}$:

$\displaystyle\sup_{[0,T]}\lVert u,\,u_{t},\,\nabla
u\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$ $\displaystyle+\lVert\nabla
u\rVert_{L^{\infty}((0,T),L^{6}(\mathbb{R}^{d}))}^{2}+\int_{0}^{T}\lVert\nabla
u,\,\nabla u_{t},\,u_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle\leqslant
M\left(\lVert(F,u_{0})\rVert_{Y_{T}(\mu_{0},\lambda_{0})}^{2}+\int_{0}^{T}\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{4}(\mathbb{R}^{d})}^{2}\lVert
F\rVert_{L^{2}\cap L^{6}(\mathbb{R}^{d})}^{2}\right)$
$\displaystyle\times\exp\left(M\int_{0}^{T}\left[\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{\infty}(\mathbb{R}^{d})}+\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{4}(\mathbb{R}^{d})}^{2}\right]\right).$
(2.13)

  2. (2)

Estimates for $u_{tt}$:

$\displaystyle\int_{0}^{T}\lVert\sqrt{\sigma}u_{tt},\;\sigma\nabla
u_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle+\lVert\sqrt{\sigma}\nabla u_{t},\;\sigma
u_{tt}\rVert^{2}_{L^{\infty}((0,T),L^{2}(\mathbb{R}^{d}))}\leqslant
M\left[\int_{0}^{\sigma(T)}\lVert\nabla
u_{t}\rVert^{2}_{L^{2}(\mathbb{R}^{d})}+\int_{0}^{T}\lVert F_{t},\,\sigma
F_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\right.$
$\displaystyle\left.+\int_{0}^{T}\lVert\mu_{t},\,\lambda_{t},\,\sigma(\mu_{tt},\,\lambda_{tt})\rVert_{L^{4}(\mathbb{R}^{d})}^{2}\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{d})}^{2}\right]$
$\displaystyle\times\exp\left(M\int_{0}^{T}\left[\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{\infty}(\mathbb{R}^{d})}+\lVert\mu_{tt},\lambda_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}\right]+M\int_{0}^{T}\sigma\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{\infty}(\mathbb{R}^{d})}^{2}\right).$
(2.14)

The norms of $u$ involved in this estimate are controlled thanks to (2.13).

The first estimate (2.13) follows from the classical energy estimate and the
so-called Hoff estimates introduced in [15, 17]. These estimates provide
$H^{1}$ regularity for the time derivative of the velocity $u$. The other
estimate (2.14) can be seen as a higher Hoff estimate and has be used in [9]
in the study of the incompressible Navier-Stokes model. The solution in the
above theorem comes with subsequent estimates:

  1. (1)

$L^{16}((0,T),L^{8}(\mathbb{R}^{d}))$ estimate for the $\nabla u$:

$\displaystyle\lVert\nabla
u\rVert_{L^{16}((0,T),L^{8}(\mathbb{R}^{d}))}\lesssim\lVert
F\rVert_{L^{16}((0,T),L^{8}(\mathbb{R}^{d}))}+\lVert F,u_{t},\nabla
u\rVert_{L^{2}\cap L^{\infty}((0,T),L^{2}(\mathbb{R}^{d}))}+\lVert\nabla
u_{t}\rVert_{L^{2}((0,T)\times\mathbb{R}^{d})}.$ (2.15)

  2. (2)

$L^{8/d}((0,T),L^{4}(\mathbb{R}^{d}))$ norm estimate for $\sqrt{\sigma}\nabla
u_{t}$:

$\int_{0}^{T}\sigma^{4/d}\lVert\nabla
u_{t}\rVert_{L^{4}(\mathbb{R}^{d})}^{8/d}\lesssim\int_{0}^{T}\sigma^{4/d}\lVert
F_{t}\rVert_{L^{4}(\mathbb{R}^{d})}^{8/d}+\int_{0}^{T}\sigma^{4/d}\lVert\nabla
u\rVert_{L^{8}(\mathbb{R}^{d})}^{8/d}\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{8}(\mathbb{R}^{d})}^{8/d}+\lVert\sigma
u_{tt}\rVert_{L^{\infty}((0,T),L^{2}(\mathbb{R}^{d}))}^{8/d}\\\
+\left[\int_{0}^{T}\lVert\nabla
u_{t},\,F_{t},\,\lambda_{t}\operatorname{div}u,\,\mu_{t}\mathbb{D}u\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\right]^{4/d}.$
(2.16)

  3. (3)

Hölder estimate for $\nabla u$:

$\displaystyle\int_{0}^{T}\sigma^{r}\lVert\nabla
u\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})}^{4}$
$\displaystyle\lesssim(1+\mathscr{P}(\mathcal{C}))\int_{0}^{T}\sigma^{r}\lVert
F\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})}+\lVert F,\,\nabla
u,\,u_{t},\,\sqrt{\sigma}\nabla
u_{t}\rVert_{L^{\infty}((0,T),L^{2}(\mathbb{R}^{d}))}^{4}$
$\displaystyle+\left[\int_{0}^{T}\lVert F,\,\nabla u,\,u_{t},\,\nabla
u_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\right]^{2}+\left[\int_{0}^{T}\sigma^{4/d}\lVert\nabla
u_{t}\rVert_{L^{4}(\mathbb{R}^{d})}^{8/d}\right]^{d/2}.$ (2.17)

####  2.2.2. Sketch of the Proof

The proof of the above theorem is done in section 2.2.5 and is based on a
homotopy argument. For this purpose, we define for $\theta\in[0,1]$

$\mu^{\theta}=(1-\theta)\widetilde{\mu}+\theta\mu\quad\text{ and
}\quad\lambda^{\theta}=(1-\theta)\widetilde{\lambda}+\theta\lambda$

and consider the equation (2.12) where the viscosities $\mu,\,\lambda$ are
replaced by $\mu^{\theta}$ and $\lambda^{\theta}$ respectively :

$\displaystyle\begin{cases}\rho_{0}u_{t}-\operatorname{div}(2\mu^{\theta}\mathbb{D}u+\lambda^{\theta}\operatorname{div}uI_{d})=\operatorname{div}F,\\\
u_{|t=0}=u_{0}.\end{cases}$ (2.18)

An important remark is that the last condition in the definition of the set
$Y_{T}(\mu_{0},\lambda_{0})$ complicates the proof. In fact, it is not obvious
that if $(F,u_{0})$ belongs to $Y_{T}(\mu^{\theta}_{0},\lambda^{\theta}_{0})$
then it also belongs to
$Y_{T}(\mu^{\theta^{\prime}}_{0},\lambda^{\theta^{\prime}}_{0})$. In general,
this is not the case unless $F_{|t=0}$, $\nabla u_{0}$, $\lambda_{|t=0}$ and
$\mu_{|t=0}$ have a Sobolev regularity. Of course, this condition does not
hold true in our context because $F$ and the viscosities are assumed to be
discontinuous across a hyper-surface so that their weak gradient are just a
measure. In order to ease the reading, we set
$Y_{T}^{\theta}:=Y_{T}(\mu^{\theta}_{0},\lambda^{\theta}_{0})$. The principle
of homotopy argument is to look for $\theta\in[0,1]$ such that (2.18) is
uniquely solvable in $X_{T}$. So we define the following set:

$\displaystyle\mathscr{E}_{T}:=\\{\theta\in[0,1]\colon\forall(F,u_{0})\in
Y_{T}^{\theta},\,\exists\,!\;u\in X_{T}\text{ solution of }\eqref{eq3.19}\\}.$
(2.19)

It is obvious that section 2.2.1 follows as soon as $1\in\mathscr{E}$. This
will be obtained by proving that $\mathscr{E}$ is at the same time a non-
empty, open and closed set of $[0,1]$. The proof is structured into three
steps: In the first one, section 2.2.3, we derive some a priori estimates on a
solution $u\in X_{T}$ of the equation (2.12). In the second step, we prove
that $0\in\mathscr{E}$ and finally, we prove in section 2.2.5 that
$\mathscr{E}$ is an open and a closed set of $[0,1]$ from which, section 2.2.1
follows. In the following, we will discuss a little bit the main difficulties
in proving that $0\in\mathscr{E}$.

When $\theta=0$, the equation (2.18) is reduced to the following:

$\displaystyle\begin{cases}\rho_{0}u_{t}-\widetilde{\mu}\Delta
u-(\widetilde{\mu}+\widetilde{\lambda})\nabla\operatorname{div}u=\operatorname{div}F,\\\
u_{|t=0}=u_{0}\end{cases}$ (2.20)

and the aim is to prove that for all $(F,u_{0})\in Y_{T}^{0}$ there exists an
unique solution of (2.20) in $X_{T}$. This is not obvious since,
$\eqref{eq3.37}$ is not really a heat equation and also the initial density is
rough. Despite, one can add $(\widetilde{\rho}-\rho_{0})u_{t}$ in both sides
of (2.20), in order to obtain a heat equation with a right hand side that
belongs to
$L^{2}((0,T),L^{2}(\mathbb{R}^{d}))+L^{2}((0,T),\dot{H}^{-1}(\mathbb{R}^{d}))$.
As in [8], we may require a smallness of $\widetilde{\rho}-\rho_{0}$ in order
to obtain a stability for approximate solution. In order to get around this
difficulty, the strategy adopted is that, we consider a mollifying sequence of
the initial density $\rho_{0}$, velocity $u_{0}$ and $F$ respectively
$\rho_{0,\delta}$ $u_{0,\delta}$, and $F_{\delta}$ and consider the equation:

$\displaystyle\begin{cases}\rho_{0,\delta}u_{t}-\widetilde{\mu}\Delta
u-(\widetilde{\mu}+\widetilde{\lambda})\nabla\operatorname{div}u=\operatorname{div}F_{\delta},\\\
u_{|t=0}=u_{0,\delta}.\end{cases}$ (2.21)

The existence and uniqueness of global solution $u^{\delta}$ of the above
system is based on, a change of unknowns, a study of the heat kernel and a
homotopy argument. Notice that the homotopy argument applied in this context
seems simpler than the one we are applying in order to deduce the proof of
section 2.2.1. Taking the advantage that for any $\delta>0$, $\rho_{0,\delta}$
is smooth enough and bounded away from vacuum, one has that $u$ solves (2.21)
if and only if $v:=\rho_{0,\delta}u$ solves the following Cauchy problem:

$\displaystyle\begin{cases}v_{t}^{j}-\operatorname{div}\left(\dfrac{\widetilde{\mu}}{\rho_{0,\delta}}\mathbb{D}^{j}v\right)-\partial_{j}\left(\dfrac{\widetilde{\lambda}}{\rho_{0,\delta}}\operatorname{div}v\right)&=\operatorname{div}F_{\delta}^{j}-\widetilde{\mu}\partial_{k}\left(\rho_{0,\delta}^{-2}\left(v^{k}\partial_{j}\rho_{0,\delta}+v^{j}\partial_{k}\rho_{0,\delta}\right)\right)-\widetilde{\lambda}\partial_{j}\left(\rho_{0,\delta}^{-2}v\cdot\nabla\rho_{0,\delta}\right),\\\
v_{|t=0}&=\rho_{0,\delta}u_{0,\delta}.\end{cases}$ (2.22)

In order to ease computations, we consider the following problem for
$\mu^{\prime}_{0},\lambda^{\prime}_{0},b_{0}=\mu^{\prime}_{0}(x),\lambda^{\prime}_{0}(x),b_{0}(x)$

$\displaystyle\begin{cases}v_{t}-\operatorname{div}(2\mu_{0}^{\prime}\mathbb{D}v+\lambda_{0}^{\prime}\operatorname{div}vI_{d})=f+\operatorname{div}(b_{0}\,\colon
v),\\\ v_{|t=0}=v_{0}\end{cases}$ (2.23)

and we have the following:
Let $0<T<\infty$ and $f\in L^{2}((0,T)\times\mathbb{R}^{d})$,
$0<\mu_{*}\leqslant\mu_{0}^{\prime},\,\lambda_{0}^{\prime}\in
W^{1,\infty}(\mathbb{R}^{d})$, $b_{0}\in W^{1,\infty}(\mathbb{R}^{d})$, and
finally $v_{0}\in H^{1}(\mathbb{R}^{d})$. Then there exists an unique solution
$v\in\mathscr{C}([0,T],H^{1}(\mathbb{R}^{d}))$ with $v_{t},\nabla^{2}v\in
L^{2}((0,T),L^{2}(\mathbb{R}^{d}))$ for the system (2.23). Moreover, there
exist constants $\Lambda,\,K>0$ that depend on the norms of
$\mu^{\prime}_{0},\,\lambda_{0}^{\prime},\,b_{0}$ and the lower and upper
bounds $\mu_{*},\mu^{*}$ such that

$\sup_{[0,T]}e^{-2\Lambda t}\left\\{\lVert
v(t)\rVert_{H^{1}(\mathbb{R}^{d})}^{2}+\int_{0}^{t}\lVert v_{t},\,\nabla
v,\,\nabla^{2}v\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\right\\}\leqslant
K\left(\lVert v_{0}\rVert^{2}_{H^{1}(\mathbb{R}^{d})}+\int_{0}^{T}\lVert
f(s)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}ds\right).$

The proof of section 2.2.2 is given in section 2.2.4.

