part of the velocity. However, a smallness condition on the kinematic
viscosity [17, Section 2.3] has been required. Alternatively, we will derive
functional $\mathcal{A}_{3}$, (see (1.35) for the definition), that controls
the following norm of $\dot{u}$

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

This implies in particular (3.10) and the bound of the energy
$\mathcal{A}_{3}$ does not require any smallness condition on the kinematic
viscosity. A similar functional was considered in [9] in the context of the
Navier-Stokes equations for incompressible fluids. To obtain this functional,
we first apply the material derivative
$\partial_{t}\cdot+\operatorname{div}(\,\cdot\,u)$ to the momentum equation
$\eqref{epq4}_{2}$ and obtain the fact that $\dot{u}$ verifies:

$\displaystyle\partial_{t}(\rho\dot{u}^{j})+\operatorname{div}(\rho\dot{u}^{j}u)=\partial_{k}(\dot{\Pi}^{jk})+\partial_{k}(\Pi^{jk}\operatorname{div}u)-\operatorname{div}(\partial_{k}u\Pi^{jk}).$
(3.11)

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

######  Proposition 3.

Assume that the hypotheses of 1 hold true for $l\in\\{2,3,5\\}$. Then, the
following bound holds true with a constant $M$ that depends on the lower and
upper bounds of the density and the viscosities, as well as $\delta$:

$\displaystyle\mathcal{A}_{3}(t)\leqslant
M\left[C_{0}+\mathcal{A}_{1}(t)+\mathcal{A}_{2}(t)+\mathcal{L}(C_{0},\mathcal{A}_{1}(t),\mathcal{A}_{2}(t),\mathcal{A}_{3}(t))\right].$
(3.12)

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

The proof is given is section 3.2.3. Let us notice that, the functionals
$\mathcal{A}_{1}$, $\mathcal{A}_{2}$ and $\mathcal{A}_{3}$, are under control
as long as the density and the viscosities are upper bounded and bounded away
from vacuum as well, the dynamic viscosity is close to $\widetilde{\mu}$. The
final step of our analysis consists in translating (3.4), (3.5), (3.12), (3.7)
and (3.8) into piecewise Hölder estimates for the density $\rho$ and the
velocity gradient $\nabla u$.

In general, the $L^{\infty}((0,T)\times\mathbb{R}^{2})$ estimate for the
density follows as soon as the effective flux belongs to
$L^{1}((0,T),L^{\infty}(\mathbb{R}^{2}))$. As it is well-explained in [1], the
algebraic structure of the Navier-Stokes equations with density-dependent
viscosity does not allow to propagate the $L^{\infty}$ norm of the density as
it has been done by Hoff [15, 16] for the isotropic case. The issue arises
from the fact that the effective flux, which plays the role of connecting the
momentum equation and the mass equation, does not belong to
$L^{1}_{\text{loc}}((0,\infty),L^{\infty}(\mathbb{R}^{2}))$ when the density
is only bounded. In particular, the effective flux and the velocity gradient
are of the same order. In fact, the effective flux $F$ is given by:

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

where $K$ is a singular integral operator, well-known to map
$L^{\infty}(\mathbb{R}^{2})$ into $BMO(\mathbb{R}^{2})$ space. Thus, even if
the velocity gradient is bounded, it is not obvious that the effective flux
will also be bounded. It is straightforward to show the following quantitative
Hölder estimate by combining 5 and appendix A.

Let $0<\alpha<1$, $1\leqslant p<\infty$ and $K$ an even higher-order Riesz
operator. We consider a curve $\mathcal{C}\in\mathscr{C}^{1+\alpha}$ that
satisfies (1.9), and for $g\in
L^{p}(\mathbb{R}^{2})\cap\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2})$ we
define the following function:

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

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

$\displaystyle\lVert
w\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2})}\leqslant
C\left[\lVert g\rVert_{L^{p}(\mathbb{R}^{2})}+\lVert\llbracket
g\rrbracket\rVert_{L^{p}(\mathcal{C})}+\lVert
g\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2})}+\lVert\llbracket
g\rrbracket\rVert_{L^{\infty}(\mathcal{C})}\mathcal{P}(\mathcal{C})\right]$
(3.13)

where

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

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

section 3.1 appears as an interpolation of the fact that singular integral
operators of Calderón-Zygmung type map linearly $L^{p}(\mathbb{R}^{2})$ into
itself for $1<p<\infty$ and $\dot{\mathscr{C}}^{\alpha}(\mathbb{R}^{2})$ into
itself. By the help of this quantitative Hölder estimate, one has the
following estimates for functionals  and .

######  Proposition 4.

Assume that the density is bounded away from vacuum and upper bounded (3.6).
There are constants $M$ and $M^{\prime}$ such that the following estimates
hold true:

$\displaystyle\text{\textbeta}(t)\leqslant
M^{\prime}e^{M^{\prime}\text{\texttheta}(t)}\left(\lVert
f(\rho_{0})\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma_{0}}(\mathbb{R}^{2})}^{4}+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\right),$
(3.14) $\displaystyle\text{\texttheta}(t)$ $\displaystyle\leqslant
M\left[C_{0}^{2}+\mathcal{A}_{1}^{2}(t)+\mathcal{A}_{2}^{2}(t)+\mathcal{A}_{3}(t)^{2}+\text{\textbeta}(t)(1+\text{\texttheta}(t))+(1+\text{\textbeta}(t))\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\right.$
$\displaystyle\left.+(1+K_{0})e^{M^{\prime}\text{\texttheta}(t)}\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{\infty}\cap
L^{2}(\mathcal{C}(0))}^{4}\left(1+\text{\texttheta}(t)+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\right)\right],$
(3.15)

with

$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau$
$\displaystyle\leqslant
M^{\prime}\left[C_{0}^{2}+\mathcal{A}_{1}(t)^{2}+\mathcal{A}_{2}(t)^{2}+\mathcal{A}_{3}(t)^{2}+\text{\textbeta}(t)\text{\texttheta}(t)\right.$
$\displaystyle\left.+(1+K_{0})\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{\infty}\cap
L^{2}(\mathcal{C}(0))}^{4}\left(1+\text{\texttheta}(t)\right)e^{M^{\prime}\text{\texttheta}(t)}\right].$
(3.16)

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

The proof of this proposition is given in section 3.2.4. It is obvious that
the estimate for  also holds for the dynamic viscosity $\mu(\rho)$. We finally
close the estimates in 1, 3 and 4 with the help of the bootstrap argument
similar to the one in [1], and we do not present the proof. Let $(\rho,u)$ be
a local solution of the equations (3.1) with initial data $(\rho_{0},u_{0})$
that verifies (1.28) and (1.29) and the compatibility condition:

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

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

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

and

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

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

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

and

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

To complete the proof of section 1.2, we have two remaining steps. First, we
need to construct a sequence $(\rho^{\delta},u^{\delta})$ that is globally
defined in time, and converges to $(\rho,u)$, which is a solution of the
Navier-Stokes equations (3.1). Usually, this sequence
$(\rho^{\delta},u^{\delta})$ corresponds to the solution of the Cauchy problem
(3.1) with initial data $(\rho_{0}^{\delta},u_{0}^{\delta})$ where
$(\rho_{0}^{\delta},u_{0}^{\delta})$ is obtained by smoothing
$(\rho_{0},u_{0})$. In our case, this does not seem to be possible. The
problem arises because smoothing the initial data would lead to a loss of
density discontinuity. The approach we employed is more subtle and it is
presented in section 3.2.5 along with the proof of the stability of the
approximate sequence. The second step will be devoted to showing that the
regularity of the limit $(\rho,u)$ is sufficient to establish it uniqueness.
As in the proof of the local well-posedness result, the proof of uniqueness
follows the same lines as in [8] since the velocity limit $u$ comes with the
following regularity:

$\displaystyle\nabla u\in
L^{1}_{\text{loc}}([0,\infty),L^{\infty}(\mathbb{R}^{2}))\quad\text{and}\quad\sqrt{\sigma}\nabla
u\in L^{2}_{\text{loc}}([0,\infty),L^{\infty}(\mathbb{R}^{2})).$ (3.17)

###  3.2. Proofs

####  3.2.1. Proof of 1

###### .

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

$\displaystyle\rho\dot{u}^{j}=\operatorname{div}\Pi^{j}$ (3.18)

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

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

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

$\displaystyle\mathcal{A}_{1}(t)\lesssim C_{0}+\sup_{[0,t]}\lVert
P(\rho)-\widetilde{P}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\int_{0}^{t}\lVert\nabla
u\rVert_{L^{3}(\mathbb{R}^{2})}^{3}+\int_{0}^{t}\lVert
P(\rho)-\widetilde{P}\rVert_{L^{p}(\mathbb{R}^{2})}\lVert\nabla
u\rVert_{L^{2p^{\prime}}(\mathbb{R}^{2})}^{2}$ (3.19)

where we have used the classical energy (3.2) and where $p\geqslant 3$. On the
other hand, we take the material derivative of the momentum equation, written
in the form (3.18), before multiplying by $\dot{u}$ and space integration
yields:

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

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

$\displaystyle\mathcal{A}_{2}(t)\lesssim
C_{0}+\mathcal{A}_{1}(\sigma(t))+\int_{0}^{t}\sigma\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{2})}^{4}+\int_{0}^{t}\sigma\lVert
P(\rho)-\widetilde{P}\rVert_{L^{4}(\mathbb{R}^{2})}^{4}.$ (3.20)

We turn to the estimation of the norm of the gradient of the velocity and the
pressure that appear in the estimate (3.19) and (3.20). First, we apply the
divergence operator to the momentum equation (3.18) in order to obtain an
elliptic equation on the so called effective flux:

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

from which we deduce that

$\displaystyle\lambda(\rho)\operatorname{div}u-P(\rho)+\widetilde{P}=(-\Delta)^{-1}\operatorname{div}\operatorname{div}(2\mu(\rho)\mathbb{D}u)-(-\Delta)^{-1}\operatorname{div}(\rho\dot{u}).$
(3.21)

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

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

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

$\displaystyle
F=(2\mu(\rho)+\lambda(\rho))\operatorname{div}u-P(\rho)+\widetilde{P}=K\\{(\mu(\rho)-\widetilde{\mu})\mathbb{D}u\\}-(-\Delta)^{-1}\operatorname{div}(\rho\dot{u}).$
(3.22)

Above, we have used the fact that

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

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

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

From which, we deduce

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

It then holds that the vorticity takes the following form:

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

Let us notice that the singular integral operators $K$ and $K^{\prime}$ are
homogeneous with symbols of degree zero, such that, in view of Calderón
Zygmung theory, see [36], they map the Lebesgue space $L^{p}(\mathbb{R}^{d})$
into itself for $1<p<\infty$, with norm $\kappa(p)$. It turns out that the
following $L^{p}$ estimates hold true for the vorticity and the effective
flux:

$\displaystyle\lVert
F\rVert_{L^{p}(\mathbb{R}^{2})}\leqslant\kappa(p)\lVert\mu(\rho)-\widetilde{\mu}\rVert_{L^{\infty}(\mathbb{R}^{2})}\lVert\mathbb{D}u\rVert_{L^{p}(\mathbb{R}^{2})}+\lVert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rVert_{L^{p}(\mathbb{R}^{2})},$
(3.23)
$\displaystyle\lVert\mu(\rho)\operatorname{curl}u\rVert_{L^{p}(\mathbb{R}^{2})}\leqslant\kappa(p)\lVert\mu(\rho)-\widetilde{\mu}\rVert_{L^{\infty}(\mathbb{R}^{2})}\lVert\mathbb{D}u\rVert_{L^{p}(\mathbb{R}^{2})}+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\dot{u})\rVert_{L^{p}(\mathbb{R}^{2})}$
(3.24)

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

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

Assume that

$\displaystyle\delta:=\dfrac{\lVert\mu(\rho)-\widetilde{\mu}\rVert_{L^{\infty}(\mathbb{R}^{2})}}{\underline{\mu}}<\dfrac{1}{\kappa(p)^{2}}$
(3.25)

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

$\displaystyle\lVert\nabla u\rVert_{L^{p}(\mathbb{R}^{2})}$
$\displaystyle\leqslant\dfrac{\kappa(p)}{\underline{\mu}(1-\delta\kappa(p)^{2})}\left(\lVert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rVert_{L^{p}(\mathbb{R}^{2})}+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\dot{u})\rVert_{L^{p}(\mathbb{R}^{2})}\right)$
$\displaystyle+\dfrac{\kappa(p)}{(2\underline{\mu}+\underline{\lambda})(1-\delta\kappa(p)^{2})}\lVert
P(\rho)-\widetilde{P}\rVert_{L^{p}(\mathbb{R}^{2})}.$ (3.26)

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

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

One can substitute the divergence of the velocity in the expression of the
effective flux (3.22) as:

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

and then obtains, after Hölder inequality:

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

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

$\displaystyle\dfrac{d}{dt}\int_{\mathbb{R}^{2}}H_{l}(\rho)+\dfrac{1}{l+1}\int_{\mathbb{R}^{2}}(2\mu(\rho)+\lambda(\rho))^{-1}\lvert
P(\rho)-\widetilde{P}\rvert^{l+1}\leqslant\dfrac{1}{l+1}\int_{\mathbb{R}^{2}}(2\mu(\rho)+\lambda(\rho))^{-1}\lvert
F\rvert^{l+1}.$ (3.27)

In order to express the $L^{l+1}$ norm of the effective flux, in the above
inequality, one goes back to (3.23) and then uses (3.26) and obtains the
following:

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

In the following lines, we will replace the above estimate for the effective
flux in (3.27) and absorb the norm of the pressure that is the first term of
the right hand side of the above estimate in the left side of (3.27). And so
that, we need for example, for $l>1$, the following:

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

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

$\displaystyle\delta\left(\dfrac{2\underline{\mu}+\underline{\lambda}}{2\overline{\mu}+\overline{\lambda}}\right)^{-\tfrac{1}{l+1}}<\text{\textkappa}(l)$
(3.28)

then, (3.26) holds for $p=l+1$ and the following inequality also:

$\displaystyle\dfrac{d}{dt}\int_{\mathbb{R}^{2}}H_{l}(\rho)+\lVert
P(\rho)-\widetilde{P}\rVert^{l+1}_{L^{l+1}(\mathbb{R}^{2})}\lesssim\lVert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rVert_{L^{l+1}(\mathbb{R}^{2})}^{l+1}+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\dot{u})\rVert_{L^{l+1}(\mathbb{R}^{2})}^{l+1}.$
(3.29)

The objective of the above computations is to know the appropriate smallness
condition on the viscosity (3.28). Let us notice that, here, in order to close
the estimates for functional $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$, we will
only need the smallness condition on the viscosity (3.28) for $l\in\\{2,3\\}$
however, latter, one will need the case $l=5$. The constants involved in
(3.26) and (3.28) and the latter estimates depend not only on the lower and
upper bounds of the density, viscosities but also on $\delta$. In particular,
when $\delta$ approaches

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

the underlying constants become large. Assuming the smallness condition (3.28)
for $l=2$, the inequality (3.29) holds true for $l=2$ and one can integrate it
in time in order to obtain:

$\displaystyle\sup_{[0,t]}\int_{\mathbb{R}^{2}}H_{2}(\rho)+\int_{0}^{t}\lVert
P(\rho)-\widetilde{P}\rVert^{3}_{L^{3}(\mathbb{R}^{2})}\lesssim\int_{\mathbb{R}^{2}}H_{2}(\rho_{0})+\int_{0}^{t}\left(\lVert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rVert_{L^{3}(\mathbb{R}^{2})}^{3}+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\dot{u})\rVert_{L^{3}(\mathbb{R}^{2})}^{3}\right).$
(3.30)

Next, we assume again that the smallness condition (3.28) holds for $l=3$,
then, one can multiply (3.29) by $\sigma$ before integrating in time and
obtains:

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

By combining the above estimate, the inequality (3.26), Gagliardo-Nirenberg
inequality and the fact that
$H_{3}(\rho)\lesssim_{\overline{\rho}}H_{1}(\rho)$, one obtains the following:

$\displaystyle\int_{0}^{t}\sigma\left(\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{2})}^{4}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{4}(\mathbb{R}^{2})}^{4}\right)\lesssim
C_{0}+\mathcal{A}_{1}(t)\left(C_{0}+\mathcal{A}_{1}(t)\right).$ (3.31)

Above, we also use the fact that

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

We now go back to (3.20) from which we deduce the following bound for the
second Hoff functional:

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

and we turn to the bound for the first Hoff functional. We may first recall
that from (3.19), one has the following:

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

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

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

For the lower time, we take $p=3$ and by Hölder and Young inequalities, we
have from (3.26) and (3.30) the following:

$\displaystyle\int_{0}^{\sigma(t)}\left(\lVert\nabla
u\rVert_{L^{3}(\mathbb{R}^{2})}^{3}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{3}(\mathbb{R}^{2})}^{3}\right)\lesssim
C_{0}+\mathcal{A}_{1}(t)^{1/2}\left(C_{0}+\mathcal{A}_{1}(t)\right).$ (3.32)

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

$\displaystyle\int_{\sigma(t)}^{t}\lVert\nabla
u\rVert_{L^{3}(\mathbb{R}^{2})}^{3}+\int_{\sigma(t)}^{t}\lVert
P(\rho)-\widetilde{P}\rVert_{L^{4}(\mathbb{R}^{2})}\lVert\nabla
u\rVert_{L^{8/3}(\mathbb{R}^{2})}^{2}\lesssim
C_{0}+\mathcal{A}_{1}(t)\left(C_{0}+\mathcal{A}_{1}(t)\right).$ (3.33)

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

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

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

####  3.2.2. Proof of 2

###### .

Let us begin the proof of 2 by rewriting the mass equation $\eqref{epq4}_{1}$.
We recall that it can be written as follows:

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

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

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

where $f(\rho)$ is:

$\displaystyle
f(\rho)=\int_{\widetilde{\rho}}^{\rho}\dfrac{2\mu(s)+\lambda(s)}{s}ds.$ (3.34)

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

$\displaystyle\dfrac{d}{dt}f(\rho(t,X(t,x)))+P(\rho(t,X(t,x)))-\widetilde{P}=-F(t,X(t,x)).$
(3.35)

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

$\displaystyle\dfrac{d}{dt}f(\rho(t,\gamma(t,s)))+P(\rho(t,\gamma(t,s)))-\widetilde{P}=-F(t,\gamma(t,s))$
(3.36)

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

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

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

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

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

$\displaystyle
0<\underline{\text{\textnu}}\leqslant\dfrac{P(\rho)-P(\rho^{\prime})}{f(\rho)-f(\rho^{\prime})}\leqslant\overline{\text{\textnu}}\quad\text{and}\quad\bigg{|}\dfrac{\mu(\rho)-\mu(\rho^{\prime})}{f(\rho)-f(\rho^{\prime})}\bigg{|}\leqslant\overline{\text{\textnu}}$
(3.37)

for some constant $0<\underline{\text{\textnu}}<\overline{\text{\textnu}}$
that depend on the lower and upper bound of the density. We observe that when
the pressure law and viscosity laws are proportional, the condition (3.37)
obviously holds. In fact the above condition states somehow that the
difference of the pressure, of the dynamic viscosity $\mu$ and of $f(\rho)$
vanish at the same state to which the density converges exponentially. The
first condition in (3.37) naturally holds due to the fact that the pressure
and $f(\rho)$ are increasing functions of $\rho$ whereas the second inequality
(3.37) holds as soon as the viscosity is an increasing function of $\rho$. The
same condition holds also for the kinematic viscosity $\lambda$. It then
follows that

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

and consequently:

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

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

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

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

$\displaystyle\lVert\llbracket
f(\rho(t))\rrbracket\rVert_{L^{p}(\mathcal{C}(t))}\leqslant\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{p}(\mathcal{C}(0))}\exp\left(-\underline{\text{\textnu}}t+(4\overline{\text{\textnu}}+1/p)\int_{0}^{t}\lVert\nabla
u(\tau)\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau\right).$ (3.38)

From the condition (3.37), a similar estimate holds for the jump of the
viscosity $\mu(\rho)$ and the pressure $P(\rho)$. Since the kinematic
viscosity $\lambda$ fulfills the same condition as the dynamic viscosity $\mu$
in (3.37) a similar estimate also holds for the jump of $\lambda(\rho)$. Next,
we go back to (1.25) and (1.26) in order to express the jump of
$\operatorname{div}u$ and $\operatorname{curl}u$ as follows:

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

From these expressions and the $L^{p}(\mathcal{C}(t))$ estimate (3.38), we
deduce the $L^{p}(\mathcal{C}(t))$ norm of the jump of $\operatorname{div}u$
and $\operatorname{curl}u$ before deriving the following estimate for the jump
of the velocity gradient using (1.22) and (1.23):

$\displaystyle\lVert\llbracket\nabla
u(t)\rrbracket\rVert_{L^{p}(\mathcal{C}(t))}\leqslant M\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{p}(\mathcal{C}(0))}\left(1+\lVert\nabla
u(t)\rVert_{L^{\infty}(\mathbb{R}^{2})}\right)\exp\left(-\underline{\text{\textnu}}t+(4\overline{\text{\textnu}}+1/p)\int_{0}^{t}\lVert\nabla
u(\tau)\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau\right),\;p\in[1,\infty].$
(3.39)

Above the constant $M$ depends only on the lower and upper bounds of the
density and viscosities. This ends the proof of 2. ∎

####  3.2.3. Proof of 3

###### .

Let us recall that the functional $\mathcal{A}_{3}$ naturally comes while
testing the equation (3.11) with $\sigma^{2}\ddot{u}$. By doing so, we obtain
the following:

$\displaystyle\int_{0}^{t}\sigma^{2}\lVert\sqrt{\rho}\ddot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}$
$\displaystyle+\sigma^{2}(t)\int_{\mathbb{R}^{2}}\left\\{\mu(\rho)\lvert\mathbb{D}\dot{u}\rvert^{2}+\dfrac{\lambda(\rho)}{2}\lvert\operatorname{div}\dot{u}\rvert^{2}\right\\}=2\int_{0}^{\sigma(t)}\sigma\int_{\mathbb{R}^{2}}\left\\{\mu(\rho)\lvert\mathbb{D}\dot{u}\rvert^{2}+\dfrac{\lambda(\rho)}{2}\lvert\operatorname{div}\dot{u}\rvert^{2}\right\\}$
$\displaystyle-\sigma^{2}\int_{\mathbb{R}^{2}}\rho
P^{\prime}(\rho)\operatorname{div}\dot{u}\operatorname{div}u+\int_{0}^{\sigma(t)}\sigma\int_{\mathbb{R}^{2}}\rho
P^{\prime}(\rho)\operatorname{div}\dot{u}\operatorname{div}u+\int_{0}^{t}\sigma^{2}\int_{\mathbb{R}^{2}}\rho
P^{\prime}(\rho)(\operatorname{div}\dot{u})^{2}$
$\displaystyle+\int_{0}^{t}\sigma^{2}\int_{\mathbb{R}^{2}}\operatorname{div}\dot{u}(\operatorname{div}u)^{2}(\rho
P^{\prime}(\rho)-P(\rho)+\widetilde{P})-\int_{0}^{t}\sigma^{2}\int_{\mathbb{R}^{2}}\partial_{l}\dot{u}^{j}\operatorname{div}u\partial_{j}u^{l}(\rho
P^{\prime}(\rho)-P(\rho)+\widetilde{P})$
$\displaystyle+\sigma^{2}(t)I_{1}(t)-2\int_{0}^{\sigma(t)}\sigma
I_{1}(s)ds+\int_{0}^{t}\sigma^{2}I_{2}(s)ds+\int_{0}^{t}\sigma^{2}I_{3}(s)ds$
(3.40)

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

$\displaystyle
I_{1}=\int_{\mathbb{R}^{2}}\varphi(\rho)\partial_{j_{1}}\dot{u}^{j_{2}}\partial_{j_{3}}u^{j_{4}}\partial_{j_{5}}u^{j_{6}}+\int_{\mathbb{R}^{2}}\partial_{j_{1}}\dot{u}^{j_{2}}\partial_{j_{3}}u^{j_{4}}(P(\rho)-\widetilde{P}),$
(3.41) $\displaystyle
I_{2}=\int_{\mathbb{R}^{2}}\varphi(\rho)\partial_{j_{1}}\dot{u}^{j_{2}}\partial_{j_{3}}\dot{u}^{j_{4}}\partial_{j_{5}}u^{j_{6}}+\int_{\mathbb{R}^{2}}\partial_{j_{1}}\dot{u}^{j_{2}}\partial_{j_{3}}\dot{u}^{j_{4}}(P(\rho)-\widetilde{P})+\int_{\mathbb{R}^{2}}\psi(\rho)\partial_{j_{1}}\dot{u}^{j_{2}}\partial_{j_{3}}u^{j_{4}}\partial_{j_{5}}\dot{u}^{j_{6}},$
(3.42) $\displaystyle
I_{3}=\int_{\mathbb{R}^{2}}\varphi(\rho)\partial_{j_{1}}\dot{u}^{j_{2}}\partial_{j_{3}}u^{j_{4}}\partial_{j_{5}}u^{j_{6}}\partial_{j_{7}}u^{j_{8}}+\int_{\mathbb{R}^{2}}\partial_{j_{1}}\dot{u}^{j_{2}}\partial_{j_{3}}u^{j_{4}}\partial_{j_{5}}u^{j_{6}}(P(\rho)-\widetilde{P}),$
(3.43)

and where $\varphi$ is either the viscosity $\mu$ or $\lambda$ or
$\rho\mu^{\prime}$ , $\rho\lambda^{\prime}$, $\rho^{2}\mu^{\prime\prime}$ or
$\rho^{2}\lambda^{\prime\prime}$ whereas $\psi$ is either $\rho P^{\prime}$ or
$\rho^{2}P^{\prime\prime}$. The computations leading to (3.40) can be found in
section B.2 and use the fact that the local solution $(\rho,u)$ given by
section 1.2 enjoys the following Rankine Hugoniot condition at the interface
(see section B.1):

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

and the fact that the second derivative of the velocity, $\ddot{u}$, is
continuous at the interface, see (3). In the following, we will estimate the
terms appearing in the left hand side of (3.40) in term of $C_{0}$ and
functional $\mathcal{A}_{1}$, $\mathcal{A}_{2}$ and $\mathcal{A}_{3}$. It is
clear that the first term on the right hand side of (3.40) is controlled by
the second Hoff functional:

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

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

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

for some positive $\eta$ small in order to absorb the first term in the right
hand side of (3.40). Next, the third term of the right hand side of (3.40) can
be bounded by the help of the initial energy and the second Hoff functional:

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

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

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

We observe the following bound for the sixth and seventh term of the right
hand side of (3.40)

$\displaystyle\bigg{|}\int_{0}^{t}\sigma^{2}\int_{\mathbb{R}^{2}}\operatorname{div}\dot{u}(\operatorname{div}u)^{2}(\rho
P^{\prime}(\rho)-P(\rho)+\widetilde{P})\bigg{|}$
$\displaystyle+\bigg{|}\int_{0}^{t}\sigma^{2}\int_{\mathbb{R}^{2}}\partial_{l}\dot{u}^{j}\operatorname{div}u\partial_{j}u^{l}(\rho
P^{\prime}(\rho)-P(\rho)+\widetilde{P})\bigg{|}$
$\displaystyle\lesssim\left[\int_{0}^{t}\sigma\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\right]^{1/2}\left[\int_{0}^{t}\sigma\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{2})}^{4}\right]^{1/2}$
$\displaystyle\lesssim\mathcal{A}_{2}(t)^{1/2}\left(C_{0}+\mathcal{A}_{1}(t)\left(C_{0}+\mathcal{A}_{1}(t)\right)\right)^{1/2},$
(3.44)

where we have used the estimate (3.31) to control the
$L^{4}((0,T)\times\mathbb{R}^{2})$ norm of the gradient of the velocity. We
now turn to the last four terms of (3.40). The first term among the underlying
terms can be bounded, thanks to Hölder and Young’s inequalities, by:

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

for some $\eta$ small in order to absorb the first term on the right hand side
in the left hand side of (3.40). On the other hand, owing to the fact that the
$L^{\infty}((0,T),L^{4}(\mathbb{R}^{2}))$ norm of the pressure is controlled
by the classical energy, we deduce from (3.26) the following:

$\displaystyle\sup_{[0,t]}\sigma\left(\lVert\nabla
u\rVert^{4}_{L^{4}(\mathbb{R}^{2})}+\lVert
P(\rho)-\widetilde{P}\rVert^{4}_{L^{4}(\mathbb{R}^{2})}\right)$
$\displaystyle\lesssim
C_{0}+\sup_{[0,t]}\sigma\left(\lVert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rVert_{L^{4}(\mathbb{R}^{2})}^{4}+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\dot{u})\rVert_{L^{4}(\mathbb{R}^{2})}^{4}\right)$
$\displaystyle\lesssim
C_{0}+\sup_{[0,t]}\sigma\lVert\rho\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\left(\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\right)$
$\displaystyle\lesssim
C_{0}+\mathcal{A}_{2}(t)\left(C_{0}+\mathcal{A}_{1}(t)\right).$ (3.45)

Thus, we have

$\displaystyle\sup_{[0,t]}\lvert\sigma
I_{1}\rvert\leqslant\eta\mathcal{A}_{3}(t)+C_{\eta}\left(C_{0}+\mathcal{A}_{2}(t)\left(C_{0}+\mathcal{A}_{1}(t)\right)\right).$
(3.46)

Following (3.44), the second term can be bounded by:

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

Next, the penultimate term of (3.40) is, thanks to Hölder and interpolation
inequalities:

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

where we have used the inequality (3.26) to control the
$L^{4}((0,T)\times\mathbb{R}^{2})$ norm of the pressure and of the velocity
gradient. Finally, with the help of the Hölder inequality, the last term of
equation (3.40) can be bounded as follows:

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

In order to control the $L^{6}((0,T)\times\mathbb{R}^{2})$ norm of the
pressure and of the velocity gradient, we will assume that the smallness
condition for the viscosity (3.28) holds for $l=5$, multiply (3.29) by
$\sigma$ and integrate in time and finally obtain, after considering (3.26):

$\displaystyle\int_{0}^{t}\sigma\left(\lVert\nabla
u\rVert_{L^{6}(\mathbb{R}^{2})}^{6}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{6}(\mathbb{R}^{2})}^{6}\right)$
$\displaystyle\lesssim
C_{0}+\int_{0}^{t}\sigma\left(\lVert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rVert_{L^{6}(\mathbb{R}^{2})}^{6}+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\dot{u})\rVert_{L^{6}(\mathbb{R}^{2})}^{6}\right)$
$\displaystyle\lesssim
C_{0}+\int_{0}^{t}\sigma\lVert\rho\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{4}\left(\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\right)$
$\displaystyle\lesssim
C_{0}+\mathcal{A}_{1}(t)\mathcal{A}_{2}(t)\left(\mathcal{A}_{1}(t)+C_{0}\right).$
(3.47)

In sum, we estimate all terms in the right hand side of (3.40) in term of
$C_{0}$, functional $\mathcal{A}_{1}$, $\mathcal{A}_{2}$, $\mathcal{A}_{3}$
and the following term

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

that we will be analyzed in the following lines. The idea is similar to what
has be done previously to estimate the $L^{3}((0,T)\times\mathbb{R}^{2})$ norm
of $\nabla u$ and $L^{4}((0,T)\times\mathbb{R}^{2}))$ norm of
$\sigma^{1/4}\nabla u$. We begin by recalling that the equation (3.11) can be
written as follows:

$\displaystyle\rho\ddot{u}^{j}=\partial_{k}(\dot{\Pi}^{jk})+\partial_{k}(\Pi^{jk}\operatorname{div}u)-\operatorname{div}(\partial_{k}u\Pi^{jk})$
(3.48)

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

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

can be expressed as follows:

$F_{*}=-(-\Delta)^{-1}\operatorname{div}(\rho\ddot{u})-(-\Delta)^{-1}\partial_{jk}\\{\mu(\rho)\partial_{j}u^{l}\partial_{l}u^{k}+\mu(\rho)\partial_{k}u^{l}\partial_{l}u^{j}+2\rho\mu^{\prime}(\rho)\mathbb{D}^{jk}u\operatorname{div}u\\}\\\
+(-\Delta)^{-1}\partial_{jk}(\Pi^{jk}\operatorname{div}u)-(-\Delta)^{-1}\partial_{j}\operatorname{div}(\partial_{k}u\Pi^{jk})+K\\{(\mu(\rho)-\widetilde{\mu})\mathbb{D}\dot{u}\\}.$
(3.49)

On the other hand, we can apply the curl to the equation (3.48) and obtain
that the curl of the material derivative $\dot{u}$ can be expressed as:

$\mu(\rho)\operatorname{curl}\dot{u}=-(-\Delta)^{-1}\operatorname{curl}(\rho\ddot{u})+K^{\prime}\\{(\mu(\rho)-\widetilde{\mu})\mathbb{D}\dot{u}\\}+(-\Delta)^{-1}\operatorname{curl}\partial_{k}(\Pi^{jk}\operatorname{div}u)\\\
-(-\Delta)^{-1}\operatorname{curl}\operatorname{div}(\partial_{k}u\Pi^{jk})-(-\Delta)^{-1}\operatorname{curl}\partial_{k}\\{\mu(\rho)\partial_{j}u^{l}\partial_{l}u^{k}+\mu(\rho)\partial_{k}u^{l}\partial_{l}u^{j}+2\rho\mu^{\prime}(\rho)\mathbb{D}^{jk}u\operatorname{div}u\\}.$
(3.50)

Since the smallness condition (3.26) holds true for $p=3$, one deduces from
singular operators theory that:

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

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

$\int_{0}^{t}\sigma^{5/2}\lVert\nabla\dot{u}\rVert_{L^{3}(\mathbb{R}^{2})}^{3}\lesssim\int_{0}^{t}\sigma^{5/2}\left(\lVert(-\Delta)^{-1}\operatorname{div}(\rho\ddot{u})\rVert_{L^{3}(\mathbb{R}^{2})}^{3}+\lVert(-\Delta)^{-1}\operatorname{curl}(\rho\ddot{u})\rVert_{L^{3}(\mathbb{R}^{2})}^{3}\right)\\\
+\int_{0}^{t}\sigma^{5/2}\lVert\nabla
u\rVert_{L^{3}(\mathbb{R}^{2})}^{3}+\int_{0}^{t}\sigma^{5/2}\left(\lVert\nabla
u\rVert_{L^{6}(\mathbb{R}^{2})}^{6}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{6}(\mathbb{R}^{2})}^{6}\right).$ (3.51)

Let us notice that the last two terms of the above inequality are controlled
by (3.47) for the last term, and (3.32), (3.33) for the previous one. The
first two terms on the right hand side of (3.51) can be bounded using the
Gagliardo-Nirenberg inequality:

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

Above we have used (3.49) and (3.50) in order to express

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

Finally, we have:

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

The norm of the pressure and the velocity gradient are controlled in (3.31)
and (3.45). The estimate of the functional $\mathcal{A}_{3}$ is then achieved.
To obtain regularity about higher-order material derivatives of velocity, one
can follow the procedure as before by applying the material derivative
$\partial_{t}\,\cdot+\operatorname{div}(\,\cdot\,u)$ to (3.48) and so on. It
is clear that the computations are quite lengthy, but they follow the same
principle. ∎

####  3.2.4. Proof of 4

###### .

The proof of 4 is divided in two parts: in the first one we obtain estimates
for the functional  and the second part is devoted to the estimates for  and
for the effective flux $F$. We begin the first part of the proof by noticing
that from the fact that the pressure and $f$ are increasing functions of the
density on $[\underline{\rho},\overline{\rho}]$ and the fact that the
viscosity $2\mu(\rho)+\lambda(\rho)$ is bounded away from vacuum, we have the
following:

$\displaystyle\forall\;\underline{\rho}\leqslant\rho\leqslant\overline{\rho},\quad\quad
0<\underline{\text{\textnu}}\leqslant\dfrac{P(\rho)-\widetilde{P}}{f(\rho)}\leqslant\overline{\text{\textnu}}.$
(3.52)

The constants $\underline{\text{\textnu}},\;\overline{\text{\textnu}}$ depend
on the lower and upper bounds of the density and viscosities. In fact, (3.52)
can be obtained, for example, with the help of the Cauchy mean value theorem.
We deduce from (3.35) the following:

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

where the function $g_{1}$ is:

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

Thus, we have:

$\displaystyle\lVert f(\rho(\tau))\rVert_{L^{\infty}(\mathbb{R}^{2})}\leqslant
e^{-\underline{\text{\textnu}}\,\tau}\lVert
f(\rho_{0})\rVert_{L^{\infty}(\mathbb{R}^{2})}+\int_{0}^{\tau}e^{\underline{\text{\textnu}}\,(\tau^{\prime}-\tau)}\lVert
F(\tau^{\prime})\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau^{\prime},$ (3.53)

from which we deduce:

$\displaystyle\sup_{[0,t]}\lVert
f(\rho)\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}\leqslant\lVert
f(\rho_{0})\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}+M\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}$ (3.54)

for all $0<r_{\alpha}<3$ with $M$ depending on $r_{\alpha}$ and
$\underline{\text{\textnu}}$. To obtain the above estimate, we express the
last term of (3.53) into two terms as follows:

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

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

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

and

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

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

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

and

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

Thus, we have the following:

$\displaystyle\int_{0}^{t}\lVert
f(\rho(\tau))\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}d\tau\lesssim\lVert
f(\rho_{0})\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F(\tau)\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}d\tau.$ (3.55)

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

$\displaystyle\dfrac{d}{dt}f(\rho(t,x_{i}(t)))\Bigg{|}_{i=1}^{i=2}+P(\rho(t,x_{i}(t)))\Bigg{|}_{i=1}^{i=2}=-F(t,x_{i}(t))\Bigg{|}_{i=1}^{i=2}.$
(3.56)

Using the fact that

$\displaystyle
g_{2}(t,x,y):=\dfrac{P(\rho(t,x))-P(\rho(t,y))}{f(\rho(t,x))-f(\rho(t,y))}\in[\underline{\text{\textnu}},\,\overline{\text{\textnu}}]$
(3.57)

we have from (3.56):

$\displaystyle
f(\rho(\tau,x_{i}(\tau)))\bigg{|}_{i=1}^{i=2}=f(\rho_{0}(x_{i}^{0}))\bigg{|}_{i=1}^{i=2}e^{-\int_{0}^{\tau}g_{2}(\tau^{\prime},x_{2}(\tau^{\prime}),x_{1}(\tau^{\prime}))d\tau^{\prime}}-\int_{0}^{\tau}e^{-\int_{\tau^{\prime}}^{\tau}g_{2}(\tau^{\prime\prime},x_{2}(\tau^{\prime\prime}),x_{1}(\tau^{\prime\prime}))d\tau^{\prime\prime}}F(\tau^{\prime},x_{i}(\tau^{\prime}))\bigg{|}_{i=1}^{i=2}d\tau^{\prime}.$
(3.58)

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

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

and thus (3.58) implies:

$\displaystyle\lVert
f(\rho(\tau))\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}$
$\displaystyle\leqslant
e^{\int_{0}^{\tau}\left[-g_{2}(\tau^{\prime},x_{2}(\tau^{\prime}),x_{1}(\tau^{\prime}))+\lVert\nabla
u(\tau^{\prime})\rVert_{L^{\infty}(\mathbb{R}^{2})}\right]d\tau^{\prime}}\lVert
f(\rho_{0})\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma_{0}}(\mathbb{R}^{2})}$
$\displaystyle+\int_{0}^{\tau}e^{\int_{\tau^{\prime}}^{\tau}\left[-g_{2}(\tau^{\prime\prime},x_{2}(\tau^{\prime\prime}),x_{1}(\tau^{\prime\prime}))+\lVert\nabla
u(\tau^{\prime\prime})\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau^{\prime\prime}\right]}\lVert
F(\tau^{\prime})\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}d\tau^{\prime}.$
(3.59)

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

$\displaystyle\int_{\tau^{\prime}}^{\tau}\lVert\nabla
u(\tau^{\prime\prime})\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau^{\prime\prime})}(\mathbb{R}^{2})}d\tau^{\prime\prime}\leqslant\varepsilon(\tau-\tau^{\prime})+C_{\varepsilon}\text{\texttheta}(t)$
(3.60)

for $\varepsilon>0$ small and a constant $C_{\varepsilon}$ depending on
$\varepsilon$ and $r_{\alpha}$. Here we choose
$\varepsilon=\dfrac{\underline{\text{\textnu}}}{4}$ and we deduce from the
lower bound of $g_{2}$ in (3.57), (3.59) and (3.60) the following estimates
for $f(\rho)$:

$\displaystyle\sup_{[0,t]}\lVert
f(\rho)\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2})}^{4}+\int_{0}^{t}\lVert
f(\rho(\tau))\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\leqslant
M^{\prime}e^{M^{\prime}\text{\texttheta}(t)}\left(\lVert
f(\rho_{0})\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma_{0}}(\mathbb{R}^{2})}^{4}+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F(\tau)\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\right).$
(3.61)

Finally, by gathering (3.54), (3.55) and (3.61) we obtain the estimates for
the functional  and this closes the first part of the proof of 4. The second
part concerns the estimates for the functional  and the Hölder norm of the
effective flux. For this purpose, we recall that from the momentum equation,
we have the following expression for the velocity gradient:

$\displaystyle\widetilde{\mu}\Delta u^{j}$
$\displaystyle=\rho\dot{u}-\partial_{j}\left((\widetilde{\mu}+\lambda(\rho))\operatorname{div}u-P(\rho)+\widetilde{P}\right)-\partial_{k}\\{(2\mu(\rho)-\widetilde{\mu})\mathbb{D}^{jk}u\\}$
$\displaystyle=\rho\dot{u}-\partial_{j}\left(\dfrac{\widetilde{\mu}+\lambda(\rho)}{2\mu(\rho)+\lambda(\rho)}F\right)+\partial_{j}\left(\dfrac{2\mu(\rho)-\widetilde{\mu}}{2\mu(\rho)+\lambda(\rho)}(P(\rho)-\widetilde{P})\right)-\partial_{k}\\{2(\mu(\rho)-\widetilde{\mu})\mathbb{D}^{jk}u\\}.$
(3.62)

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

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

and consequently, we have:

$\displaystyle\text{\texttheta}(t)$
$\displaystyle\leqslant\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u_{*}(\tau)\rVert_{\mathscr{C}^{\alpha}(\mathbb{R}^{2})}^{4}d\tau+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u_{P}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau$
$\displaystyle+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u_{F}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u_{\delta}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau.$
(3.63)

Notice that $u_{*}$ is the regular part of the velocity associated to the
first term of the right hand side of (3.62). The gradient of this regular part
is Hölder continuous in the whole $\mathbb{R}^{2}$ with a norm that can be
estimated as follows:

$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u_{*}\rVert_{\mathscr{C}^{\alpha}(\mathbb{R}^{2})}^{4}$
$\displaystyle\lesssim\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\rho\dot{u}\rVert_{L^{6}(\mathbb{R}^{2})}^{12/5}\left(\lVert\nabla
u\rVert_{L^{2}(\mathbb{R}^{2})}^{8/5}+\lVert
P(\rho)-\widetilde{P}\rVert_{L^{2}(\mathbb{R}^{2})}^{8/5}\right)+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{4\alpha}\lVert\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{4(1-\alpha)}$
$\displaystyle\lesssim\left(\mathcal{A}_{1}(t)^{4/5}+E_{0}^{4/5}\right)\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{8/5}\lVert\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{4/5}$
$\displaystyle+\left(\int_{0}^{t}\sigma^{1+(r_{\alpha}-1)/\alpha}\lVert\nabla\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{4}\right)^{\alpha}\left(\int_{0}^{t}\sigma\lVert\dot{u}\rVert_{L^{2}(\mathbb{R}^{2})}^{4}\right)^{1-\alpha}.$
(3.64)

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

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

and, in order to control the singularity at initial time for the second term
of (3.64), we choose $r_{\alpha}\geqslant 2\alpha+1$ and then, we obtain:

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

Finally, we choose

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

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

$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u_{*}(\tau)\rVert_{\mathscr{C}^{\alpha}(\mathbb{R}^{2})}^{4}d\tau\leqslant
M^{\prime}\left[C_{0}^{2}+\mathcal{A}_{1}(t)^{2}+\mathcal{A}_{2}(t)^{2}+\mathcal{A}_{3}(t)^{2}\right].$
(3.65)

The others terms of the expression of the velocity gradient (3.62) are less
regular than $\nabla u_{*}$, in particular, they are discontinuous across the
interface $\gamma$ and we need the quantitative Hölder estimate section 3.1 to
bound their piecewise Hölder norms. For this purpose, we will first estimate
$\mathcal{P}(\mathcal{C}(\tau))$. Obviously, we have:

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

Thus, the following estimate holds:

$\displaystyle\mathcal{P}(\mathcal{C}(\tau))\leqslant
K_{0}\left(1+\int_{0}^{\tau}\lVert\nabla
u(\tau^{\prime})\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma(\tau^{\prime})}(\mathbb{R}^{2})}d\tau^{\prime}\right)\exp\left(m\int_{0}^{\tau}\lVert\nabla
u(\tau^{\prime})\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau^{\prime}\right)$
(3.66)

with the constant $K_{0}$ depending on the regularity of the initial interface
and $m$ a positive constant that depends on polynomial in the definition of
$\mathcal{P}(\mathcal{C})$ in section 3.1.

By applying the quantitative Hölder estimate, the last term of (3.63) can be
bounded as follows:

$\displaystyle\lVert\nabla
u_{\delta}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}$
$\displaystyle\lesssim\lVert\nabla
u(\tau)\rVert_{L^{2}(\mathbb{R}^{2})}+\lVert\llbracket(\mu(\rho(\tau))-\widetilde{\mu})\mathbb{D}u(\tau)\rrbracket\rVert_{L^{2}(\mathcal{C}(\tau))}+\lVert
2(\mu(\rho(\tau))-\widetilde{\mu})\mathbb{D}u\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}$
$\displaystyle+\lVert\llbracket(\mu(\rho(\tau))-\widetilde{\mu})\mathbb{D}u(\tau)\rrbracket\rVert_{L^{\infty}(\mathcal{C}(\tau))}\mathcal{P(\mathcal{C}}(\tau)).$
(3.67)

Obviously, we have

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

and on the other hand we consider (3.60) with
$\varepsilon=\dfrac{\underline{\text{\textnu}}}{4(4\overline{\text{\textnu}}+1/2)}$
in order to obtain a decay rate for the $L^{2}$ norm of the jump of $f(\rho)$
(3.38), and for the $L^{2}$ norm of the jump of the velocity gradient (3.39)
from which we deduce the following:

$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\llbracket(\mu(\rho(\tau))-\widetilde{\mu})\mathbb{D}u(\tau)\rrbracket\rVert_{L^{2}(\mathcal{C}(\tau))}^{4}d\tau\leqslant
M^{\prime}\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{2}(\mathcal{C}(0))}^{4}(1+\text{\texttheta}(t))e^{M^{\prime}\text{\texttheta}(t)}.$
(3.68)

In a similar way, the following estimate holds by gathering (3.38), (3.39),
(3.60) and (3.66):

$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\llbracket(\mu(\rho(\tau))-\widetilde{\mu})\mathbb{D}u(\tau)\rrbracket\rVert_{L^{\infty}(\mathcal{C}(\tau))}^{4}\mathcal{P(\mathcal{C}}(\tau))^{4}d\tau\leqslant
M^{\prime}K_{0}\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{\infty}(\mathcal{C}(0))}^{4}\left(1+\text{\texttheta}(t)\right)e^{M^{\prime}\text{\texttheta}(t)}.$
(3.69)

Finally, the penultimate term of (3.67) is bounded by:

$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
2(\mu(\rho(\tau))-\widetilde{\mu})\mathbb{D}u(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\leqslant
M^{\prime}\text{\textbeta}(t)\text{\texttheta}(t).$ (3.70)

All the previous computations give the following estimate:

$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u_{\delta}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\leqslant
M^{\prime}\left[C_{0}+\mathcal{A}_{1}(t)^{2}+\text{\textbeta}(t)\text{\texttheta}(t)+(1+K_{0})\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{\infty}(\mathcal{C}(0))\cap
L^{2}(\mathcal{C}(0))}^{4}\left(1+\text{\texttheta}(t)\right)e^{M^{\prime}\text{\texttheta}(t)}\right].$
(3.71)

To estimate the third term of the right-hand side of (3.63), we make use of
the quantitative Hölder estimate section 3.1 and we obtain:

$\displaystyle\lVert\nabla
u_{F}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}\lesssim\lVert\mathscr{F}(\tau)\rVert_{L^{4}(\mathbb{R}^{2})}+\lVert\llbracket\mathscr{F}(\tau)\rrbracket\rVert_{L^{4}(\mathcal{C}(\tau))}+\lVert\mathscr{F}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}+\lVert\llbracket\mathscr{F}(\tau)\rrbracket\rVert_{L^{\infty}(\mathcal{C}(\tau))}\mathcal{P}(\mathcal{C}(\tau))$
(3.72)

with

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

and we recall that the effective flux is:

$\displaystyle
F=-(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})+K\\{(\mu(\rho)-\widetilde{\mu})\mathbb{D}u\\}.$
(3.73)

Obviously from (3.31) and Gagliardo-Nirenberg inequality, we have:

$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\mathscr{F}\rVert_{L^{4}(\mathbb{R}^{2})}^{4}\lesssim\int_{0}^{t}\sigma^{r_{\alpha}}\left(\lVert(-\Delta)^{-1}\operatorname{div}(\rho\dot{u})\rVert_{L^{4}(\mathbb{R}^{2})}^{4}+\lVert\nabla
u\rVert_{L^{4}(\mathbb{R}^{2})}^{4}\right)\leqslant
M\left(C_{0}+\left(C_{0}+\mathcal{A}_{1}(t)\right)\mathcal{A}_{1}(t)\right).$
(3.74)

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

$\displaystyle\lvert\llbracket\mathscr{F}\rrbracket\rvert\lesssim\lvert\llbracket
f(\rho)\rrbracket\rvert\lvert<F>\rvert+\lvert\llbracket
F\rrbracket\rvert\quad\text{ and }\quad\llbracket
F\rrbracket=2\llbracket\mu(\rho)\rrbracket\left(<\operatorname{div}u>-<\mathbb{D}^{jk}u>n^{j}_{x}n^{k}_{x}\right).$
(3.75)

Thus, we have:

$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\llbracket\mathscr{F}(\tau)\rrbracket\rVert_{L^{4}(\mathcal{C}(\tau))}^{4}d\tau\leqslant
M^{\prime}\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{4}(\mathcal{C}(0))}^{4}\left(\text{\texttheta}(t)+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}\right)e^{M^{\prime}\text{\texttheta}(t)}.$
(3.76)

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

$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\mathscr{F}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\leqslant
M^{\prime}\int_{0}^{t}\sigma^{r_{\alpha}}\left(\lVert
F(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau+\lVert
f(\rho(\tau))\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}\lVert
F(\tau)\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}d\tau\right).$ (3.77)

Finally, as it has been done previously, we combine from (3.75), (3.66),
(3.60) and (3.38) in order to obtain the following:

$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\llbracket\mathscr{F}(\tau)\rrbracket\rVert_{L^{\infty}(\mathcal{C}(\tau))}^{4}\mathcal{P}(\mathcal{C}(\tau))^{4}d\tau\leqslant
M^{\prime}K_{0}\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{\infty}(\mathcal{C}(0))}^{4}e^{M^{\prime}\text{\texttheta}(t)}\left(\text{\texttheta}(t)+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}\right).$ (3.78)

We sum (3.74), (3.76), (3.77) and (3.78) and we obtain the following estimate
for the third term of (3.63):

$\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u_{F}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\leqslant
M\left[C_{0}+\left(C_{0}+\mathcal{A}_{1}(t)\right)\mathcal{A}_{1}(t)+(1+\text{\textbeta}(t))\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\right.\\\
\left.+\lVert\llbracket f(\rho_{0})\rrbracket\rVert_{L^{2}\cap
L^{\infty}(\mathcal{C}(0))}^{4}(1+K_{0})e^{M^{\prime}\text{\texttheta}(t)}\left(\text{\texttheta}(t)+\int_{0}^{t}\sigma^{r_{\alpha}}\lVert
F\rVert_{L^{\infty}(\mathbb{R}^{2})}^{4}\right)\right].$ (3.79)

To obtain an estimate for the remainder term of (3.63), we apply again the
quantitative Hölder estimate section 3.1:

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

where

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

On the one hand, from (3.31), the following estimate holds:

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

and on the other hand from (3.60) and (3.38) we have:

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

As it has been done previously, we have from (3.66), (3.60) and (3.38) the
following estimate:

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

Finally, we have:

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

Gathering all of these estimates, we have the following estimate for the
remainder term of (3.63):

$\displaystyle\int_{0}^{t}\sigma^{r_{\alpha}}\lVert\nabla
u_{P}(\tau)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma(\tau)}(\mathbb{R}^{2})}^{4}d\tau\leqslant
M\left[C_{0}+\mathcal{A}_{1}(t)(C_{0}+\mathcal{A}_{1}(t))+(1+K_{0})e^{M^{\prime}\text{\texttheta}(t)}\lVert\llbracket
f(\rho_{0})\rrbracket\rVert_{L^{2}\cap
L^{\infty}(\mathcal{C}(0))}^{4}+\text{\textbeta}(t)\right].$ (3.80)

The Hölder norm of the effective flux that appears in (3.79) and (3.80) can be
estimaed by summing (3.65) and (3.71) since the effective flux is given by
(3.73):

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

Finally the proof of 4 follows by summing (3.65), (3.71), (3.79) and (3.80):

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

∎

####  3.2.5. Construction and stability of approximate sequence

We begin the last step of the proof of section 1.2 by identifying which
initial datum to smooth and which not to smooth. First, the initial interface
$\gamma_{0}$ and the initial density and velocity have the required regularity
as stated in the local well-posedness theorem, section 1.2. Therefore, it is
not necessary to smooth them. However, the stress tensor at the initial time
only fulfills

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

In order to preserve the discontinuity in the initial density, we will
consider, for each $\delta>0$, $(\rho_{0},u_{0}^{\delta})$ as initial data for
the equations (3.1) where $u_{0}^{\delta}$ solves the following elliptic
equation:

$\displaystyle-\operatorname{div}(2\mu(\rho_{0})\mathbb{D}u_{0}^{\delta}+(\lambda(\rho_{0})\operatorname{div}u_{0}^{\delta}-P(\rho_{0})+\widetilde{P})I_{2})+c^{\delta}u_{0}^{\delta}=-\operatorname{div}(w_{\delta}*\Pi_{0}).$
(3.81)

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

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

Since the viscosities are bounded away from vacuum and upper bounded and the
pressure $P(\rho_{0})-\widetilde{P}$ belongs to $L^{2}(\mathbb{R}^{2})$, the
existence of unique solution $u_{0}^{\delta}\in H^{1}(\mathbb{R}^{2})$ for the
elliptic equation (3.81) follows from the Lax-Milgram theorem. Moreover the
sequence $(u_{0}^{\delta})_{\delta}$ comes with the following estimate:

$\displaystyle M^{-1}\lVert\nabla
u_{0}^{\delta}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+c^{\delta}\lVert
u_{0}^{\delta}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\leqslant
M\left(\lVert\Pi_{0}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\lVert
P(\rho_{0})-\widetilde{P}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\right)$ (3.82)

with a constant $M>1$ depending on the lower bound of the initial viscosities.
To show that the sequence $(u^{\delta}_{0})_{\delta}$ converges strongly to
$u_{0}$ in $H^{1}(\mathbb{R}^{2})$, we add $\Pi_{0}$ to both sides of (3.81)
and we obtain:

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

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

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

The following estimate holds from Hölder and Young inequalities:

$\displaystyle
M^{-1}\lVert\nabla(u_{0}^{\delta}-u_{0})\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\dfrac{1}{2}c^{\delta}\lVert
u_{0}^{\delta}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\leqslant M\lVert
w_{\delta}*\Pi_{0}-\Pi_{0}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}+\dfrac{1}{2}c^{\delta}\lVert
u_{0}\rVert_{L^{2}(\mathbb{R}^{2})}^{2},$ (3.83)

from which we deduce, owing to the definition of $c^{\delta}$, see (3.82), the
following convergence:

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

As well, we have from (3.83) that

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

from which we deduce:

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

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

$\displaystyle\operatorname{div}(2\mu(\rho_{0})\mathbb{D}u_{0}^{\delta}+(\lambda(\rho_{0})\operatorname{div}u_{0}^{\delta}-P(\rho_{0})+\widetilde{P})I_{2})=-c^{\delta}u_{0}^{\delta}+\operatorname{div}(w_{\delta}*\Pi_{0})\in
L^{2}(\mathbb{R}^{2}).$ (3.84)

With regard to the regularity of the initial density and interface
$\gamma_{0}$, the fact that the initial viscosity $\mu(\rho_{0})$ is a small
perturbation around the constant state $\widetilde{\mu}$, along with the
regularity of $u_{0}^{\delta}$ and (3.84) for each $\delta>0$, section 1.2
guarantee the existence of a local solution $(\rho^{\delta},u^{\delta})$ for
the equation (3.1) with initial data:

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

The solution is defined up to a time $T_{\delta}>0$ and enjoys the regularity
in section 1.2 and section 1.2. This regularity is sufficient for the
computations carried out in the previous sections to make sense. As a
consequence, section 3.1 holds true for solution $(\rho^{\delta},u^{\delta})$.
In particular, all the conditions outlined in section 1.2 are fulfilled,
resulting in $T_{\delta}=+\infty$. We turn to the final step of this section,
which consists of showing the stability of the sequence
$(\rho^{\delta},u^{\delta})$.

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

$\displaystyle\begin{cases}\partial_{t}\rho^{\delta}+\operatorname{div}(\rho^{\delta}u^{\delta})=0,\\\
\partial_{t}(\rho^{\delta}u^{\delta})+\operatorname{div}(\rho^{\delta}u^{\delta}\otimes
u^{\delta})+\nabla
P(\rho^{\delta})=\operatorname{div}(2\mu(\rho^{\delta})\mathbb{D}u^{\delta})+\nabla(\lambda(\rho^{\delta})\operatorname{div}u^{\delta}).\end{cases}$
(3.85)

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

$\displaystyle\lVert\rho^{\delta}-\widetilde{\rho}\rVert_{L^{\infty}((0,T),L^{2}(\mathbb{R}^{2}))}^{2}+\lVert\rho^{\delta}-\widetilde{\rho}\rVert_{L^{\infty}((0,T)\times\mathbb{R}^{2})}^{2}+\lVert
u^{\delta}\rVert_{L^{\infty}((0,T),H^{1}(\mathbb{R}^{2}))}^{2}+\lVert\nabla
u^{\delta}\rVert_{L^{3}((0,T)\times\mathbb{R}^{2})}^{2}+\lVert\dot{u}^{\delta}\rVert_{L^{2}((0,T)\times\mathbb{R}^{2})}^{2}\leqslant
Cc_{0},$ (3.86)

thus, there exist $\rho-\widetilde{\rho}\in
L^{\infty}((0,\infty),L^{2}(\mathbb{R}^{2}))\cap
L^{\infty}((0,\infty)\times\mathbb{R}^{2})$ and $u\in
L^{\infty}((0,\infty),H^{1}(\mathbb{R}^{2}))$ such that:

$\displaystyle\begin{cases}\rho^{\delta}-\widetilde{\rho}\;\;&\rightharpoonup^{*}\;\;\rho-\widetilde{\rho}\quad\text{
in }\quad L^{\infty}((0,T)\times\mathbb{R}^{2}),\\\
\rho^{\delta}-\widetilde{\rho}\;\;&\xrightarrow{}\quad\rho-\widetilde{\rho}\quad\text{
strongly in }\quad\mathscr{C}([0,T],L^{2}_{w}(\mathbb{R}^{2})),\\\
u^{\delta}\;\;&\rightharpoonup^{*}\;\;u\quad\text{ in
}L^{\infty}((0,T),H^{1}(\mathbb{R}^{2})),\\\ u^{\delta}&\xrightarrow{}\quad
u\quad\text{ strongly in
}\mathscr{C}([0,T],L^{2}_{\text{loc}}(\mathbb{R}^{2})).\end{cases}$ (3.87)

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

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

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

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

with

$\displaystyle\gamma^{\delta}(t,s)=\gamma_{0}(s)+\int_{0}^{t}u^{\delta}(\tau,\gamma^{\delta}(\tau,s))d\tau.$
(3.88)

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

$\displaystyle\sup_{[0,T]}\sigma\lVert\nabla
u^{\delta}\rVert_{L^{4}(\mathbb{R}^{2})}^{4}+\int_{0}^{T}\sigma^{r_{\alpha}}\lVert\nabla
u^{\delta}\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma^{\delta}}(\mathbb{R}^{2})}^{4}\leqslant
C_{0}$ (3.89)

from which we deduce, together with (3.86), that the sequence
$(\gamma^{\delta})_{\delta}$ verifies: for all $T>0$ and $n\in\mathbb{N}^{*}$,

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

The constant $C_{0}(T)$ depends on $c_{0}$ and $T$, whereas the constant
$C_{0}(n)$ depends on $c_{0}$ and $n$. So, up to a subsequence,
$(\gamma^{\delta})_{\delta}$ convergences uniformly in $[0,T]\times[0,2\pi]$
to some $\gamma\in W^{1,\infty}((0,T)\times(0,2\pi))$. Moreover, from (3.86)
and embedding, the velocity sequence verifies:

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

such that we can let $\delta$ goes to zero in (3.88) and obtain the fact that
the interface limit $\gamma$ verifies:

$\displaystyle\gamma(t,s)=\gamma_{0}(s)+\int_{0}^{t}u(\tau,\gamma(\tau,s))d\tau.$
(3.90)

On the other hand, we have:

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

in such way that, by lower semi-continuity of the norm and the fact that
$r_{\alpha}<3$ we have $\nabla u\in
L^{1}_{\text{loc}}([0,\infty),\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2}))$.
Thus, the regularity of the velocity ensures the uniqueness of $\gamma$ that
satisfies (3.90). Moreover, the interface has the following regularity,

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

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

$c_{\gamma}(t)\leqslant c_{\gamma_{0}}\exp\left(\int_{0}^{t}\lVert\nabla
u(\tau)\rVert_{L^{\infty}(\mathbb{R}^{2})}d\tau\right).$

We turn to the proof of the strong convergence of the density sequence
$(\rho^{\delta})_{\delta}$. For this purpose, we define, for all $\delta>0$

$\varrho^{\delta}(t,x)=\rho^{\delta}(t,X^{\delta}(t,x))$

where $X^{\delta}$ is the flow of the velocity $u^{\delta}$. As above, we can
show that the sequence $(X^{\delta})_{\delta}$ converges uniformly on compact
set in $[0,\infty)\times\mathbb{R}^{2}$ to
$X\in\mathscr{C}([0,\infty),W^{1,\infty}(\mathbb{R}^{2}))$, the flow of the
limit velocity $u$, given by:

$X(t,x)=x+\int_{0}^{t}u(\tau,X(\tau,x))d\tau.$

We notice that the density sequence $(\rho^{\delta})_{\delta}$ comes with the
following bound

$\sup_{\delta}\lVert\rho^{\delta}-\widetilde{\rho}\rVert_{L^{\infty}((0,\infty),\mathscr{C}^{\alpha}_{pw,\gamma^{\delta}}(\mathbb{R}^{2}))}^{4}\leqslant
Cc_{0}$

from which, we deduce, together with (3.89) and (3.90), the following: for all
$T>0$,

$\sup_{\delta}\lVert\varrho^{\delta}-\widetilde{\rho}\rVert_{L^{\infty}((0,T),\mathscr{C}^{\alpha}_{pw,\gamma_{0}}(\mathbb{R}^{2}))}^{4}+\sup_{\delta}\lVert\partial_{t}\varrho^{\delta}\rVert_{L^{q}((0,T),L^{\infty}(\mathbb{R}^{2}))}^{4}\leqslant
C_{0}(T,q),$

for all $1<q<4/(1+r_{\alpha})$. Since
$\mathscr{C}^{\alpha}_{pw,\gamma_{0}}(\mathbb{R}^{2})$ embeds compactly in
$L^{\infty}_{\text{loc}}(\mathbb{R}^{d})$, we deduce from Aubin-Lions Lemma
that $(\varrho^{\delta})_{\delta}$ converges uniformly on compact set in
$\\{(t,x)\colon x\in\mathbb{R}^{2}\setminus\mathcal{C}(0)\\}$ to some
$\varrho\in\mathscr{C}([0,\infty),\mathscr{C}^{\alpha}_{pw,\gamma_{0}}(\mathbb{R}^{2}))$.
To deduce the strong convergence of the density, from the above analysis, we
will show that $(\rho^{\delta})_{\delta}$ converges to $\overline{\varrho}$
strongly in $L^{2}_{\text{loc}}((0,\infty)\times\mathbb{R}^{2})$ where

$\overline{\varrho}(t,x)=\varrho(t,X^{-1}(t,x)).$

Above, $X^{-1}\in\mathscr{C}([0,\infty),W^{1,\infty}(\mathbb{R}^{2}))$
satisfies, for all $t\geqslant 0$ and $x\in\mathbb{R}^{2}$:

$\displaystyle
X^{-1}(t,x)=x-\int_{0}^{t}u(\tau,X^{-1}(\tau,x))d\tau,\quad\;X(t,X^{-1}(t,x))=x\quad\text{
and }\quad X^{-1}(t,X(t,x))=x.$ (3.91)

In fact, for every $T>0$ and bounded set $B$, we have:

$\int_{0}^{T}\int_{B}\lvert\rho^{\delta}(t,x)-\overline{\varrho}(t,x)\rvert^{2}dtdx=\int_{0}^{T}\int_{X^{\delta}(t)B}\lvert\rho^{\delta}(t,X^{\delta}(t,x))-\overline{\varrho}(t,X^{\delta}(t,x))\rvert^{2}J^{\delta}(t,x)dtdx\\\
\leqslant
M\int_{0}^{T}\int_{X^{\delta}(t)B}\lvert\varrho^{\delta}(t,x)-\varrho(t,x)\rvert^{2}dtdx+M\int_{0}^{T}\int_{X^{\delta}(t)B}\lvert\overline{\varrho}(t,X^{\delta}(t,x))-\varrho(t,x)\rvert^{2}dtdx$
(3.92)

where $J^{\delta}$ comes from the change of variable $x\mapsto
X^{\delta}(t,x)$ and satisfies555we refer to [15, Lemma 3.2]:

$\sup_{[0,T]}\lVert J^{\delta}\rVert_{L^{\infty}(\mathbb{R}^{2})}\leqslant M.$

On the other hand, since the sequence $(X^{\delta})_{\delta}$ converges to
$X$, uniformly on compact set in $[0,\infty)\times\mathbb{R}^{2}$ , there
exists $\delta_{0}>0$ such that for all $\delta<\delta_{0}$, we have:

$X^{\delta}(t)B\subset X(t)B+B(0,1).$

Moreover, due to the uniform continuity of $\varrho$ on the both sides of
$\mathcal{C}(0)$, and (3.91) we have:

$\overline{\varrho}(t,X^{\delta}(t,x))=\varrho(t,X^{-1}(t,X^{\delta}(t,x)))\quad\xrightarrow{\delta\to
0}\quad\varrho(t,x)\quad\text{ a.e}.$

As a result, thanks to the dominated convergence theorem, we have:

$\displaystyle\lim_{\delta\to
0}\int_{0}^{T}\int_{X^{\delta}(t)B}\lvert\overline{\varrho}(t,X^{\delta}(t,x))-\varrho(t,x)\rvert^{2}dtdx=0.$
(3.93)

As well, since the sequence $(\varrho^{\delta})_{\delta}$ converges to
$\varrho$, uniformly in compact set, we have:

$\displaystyle\lim_{\delta\to
0}\int_{0}^{T}\int_{X^{\delta}(t)B}\lvert\varrho^{\delta}(t,x)-\varrho(t,x)\rvert^{2}dtdx=0.$
(3.94)

Thus, from (3.92), (3.93) and (3.94), the strong convergence of the sequence
$(\rho^{\delta})_{\delta}$ to $\overline{\varrho}$ in
$L^{2}_{\text{loc}}((0,\infty)\times\mathbb{R}^{2})$ follows. In particular we
have
$\rho=\overline{\varrho}\in\mathscr{C}([0,\infty),\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2}))$
and by interpolation, it holds that the sequence of density
$(\rho^{\delta})_{\delta}$ converges to $\rho$ strongly in
$L^{q}_{\text{loc}}((0,\infty)\times\mathbb{R}^{2})$ for all $1\leqslant
q<\infty$.

Gathering all of these strong and weak convergences of the sequence
$(\rho^{\delta},u^{\delta})_{\delta}$ and classical arguments, we can let
$\delta$ goes to zero in (3.85) and we obtain that $(\rho,u)$ solves the
following:

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

This ends the proof of section 1.2.

### Acknowledgment

This project has received funding from the European Union’s Horizon 2020
research and innovation program under the Marie Skłodowska-Curie grant
agreement No 945322. I am grateful for the support of the SingFlows project
grant (ANR-18- CE40-0027) of the French National Research Agency (ANR). This
work has been partially supported by the project CRISIS (ANR-20-CE40-0020-01),
operated by the French National Research Agency (ANR). I would like to
acknowledge my PhD advisors Cosmin Burtea and David Gérard-Varet for the
fruitful discussions and careful reading of this work.