One deduces from section 2.2.2, (2.22) and then $\eqref{eq3.22}$ that the
Cauchy problem (2.21) admits an unique solution
$u^{\delta}\in\mathscr{C}([0,T],H^{1}(\mathbb{R}^{d}))$ with
$u_{t}^{\delta},\nabla^{2}u^{\delta}\in L^{2}((0,T)\times\mathbb{R}^{d})$.
Moreover, by deriving the equation (2.21) in space, and noticing that
$\rho_{0}^{\delta}$ and $u_{0}^{\delta}$ are smooth, one obtains that for all
$s\geqslant 2$, $u^{\delta}\in\mathscr{C}([0,T],H^{s}(\mathbb{R}^{d})),$ and
$u_{t}\,,\nabla^{2}u^{\delta}\in L^{2}((0,T),H^{s-1}(\mathbb{R}^{d}))$. In the
same manner, one can derive the equation (2.21) with respect to time, in order
to obtain the equation satisfied by $u_{t}^{\delta}$. Of course,
$F_{t}^{\delta}$ is regular enough and the regularity $u_{t}^{\delta}\in
L^{2}((0,T),H^{s-1}(\mathbb{R}^{d}))$ is sufficient in order to derive the
equation in a classical sense. Thus, applying again section 2.2.2 one deduces
from the previous discussion that
$u_{t}^{\delta}\in\mathscr{C}([0,T],H^{s}(\mathbb{R}^{d}))$ and
$u_{tt}^{\delta},\nabla u_{t}^{\delta}\in
L^{2}((0,T),H^{s-1}(\mathbb{R}^{d}))$. This closes the existence of global
smooth solution of the Cauchy problem (2.21). We continue the sketch of the
proof of section 2.2.1 in the following.

The sequence of solution $(u^{\delta})_{\delta}$ of the system (2.21) verifies
the estimates for $u$, $u_{t}$, (in (2.13)) $u_{tt}$ (in (2.14)) and the
$L^{p}$ estimates in section 2.2.1 for $u$ and $u_{t}$, uniformly with respect
to the regularisation parameter $\delta>0$ with the exception of the piecewise
Hölder estimate. Hence, one can let $\delta$ go to zero in order to obtain an
solution of the Cauchy problem (2.20). Since $(u^{\delta})_{\delta}$ is
bounded in $L^{\infty}((0,T),H^{1}(\mathbb{R}^{d}))$ and its time derivative
$(u_{t}^{\delta})_{\delta}$ is bounded in
$L^{2}((0,T),H^{1}(\mathbb{R}^{d}))$, it then turns out that
$u\in\mathscr{C}([0,T],H^{1}(\mathbb{R}^{d}))$. On the other hand, one recalls
that from the second Hoff estimate $(\rho_{0}u_{t}^{\delta})_{\delta}$ is
bounded in $L^{\infty}((0,T),L^{2}(\mathbb{R}^{d}))$ and the time derivative
of the equation (2.21) implies that $((\rho_{0}u_{t}^{\delta})_{t})_{\delta}$
is bounded in $L^{2}((0,T),H^{-1}(\mathbb{R}^{d}))$. Then, from [27, Appendix
C], $\rho_{0}u_{t}$ belongs to $\mathscr{C}([0,T],L^{2}_{w}(\mathbb{R}^{d}))$
and since $\rho_{0}$ is independent from time, one deduces that $u_{t}$ is
continuous in time with values in the weak topology of
$L^{2}(\mathbb{R}^{d})$. We will obtain the continuity of $u_{t}$ in time with
values in the strong topology in two steps. In the first step, we notice that
$(u_{t}^{\delta})_{\delta}$ is bounded in
$L^{\infty}((0,T),L^{2}(\mathbb{R}^{d}))\cap
L^{2}((0,T),\dot{H}^{1}(\mathbb{R}^{d}))$ and $(u_{tt}^{\delta})_{\delta}$ is
bounded in $L^{2}((\tau,T)\times\mathbb{R}^{d})$ for all $0<\tau<T$, and since
$H^{1}(\mathbb{R}^{d})$ embeds compactly in $L^{2}(\mathbb{R}^{d})$, one
deduces from the Aubin-Lions theorem that
$u_{t}\in\mathscr{C}((0,T],L^{2}(\mathbb{R}^{d})).$ It only remains the
continuity of $u_{t}$ at the initial time. At this level, we know that $u$
already solves (2.20), the regularity of $u_{t}$ and $u_{tt}$ ensures that one
can derive (2.20) in time and use $u_{t}$ as a test function in the weak
formulation of the obtained equation in order to obtain:

$\displaystyle\dfrac{1}{2}\int_{\mathbb{R}^{d}}\rho_{0}\lvert
u_{t}(\tau)\rvert^{2}+\int_{0}^{\tau}2\widetilde{\mu}\int_{\mathbb{R}^{d}}\lvert\mathbb{D}u_{t}\rvert_{L^{2}(\mathbb{R}^{d})}^{2}+\widetilde{\lambda}\int_{0}^{\tau}\int_{\mathbb{R}^{d}}(\operatorname{div}u_{t})^{2}=\dfrac{1}{2}\int_{\mathbb{R}^{d}}\rho_{0}\lvert{u_{t}}_{|t=0}\rvert^{2}-\int_{0}^{\tau}\int_{\mathbb{R}^{d}}\partial_{t}F^{jk}\partial_{k}u^{j}_{t}.$

From which we deduce that

$\limsup_{\tau\to
0}\lVert\sqrt{\rho_{0}}u_{t}(\tau)\rVert_{L^{2}(\mathbb{R}^{d})}\leqslant\lVert\sqrt{\rho_{0}}{u_{t}}_{|t=0}\rVert_{L^{2}(\mathbb{R}^{d})}.$

On the other hand, we have

$\lVert\sqrt{\rho_{0}}{u_{t}}_{|t=0}\rVert_{L^{2}(\mathbb{R}^{d})}\leqslant\liminf_{\tau\to
0}\lVert\sqrt{\rho_{0}}u_{t}(\tau)\rVert_{L^{2}(\mathbb{R}^{d})},$

and thus, we deduce that $\sqrt{\rho_{0}}u_{t}(\tau)$ convergences to
$\sqrt{\rho_{0}}{u_{t}}_{|t=0}$ strongly in $L^{2}(\mathbb{R}^{d})$. From the
fact that $\rho_{0}$ is bounded away from vacuum, we finally deduce the
continuity of $u_{t}$ at the initial time. Moreover, this solution is unique,
since, the difference $\delta u$ of two solutions $u_{1}$ and $u_{2}$ of
(2.20) verifies:

$\begin{cases}\rho_{0}\delta u_{t}-\widetilde{\mu}\Delta\delta
u-(\widetilde{\mu}+\widetilde{\lambda})\nabla\delta u=0,\\\ \delta
u_{|t=0}=0\end{cases}$

and one has, from the classical energy computation, that $\delta u=0$.
Furthermore the gradient of $u$ is given by:

$\nabla
u=-\frac{1}{\widetilde{\mu}}(-\Delta)^{-1}\nabla\left[\mathcal{P}\left(\rho_{0}u_{t}\right)-\mathcal{P}\operatorname{div}F\right]\\\
-\dfrac{1}{2\widetilde{\mu}+\widetilde{\lambda}}(-\Delta)^{-1}\nabla\left[\mathcal{Q}(\rho_{0}u_{t})-\mathcal{Q}\operatorname{div}F\right].$

where $\mathcal{P}$ is the Leray projector onto the space of divergence-free
vector fields and $\mathcal{Q}$ is the projector onto the space of curl-free
vector fields. Finally, using the regularity of $u_{t}$ and the Hölder
regularity of $F$, one has the last estimate of $\nabla u$ that is (2.17). All
these discussions establish that $\mathscr{E}$ is a non-empty set. The only
thing left to prove is that $\mathscr{E}$ is both an open and a closed subset
of $[0,1]$. This objective is purpose of the final section section 2.2.5.

####  2.2.3. A priori estimates

###### Proof.

This section consists in performing five kind of energy estimates on the
solution $u\in X_{T}$ of the problem (2.12). We notice that the regularity of
$u$ is enough in order to use $u$ or $u_{t}$ as test function in subsequent
calculations. Let us begin by the first energy estimate:

#####  First step: Estimates for $u$

This energy consists in taking the scalar product of the equation (2.12) with
the velocity $u$, in order to obtain the following:

$\displaystyle\dfrac{1}{2}\dfrac{d}{dt}\lVert\sqrt{\rho_{0}}u\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{\mathbb{R}^{d}}2\mu\lvert\mathbb{D}u\rvert^{2}+\int_{\mathbb{R}^{d}}\lambda\lvert\operatorname{div}u\rvert^{2}=-\int_{\mathbb{R}^{d}}F^{jk}\partial_{k}u^{j}.$

We then integrate the above in time in order to obtain the following:

$\displaystyle\dfrac{1}{2}\lVert\sqrt{\rho_{0}}u(t)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{t}\int_{\mathbb{R}^{d}}2\mu\lvert\mathbb{D}u\rvert^{2}+\int_{0}^{t}\int_{\mathbb{R}^{d}}\lambda\lvert\operatorname{div}u\rvert^{2}=\dfrac{1}{2}\lVert\sqrt{\rho_{0}}u_{0}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}-\int_{0}^{t}\int_{\mathbb{R}^{d}}F^{jk}\partial_{k}u^{j}$
(2.24)

and finally Hölder and Young inequalities yield:

$\displaystyle\sup_{[0,T]}\lVert\sqrt{\rho_{0}}u\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{T}\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\lesssim\lVert\sqrt{\rho_{0}}u_{0}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{T}\lVert
F\rVert_{L^{2}(\mathbb{R}^{d})}^{2}.$

#####  Second Step: Estimates for $u_{t}$

We use $u_{t}$ as a test function in the weak formulation of (2.8) in order to
obtain:

$\displaystyle\int_{0}^{t}\lVert\sqrt{\rho_{0}}u_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle+\int_{\mathbb{R}^{d}}\mu(t)\lvert\mathbb{D}u(t)\rvert^{2}+\dfrac{1}{2}\int_{\mathbb{R}^{d}}\lambda(t)(\operatorname{div}u(t))^{2}=\int_{\mathbb{R}^{d}}\mu_{0}\lvert\mathbb{D}u_{0}\rvert^{2}+\dfrac{1}{2}\int_{\mathbb{R}^{d}}\lambda_{0}(\operatorname{div}u_{0})^{2}$
$\displaystyle-\int_{\mathbb{R}^{d}}F^{jk}(s)\partial_{k}u^{j}(s)\bigg{|}_{s=0}^{s=t}+\int_{0}^{t}\int_{\mathbb{R}^{d}}F^{jk}_{t}\partial_{k}u^{j}+\int_{0}^{t}\int_{\mathbb{R}^{d}}\mu_{t}\lvert\mathbb{D}u\rvert^{2}+\dfrac{1}{2}\int_{0}^{t}\int_{\mathbb{R}^{d}}\lambda_{t}(\operatorname{div}u)^{2}$

In the following lines, we will estimate the right hand side in order to use
Grönwall’s lemma. The third and the fourth terms are:

$\displaystyle\Bigg{|}\int_{\mathbb{R}^{d}}F^{jk}(s)\partial_{k}u^{j}(s)\bigg{|}_{s=0}^{s=t}\Bigg{|}\leqslant\eta\lVert\nabla
u(t)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+C_{\eta}\lVert
F(t)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\lVert\nabla
u(0)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\lVert
F(0)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$

for some positive $\eta>0$ small in order to absorb the norm of the gradient
in the left hand side of the previous above equality. Next, the fifth term can
be bounded as follows:

$\bigg{|}\int_{0}^{t}\int_{\mathbb{R}^{d}}F^{jk}_{t}\partial_{k}u^{j}\bigg{|}\leqslant\int_{0}^{t}\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{t}\lVert
F_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}.$

Finally, the last two terms are controlled by:

$\displaystyle\int_{0}^{t}\int_{\mathbb{R}^{d}}\mu_{t}\lvert\mathbb{D}u\rvert^{2}+\dfrac{1}{2}\int_{0}^{t}\int_{\mathbb{R}^{d}}\lambda_{t}(\operatorname{div}u)^{2}\lesssim\int_{0}^{t}\left(\lVert\mu_{t}\rVert_{L^{\infty}(\mathbb{R}^{d})}+\lVert\lambda_{t}\rVert_{L^{\infty}(\mathbb{R}^{d})}\right)\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{d})}^{2}.$

Gathering all of these estimates, and choosing $\eta$ small, one has:

$\displaystyle\int_{0}^{t}\lVert\sqrt{\rho_{0}}u_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\sup_{[0,t]}\lVert\nabla
u\rVert^{2}_{L^{2}(\mathbb{R}^{d})}\lesssim\lVert\nabla
u_{0}\rVert^{2}_{L^{2}(\mathbb{R}^{d})}+\sup_{[0,t]}\lVert
F\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{t}\lVert
F_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{t}\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{\infty}(\mathbb{R}^{d})}\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{d})}^{2}.$

We turn to the computations of the second energy that consists in deriving the
equation (2.12) in time and use $u_{t}$ as test function in the weak
formulation of the obtained equation. By doing so, we obtain:

$\displaystyle\dfrac{1}{2}\int_{\mathbb{R}^{d}}\rho_{0}\lvert
u_{t}(t)\rvert^{2}+\int_{0}^{t}2\int_{\mathbb{R}^{d}}\mu\lvert\mathbb{D}u_{t}\rvert_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{R}^{d}}\lambda(\operatorname{div}u_{t})^{2}=\dfrac{1}{2}\int_{\mathbb{R}^{d}}\rho_{0}\lvert{u_{t}}_{|t=0}\rvert^{2}$
$\displaystyle-\int_{0}^{t}\int_{\mathbb{R}^{d}}2\mu_{t}\mathbb{D}^{jk}u\partial_{k}u^{j}_{t}-\int_{0}^{t}\int_{\mathbb{R}^{d}}\lambda_{t}\operatorname{div}u\operatorname{div}u_{t}+\int_{0}^{t}\int_{\mathbb{R}^{d}}F^{jk}_{t}\partial_{k}u^{j}_{t}$
$\displaystyle\leqslant\int_{\mathbb{R}^{d}}\rho_{0}\lvert{u_{t}}_{|t=0}\rvert^{2}+\eta\int_{0}^{t}\lVert\nabla
u_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+C_{\eta}\int_{0}^{t}\lVert
F_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle+C_{\eta}\int_{0}^{t}\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{4}(\mathbb{R}^{d})}^{2}\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{d})}^{2}$ (2.25)

with some positive $\eta$ small related to the lower and upper bounds of the
viscosities. We now turn to the estimate of the $L^{4}(\mathbb{R}^{d})$ norm
of $\nabla u$ that appears in the above inequality. We begin by observing that
from (2.12), one has:

$-\widetilde{\mu}\Delta
u-\nabla\\{(\widetilde{\mu}+\lambda)\operatorname{div}u\\}=\operatorname{div}\\{2(\mu-\widetilde{\mu})\mathbb{D}u\\}-\rho_{0}u_{t}+\operatorname{div}F$

and so, we can express:

$\displaystyle\nabla u$
$\displaystyle=\nabla\mathcal{P}u+\nabla(\mathcal{I-P})u$
$\displaystyle=\frac{1}{\widetilde{\mu}}(-\Delta)^{-1}\nabla\left[\mathcal{P}\operatorname{div}\\{2(\mu-\widetilde{\mu})\mathbb{D}u\\}-\mathcal{P}\left(\rho_{0}\partial_{t}u\right)+\mathcal{P}\operatorname{div}F\right]$
$\displaystyle+\dfrac{1}{2\widetilde{\mu}+\lambda}(-\Delta)^{-1}\nabla\left[\mathcal{Q}\operatorname{div}\\{2(\mu-\widetilde{\mu})\mathbb{D}u\\}-\mathcal{Q}\left(\rho_{0}\partial_{t}u\right)+\mathcal{Q}\left(\operatorname{div}F\right)\right].$
(2.26)

Above, $\mathcal{P}$ is the Leray projector onto the space of divergence-free
vector fields and where $\mathcal{Q}$ is the projector onto the space of curl-
free vector fields. Since $F\in L^{4}(\mathbb{R}^{d})$ a.e in time, singular
integral theory helps us to obtain:

$\displaystyle\lVert\nabla u\rVert_{L^{4}(\mathbb{R}^{d}))}$
$\displaystyle\lesssim_{d}\dfrac{1}{\widetilde{\mu}}\lVert\mu-\widetilde{\mu}\rVert_{L^{\infty}(\mathbb{R}^{d})}\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{d}))}+\lVert F\rVert_{L^{4}(\mathbb{R}^{d}))}$
$\displaystyle+\dfrac{1}{\widetilde{\mu}}\lVert(-\Delta)^{-1}\nabla\mathcal{P}(\rho_{0}u_{t}),\,(-\Delta)^{-1}\nabla\mathcal{Q}(\rho_{0}u_{t})\rVert_{L^{4}(\mathbb{R}^{d})}.$
(2.27)

Therefore, for small perturbation of the viscosity $\mu$ around the constant
state $\widetilde{\mu}$, one can absorb the first term of the right hand side
in the left one. By Gagliardo-Nirenberg inequality, the last terms above can
be bounded by:

$\lVert(-\Delta)^{-1}\nabla\mathcal{P}(\rho_{0}u_{t}),\,(-\Delta)^{-1}\nabla\mathcal{Q}(\rho_{0}u_{t})\rVert_{L^{4}(\mathbb{R}^{d}))}\lesssim\lVert\rho_{0}u_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{d/4}\lVert\nabla
u,F\rVert_{L^{2}(\mathbb{R}^{d})}^{1-d/4}.$

In consequence, the last term of (2.25) can be estimated as:

$\displaystyle\int_{0}^{t}\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{4}(\mathbb{R}^{d}))}^{2}\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{d}))}^{2}\lesssim\int_{0}^{t}\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{4}(\mathbb{R}^{d}))}^{2}\lVert
F\rVert_{L^{4}(\mathbb{R}^{d}))}^{2}+\int_{0}^{t}\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{4}(\mathbb{R}^{d}))}^{2}\lVert
F,\;\nabla u,\;u_{t}\rVert_{L^{2}(\mathbb{R}^{d}))}^{2}.$ (2.28)

We now add (2.28) to (2.25) and choose $\eta$ related to the lower bound and
or upper bounds of the viscosities and density in order to absorb the second
term of the right hand side of (2.25) in the left hand side and then obtains,
after Gronwall’s Lemma:

$\sup_{[0,T]}\lVert u_{t},\,\nabla
u\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{T}\lVert\nabla
u_{t},\,u_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\leqslant
M\left(\lVert(F,u_{0})\rVert_{Y_{T}(\mu_{0},\lambda_{0})}^{2}+\int_{0}^{T}\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{4}(\mathbb{R}^{d}))}^{2}\lVert
F\rVert_{L^{2}\cap L^{6}(\mathbb{R}^{d}))}^{2}\right)\\\
\times\exp\left(M\int_{0}^{T}\left[\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{\infty}(\mathbb{R}^{d})}+\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{4}(\mathbb{R}^{d})}^{2}\right]\right).$
(2.29)

We turn to the estimate for the second time derivative of $u$.

#####  Third step: Estimates for $u_{tt}$

They consist in two kinds of estimates and they are obtained from the equation
satisfied by $u_{t}$. In fact, when we derive equations (2.12) in time, we
have:

$\displaystyle\rho_{0}u_{tt}-\operatorname{div}(2\mu\mathbb{D}u_{t}+\lambda\operatorname{div}u_{t}I_{d})=\operatorname{div}F_{t}+\operatorname{div}(2\mu_{t}\mathbb{D}u+\lambda_{t}\operatorname{div}uI_{d}).$
(2.30)

The first energy consists in multiplying the above equation by $\sigma u_{tt}$
whereas, the second one consists in deriving the above equation with respect
to time before testing with $\sigma^{2}u_{tt}$. Testing the equation (2.30)
with $\sigma u_{tt}$ leads to:

$\displaystyle\dfrac{1}{2}\int_{0}^{t}\sigma\lVert\sqrt{\rho_{0}}u_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle+\sigma(t)\int_{\mathbb{R}^{d}}\mu\lvert\mathbb{D}u_{t}\rvert^{2}+\dfrac{\sigma(t)}{2}\int_{\mathbb{R}^{d}}\lambda\lvert\operatorname{div}u_{t}\rvert^{2}=\dfrac{1}{2}\int_{0}^{\sigma(t)}\int_{\mathbb{R}^{d}}\mu\lvert\mathbb{D}u_{t}\rvert^{2}+\dfrac{1}{2}\int_{0}^{\sigma(t)}\int_{\mathbb{R}^{d}}\lambda\lvert\operatorname{div}u_{t}\rvert^{2}$
$\displaystyle+\int_{0}^{t}\sigma\int_{\mathbb{R}^{d}}\mu_{t}\lvert\mathbb{D}u_{t}\rvert^{2}+\dfrac{1}{2}\int_{0}^{t}\sigma\int_{\mathbb{R}^{d}}\lambda_{t}\lvert\operatorname{div}u_{t}\rvert^{2}-2\int_{0}^{t}\sigma\int_{\mathbb{R}^{d}}\mu_{t}\mathbb{D}^{jk}u\partial_{k}u^{j}_{tt}$
$\displaystyle-\int_{0}^{t}\sigma\int_{\mathbb{R}^{d}}\lambda_{t}\operatorname{div}u\operatorname{div}u_{tt}-\int_{0}^{t}\sigma\int_{\mathbb{R}^{d}}F_{t}^{jk}\partial_{k}u^{j}_{tt}.$
(2.31)

By Hölder’s inequality, the last term can be estimated as:

$\bigg{|}\int_{0}^{t}\sigma\int_{\mathbb{R}^{d}}F_{t}^{jk}\partial_{k}u^{j}_{tt}\bigg{|}\lesssim\eta\int_{0}^{t}\sigma^{2}\lVert\nabla
u_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+C_{\eta}\int_{0}^{t}\lVert
F_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$

and on the other hand, the remainder terms are bounded by:

$\displaystyle\bigg{|}\dfrac{1}{2}\int_{0}^{t}\sigma\int_{\mathbb{R}^{d}}\lambda_{t}\lvert\operatorname{div}u_{t}\rvert^{2}$
$\displaystyle-\int_{0}^{t}\sigma\int_{\mathbb{R}^{d}}\lambda_{t}\operatorname{div}u\operatorname{div}u_{tt}+\int_{0}^{t}\sigma\int_{\mathbb{R}^{d}}\mu_{t}\lvert\mathbb{D}u_{t}\rvert^{2}-2\int_{0}^{t}\sigma\int_{\mathbb{R}^{d}}\mu_{t}\mathbb{D}^{jk}u\partial_{k}u^{j}_{tt}\bigg{|}$
$\displaystyle\lesssim\eta\int_{0}^{t}\sigma^{2}\lVert\nabla
u_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+C_{\eta}\int_{0}^{t}\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{4}(\mathbb{R}^{d})}^{2}\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{d})}^{2}$
$\displaystyle+\int_{0}^{t}\sigma\lVert\nabla
u_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{\infty}(\mathbb{R}^{d})}.$

Gathering all of these estimates, we have:

$\displaystyle\int_{0}^{t}\sigma\lVert\sqrt{\rho_{0}}u_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle+\sigma(t)\lVert\nabla
u_{t}\rVert^{2}_{L^{2}(\mathbb{R}^{d})}\lesssim\int_{0}^{\sigma(t)}\lVert\nabla
u_{t}\rVert^{2}_{L^{2}(\mathbb{R}^{d})}+\eta\int_{0}^{t}\sigma^{2}\lVert\nabla
u_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+C_{\eta}\int_{0}^{t}\lVert
F_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle+C_{\eta}\int_{0}^{t}\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{4}(\mathbb{R}^{d}))}^{2}\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{d}))}^{2}+\int_{0}^{t}\sigma\lVert\mu_{t},\lambda_{t}\rVert_{L^{\infty}(\mathbb{R}^{d})}\lVert\nabla
u_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}.$ (2.32)

The second term on the right hand side above will be reduced with the terms
coming from the second energy which we turn to, in the following lines.

We first derive (2.30) with respect to time before multiplying the obtained
equation by $\sigma^{2}u_{tt}$. By doing so, one obtains:

$\displaystyle\dfrac{\sigma^{2}}{2}\lVert\sqrt{\rho_{0}}u_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle+2\int_{0}^{t}\sigma^{2}\lVert\sqrt{\mu}\mathbb{D}u_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{t}\sigma^{2}\lVert\sqrt{\lambda}\operatorname{div}u_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}=\int_{0}^{\sigma(t)}\sigma\lVert\sqrt{\rho_{0}}u_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle-4\int_{0}^{t}\sigma^{2}\mu_{t}\mathbb{D}^{jk}u_{t}\partial_{k}u_{tt}^{j}-2\int_{0}^{t}\sigma^{2}\int_{\mathbb{R}^{d}}\lambda_{t}\operatorname{div}u_{t}\operatorname{div}u_{tt}-\int_{0}^{t}\sigma^{2}\int_{\mathbb{R}^{d}}F^{jk}_{tt}\partial_{k}u^{j}_{tt}$
$\displaystyle-2\int_{0}^{t}\sigma^{2}\int_{\mathbb{R}^{d}}\mu_{tt}\mathbb{D}^{jk}u\partial_{k}u^{j}_{tt}-\int_{0}^{t}\sigma^{2}\int_{\mathbb{R}^{d}}\lambda_{tt}\operatorname{div}u\operatorname{div}u_{tt}.$

We estimate the second and the third terms of the right hand side using
Young’s inequality, as follows:

$\bigg{|}4\int_{0}^{t}\sigma^{2}\mu_{t}\mathbb{D}^{jk}u_{t}\partial_{k}u_{tt}^{j}+2\int_{0}^{t}\sigma^{2}\int_{\mathbb{R}^{d}}\lambda_{t}\operatorname{div}u_{t}\operatorname{div}u_{tt}\bigg{|}\lesssim\eta\int_{0}^{t}\sigma^{2}\lVert\nabla
u_{tt}\rVert^{2}_{L^{2}(\mathbb{R}^{d})}+C_{\eta}\int_{0}^{t}\sigma^{2}\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{\infty}(\mathbb{R}^{d})}^{2}\lVert\nabla
u_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{2},$

whereas the last two terms are bounded by

$\displaystyle\bigg{|}2\int_{0}^{t}\sigma^{2}\int_{\mathbb{R}^{d}}\mu_{tt}\mathbb{D}^{jk}u\partial_{k}u^{j}_{tt}+\int_{0}^{t}\sigma^{2}\int_{\mathbb{R}^{d}}\lambda_{tt}\operatorname{div}u\operatorname{div}u_{tt}\bigg{|}\lesssim\eta\int_{0}^{t}\sigma^{2}\lVert\nabla
u_{tt}\rVert^{2}_{L^{2}(\mathbb{R}^{d})}+C_{\eta}\int_{0}^{t}\sigma^{2}\lVert\mu_{tt},\,\lambda_{tt}\rVert_{L^{4}(\mathbb{R}^{d})}^{2}\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{d})}^{2}.$

Finally, the fourth term of the right hand side is:

$\bigg{|}\int_{0}^{t}\sigma^{2}\int_{\mathbb{R}^{d}}F^{jk}_{tt}\partial_{k}u^{j}_{tt}\bigg{|}\lesssim\eta\int_{0}^{t}\sigma^{2}\lVert\nabla
u_{tt}\rVert^{2}_{L^{2}(\mathbb{R}^{d})}+C_{\eta}\int_{0}^{t}\sigma^{2}\lVert
F_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}.$

Gathering all of these estimates, and adding (2.32) before choosing $\eta$
small related to the lower bounds of the viscosities, one finally applies
Grönwall’s lemma in order to obtain the following:

$\displaystyle\int_{0}^{T}\lVert\sigma u_{tt},\;\sigma^{2}\nabla
u_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$ $\displaystyle+\lVert\sigma\nabla
u_{t},\;\sigma^{2}u_{tt}\rVert^{2}_{L^{\infty}((0,T),L^{2}(\mathbb{R}^{d}))}\leqslant
M\left[\int_{0}^{\sigma(T)}\lVert\nabla
u_{t}\rVert^{2}_{L^{2}(\mathbb{R}^{d})}+\int_{0}^{T}\lVert
F_{t},\,\sigma^{2}F_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\right.$
$\displaystyle\left.+\int_{0}^{T}\lVert\mu_{t},\,\lambda_{t},\,\sigma(\mu_{tt},\,\lambda_{tt})\rVert_{L^{4}(\mathbb{R}^{d})}^{2}\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{d})}^{2}\right]$
$\displaystyle\times\exp\left(M\int_{0}^{T}\left[\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{\infty}(\mathbb{R}^{d})}+\lVert\mu_{tt},\lambda_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}\right]+M\int_{0}^{T}\sigma\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{\infty}(\mathbb{R}^{d})}^{2}\right).$
(2.33)

This ends the computations of the last energy.

#####  Fourth Step: $L^{p}((0,T),L^{q}(\mathbb{R}^{d}))$ norm estimate for
$\nabla u$

Starting from (2.26), one has:

$\displaystyle\lVert\nabla
u\rVert_{L^{q}(\mathbb{R}^{d}))}\lesssim_{d,q}\dfrac{1}{\widetilde{\mu}}\lVert\mu-\widetilde{\mu}\rVert_{L^{\infty}(\mathbb{R}^{d})}\lVert\nabla
u\rVert_{L^{q}(\mathbb{R}^{d})}+\lVert
F\rVert_{L^{q}(\mathbb{R}^{d})}+\dfrac{1}{\widetilde{\mu}}\lVert(-\Delta)^{-1}\nabla\mathcal{P}(\rho_{0}u_{t}),\,(-\Delta)^{-1}\nabla\mathcal{Q}(\rho_{0}u_{t})\rVert_{L^{q}(\mathbb{R}^{d})}.$

Therefore, for small perturbation of the viscosity $\mu$ around the constant
state $\widetilde{\mu}$, one can absorb the first term of the right hand side
in the left one. We only have to estimate the last terms, by Gagliardo-
Nirenberg inequality:

$\lVert(-\Delta)^{-1}\nabla\mathcal{P}(\rho_{0}u_{t}),\,(-\Delta)^{-1}\nabla\mathcal{Q}(\rho_{0}u_{t})\rVert_{L^{6}(\mathbb{R}^{d}))}\lesssim\lVert\rho_{0}u_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{\beta}\lVert\nabla
u,F\rVert_{L^{2}(\mathbb{R}^{d})}^{1-\beta}$

with $\beta=d/2-d/6\in\\{2/3,1\\}$. In consequence, we have:

$\displaystyle\lVert\nabla
u\rVert_{L^{\infty}((0,T),L^{6}(\mathbb{R}^{d}))}\lesssim\lVert
F\rVert_{L^{\infty}((0,T),L^{6}(\mathbb{R}^{d}))}+\sup_{[0,T]}\lVert
F,u_{t},\nabla u\rVert_{L^{2}(\mathbb{R}^{d})}.$ (2.34)

On the other hand, for $q=8$, in two space dimension one has:

$\lVert(-\Delta)^{-1}\nabla\mathcal{P}(\rho_{0}u_{t}),\,(-\Delta)^{-1}\nabla\mathcal{Q}(\rho_{0}u_{t})\rVert_{L^{8}(\mathbb{R}^{2}))}\lesssim\lVert\rho_{0}u_{t}\rVert_{L^{2}(\mathbb{R}^{2})}^{3/4}\lVert\nabla
u,F\rVert_{L^{2}(\mathbb{R}^{2})}^{1/4}$

whereas in three space dimension, Sobolev and interpolation inequalities
yield:

$\lVert(-\Delta)^{-1}\nabla\mathcal{P}(\rho_{0}u_{t}),\,(-\Delta)^{-1}\nabla\mathcal{Q}(\rho_{0}u_{t})\rVert_{L^{8}(\mathbb{R}^{3}))}\lesssim\lVert\rho_{0}u_{t}\rVert_{L^{24/11}(\mathbb{R}^{3})}\lesssim\lVert\nabla
u_{t}\rVert_{L^{2}(\mathbb{R}^{3})}^{1/8}\lVert
u_{t}\rVert_{L^{2}(\mathbb{R}^{3})}^{7/8},$

such that,

$\displaystyle\lVert\nabla
u\rVert_{L^{16}((0,T),L^{8}(\mathbb{R}^{d}))}\lesssim\lVert
F\rVert_{L^{16}((0,T),L^{8}(\mathbb{R}^{d}))}+\lVert F,u_{t},\nabla
u\rVert_{L^{2}\cap L^{\infty}((0,T),L^{2}(\mathbb{R}^{d}))}+\lVert\nabla
u_{t}\rVert_{L^{2}((0,T)\times\mathbb{R}^{d})}.$ (2.35)

#####  Fifth Step: $L^{4}((0,T)\times\mathbb{R}^{d})$ norm estimate for
$\nabla u_{t}$

As above, we begin by recalling that, from (2.30), one has:

$\displaystyle\nabla u_{t}$
$\displaystyle=\nabla\mathcal{P}u_{t}+\nabla\mathcal{Q}u_{t}$
$\displaystyle=\frac{1}{\widetilde{\mu}}(-\Delta)^{-1}\nabla\left[\mathcal{P}\operatorname{div}\\{2(\mu-\widetilde{\mu})\mathbb{D}u_{t}\\}-\mathcal{P}\left(\rho_{0}u_{tt}\right)+\mathcal{P}\operatorname{div}F_{t}+\mathcal{P}\operatorname{div}(2\mu_{t}\mathbb{D}u+\lambda_{t}\operatorname{div}uI_{d})\right]$
$\displaystyle+\dfrac{1}{2\widetilde{\mu}+\lambda}(-\Delta)^{-1}\nabla\left[\mathcal{Q}\operatorname{div}\\{2(\mu-\widetilde{\mu})\mathbb{D}u_{t}\\}-\mathcal{Q}\left(\rho_{0}u_{tt}\right)+\mathcal{Q}\operatorname{div}F_{t}+\mathcal{Q}\operatorname{div}(2\mu_{t}\mathbb{D}u+\lambda_{t}\operatorname{div}uI_{d})\right]$

so it holds that, under the smallness of the viscosity, one has:

$\lVert\nabla u_{t}\rVert_{L^{4}(\mathbb{R}^{d})}\lesssim\lVert
F_{t}\rVert_{L^{4}(\mathbb{R}^{d})}+\lVert\nabla
u\rVert_{L^{8}(\mathbb{R}^{d})}\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{8}(\mathbb{R}^{d})}+\lVert(-\Delta)^{-1}\nabla\mathcal{P}(\rho_{0}u_{tt}),\,(-\Delta)^{-1}\nabla\mathcal{Q}(\rho_{0}u_{tt})\rVert_{L^{4}(\mathbb{R}^{d})}$

and we only need to compute

$\int_{0}^{T}\sigma^{4/d}\lVert(-\Delta)^{-1}\nabla\mathcal{P}(\rho_{0}u_{tt}),\,(-\Delta)^{-1}\nabla\mathcal{Q}(\rho_{0}u_{tt})\rVert_{L^{4}(\mathbb{R}^{d})}^{8/d}.$

By Gagliardo-Nirenberg inequality, one has:

$\displaystyle\int_{0}^{T}\sigma^{4/d}\lVert(-\Delta)^{-1}\nabla\mathcal{P}(\rho_{0}u_{tt}),\,(-\Delta)^{-1}\nabla\mathcal{Q}(\rho_{0}u_{tt})\rVert_{L^{4}(\mathbb{R}^{d})}^{8/d}\lesssim\int_{0}^{T}\sigma^{\tfrac{4}{d}}\lVert\nabla
u_{t},\,F_{t},\,\lambda_{t}\operatorname{div}u,\,\mu_{t}\mathbb{D}u\rVert_{L^{2}(\mathbb{R}^{d})}^{\tfrac{8}{d}-2}\lVert
u_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}.$

When $d=2$, one has:

$\displaystyle\int_{0}^{T}\sigma^{2}\lVert(-\Delta)^{-1}\nabla\mathcal{P}(\rho_{0}u_{tt}),\,(-\Delta)^{-1}\nabla\mathcal{Q}(\rho_{0}u_{tt})\rVert_{L^{4}(\mathbb{R}^{2})}^{4}\lesssim\lVert\sigma
u_{tt}\rVert_{L^{\infty}((0,T),L^{2}(\mathbb{R}^{2}))}^{2}\int_{0}^{T}\lVert\nabla
u_{t},\,F_{t},\,\lambda_{t}\operatorname{div}u,\,\mu_{t}\mathbb{D}u\rVert_{L^{2}(\mathbb{R}^{2})}^{2}$

whereas, in $d=3$, one has:

$\displaystyle\int_{0}^{T}\sigma^{4/3}\lVert(-\Delta)^{-1}\nabla\mathcal{P}(\rho_{0}u_{tt}),$
$\displaystyle\,(-\Delta)^{-1}\nabla\mathcal{Q}(\rho_{0}u_{tt})\rVert_{L^{4}(\mathbb{R}^{3})}^{8/3}\lesssim\int_{0}^{T}\sigma^{\tfrac{4}{3}}\lVert\nabla
u_{t},\,F_{t},\,\lambda_{t}\operatorname{div}u,\,\mu_{t}\mathbb{D}u\rVert_{L^{2}(\mathbb{R}^{3})}^{\tfrac{2}{3}}\lVert
u_{tt}\rVert_{L^{2}(\mathbb{R}^{3})}^{2}$
$\displaystyle\lesssim\left[\int_{0}^{T}\sigma^{2}\lVert
u_{tt}\rVert_{L^{2}(\mathbb{R}^{3})}^{3}\right]^{2/3}\left[\int_{0}^{T}\lVert\nabla
u_{t},\,F_{t},\,\lambda_{t}\operatorname{div}u,\,\mu_{t}\mathbb{D}u\rVert_{L^{2}(\mathbb{R}^{3})}^{2}\right]^{1/3}.$

Finally, one has:

$\int_{0}^{T}\sigma^{4/d}\lVert\nabla
u_{t}\rVert_{L^{4}(\mathbb{R}^{d})}^{8/d}\lesssim\int_{0}^{T}\sigma^{4/d}\lVert
F_{t}\rVert_{L^{4}(\mathbb{R}^{d})}^{8/d}+\int_{0}^{T}\sigma^{4/d}\lVert\nabla
u\rVert_{L^{8}(\mathbb{R}^{d})}^{8/d}\lVert\mu_{t},\,\lambda_{t}\rVert_{L^{8}(\mathbb{R}^{d})}^{8/d}+\sigma^{8/d}\lVert\sigma
u_{tt}\rVert_{L^{\infty}((0,T),L^{2}(\mathbb{R}^{d}))}^{8/d}\\\
+\left[\int_{0}^{T}\lVert\nabla
u_{t},\,F_{t},\,\lambda_{t}\operatorname{div}u,\,\mu_{t}\mathbb{D}u\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\right]^{4/d}.$
(2.36)

#####  Sixth step: Piece-wise Hölder norm estimate for $\nabla u$

In this step, we apply the quantitative Hölder estimate 5 to (2.26), in order
to obtain:

$\displaystyle\lVert\nabla
u\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})}$
$\displaystyle\lesssim\dfrac{1}{\widetilde{\mu}}\lVert(\mu-\widetilde{\mu})\mathbb{D}u\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})}+\lVert
F\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})}+\eta^{-\tfrac{d}{2}}\left(\lVert\mathbb{D}u\rVert_{L^{2}(\mathbb{R}^{d})}+\lVert
F\rVert_{L^{2}(\mathbb{R}^{d})}\right)$
$\displaystyle+\lVert(-\Delta)^{-1}\nabla\mathcal{P}(\rho_{0}u_{t})\rVert_{\mathscr{C}^{\alpha}(\mathbb{R}^{d})}+\lVert(-\Delta)^{-1}\nabla\mathcal{Q}(\rho_{0}u_{t})\rVert_{\mathscr{C}^{\alpha}(\mathbb{R}^{d})}+\bigg{\|}\dfrac{1}{\widetilde{\mu}}(\mu-\widetilde{\mu})\mathbb{D}u,\,F\bigg{\|}_{L^{\infty}(\mathbb{R}^{d})}\mathscr{P}(\mathcal{C}).$

Thus, there exists a constant $[\mu]>0$ that depends on $d$ and the regularity
of the interface such that if

$\dfrac{1}{\widetilde{\mu}}\lVert\mu-\widetilde{\mu}\rVert_{L^{\infty}((0,T),\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d}))}\leqslant[\mu]\approx\dfrac{1}{1+\mathscr{P}(\mathcal{C})}$

then the first and last terms of the right hand side above can be absorbed in
the left hand side and this will lead to:

$\displaystyle\lVert\nabla
u\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})}$
$\displaystyle\lesssim\lVert\mathbb{D}u\rVert_{L^{2}(\mathbb{R}^{d})}+\lVert
F\rVert_{L^{2}(\mathbb{R}^{d})}+(1+\mathscr{P}(\mathcal{C}))\lVert
F\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})}$
$\displaystyle+\lVert(-\Delta)^{-1}\nabla\mathcal{P}(\rho_{0}u_{t})\rVert_{\mathscr{C}^{\alpha}(\mathbb{R}^{d})}+\lVert(-\Delta)^{-1}\nabla\mathcal{Q}(\rho_{0}u_{t})\rVert_{\mathscr{C}^{\alpha}(\mathbb{R}^{d})}.$

We only have to compute the time norm of the two last term. By interpolation,
one has:

$\displaystyle\int_{0}^{T}\sigma^{\tfrac{d\beta}{4}}\lVert(-\Delta)^{-1}\nabla\mathcal{P}(\rho_{0}u_{t})$
$\displaystyle;\,(-\Delta)^{-1}\nabla\mathcal{Q}(\rho_{0}u_{t})\rVert_{L^{\infty}(\mathbb{R}^{d})}^{4}\lesssim\int_{0}^{T}\sigma^{\tfrac{d\beta}{4}}\lVert\rho_{0}u_{t}\rVert_{L^{4}(\mathbb{R}^{d})}^{4\beta}\lVert\nabla(-\Delta)^{-1}(\rho_{0}u_{t})\rVert_{L^{2}(\mathbb{R}^{d})}^{4(1-\beta)}$
$\displaystyle\lesssim(\rho^{*})^{4\beta}\int_{0}^{T}\sigma^{\tfrac{d\beta}{4}}\lVert\nabla
u_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{d\beta}\lVert
u_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{(4-d)\beta}\lVert F,\nabla
u\rVert_{L^{2}(\mathbb{R}^{d})}^{4(1-\beta)}$
$\displaystyle\lesssim\left[\int_{0}^{T}\sigma\lVert\nabla
u_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{4}\right]^{\tfrac{d\beta}{4}}\left[\int_{0}^{T}\lVert
u_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{4}\right]^{(1-\tfrac{d}{4})\beta}\left[\int_{0}^{T}\lVert\nabla(-\Delta)^{-1}(\rho_{0}u_{t})\rVert_{L^{2}(\mathbb{R}^{d})}^{4}\right]^{1-\beta}$
(2.37)

with

$\beta=\dfrac{2d}{4+d}.$

On the other hand, one has:

$\displaystyle\int_{0}^{T}\sigma^{r}\lVert(-\Delta)^{-1}\nabla\mathcal{P}(\rho_{0}u_{t});\,(-\Delta)^{-1}\nabla\mathcal{Q}(\rho_{0}u_{t})\rVert_{\dot{\mathscr{C}}^{\alpha}(\mathbb{R}^{d})}^{4}\lesssim\int_{0}^{T}\sigma^{r}\lVert
u_{t}\rVert_{L^{d/(1-\alpha)}(\mathbb{R}^{d})}^{4}.$ (2.38)

In two space dimension, one has:

$\displaystyle\int_{0}^{T}\sigma^{\alpha}\lVert
u_{t}\rVert_{L^{2/(1-\alpha)}(\mathbb{R}^{2})}^{4}$
$\displaystyle\lesssim\int_{0}^{T}\sigma^{\alpha}\lVert\nabla
u_{t}\rVert_{L^{2}(\mathbb{R}^{2})}^{4\alpha}\lVert
u_{t}\rVert_{L^{2}(\mathbb{R}^{2})}^{4(1-\alpha)}$
$\displaystyle\leqslant\left[\int_{0}^{T}\sigma\lVert\nabla
u_{t}\rVert_{L^{2}(\mathbb{R}^{2}))}^{4}\right]^{\alpha}\left[\int_{0}^{T}\lVert
u_{t}\rVert_{L^{2}(\mathbb{R}^{2})}^{4}\right]^{1-\alpha}$

whereas in three space dimension, we only have for all $0<\alpha<1/2$

$\displaystyle\int_{0}^{T}\sigma^{2\alpha}\lVert
u_{t}\rVert_{L^{3/(1-\alpha)}(\mathbb{R}^{3})}^{4}$
$\displaystyle\leqslant\int_{0}^{T}\sigma^{2\alpha}\lVert\nabla
u_{t}\rVert_{L^{2}(\mathbb{R}^{3})}^{2(1+2\alpha)}\lVert
u_{t}\rVert_{L^{2}(\mathbb{R}^{3})}^{2(1-2\alpha)}$
$\displaystyle\leqslant\sup_{[0,T]}\lVert
u_{t}\rVert_{L^{2}(\mathbb{R}^{3})}^{2(1-2\alpha)}\int_{0}^{T}\sigma^{2\alpha}\lVert\nabla
u_{t}\rVert_{L^{2}(\mathbb{R}^{3})}^{2(1+2\alpha)}.$

For $1/2\leqslant\alpha<1$, one has:

$\displaystyle\int_{0}^{T}\sigma^{2\alpha}\lVert
u_{t}\rVert_{L^{3/(1-\alpha)}(\mathbb{R}^{3})}^{4}$
$\displaystyle\lesssim\int_{0}^{T}\sigma^{2\alpha}\lVert\nabla
u_{t}\rVert_{L^{4}(\mathbb{R}^{3})}^{\tfrac{16}{3}(\alpha-\tfrac{1}{2})}\lVert\nabla
u_{t}\rVert_{L^{2}(\mathbb{R}^{3})}^{\tfrac{4}{3}(5-4\alpha)}$
$\displaystyle\lesssim\left[\lVert\sqrt{\sigma}\nabla
u_{t}\rVert_{L^{\infty}((0,T),L^{2}(\mathbb{R}^{3}))}^{2}\right]^{\tfrac{2}{3}(2-\alpha)}\left[\int_{0}^{T}\sigma^{4/3}\lVert\nabla
u_{t}\rVert_{L^{4}(\mathbb{R}^{3})}^{8/3}\right]^{2\alpha-1}\left[\int_{0}^{T}\lVert\nabla
u_{t}\rVert^{2}_{L^{2}(\mathbb{R}^{3})}\right]^{2(1-\alpha)}.$

In conclusion, we choose $r$ to be

$\displaystyle r=\max\left\\{\dfrac{d^{2}}{2(4+d)},\;2\alpha\right\\}<2$
(2.39)

and we have:

$\displaystyle\int_{0}^{T}\sigma^{r}\lVert\nabla
u\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})}^{4}$
$\displaystyle\lesssim(1+\mathscr{P}(\mathcal{C}))\int_{0}^{T}\sigma^{r}\lVert
F\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})}^{4}+\lVert
F,\,\nabla u,\,u_{t},\,\sqrt{\sigma}\nabla
u_{t}\rVert_{L^{\infty}((0,T),L^{2}(\mathbb{R}^{d}))}^{4}$
$\displaystyle+\left[\int_{0}^{T}\lVert F,\,\nabla u,\,u_{t},\,\nabla
u_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\right]^{2}+\left[\int_{0}^{T}\sigma^{4/d}\lVert\nabla
u_{t}\rVert_{L^{4}(\mathbb{R}^{d})}^{8/d}\right]^{d/2}.$ (2.40)

This completes the first part of the proof of section 1.2. ∎

We turn to the second step in the proof of section 1.2 that consists in
proving that the set $\mathscr{E}$, defined in (2.19), is not empty, that is
$0\in\mathscr{E}$. We have already discussed this fact linearly on page
2.2.2333from (2.20), and now it only remains to prove section 2.2.2, which is
the purpose of the following section.

####  2.2.4. Proof of section 2.2.2

The proof of section 2.2.2 is based on a homotopy argument and for this
purpose, we define

${\mu_{0}^{\prime}}^{\theta}=(1-\theta)\widetilde{\mu^{\prime}}+\theta\mu_{0}^{\prime}\quad\text{and}\quad{\lambda_{0}^{\prime}}^{\theta}=(1-\theta)\widetilde{\lambda^{\prime}}+\theta\lambda_{0}^{\prime},$

and we consider the equation (2.23) where the viscosities
$\mu_{0}^{\prime},\lambda_{0}^{\prime}$ are replaced by
${\mu_{0}^{\prime}}^{\theta},\,{\lambda_{0}^{\prime}}^{\theta}$ respectively.
Also, we define

$X_{T}^{\prime}:=\\{v\in\mathscr{C}([0,T],H^{1}(\mathbb{R}^{d}))\colon
v_{t},\,\nabla^{2}v\in L^{2}((0,T)\times\mathbb{R}^{d})\\}$

and the map:

$\displaystyle\Pi_{\theta}\colon\quad$ $\displaystyle X_{T}^{\prime}\quad\to
L^{2}((0,T)\times\mathbb{R}^{d})\times H^{1}(\mathbb{R}^{d})$ $\displaystyle
v\quad\quad\mapsto\left(v_{t}-\operatorname{div}(2{\mu_{0}^{\prime}}^{\theta}\mathbb{D}v+{\lambda_{0}^{\prime}}^{\theta}\operatorname{div}vI_{d})-\operatorname{div}(b_{0}\,\colon
v),\;v_{|t=0}\right)$

and finally we consider the set:

$\mathscr{E}_{T}^{\prime}:=\\{\theta\in[0,1]\colon\Pi_{\theta}\text{ is one to
one}\\}.$

Obviously, the map $\Pi_{\theta}$ is well-defined and continuous, we will only
have to show that $1\in\mathscr{E}^{\prime}_{T}$ to conclude the proof of
section 2.2.2. To obtain $0\in\mathscr{E}$, one can observe that for
$\theta=0$, the equation $\eqref{eq3.23}_{\theta}$ is reduced to the Lamé
system:

$\displaystyle v_{t}-\widetilde{\mu^{\prime}}\Delta
u-(\widetilde{\mu^{\prime}}+\widetilde{\lambda^{\prime}})\nabla\operatorname{div}v=f+\operatorname{div}(b_{0}\,\colon
v)$ (2.41)

in such a way that $v=\mathcal{P}v+\mathcal{Q}v$ with

$\begin{cases}\partial_{t}\mathcal{A}v-k\mathcal{A}v=\mathcal{A}f+\mathcal{A}\operatorname{div}(b_{0}\,\colon
v),\\\ \mathcal{A}v_{|t=0}=\mathcal{A}v_{0}\end{cases}$

where $k=\widetilde{\mu^{\prime}}$ if $\mathcal{A}=\mathcal{P}$ and
$k=2\widetilde{\mu^{\prime}}+\widetilde{\lambda^{\prime}}$ if
$\mathcal{A}=\mathcal{Q}$. In the following theorem we address the global
well-posedness of (2.41).

Let $0<T<\infty$, $v_{0}\in H^{1}(\mathbb{R}^{d})$, $b_{0}\in
W^{1,\infty}(\mathbb{R}^{d})$ and $f\in L^{2}((0,T)\times\mathbb{R}^{d})$.
Then, the Cauchy problem associated to the equation (2.41) with initial datum
$v_{0}$ admits an unique solution
$v\in\mathscr{C}([0,T],H^{1}(\mathbb{R}^{d}))$. The solution $v$ verifies the
following integral formulation

$\mathcal{A}v(t)=\mathcal{T}(kt)\mathcal{A}v_{0}+\int_{0}^{t}\mathcal{T}(k(t-s))\mathcal{A}\\{f+\operatorname{div}(b_{0}v)\\}(s)ds$

where $(\mathcal{T}(t))_{t}$ is the heat semi-group. Moreover, there are
constants $K,\Lambda$ depending on $b_{0}$ and
$\widetilde{\mu},\widetilde{\lambda}$ such that:

$\sup_{t\in[0,T]}e^{-\Lambda t}\lVert
v(t)\rVert_{H^{1}(\mathbb{R}^{d})}\leqslant KR_{T}$

where

$R_{T}:=\sup_{[0,T]}\left\|\mathcal{T}(kt)\mathcal{A}v_{0}+\int_{0}^{t}\mathcal{T}(k(t-s))\mathcal{A}f(s)ds\right\|_{H^{1}(\mathbb{R}^{d})}.$

As well, $v_{t},\,\nabla^{2}v\in L^{2}((0,T)\times\mathbb{R}^{d})$ with the
following estimate:

$\sup_{[0,T]}e^{-2\Lambda t}\int_{0}^{t}\lVert
v_{t}(s),\nabla^{2}v(s)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}ds\leqslant
K^{2}R_{T}^{2}.$

The proof of this theorem is done in appendix C and is based on some
appropriate estimate on the heat kernel. From this discussion, it turns out
that $\mathscr{E}^{\prime}_{T}$ is a non-empty set. The remainder of section
2.2.2 consists in first deriving some a priori estimates for the solution
$v\in X^{\prime}_{T}$ of the equation (2.23) in section 2.2.4. Then, in
section 2.2.4, we prove that $\mathscr{E}^{\prime}_{T}$ is at the same time
open and closed in much easier way.

##### A priori estimates

We establish three types estimates for the solution $v\in X^{\prime}_{T}$ of
the equation (2.23). The first one is the classical energy that consists in
multiplying the equation (2.23) by $v$ and integrating in time and in space in
order to obtain:

$\displaystyle\dfrac{1}{2}\lVert v(t)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{R}^{d}}2\mu_{0}^{\prime}\lvert\mathbb{D}v\rvert^{2}+\int_{0}^{t}\int_{\mathbb{R}^{d}}\lambda_{0}^{\prime}\lvert\operatorname{div}v\rvert^{2}=\dfrac{1}{2}\lVert
v_{0}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{t}\int_{\mathbb{R}^{d}}fv-\int_{0}^{t}\int_{\mathbb{R}^{d}}b_{0}\,\colon
v\nabla v$ $\displaystyle\leqslant\eta\int_{0}^{t}\lVert\nabla
v\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\dfrac{1}{2}\lVert
v_{0}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{t}\lVert
f\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+C_{\eta}\int_{0}^{t}(1+\lVert
b_{0}\rVert_{L^{\infty}(\mathbb{R}^{d})}^{2})\lVert
v\rVert_{L^{2}(\mathbb{R}^{d})}^{2})$

for some positive constant $\eta$ small related to the lower bound of the
viscosities, in such way that the first term of the right hand side can be
absorbed in the left hand side in order to obtain:

$\displaystyle\lVert
v(t)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{t}\lVert\nabla
v\rVert^{2}_{L^{2}(\mathbb{R}^{d})}\lesssim\lVert
v_{0}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{t}\lVert
f\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{t}(1+\lVert
b_{0}\rVert_{L^{\infty}(\mathbb{R}^{d})}^{2})\lVert
v\rVert_{L^{2}(\mathbb{R}^{d})}^{2}.$ (2.42)

The second estimate is obtained by testing the equation (2.23) with $\Delta
v$. The first term is:

$\int_{\mathbb{R}^{d}}v_{t}\Delta v=-\int_{\mathbb{R}^{d}}\nabla
v_{t}\cdot\nabla v=-\dfrac{1}{2}\dfrac{d}{dt}\int_{\mathbb{R}^{d}}\lvert\nabla
v\rvert^{2}$

and the second one is:

$\displaystyle-2\int_{\mathbb{R}^{d}}\partial_{k}(\mu_{0}^{\prime}\mathbb{D}^{jk}v)\Delta
v^{j}$
$\displaystyle-\int_{\mathbb{R}^{d}}\partial_{j}(\lambda_{0}^{\prime}\operatorname{div}v)\Delta
v^{j}=-\int_{\mathbb{R}^{d}}\mu_{0}^{\prime}\lvert\Delta
v\rvert^{2}-\int_{\mathbb{R}^{d}}\mu_{0}^{\prime}\partial_{j}\operatorname{div}v\Delta
v^{j}-\int_{\mathbb{R}^{d}}\partial_{k}\mu_{0}^{\prime}\mathbb{D}^{jk}v\Delta
v^{j}$
$\displaystyle-\int_{\mathbb{R}^{d}}\lambda_{0}^{\prime}\lvert\nabla\operatorname{div}v\rvert^{2}-\int_{\mathbb{R}^{d}}\operatorname{div}v\nabla\lambda_{0}^{\prime}\cdot\nabla\operatorname{div}v$
$\displaystyle=-\int_{\mathbb{R}^{d}}\mu_{0}^{\prime}\lvert\Delta
v\rvert^{2}-\int_{\mathbb{R}^{d}}(\mu_{0}^{\prime}+\lambda_{0}^{\prime})\lvert\nabla\operatorname{div}v\rvert^{2}+\int_{\mathbb{R}^{d}}\operatorname{div}v\partial_{j}\mu_{0}^{\prime}\Delta
v^{j}$
$\displaystyle-\int_{\mathbb{R}^{d}}\operatorname{div}v\nabla\mu_{0}^{\prime}\cdot\nabla\operatorname{div}v-\int_{\mathbb{R}^{d}}\partial_{k}\mu_{0}^{\prime}\mathbb{D}^{jk}v\Delta
v^{j}-\int_{\mathbb{R}^{d}}\operatorname{div}v\nabla\lambda_{0}^{\prime}\cdot\nabla\operatorname{div}v.$

Gathering all of theses computations, one has:

$\displaystyle\dfrac{1}{2}\dfrac{d}{dt}\lVert\nabla
v\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle+\int_{\mathbb{R}^{d}}\mu_{0}^{\prime}\lvert\Delta
v\rvert^{2}+\int_{\mathbb{R}^{d}}(\mu_{0}^{\prime}+\lambda_{0}^{\prime})\lvert\nabla\operatorname{div}v\rvert^{2}=\int_{\mathbb{R}^{d}}f^{j}\Delta
v^{j}+\int_{\mathbb{R}^{d}}\operatorname{div}(b_{0}\,:v)\Delta v^{j}$
$\displaystyle+$
$\displaystyle\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\operatorname{div}v\partial_{j}\mu_{0}^{\prime}\Delta
v^{j}-\int_{\mathbb{R}^{d}}\operatorname{div}v\nabla\mu_{0}^{\prime}\cdot\nabla\operatorname{div}v-\int_{\mathbb{R}^{d}}\partial_{k}\mu_{0}^{\prime}\mathbb{D}^{jk}v\Delta
v^{j}-\int_{\mathbb{R}^{d}}\operatorname{div}v\nabla\lambda_{0}^{\prime}\cdot\nabla\operatorname{div}v.$

Thus, Hölder and Young inequalities yield:

$\displaystyle\lVert\nabla
v(t)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{t}\lVert\Delta
v\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\lesssim\lVert\nabla
v_{0}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{t}\lVert
f\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{t}\lVert\nabla
v\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\lVert\nabla\mu_{0}^{\prime},\nabla\lambda_{0}^{\prime},b_{0},\nabla
b_{0}\rVert_{L^{\infty}(\mathbb{R}^{d})}^{2}.$ (2.43)

Finally, we gather (2.42) and (2.43), and we make use of Grönwall’s lemma and
equation (2.23), in order to obtain:

$\displaystyle\sup_{[0,T]}e^{-\Lambda t}\left\\{\lVert
v(t)\rVert_{H^{1}(\mathbb{R}^{d})}^{2}+\int_{0}^{t}\lVert v_{t},\,\nabla
v,\,\nabla^{2}v\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\right\\}\lesssim\lVert\nabla
v_{0}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{T}\lVert
f\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$

with

$\Lambda=C(1+\lVert\nabla\mu_{0}^{\prime},\nabla\lambda_{0}^{\prime},b_{0},\nabla
b_{0}\rVert_{L^{\infty}(\mathbb{R}^{d})}^{2})$

and constant $C$ that depends on the lower and upper bounds of the viscosities
$\mu_{0}^{\prime},\lambda_{0}^{\prime}$. The
$L^{2}((0,T)\times\mathbb{R}^{d})$ norm estimate for $v_{t}$ can be obtained
differently by testing the equation (2.23) with $v_{t}$. This ends the
computations of the a priori estimates and we turn to the last step.

##### Closing argument

In this section, we will show that $\mathscr{E}_{T}^{\prime}=[0,1]$. We have
already showed that $\mathscr{E}_{T}^{\prime}$ is not empty. Also, we may
notice that if $\Pi_{\theta}$ is invertible then, from the previous
subsection, $\Pi_{\theta}^{-1}$ is linear continuous with a norm that depends
on $\Lambda$ and time $T$. In particular, one has:

$K_{T}:=\sup_{\theta\in\mathscr{E}_{T}^{\prime}}\lVert\Pi_{\theta}^{-1}\rVert_{\mathcal{L}}<\infty.$

Let us prove that $\mathscr{E}_{T}^{\prime}$ is an open set. For this purpose,
let $\theta_{0}\in\mathscr{E}_{T}^{\prime}$, then, $\Pi_{\theta_{0}}$ is a one
to one map and one can write, for all $\theta\in[0,1]$,

$\Pi_{\theta}=\Pi_{\theta_{0}}+(\theta-\theta_{0})(\Pi_{1}-\Pi_{0})=\Pi_{\theta_{0}}\left(I+(\theta-\theta_{0})\Pi_{\theta_{0}}^{-1}(\Pi_{1}-\Pi_{0})\right).$

Thus, all $\theta$ such that
$2\lvert\theta-\theta_{0}\rvert\lVert\Pi_{\theta_{0}}^{-1}(\Pi_{1}-\Pi_{0})\rVert_{\mathcal{L}(X_{T}^{\prime})}<1$
belongs to $\mathscr{E}_{T}^{\prime}$, therefore $\mathscr{E}_{T}^{\prime}$ is
an open set of $[0,1]$. We turn to the last part. Let $(\theta_{n})_{n}$ be a
sequence of element of $\mathscr{E}_{T}^{\prime}$ that converges to some
$\theta\in[0,1]$. As above, one can write, for some $n_{0}$

$\Pi_{\theta}=\Pi_{\theta_{n_{0}}}\left(I+(\theta-\theta_{n_{0}})\Pi_{\theta_{n_{0}}}^{-1}(\Pi_{1}-\Pi_{0})\right).$

Since $K_{T}<\infty$, one can choose large $n_{0}$ such that

$2\lvert\theta-\theta_{n_{0}}\rvert
K_{T}\lVert\Pi_{1}-\Pi_{0}\rVert_{\mathcal{L}}<1$

thus $\Pi_{\theta}$ is one to one as composition of two one to one operators.
Therefore $\theta\in\mathscr{E}_{T}^{\prime}$ and consequently
$\mathscr{E}^{\prime}_{T}$ is a closed set of $[0,1]$. One then concludes that
$1\in\mathscr{E}_{T}^{\prime}$ and section 2.2.2 then follows.

####  2.2.5. Remainder of the proof of section 2.2.1

###### Proof.

We recall that section 2.2.1 follows as soon as we will prove that the set
$\mathscr{E}=[0,1]$. What remains to be proved is that h $\mathscr{E}$ is at
the same time open and closed. We begin by proving that it is an open subset
of $[0,1]$. So let $\theta_{0}\in\mathscr{E}$, the goal is to find some
$\eta>0$ such that

$\displaystyle B(\theta_{0},\eta)\cap[0,1]\subset\mathscr{E}.$ (2.44)

For this purpose, let $\theta\in[0,1]$, $(F,u_{0})\in Y_{T}^{\theta}$ and
consider the Cauchy problem (2.18) that can be written as:

$\displaystyle\begin{cases}\rho_{0}u_{t}-\operatorname{div}(2\mu^{\theta_{0}}\mathbb{D}u+\lambda^{\theta_{0}}\operatorname{div}uI_{d})=\operatorname{div}F+(\theta-\theta_{0})\operatorname{div}\\{2(\mu-\widetilde{\mu})\mathbb{D}u+(\lambda-\widetilde{\lambda})\operatorname{div}uI_{d}\\},\\\
u_{|t=0}=u_{0}.\end{cases}$ (2.45)

We will apply a fixed point theorem in order to solve the above Cauchy problem
with a small $\eta$ that does not depend on $F$ nor $u_{0}$. In fact, the
proof shows that the constant $\eta$ only depends on the norm of the
viscosities and initial density as required in (2.9), as well on
$\mathscr{P}(\mathcal{C})$444see (2.17). The fixed point theorem will be
applied in $X_{T}(u_{0})$ defined as:

$X_{T}(u_{0}):=\\{v\in X_{T}\colon v_{|t=0}=u_{0}\\}$

a complete subset of the Banach space $X_{T}$. So for $v\in X_{T}(u_{0})$, we
consider the following equation:

$\displaystyle\begin{cases}\rho_{0}u_{t}-\operatorname{div}(2\mu^{\theta_{0}}\mathbb{D}u+\lambda^{\theta_{0}}\operatorname{div}uI_{d})=\operatorname{div}F+(\theta-\theta_{0})\operatorname{div}\\{2(\mu-\widetilde{\mu})\mathbb{D}v+(\lambda-\widetilde{\lambda})\operatorname{div}vI_{d}\\},\\\
u_{|t=0}=u_{0}.\end{cases}$ (2.46)

The regularity of $v$ and $\mu,\lambda$ ensures that the right hand side can
be written as $\operatorname{div}\widetilde{F}(v)$ with

$\displaystyle\widetilde{F}(v):=F+(\theta-\theta_{0})\\{2(\mu-\widetilde{\mu})\mathbb{D}v+(\lambda-\widetilde{\lambda})\operatorname{div}vI_{d}\\}$
(2.47)

that verifies $(\widetilde{F}(v),u_{0})\in Y_{T}^{\theta_{0}}$. The last
condition in the definition of $Y_{T}^{\theta_{0}}$ is valid since
$v_{|t=0}=u_{0}$. Now, owing to the fact that $\theta_{0}\in\mathscr{E}$, the
underlying equation admits an unique solution $\Phi(v)\in X_{T}(u_{0})$ that
verifies all of the estimates in section 2.2.1 where $F$ is replaced by
$\widetilde{F}(v)$. In the following we are looking for some small
$\theta-\theta_{0}$ and some $R_{T}$ that depends on the initial data such
that the condition $\lVert v\rVert_{X_{T}}\leqslant R_{T}$ implies the same
for $\Phi(v)$. In order to facilitate the computations and the reading, the
estimates will be modulo a constant that depends on, not only of the lower and
upper bound of $\mu,\lambda,\rho_{0}$, but also on all of the norms of
$\mu,\lambda$ as required in (2.9).