##  Appendix A Quantitative Hölder estimates

In this section, we present three results concerning the regularity of even
higher-order Riesz transforms of discontinuous functions. The first result is
a recent contribution by Gancedo and García-Juárez [12] regarding quantitative
Hölder estimates for even singular integral operators on patches. This is the
key element for their recent result [13] concerning the global well-posedness
for the 2d Navier-Stokes free boundary problem for incompressible fluids. 5
summarizes their computations in step 4 and step 5. Finally, the last result
we state in this section is from [17] and addresses the extension of a
piecewise Hölder continuous function to a function that is Hölder continuous
over the whole space.

We consider a singular integral operator $T$ of even order $2l$ $(l\geqslant
1)$, given by:

$\displaystyle T(g)(x)=p.v.\int_{\mathbb{R}^{d}}K(x-y)g(y)dy$ (A.1)

where the kernel $K$ is given by:

$K(x)=\dfrac{P_{2l}(x)}{\lvert x\rvert^{d+2l}}$

with $P$ an harmonic homogeneous polynomial of degree $2l$. We also consider
an open bounded simply connected subset $D\subset\mathbb{R}^{d}$ such that its
boundary is an hyper-surface

$\displaystyle\partial
D=:\mathcal{C}=\left\\{\gamma(s)\in\mathbb{R}^{d}\;\colon\;s\in\mathbb{T}^{d-1}\right\\}\in\mathscr{C}^{1+\alpha}$
(A.2)

and verifies the following non self-intersection condition:

$\displaystyle\exists\;c_{\gamma}>0,\;\mid\forall\;\;(s,s^{\prime})\in\mathbb{T}^{d-1}\colon\quad\lvert\gamma(s)-\gamma(s^{\prime})\rvert\geqslant
c_{\gamma}^{-1}\lvert s-s^{\prime}\rvert.$ (A.3)

We finally consider $\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})$, the
space of piecewise Hölder continuous function defined in section 1.1. The
first result by Gancedo and García-Juárez [12] reads as follows.

Let $0<\alpha<1$, $T$ be a singular integral operator in (A.1) and let
$\mathcal{C}$ be an interface verifying (A.2) and (A.3). There exists a
constant $C=C(d,\alpha)$ and a polynomial $\mathrm{P}$ depending on $T$ such
that the following estimate holds true:

$\displaystyle\lVert
T(\mathbb{1}_{D})\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})}\leqslant
C(1+\lvert\partial
D\rvert)\mathrm{P}(c_{\gamma}+\lVert\nabla\gamma\rVert_{L^{\infty}})(1+\lVert\nabla\gamma\rVert_{\dot{\mathscr{C}}^{\alpha}}).$
(A.4)

The second result from [13, Step 4 & Step 5] reads as follows.

######  Proposition 5.

Assume that the hypotheses in appendix A hold true for singular integral
operator $T$ and interface $\mathcal{C}$ and let $1\leqslant p<\infty$. There
exists a constant $C=C(d,\alpha,p)$ such that for all $g\in
L^{p}(\mathbb{R}^{d})\cap\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})$, we
have:

$\lVert T(g)\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})}\leqslant
C\left[\eta^{-\tfrac{d}{p}}\lVert g\rVert_{L^{p}(\mathbb{R}^{d})}+\lVert
g\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{d})}+\mathscr{P}(\mathcal{C})\lVert
g\rVert_{L^{\infty}(\mathbb{R}^{d})}\right],$

where

$\mathscr{P}(\partial D)=(1+\lvert\partial
D\rvert)P(c_{\gamma}+\lVert\nabla\gamma\rVert_{L^{\infty}})\left(1+\lVert\nabla\gamma\rVert_{\dot{\mathscr{C}}^{\alpha}}\right)\quad\text{and}\quad\eta^{\alpha}:=\min\left\\{1,\;\dfrac{1}{16c_{\gamma}\lVert\nabla\gamma\rVert_{\dot{\mathscr{C}}^{\alpha}}}\right\\}.$

This result was used to obtain the piecewise Hölder estimate for the velocity
gradient in the proof of local well-posedness result. Due to some technical
reasons 666exponential growth of $c_{\gamma}$ and
$\lVert\nabla\gamma\rVert_{\mathscr{C}^{\alpha}}$ in time, we gather appendix
A and the following result from [17, Lemma 5.2] in order to obtain the global
well-posedness result in $\mathbb{R}^{2}$.

Let $d=2$ and $\mathcal{C}$ be an interface that verifies (A.2), (A.3). For
all $g\in\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2})$, there exists a
function $g^{e}\in\mathscr{C}^{\alpha}(\mathbb{R}^{2})$ such that $g^{e}=g$ on
$D$ (or $D^{c}$) with the following estimates:

$\displaystyle\begin{cases}\lVert
g^{e}\rVert_{L^{\infty}(\mathbb{R}^{2})}&\leqslant C\lVert
g\rVert_{L^{\infty}(\mathbb{R}^{2})},\\\ \lVert
g^{e}\rVert_{\dot{\mathscr{C}}^{\alpha}(\mathbb{R}^{2})}+\lVert
g^{e}-g\rVert_{\dot{\mathscr{C}}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2})}&\leqslant
C\left[\lVert
g\rVert_{\mathscr{C}^{\alpha}_{pw,\gamma}(\mathbb{R}^{2})}+\lVert\llbracket
g\rrbracket\rVert_{L^{\infty}(\mathcal{C})}c_{\gamma}^{2}\lVert\nabla\gamma\rVert_{\dot{\mathscr{C}}^{\alpha}}^{1/\alpha}\right],\\\
\lVert g^{e}-g\rVert_{L^{p}(\mathbb{R}^{2})}&\leqslant C\lVert\llbracket
g\rrbracket\rVert_{L^{p}(\mathcal{C})},\quad\quad 1\leqslant
p\leqslant\infty,\end{cases}$ (A.5)

with a universal constant $C>0$ independent of $\alpha$ and $\mathcal{C}$.

The above result does not require the interface to be bounded; that is, $D$ or
$D^{c}$ could be an unbounded domain.

##  Appendix B Energy computations

In this section, we will provide details of the computations of two estimates
for solution $(\rho,u)$ of the system:

$\displaystyle\begin{cases}\partial_{t}\rho+\operatorname{div}(\rho u)=0,\\\
\partial_{t}(\rho u)+\operatorname{div}(\rho u\otimes
u)=\operatorname{div}\Pi,\\\ \llbracket\Pi\rrbracket\cdot n=0\quad\text{ in
}\quad\mathcal{C}\end{cases}$ (B.1)

where the stress tensor $\Pi$ is given by:

$\Pi=2\mu(\rho)\mathbb{D}u+(\lambda(\rho)\operatorname{div}u-P(\rho)+\widetilde{P})I_{d}.$

###  B.1. Second Hoff energy

We start by investigating the Rankine Hugoniot conditions for $\dot{u}$ at the
surface of discontinuity $\mathcal{C}$. We first notice that after applying
the derivative $\mathcal{A}$ to the momentum equation, we obtain that
$\dot{u}$ solves the equation

$\displaystyle\partial_{t}(\rho\dot{u}^{j})+\operatorname{div}(\rho\dot{u}^{j}u)=\partial_{k}(\dot{\Pi}^{jk})+\partial_{k}(\Pi^{jk}\operatorname{div}u)-\operatorname{div}(\partial_{k}u\Pi^{jk}).$
(B.2)

So by Rankine Hugoniot conditions,

$\llbracket\rho\dot{u}^{j}\rrbracket
n_{t}+\llbracket\rho\dot{u}^{j}u^{k}\rrbracket
n_{x}^{k}=\llbracket\dot{\Pi}^{jk}\rrbracket
n_{x}^{k}+\llbracket\Pi^{jk}\operatorname{div}u\rrbracket
n_{x}^{k}-\llbracket\partial_{k}u^{l}\Pi^{jk}\rrbracket n_{x}^{l}.$

Also, thanks to the Rankine Hugoniot condition applied, this time, to the mass
equation $\eqref{eqA.1}_{1}$ the following jump condition holds true:

$\llbracket\rho\rrbracket n_{t}+\llbracket\rho u^{k}\rrbracket n_{x}^{k}=0$

and since the material derivative of the velocity is continuous, we finally
obtain that:

$\displaystyle\llbracket\dot{\Pi}^{jk}\rrbracket
n_{x}^{k}+\llbracket\Pi^{jk}\operatorname{div}u\rrbracket
n_{x}^{k}-\llbracket\partial_{k}u^{l}\Pi^{jk}\rrbracket n_{x}^{l}=0.$ (B.3)

This relation will be used in the subsequent computations. We recall that the
second Hoff estimate consists in multiplying (B.2) by the material derivative
of the velocity before integrating in space. By doing so, we have:

$\displaystyle\int_{\mathbb{R}^{2}}\dot{u}^{j}\\{\partial_{t}(\rho\dot{u}^{j})+\operatorname{div}(\rho\dot{u}^{j}u)\\}=\int_{\mathbb{R}^{2}}\dot{u}^{j}\\{\partial_{k}(\dot{\Pi}^{jk})+\partial_{k}(\Pi^{jk}\operatorname{div}u)-\operatorname{div}(\partial_{k}u\Pi^{jk})\\}.$
(B.4)

The right hand side of the above equality, is:

$\displaystyle\int_{\mathbb{R}^{2}}\dot{u}^{j}\\{\partial_{t}(\rho\dot{u}^{j})+\operatorname{div}(\rho\dot{u}^{j}u)\\}$
$\displaystyle=\int_{\mathbb{R}^{2}}\partial_{t}(\rho\lvert\dot{u}^{j}\rvert^{2})-\int_{\mathbb{R}^{2}}(\rho\dot{u}^{j}\partial_{t}\dot{u}^{j})+\int_{\mathbb{R}^{2}}\operatorname{div}(\rho\lvert\dot{u}^{j}\rvert^{2}u)(s,x)-\int_{\mathbb{R}^{2}}\rho\dot{u}^{j}u\cdot\nabla\dot{u}^{j}$
$\displaystyle=\dfrac{1}{2}\int_{\mathbb{R}^{2}}\partial_{t}(\rho\lvert\dot{u}^{j}\rvert^{2})+\dfrac{1}{2}\int_{\mathbb{R}^{2}}\lvert\dot{u}^{j}\rvert^{2}\partial_{t}\rho+\dfrac{1}{2}\int_{\mathbb{R}^{2}}\operatorname{div}(\rho\lvert\dot{u}^{j}\rvert^{2}u)+\dfrac{1}{2}\int_{\mathbb{R}^{2}}\lvert\dot{u}^{j}\rvert^{2}\operatorname{div}(\rho
u)$
$\displaystyle=\dfrac{1}{2}\dfrac{d}{dt}\int_{\mathbb{R}^{2}}\rho\lvert\dot{u}^{j}\rvert^{2},$
(B.5)

where we have used the Liouville transport equation and the mass equation
$\eqref{eqA.1}_{1}$. We turn now to the computations of the right hand side of
(B.4):

$\displaystyle\int_{\mathbb{R}^{2}}\dot{u}^{j}\\{\partial_{k}(\dot{\Pi}^{jk})+\partial_{k}(\Pi^{jk}\operatorname{div}u)-\operatorname{div}(\partial_{k}u\Pi^{jk})\\}$
$\displaystyle=\int_{\mathbb{R}^{2}}\left\\{\partial_{k}\\{\dot{u}^{j}\dot{\Pi}^{jk}\\}+\partial_{k}\\{\dot{u}^{j}\Pi^{jk}\operatorname{div}u\\}-\partial_{l}\\{\dot{u}^{j}\Pi^{jk}\partial_{k}u^{l}\right\\}$
$\displaystyle-\int_{\mathbb{R}^{2}}\partial_{k}\dot{u}^{j}\dot{\Pi}^{jk}-\int_{\mathbb{R}^{2}}\partial_{k}\dot{u}^{j}\Pi^{jk}\operatorname{div}u+\int_{\mathbb{R}^{2}}\partial_{l}\dot{u}^{j}\partial_{k}u^{l}\Pi^{jk}.$

Since

$\displaystyle\dot{\Pi}^{jk}$
$\displaystyle=2\mu(\rho)\mathbb{D}^{jk}\dot{u}-\mu(\rho)\partial_{j}u^{l}\partial_{l}u^{k}-\mu(\rho)\partial_{k}u^{l}\partial_{l}u^{j}-2\rho\mu^{\prime}(\rho)\mathbb{D}^{jk}u\operatorname{div}u$
$\displaystyle+\left(\lambda(\rho)\operatorname{div}\dot{u}-\lambda(\rho)\nabla
u^{l}\partial_{l}u-\rho\lambda^{\prime}(\rho)(\operatorname{div}u)^{2}+\rho
P^{\prime}(\rho)\operatorname{div}u\right)\delta^{jk},$ (B.6)

then, the terms in the right hand side of (B.4) are:

$\displaystyle\int_{\mathbb{R}^{2}}\dot{u}^{j}\\{\partial_{k}(\dot{\Pi}^{jk})$
$\displaystyle+\partial_{k}(\Pi^{jk}\operatorname{div}u)-\operatorname{div}(\partial_{k}u\Pi^{jk})\\}=-\int_{\mathbb{R}^{2}}2\mu(\rho)\lvert\mathbb{D}^{jk}\dot{u}\rvert^{2}+\int_{\mathbb{R}^{2}}\left\\{\partial_{k}\\{\dot{u}^{j}\dot{\Pi}^{jk}\\}+\partial_{k}\\{\dot{u}^{j}\Pi^{jk}\operatorname{div}u\\}-\partial_{l}\\{\dot{u}^{j}\Pi^{jk}\partial_{k}u^{l}\\}\right\\}$
$\displaystyle+\int_{\mathbb{R}^{2}}\partial_{k}\dot{u}^{j}\left\\{\mu(\rho)\partial_{j}u^{l}\partial_{l}u^{k}+\mu(\rho)\partial_{k}u^{l}\partial_{l}u^{j}+2\rho\mu^{\prime}(\rho)\mathbb{D}^{jk}u\operatorname{div}u\right\\}-\int_{\mathbb{R}^{2}}\lambda(\rho)\lvert\operatorname{div}\dot{u}\rvert^{2}$
$\displaystyle+\int_{\mathbb{R}^{2}}\operatorname{div}\dot{u}\left\\{\lambda(\rho)\nabla
u^{l}\partial_{l}u+\rho\lambda^{\prime}(\rho)(\operatorname{div}u)^{2}-\rho
P^{\prime}(\rho)\operatorname{div}u\right\\}-\int_{\mathbb{R}^{2}}\partial_{k}\dot{u}^{j}\Pi^{jk}\operatorname{div}u+\int_{\mathbb{R}^{2}}\partial_{l}\dot{u}^{j}\partial_{k}u^{l}\Pi^{jk}.$
(B.7)

We can combine (B.4), (B.5) and (B.7) and use the jump condition (B.3) and the
continuity of $\dot{u}$ in order to obtain:

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

###  B.2. Third Hoff estimate

While computing the second Hoff energy, one notices that the material
derivative of the velocity solves a parabolic equation like the velocity. The
goal is to perform the first Hoff energy to this equation (B.2) just by
testing with the material derivative of $\dot{u}$. For this purpose, we write
(B.2) as follows:

$\displaystyle\rho\ddot{u}^{j}=\partial_{k}(\dot{\Pi}^{jk})+\partial_{k}(\Pi^{jk}\operatorname{div}u)-\operatorname{div}(\partial_{k}u\Pi^{jk})$
(B.8)

where $\ddot{u}$ is the material derivative of $\dot{u}$, that is:

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

One then multiplies the above by $\ddot{u}^{j}$ in order to obtain the
following:

$\displaystyle\int_{\mathbb{R}^{2}}\rho\lvert\ddot{u}\rvert^{2}$
$\displaystyle=\int_{\mathbb{R}^{d}}\ddot{u}^{j}\partial_{k}(\dot{\Pi}^{jk})+\int_{\mathbb{R}^{d}}\ddot{u}^{j}\partial_{k}(\Pi^{jk}\operatorname{div}u)-\int_{\mathbb{R}^{d}}\ddot{u}^{j}\operatorname{div}(\partial_{k}u\Pi^{jk})$
$\displaystyle=\int_{\Gamma}\left\\{\llbracket\ddot{u}^{j}(\dot{\Pi}^{jk}+\Pi^{jk}\operatorname{div}u)\rrbracket
n^{k}_{x}-\llbracket\ddot{u}^{j}\partial_{k}u^{l}\Pi^{jk}\rrbracket
n_{x}^{l}\right\\}$
$\displaystyle-\int_{\mathbb{R}^{d}}\partial_{k}\ddot{u}^{j}\dot{\Pi}^{jk}-\int_{\mathbb{R}^{d}}\partial_{k}\ddot{u}^{j}\Pi^{jk}\operatorname{div}u+\int_{\mathbb{R}^{d}}\partial_{l}\ddot{u}^{j}\partial_{k}u^{l}\Pi^{jk}.$
(B.9)

The first term in the right hand side above vanishes since $\ddot{u}$ is
continuous through the interface and due to (B.3). Next, the second term is,
thanks to (B.6):

$\displaystyle-\int_{\mathbb{R}^{d}}\partial_{k}\ddot{u}^{j}\dot{\Pi}^{jk}$
$\displaystyle=-\int_{\mathbb{R}^{d}}\partial_{k}\ddot{u}^{j}\left\\{2\mu(\rho)\mathbb{D}^{jk}\dot{u}-\mu(\rho)\partial_{j}u^{l}\partial_{l}u^{k}-\mu(\rho)\partial_{k}u^{l}\partial_{l}u^{j}-2\rho\mu^{\prime}(\rho)\mathbb{D}^{jk}u\operatorname{div}u\right.$
$\displaystyle\left.+\left(\lambda(\rho)\operatorname{div}\dot{u}-\lambda(\rho)\nabla
u^{l}\partial_{l}u-\rho\lambda^{\prime}(\rho)(\operatorname{div}u)^{2}+\rho
P^{\prime}(\rho)\operatorname{div}u\right)\delta^{jk}\right\\}.$ (B.10)

The first term in the right hand side above is:

$\displaystyle-\int_{\mathbb{R}^{d}}2\mu(\rho)\partial_{k}\ddot{u}^{j}\mathbb{D}^{jk}\dot{u}$
$\displaystyle=-2\int_{\mathbb{R}^{d}}\mu(\rho)\partial_{tk}\dot{u}^{j}\mathbb{D}^{jk}\dot{u}-2\int_{\mathbb{R}^{d}}\mu(\rho)u^{l}\partial_{lk}\dot{u}^{j}\mathbb{D}^{jk}\dot{u}-2\int_{\mathbb{R}^{d}}\mu(\rho)\partial_{k}u^{l}\partial_{l}\dot{u}^{j}\mathbb{D}^{jk}\dot{u}$
$\displaystyle=-\int_{\mathbb{R}^{d}}\left[\partial_{t}\\{\mu(\rho)\lvert\mathbb{D}^{jk}\dot{u}\rvert^{2}\\}+\operatorname{div}\\{\mu(\rho)u\lvert\mathbb{D}^{jk}\dot{u}\rvert^{2}\\}\right]-2\int_{\mathbb{R}^{d}}\mu(\rho)\partial_{k}u^{l}\partial_{l}\dot{u}^{j}\mathbb{D}^{jk}\dot{u}$
$\displaystyle+\int_{\mathbb{R}^{d}}\lvert\mathbb{D}^{jk}\dot{u}\rvert^{2}\\{\partial_{t}\mu(\rho)+\operatorname{div}(\mu(\rho)u)\\}$
$\displaystyle=-\dfrac{d}{dt}\int_{\mathbb{R}^{d}}\mu(\rho)\lvert\mathbb{D}^{jk}\dot{u}\rvert^{2}-2\int_{\mathbb{R}^{d}}\mu(\rho)\partial_{k}u^{l}\partial_{l}\dot{u}^{j}\mathbb{D}^{jk}\dot{u}+\int_{\mathbb{R}^{d}}\lvert\mathbb{D}^{jk}\dot{u}\rvert^{2}\\{\rho\mu^{\prime}(\rho)-\mu(\rho)\\}\operatorname{div}u.$

As for the second term of the right hand side of (B.10), one has:

$\displaystyle\int_{\mathbb{R}^{d}}\mu(\rho)\partial_{k}\ddot{u}^{j}\partial_{j}u^{l}\partial_{l}u^{k}$
$\displaystyle=\int_{\mathbb{R}^{2}}\partial_{t}\\{\mu(\rho)\partial_{k}\dot{u}^{j}\partial_{j}u^{l}\partial_{l}u^{k}\\}+\int_{\mathbb{R}^{2}}\partial_{m}\\{\mu(\rho)u^{m}\partial_{k}\dot{u}^{j}\partial_{j}u^{l}\partial_{l}u^{k}\\}$
$\displaystyle+\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{k}u^{m}\partial_{m}\dot{u}^{j}\partial_{j}u^{l}\partial_{l}u^{k}+\int_{\mathbb{R}^{2}}(\rho\mu^{\prime}(\rho)-\mu(\rho))\partial_{k}\dot{u}^{j}\operatorname{div}u\partial_{j}u^{l}\partial_{l}u^{k}$
$\displaystyle-\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{k}\dot{u}^{j}\partial_{j}\dot{u}^{l}\partial_{l}u^{k}+\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{k}\dot{u}^{j}\partial_{j}u^{m}\partial_{m}u^{l}\partial_{l}u^{k}$
$\displaystyle-\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{k}\dot{u}^{j}\partial_{j}u^{l}\partial_{l}\dot{u}^{k}+\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{k}\dot{u}^{j}\partial_{j}u^{l}\partial_{l}u^{m}\partial_{m}u^{k}$
$\displaystyle=\dfrac{d}{dt}\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{k}\dot{u}^{j}\partial_{j}u^{l}\partial_{l}u^{k}+\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{k}u^{m}\partial_{m}\dot{u}^{j}\partial_{j}u^{l}\partial_{l}u^{k}$
$\displaystyle+\int_{\mathbb{R}^{2}}(\rho\mu^{\prime}(\rho)-\mu(\rho))\partial_{k}\dot{u}^{j}\operatorname{div}u\partial_{j}u^{l}\partial_{l}u^{k}-\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{k}\dot{u}^{j}\partial_{j}\dot{u}^{l}\partial_{l}u^{k}$
$\displaystyle-\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{k}\dot{u}^{j}\partial_{j}u^{l}\partial_{l}\dot{u}^{k}+\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{k}\dot{u}^{j}\partial_{j}u^{m}\partial_{m}u^{l}\partial_{l}u^{k}$
$\displaystyle+\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{k}\dot{u}^{j}\partial_{j}u^{l}\partial_{l}u^{m}\partial_{m}u^{k}.$

The third term in the right hand side of (B.10) can be deduced from the above
computations just by interchanging $j$ and $k$. On the other hand, the fourth
term, is:

$\displaystyle
2\int_{\mathbb{R}^{d}}\rho\mu^{\prime}(\rho)\partial_{k}\ddot{u}^{j}\mathbb{D}^{jk}u\operatorname{div}u$
$\displaystyle=2\dfrac{d}{dt}\int_{\mathbb{R}^{2}}\rho\mu^{\prime}(\rho)\partial_{k}\dot{u}^{j}\mathbb{D}^{jk}u\operatorname{div}u+2\int_{\mathbb{R}^{d}}\rho\mu^{\prime}(\rho)\partial_{k}u^{m}\partial_{m}\dot{u}^{j}\mathbb{D}^{jk}u\operatorname{div}u$
$\displaystyle+2\int_{\mathbb{R}^{d}}\rho^{2}\mu^{\prime\prime}(\rho)\partial_{k}\dot{u}^{j}\operatorname{div}u\mathbb{D}^{jk}u\operatorname{div}u-2\int_{\mathbb{R}^{2}}\rho\mu^{\prime}(\rho)\partial_{k}\dot{u}^{j}\mathbb{D}^{jk}\dot{u}\operatorname{div}u$
$\displaystyle+\int_{\mathbb{R}^{2}}\rho\mu^{\prime}(\rho)\partial_{k}\dot{u}^{j}\left(\partial_{j}u^{m}\partial_{m}u^{k}+\partial_{k}u^{m}\partial_{m}u^{j}\right)\operatorname{div}u$
$\displaystyle-2\int_{\mathbb{R}^{2}}\rho\mu^{\prime}(\rho)\partial_{k}\dot{u}^{j}\mathbb{D}^{jk}u\operatorname{div}\dot{u}+2\int_{\mathbb{R}^{2}}\rho\mu^{\prime}(\rho)\partial_{k}\dot{u}^{j}\mathbb{D}^{jk}u\nabla
u^{m}\cdot\partial_{m}u.$

Next, the fifth term of the right hand side of (B.10) can be computed as
follows:

$\displaystyle-\int_{\mathbb{R}^{d}}\lambda(\rho)\operatorname{div}\ddot{u}\operatorname{div}\dot{u}$
$\displaystyle=-\dfrac{1}{2}\dfrac{d}{dt}\int_{\mathbb{R}^{2}}\lambda(\rho)\lvert\operatorname{div}\dot{u}\rvert^{2}-\int_{\mathbb{R}^{2}}\lambda(\rho)\nabla
u^{m}\cdot\partial_{m}\dot{u}\operatorname{div}\dot{u}$
$\displaystyle-\dfrac{1}{2}\int_{\mathbb{R}^{2}}(\rho\lambda^{\prime}(\rho)-\lambda(\rho))\lvert\operatorname{div}\dot{u}\rvert^{2}\operatorname{div}u.$

The sixth term is:

$\displaystyle\int_{\mathbb{R}^{d}}\lambda(\rho)\operatorname{div}\ddot{u}\partial_{k}u^{l}\partial_{l}u^{k}$
$\displaystyle=\dfrac{d}{dt}\int_{\mathbb{R}^{2}}\lambda(\rho)\operatorname{div}\dot{u}\partial_{k}u^{l}\partial_{l}u^{k}+\int_{\mathbb{R}^{2}}\lambda(\rho)\nabla
u^{m}\partial_{m}\dot{u}\partial_{k}u^{l}\partial_{l}u^{k}$
$\displaystyle+\int_{\mathbb{R}^{2}}\left(\rho\lambda^{\prime}(\rho)-\lambda(\rho)\right)\operatorname{div}u\operatorname{div}\dot{u}\partial_{k}u^{l}\partial_{l}u^{k}-\int_{\mathbb{R}^{2}}\lambda(\rho)\operatorname{div}\dot{u}\partial_{k}\dot{u}^{l}\partial_{l}u^{k}$
$\displaystyle+\int_{\mathbb{R}^{2}}\lambda(\rho)\operatorname{div}\dot{u}\partial_{k}u^{m}\partial_{m}u^{l}\partial_{l}u^{k}-\int_{\mathbb{R}^{2}}\lambda(\rho)\operatorname{div}\dot{u}\partial_{k}u^{l}\partial_{l}\dot{u}^{k}$
$\displaystyle+\int_{\mathbb{R}^{2}}\lambda(\rho)\operatorname{div}\dot{u}\partial_{k}u^{l}\partial_{l}u^{m}\partial_{m}u^{k}.$

The previous last term is:

$\displaystyle\int_{\mathbb{R}^{2}}\operatorname{div}\ddot{u}\rho\lambda^{\prime}(\rho)(\operatorname{div}u)^{2}$
$\displaystyle=\dfrac{d}{dt}\int_{\mathbb{R}^{2}}\rho\lambda^{\prime}(\rho)\operatorname{div}\dot{u}(\operatorname{div}u)^{2}+\int_{\mathbb{R}^{2}}\rho\lambda^{\prime}(\rho)\partial_{j}u^{m}\partial_{m}\dot{u}^{j}(\operatorname{div}u)^{2}$
$\displaystyle+\int_{\mathbb{R}^{2}}\rho^{2}\lambda^{\prime\prime}(\rho)\operatorname{div}\dot{u}(\operatorname{div}u)^{3}-2\int_{\mathbb{R}^{2}}\rho\lambda^{\prime}(\rho)\operatorname{div}\dot{u}\operatorname{div}\dot{u}\operatorname{div}u$
$\displaystyle+2\int_{\mathbb{R}^{2}}\rho\lambda^{\prime}(\rho)\operatorname{div}\dot{u}\partial_{j}u^{m}\partial_{m}u^{j}\operatorname{div}u.$

and finally the last term is:

$\displaystyle-\int_{\mathbb{R}^{2}}\rho
P^{\prime}(\rho)\operatorname{div}\ddot{u}\operatorname{div}u$
$\displaystyle=-\dfrac{d}{dt}\int_{\mathbb{R}^{2}}\rho
P^{\prime}(\rho)\operatorname{div}\dot{u}\operatorname{div}u-\int_{\mathbb{R}^{2}}\rho
P^{\prime}(\rho)\nabla u^{m}\partial_{m}\dot{u}\operatorname{div}u$
$\displaystyle-\int_{\mathbb{R}^{2}}\rho^{2}P^{\prime\prime}(\rho)\operatorname{div}\dot{u}(\operatorname{div}u)^{2}+\int_{\mathbb{R}^{2}}\rho
P^{\prime}(\rho)(\operatorname{div}\dot{u})^{2}$
$\displaystyle-\int_{\mathbb{R}^{2}}\rho
P^{\prime}(\rho)\operatorname{div}\dot{u}\nabla u^{m}\partial_{m}u.$

This ends the computations of the second term of the right hand side of (B.9).
We now turn to the computations of the third term that we express as follows:

$\displaystyle-\int_{\mathbb{R}^{d}}\partial_{k}\ddot{u}^{j}\Pi^{jk}\operatorname{div}u=-\int_{\mathbb{R}^{d}}\partial_{k}\ddot{u}^{j}\left(2\mu(\rho)\mathbb{D}^{jk}u+\\{\lambda(\rho)\operatorname{div}u-P(\rho)+\widetilde{P}\\}\delta^{jk}\right)\operatorname{div}u.$
(B.11)

The first term of the right hand side above is:

$\displaystyle-2\int_{\mathbb{R}^{d}}\partial_{k}\ddot{u}^{j}\mu(\rho)\mathbb{D}^{jk}u\operatorname{div}u$
$\displaystyle=-2\dfrac{d}{dt}\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{k}\dot{u}^{j}\mathbb{D}^{jk}u\operatorname{div}u-2\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{k}u^{m}\partial_{m}\dot{u}^{j}\mathbb{D}^{jk}u\operatorname{div}u$
$\displaystyle-2\int_{\mathbb{R}^{2}}(\rho\mu^{\prime}(\rho)-\mu(\rho))\partial_{k}\dot{u}^{j}\mathbb{D}^{jk}u(\operatorname{div}u)^{2}+2\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{k}\dot{u}^{j}\mathbb{D}^{jk}\dot{u}\operatorname{div}u$
$\displaystyle-\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{k}\dot{u}^{j}\left(\partial_{j}u^{m}\partial_{m}u^{k}+\partial_{k}u^{m}\partial_{m}u^{j}\right)\operatorname{div}u$
$\displaystyle+2\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{k}\dot{u}^{j}\mathbb{D}^{jk}u\operatorname{div}\dot{u}-2\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{k}\dot{u}^{j}\mathbb{D}^{jk}u\nabla
u^{m}\partial_{m}u.$

Regarding the second term of the right hand side of (B.11), one has

$\displaystyle-\int_{\mathbb{R}^{2}}\lambda(\rho)\operatorname{div}\ddot{u}(\operatorname{div}u)^{2}$
$\displaystyle=-\dfrac{d}{dt}\int_{\mathbb{R}^{2}}\lambda(\rho)\operatorname{div}\dot{u}(\operatorname{div}u)^{2}-\int_{\mathbb{R}^{2}}\lambda(\rho)\nabla
u^{m}\partial_{m}\dot{u}(\operatorname{div}u)^{2}$
$\displaystyle-\int_{\mathbb{R}^{2}}(\rho\lambda^{\prime}(\rho)-\lambda(\rho))\operatorname{div}\dot{u}(\operatorname{div}u)^{3}+2\int_{\mathbb{R}^{2}}\lambda(\rho)(\operatorname{div}\dot{u})^{2}\operatorname{div}u$
$\displaystyle-2\int_{\mathbb{R}^{2}}\lambda(\rho)\operatorname{div}\dot{u}\nabla
u^{m}\partial_{m}u\operatorname{div}u$

and finally, the last term is:

$\displaystyle\int_{\mathbb{R}^{2}}\operatorname{div}\ddot{u}(P(\rho)-\widetilde{P})\operatorname{div}u$
$\displaystyle=\dfrac{d}{dt}\int_{\mathbb{R}^{2}}\operatorname{div}\dot{u}(P(\rho)-\widetilde{P})\operatorname{div}u+\int_{\mathbb{R}^{2}}\nabla
u^{m}\partial_{m}\dot{u}(P(\rho)-\widetilde{P})\operatorname{div}u$
$\displaystyle+\int_{\mathbb{R}^{2}}\operatorname{div}\dot{u}(\operatorname{div}u)^{2}(\rho
P^{\prime}(\rho)-P(\rho)+\widetilde{P})-\int_{\mathbb{R}^{2}}\operatorname{div}\dot{u}(P(\rho)-\widetilde{P})\operatorname{div}\dot{u}$
$\displaystyle+\int_{\mathbb{R}^{2}}\operatorname{div}\dot{u}(P(\rho)-\widetilde{P})\nabla
u^{m}\partial_{m}u.$

These completes the computations of the terms in the expression (B.11), that
are the third term of the right hand side of (B.9). We turn to the
computations of the last term of (B.9) that we express as:

$\displaystyle\int_{\mathbb{R}^{d}}\partial_{l}\ddot{u}^{j}\partial_{k}u^{l}\Pi^{jk}=\int_{\mathbb{R}^{d}}\partial_{l}\ddot{u}^{j}\partial_{k}u^{l}\left(2\mu(\rho)\mathbb{D}^{jk}u+\\{\lambda(\rho)\operatorname{div}u-P(\rho)+\widetilde{P}\\}\delta^{jk}\right).$
(B.12)

The first term of the right hand side above is:

$\displaystyle\int_{\mathbb{R}^{d}}2\mu(\rho)\partial_{l}\ddot{u}^{j}\partial_{k}u^{l}\mathbb{D}^{jk}u$
$\displaystyle=2\dfrac{d}{dt}\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{l}\dot{u}^{j}\partial_{k}u^{l}\mathbb{D}^{jk}u+2\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{l}u^{m}\partial_{m}\dot{u}^{j}\partial_{k}u^{l}\mathbb{D}^{jk}u$
$\displaystyle+2\int_{\mathbb{R}^{2}}(\rho\mu^{\prime}(\rho)-\mu(\rho))\operatorname{div}u\partial_{l}\dot{u}^{j}\partial_{k}u^{l}\mathbb{D}^{jk}u-2\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{l}\dot{u}^{j}\partial_{k}\dot{u}^{l}\mathbb{D}^{jk}u$
$\displaystyle+2\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{l}\dot{u}^{j}\partial_{k}u^{m}\partial_{m}u^{l}\mathbb{D}^{jk}u-2\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{l}\dot{u}^{j}\mathbb{D}^{jk}\dot{u}\partial_{k}u^{l}$
$\displaystyle+\int_{\mathbb{R}^{2}}\mu(\rho)\partial_{l}\dot{u}^{j}\partial_{k}u^{l}\left(\partial_{j}u^{m}\partial_{m}u^{k}+\partial_{k}u^{m}\partial_{m}u^{j}\right)$

and the second term is:

$\displaystyle\int_{\mathbb{R}^{d}}\lambda(\rho)\partial_{l}\ddot{u}^{j}\partial_{j}u^{l}\operatorname{div}u$
$\displaystyle=\dfrac{d}{dt}\int_{\mathbb{R}^{2}}\lambda(\rho)\partial_{l}\dot{u}^{j}\partial_{j}u^{l}\operatorname{div}u+\int_{\mathbb{R}^{2}}\lambda(\rho)\partial_{l}u^{m}\partial_{m}\dot{u}^{j}\partial_{j}u^{l}\operatorname{div}u$
$\displaystyle+\int_{\mathbb{R}^{2}}(\rho\lambda^{\prime}(\rho)-\lambda(\rho))(\operatorname{div}u)^{2}\partial_{l}\dot{u}^{j}\partial_{j}u^{l}-\int_{\mathbb{R}^{2}}\lambda(\rho)\partial_{l}\dot{u}^{j}\partial_{j}\dot{u}^{l}\operatorname{div}u$
$\displaystyle+\int_{\mathbb{R}^{2}}\lambda(\rho)\partial_{l}\dot{u}^{j}\partial_{j}u^{m}\partial_{m}u^{l}\operatorname{div}u-\int_{\mathbb{R}^{2}}\lambda(\rho)\partial_{l}\dot{u}^{j}\operatorname{div}\dot{u}\partial_{j}u^{l}$
$\displaystyle+\int_{\mathbb{R}^{2}}\lambda(\rho)\partial_{l}\dot{u}^{j}\nabla
u^{m}\partial_{m}u\partial_{j}u^{l}.$

Finally, the last term of (B.12) is:

$\displaystyle-\int_{\mathbb{R}^{d}}\partial_{l}\ddot{u}^{j}\partial_{j}u^{l}(P(\rho)-\widetilde{P})$
$\displaystyle=-\dfrac{d}{dt}\int_{\mathbb{R}^{2}}\partial_{l}\dot{u}^{j}\partial_{j}u^{l}(P(\rho)-\widetilde{P})-\int_{\mathbb{R}^{2}}\partial_{l}u^{m}\partial_{m}\dot{u}^{j}\partial_{j}u^{l}(P(\rho)-\widetilde{P})$
$\displaystyle-\int_{\mathbb{R}^{2}}\partial_{l}\dot{u}^{j}\operatorname{div}u\partial_{j}u^{l}(\rho
P^{\prime}(\rho)-P(\rho)+\widetilde{P})+\int_{\mathbb{R}^{2}}\partial_{l}\dot{u}^{j}(P(\rho)-\widetilde{P})\partial_{j}\dot{u}^{l}$
$\displaystyle-\int_{\mathbb{R}^{2}}\partial_{l}\dot{u}^{j}(P(\rho)-\widetilde{P})\partial_{j}u^{m}\partial_{m}u^{l}.$

Theses completes the computations of the third Hoff energy.

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

Many terms appearing on the left-hand side above can be grouped into three
categories: $I_{1}$, $I_{2}$, and $I_{3}$, each having the respective forms
(3.41), (3.42) and (3.43).

##  Appendix C Study of linear non homogeneous heat equation

This section is devoted to the solvability of the following system

$\displaystyle v_{t}-\widetilde{\mu}\Delta
u-(\widetilde{\mu}+\widetilde{\lambda})\nabla\operatorname{div}v=f+\operatorname{div}(b_{0}\,\colon
v)$ (C.1)

by the help of the Banach fixed point theorem. We first notice that using the
Helmholtz decomposition, one can write $v=\mathcal{P}v+\mathcal{Q}v$ where

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

with $k=\widetilde{\mu}$ if $\mathcal{A}=\mathcal{P}$ and
$k=2\widetilde{\mu}+\widetilde{\lambda}$ if $\mathcal{A}=\mathcal{Q}$. For
$w\in\mathscr{C}(\mathbb{R}_{+},H^{1}(\mathbb{R}^{d}))$, the Cauchy problem

$\displaystyle\begin{cases}\partial_{t}\mathcal{A}v-k\mathcal{A}v=\mathcal{A}f+\mathcal{A}\operatorname{div}(b_{0}\,\colon
w),\\\ \mathcal{A}v_{|t=0}=\mathcal{A}v_{0}\end{cases}$ (C.2)

is just a linear non homogeneous heat equation, so this equation is uniquely
solvable and the solution is given by the Duhamel formula:

$\displaystyle\mathcal{A}v=\mathcal{T}(kt)\mathcal{A}v_{0}+\int_{0}^{t}\mathcal{T}(k(t-s))\mathcal{A}f(s)ds+\int_{0}^{t}\mathcal{T}(k(t-s))\mathcal{A}\operatorname{div}(b_{0}\,\colon
w)(s)ds.$ (C.3)

Above $(\mathcal{T}(t))_{t}$ is the heat semi group. From the above, we deduce
that $v$ belongs at least to $\mathscr{C}([0,T],H^{1}(\mathbb{R}^{d}))$ with
the following estimate:

$\displaystyle\sup_{[0,T]}\lVert
v\rVert_{H^{1}(\mathbb{R}^{d})}\leqslant\lVert
v_{0}\rVert_{H^{1}(\mathbb{R}^{d})}+\int_{0}^{t}\lVert\mathcal{T}(k(t-s))\mathcal{A}\left[f(s)+\operatorname{div}(b_{0}\,\colon
w)\right](s)\rVert_{H^{1}(\mathbb{R}^{d})}ds.$ (C.4)

Moreover $\partial_{t}v,\,\nabla^{2}v\in L^{2}((0,T)\times\mathbb{R}^{d})$ and

$\displaystyle\int_{0}^{T}\lVert\partial_{t}v,\nabla^{2}v\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\leqslant\lVert
v_{0}\rVert_{H^{1}(\mathbb{R}^{d})}^{2}+\int_{0}^{T}\left\|\int_{0}^{t}\Delta\mathcal{T}(k(t-s))\mathcal{A}\left[f+\operatorname{div}(b_{0}\,\colon
w)\right](s)ds\right\|_{L^{2}(\mathbb{R}^{d})}^{2}dt.$ (C.5)

In consequence, for all $w\in\mathscr{C}([0,T],H^{1}(\mathbb{R}^{d}))$ we can
define $\Phi(w)=\mathcal{P}\Phi(w)+\mathcal{Q}\Phi(w)$ the unique solution of
the Cauchy problem (C.2), given by (C.3) and come with the estimate (C.4) and
$\eqref{eq3.27}$. This defines an application