#####  Step 1: Estimates for $\Phi(v)$ and $\partial_{t}\Phi(v)$

Since $\Phi(v)$ solves the Cauchy problem (2.46), $(\widetilde{F}(v),u_{0})\in
Y_{T}^{\theta_{0}}$, and $\theta_{0}\in\mathscr{E}$, one has from the a priori
estimate the following estimate for $\Phi(v)$:

$\displaystyle\sup_{[0,T]}\lVert\Phi(v),\,\partial_{t}\Phi(v),\,\nabla\Phi(v)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle+\int_{0}^{T}\lVert\nabla\Phi(v),\,\nabla\partial_{t}\Phi(v),\,\partial_{t}\Phi(v)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\lesssim\lVert
u_{0}\rVert_{H^{1}(\mathbb{R}^{d})}^{2}+\lVert\partial_{t}\Phi(v)_{|t=0}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle+\sup_{[0,T]}\lVert\widetilde{F}(v)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{T}\lVert\partial_{t}\widetilde{F}(v)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{T}\lVert\widetilde{F}(v)\rVert_{L^{2}\cap
L^{6}(\mathbb{R}^{d}))}^{2}.$ (2.48)

On the first hand, from (2.47), one deduces that:

$\displaystyle\sup_{[0,T]}\lVert\widetilde{F}(v)-F\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{T}\lVert\widetilde{F}(v)-F\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle\lesssim\lvert\theta-\theta_{0}\rvert^{2}\left\\{\sup_{[0,T]}\lVert\nabla
v\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{T}\lVert\nabla
v\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\right\\}$
$\displaystyle\lesssim\lvert\theta-\theta_{0}\rvert^{2}\lVert
v\rVert_{X_{T}}^{2}$ (2.49)

and

$\int_{0}^{T}\lVert\widetilde{F}(v)-F\rVert_{L^{6}(\mathbb{R}^{d}))}^{2}\lesssim\lvert\theta-\theta_{0}\rvert^{2}\lVert
v\rVert_{X_{T}}^{2}.$

On the other hand, from (2.47) one can express the time derivative of
$\widetilde{F}(v)$ as:

$\displaystyle\partial_{t}\widetilde{F}(v)=F_{t}+(\theta-\theta_{0})\\{2(\mu-\widetilde{\mu})\mathbb{D}v_{t}+(\lambda-\widetilde{\lambda})\operatorname{div}v_{t}I_{d}+2\mu_{t}\mathbb{D}v+\lambda_{t}\operatorname{div}vI_{d}\\}$
(2.50)

and consequently, it holds:

$\displaystyle\int_{0}^{T}\lVert\partial_{t}\widetilde{F}(v)-F_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\lesssim\lvert\theta-\theta_{0}\rvert^{2}\left\\{\lVert\nabla
v\rVert_{L^{4}((0,T)\times\mathbb{R}^{d})}^{2}+\int_{0}^{T}\lVert\nabla
v_{t}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\right\\}\lesssim\lvert\theta-\theta_{0}\rvert^{2}\lVert
v\rVert_{X_{T}}^{2}.$ (2.51)

Gathering all of these estimates, one has the following:

$\displaystyle\sup_{[0,T]}\lVert\Phi(v),\,\partial_{t}\Phi(v),\,\nabla\Phi(v)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{T}\lVert\nabla\Phi(v),\,\nabla\partial_{t}\Phi(v),\,\partial_{t}\Phi(v)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\lesssim\lVert(F,u_{0})\rVert_{Y_{T}^{\theta}}^{2}+\lvert\theta-\theta_{0}\rvert^{2}\lVert
v\rVert_{X_{T}}^{2}.$ (2.52)

Also, one has the following estimate for $\nabla\Phi(v)$:

$\displaystyle\lVert\nabla\Phi(v)\rVert_{L^{\infty}((0,T),L^{6}(\mathbb{R}^{d}))}\lesssim\lVert\widetilde{F}(v)\rVert_{L^{\infty}((0,T),L^{6}(\mathbb{R}^{d}))}+\sup_{[0,T]}\lVert\widetilde{F}(v),\partial_{t}\Phi(v),\nabla\Phi(v)\rVert_{L^{2}(\mathbb{R}^{d})}.$
(2.53)

Owing to the fact that the estimate (2.52) provides a bound for
$\partial_{t}\Phi(v)$ in $L^{2}((0,T)\times\mathbb{R}^{d})$ and for $\Phi(v)$
in $L^{\infty}((0,T),H^{1}(\mathbb{R}^{d}))$ and owing to the estimate (2.49),
we only need to estimate the $L^{\infty}((0,T),L^{6}(\mathbb{R}^{d}))$ norm of
$\widetilde{F}(v)$. In fact, this can be easily obtained from (2.47) as:

$\lVert\widetilde{F}(v)-F\rVert_{L^{\infty}((0,T),L^{6}(\mathbb{R}^{d}))}\lesssim\lvert\theta-\theta_{0}\rvert\lVert\nabla
v\rVert_{L^{\infty}((0,T),L^{6}(\mathbb{R}^{d}))}\lesssim\lvert\theta-\theta_{0}\rvert\lVert
v\rVert_{X_{T}}.$

Gathering all of theses estimates, one has:

$\displaystyle\lVert\nabla\Phi(v)\rVert_{L^{\infty}((0,T),L^{6}(\mathbb{R}^{d}))}\lesssim\lVert(F,u_{0})\rVert_{Y_{T}^{\theta}}+\lvert\theta-\theta_{0}\rvert\lVert
v\rVert_{X_{T}}.$ (2.54)

We turn to the $L^{16}((0,T),L^{8}(\mathbb{R}^{d}))$ estimate for
$\nabla\Phi(v)$. For this purpose, we recall from (2.15) that:

$\int_{0}^{T}\lVert\nabla\Phi(v)\rVert_{L^{8}(\mathbb{R}^{d})}^{16}\lesssim\int_{0}^{T}\lVert\widetilde{F}(v)\rVert_{L^{8}(\mathbb{R}^{d})}^{16}+\lVert\widetilde{F}(v),\partial_{t}\Phi(v),\nabla\Phi(v)\rVert_{L^{2}\cap
L^{\infty}((0,T),L^{2}(\mathbb{R}^{d}))}^{16}+\lVert\nabla\partial_{t}\Phi(v)\rVert_{L^{2}((0,T)\times\mathbb{R}^{d})}^{16}.$

We have already obtained bound for the last two terms of the right hand side
above in (2.52) and (2.49) and we have only to estimate the first one. So from
(2.47), one has:

$\int_{0}^{T}\lVert\widetilde{F}(v)-F\rVert_{L^{8}(\mathbb{R}^{d}))}^{16}\lesssim\lvert\theta-\theta_{0}\rvert^{16}\int_{0}^{T}\lVert\nabla
v\rVert_{L^{8}(\mathbb{R}^{d}))}^{16}\lesssim\lvert\theta-\theta_{0}\rvert^{p}\lVert
v\rVert_{X_{T}}^{16}.$

In consequence, one has:

$\displaystyle\int_{0}^{T}\lVert\nabla\Phi(v)\rVert_{L^{8}(\mathbb{R}^{d})}^{16}\lesssim\lVert(F,u_{0})\rVert_{Y_{T}^{\theta}}^{p}+\lvert\theta-\theta_{0}\rvert^{16}\lVert
v\rVert_{X_{T}}^{16}.$ (2.55)

#####  Step 2: Estimates for $\partial_{tt}\Phi(v)$ and $L^{4}$ norm estimate
for $\nabla\partial_{t}\Phi(v)$

As in the previous step, one first recalls that $\Phi(v)$ verifies:

$\int_{0}^{T}\lVert\sqrt{\sigma}\partial_{tt}\Phi(v),\;\sigma\nabla\partial_{tt}\Phi(v)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\lVert\sqrt{\sigma}\partial_{t}\nabla\Phi(v),\;\sigma\partial_{tt}\Phi(v)\rVert^{2}_{L^{\infty}((0,T),L^{2}(\mathbb{R}^{d}))}\\\
\lesssim\int_{0}^{\sigma(T)}\lVert\partial_{t}\nabla\Phi(v)\rVert^{2}_{L^{2}(\mathbb{R}^{d})}+\int_{0}^{T}\lVert\partial_{t}\widetilde{F}(v),\,\sigma\partial_{tt}\widetilde{F}(v)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\lVert\nabla\Phi(v)\rVert_{L^{\infty}((0,T),L^{4}(\mathbb{R}^{d}))}^{2}.$

All terms that appear in the right hand side above are controlled in (2.52),
(2.54), (2.49) except the $L^{2}((0,T)\times\mathbb{R}^{d})$ norm estimate for
$\sigma\partial_{tt}\widetilde{F}(v)$ that is:

$\int_{0}^{T}\sigma^{2}\lVert\partial_{tt}\widetilde{F}(v)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\leqslant\int_{0}^{T}\sigma^{2}\lVert
F_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\int_{0}^{T}\sigma^{2}\lVert\partial_{tt}\widetilde{F}(v)-F_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}.$

In order to link the last term in terms of the norm of $v$, we first express
$\partial_{tt}\widetilde{F}(v)-F_{tt}$, from (2.50), as

$\partial_{tt}\widetilde{F}(v)-F_{tt}=(\theta-\theta_{0})\\{2(\mu-\widetilde{\mu})\mathbb{D}v_{tt}+(\lambda-\widetilde{\lambda})\operatorname{div}v_{tt}I_{d}+4\mu_{t}\mathbb{D}v_{t}+2\lambda_{t}\operatorname{div}v_{t}I_{d}+2\mu_{tt}\mathbb{D}v+\lambda_{tt}\operatorname{div}vI_{d}\\}$

and then

$\displaystyle\int_{0}^{T}\sigma^{2}\lVert\partial_{tt}\widetilde{F}(v)-F_{tt}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle\lesssim\lvert\theta-\theta_{0}\rvert^{2}\lVert
v\rVert_{X_{T}}^{2}$

from which, we deduce that

$\displaystyle\int_{0}^{T}\lVert\sqrt{\sigma}\partial_{tt}\Phi(v),\;\sigma\nabla\partial_{tt}\Phi(v)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}+\lVert\sqrt{\sigma}\partial_{t}\nabla\Phi(v),\;\sigma\partial_{tt}\Phi(v)\rVert^{2}_{L^{\infty}((0,T),L^{2}(\mathbb{R}^{d}))}\lesssim\lVert(F,u_{0})\rVert_{Y_{T}^{\theta}}^{2}+\lvert\theta-\theta_{0}\rvert^{2}\lVert
v\rVert_{X_{T}}^{2}.$ (2.56)

This ends the computations of the higher Hoff estimates. Now, we may turn to
the $L^{8/d}((0,T),L^{4}(\mathbb{R}^{d}))$ norm estimate for
$\sqrt{\sigma}\nabla\partial_{t}\Phi(v)$. Thanks to (2.16), one has the
following:

$\int_{0}^{T}\sigma^{4/d}\lVert\nabla\partial_{t}\Phi(v)\rVert_{L^{4}(\mathbb{R}^{d})}^{8/d}\lesssim\int_{0}^{T}\sigma^{4/d}\lVert\partial_{t}\widetilde{F}(v)\rVert_{L^{4}(\mathbb{R}^{d})}^{8/d}+\left[\int_{0}^{T}\lVert\nabla\Phi(v)\rVert_{L^{8}(\mathbb{R}^{d})}^{16}\right]^{1/(2d)}+\lVert\sigma\partial_{tt}\Phi(v)\rVert_{L^{\infty}((0,T),L^{2}(\mathbb{R}^{d}))}^{8/d}\\\
+\lVert\nabla\Phi(v)\rVert_{L^{4}((0,T)\times\mathbb{R}^{d})}^{8/d}+\left[\int_{0}^{T}\lVert\nabla\partial_{t}\Phi(v),\,\partial_{t}\widetilde{F}(v)\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\right]^{4/d}.$

The higher estimate (2.56) and (2.55), (2.54) and (2.51) control all of the
terms that appear in the left hand side above, except the norm of
$\sqrt{\sigma}\partial_{t}\widetilde{F}(v)$ in
$L^{8/d}((0,T),L^{4}(\mathbb{R}^{d}))$. Starting from (2.50), one has:

$\displaystyle\int_{0}^{T}\sigma^{4/d}\lVert\partial_{t}\widetilde{F}(v)-F_{t}\rVert_{L^{4}(\mathbb{R}^{d})}^{8/d}$