Step 5: Estimates for $\partial_{tt}\delta\Phi(v)$ and $L^4$ norm estimates for $\nabla\partial_t\delta\Phi(v)$. In a similar way as is step 2, the following estimate holds for $\delta\Phi(v)$:

$$ \int_0^T \|\sqrt{\sigma}\partial_{tt}\delta\Phi(v), \sigma\nabla\partial_{tt}\delta\Phi(v)\|_{L^2(\mathbb{R}^d)}^2 + \|\sqrt{\sigma}\partial_t\nabla\delta\Phi(v), \sigma\partial_{tt}\delta\Phi(v)\|_{L^\infty((0,T),L^2(\mathbb{R}^d))}^2 \\ \lesssim \int_0^{\sigma^{(T)}} \|\partial_t\nabla\delta\Phi(v)\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|\partial_t\delta\tilde{F}(v), \sigma\partial_{tt}\delta\tilde{F}(v)\|_{L^2(\mathbb{R}^d)}^2 + \|\nabla\delta\Phi(v)\|_{L^\infty((0,T),L^4(\mathbb{R}^d))}^2. $$

Except the norm of $\partial_{tt}\delta\tilde{F}(v)$, all norms that appear in the right hand side above follows from (3.52) and (3.51). So, in order to estimate the remainder term of the above inequality, we express:

$$ \partial_{tt}\delta\tilde{F}(v) = (\theta - \theta_0)\{2(\mu - \tilde{\mu})\mathbb{D}\delta v_{tt} + (\lambda - \tilde{\lambda})\operatorname{div}\delta v_{tt}I_d \\ + 4\mu_t\mathbb{D}\delta v_t + 2\lambda_t\operatorname{div}\delta v_tI_d + 2\mu_{tt}\mathbb{D}\delta v + \lambda_{tt}\operatorname{div}\delta vI_d\} $$

from which, we deduce

$$ \int_0^T \sigma^2 \|\partial_{tt} \tilde{F}(v) - \partial_{tt} \tilde{F}(w)\|_{L^2(\mathbb{R}^d)}^2 \lesssim |\theta - \theta_0|^2 \|\delta v\|_{X_T}^2. $$

In consequence,

$$ (3.54) \quad \int_0^T \|\sqrt{\sigma}\partial_{tt}\delta\Phi(v), \sigma\nabla\partial_{tt}\delta\Phi(v)\|_{L^2(\mathbb{R}^d)}^2 + \|\sqrt{\sigma}\partial_t\nabla\delta\Phi(v), \sigma\partial_{tt}\delta\Phi(v)\|_{L^\infty((0,T),L^2(\mathbb{R}^d))}^2 \lesssim |\theta - \theta_0|^2 \|\delta v\|_{X_T}^2 $$

and we turn to the estimate of the $L^{8/d}((0,T), L^4(\mathbb{R}^d))$ norm of $\sqrt{\sigma}\nabla\partial_t\delta\Phi(v)$. We first notice, that:

$$ \int_0^T \sigma^{4/d} \|\nabla \partial_t \delta \Phi(v)\|_{L^4(\mathbb{R}^d)}^{8/d} \lesssim \int_0^T \sigma^{4/d} \|\partial_t \delta \tilde{F}(v)\|_{L^4(\mathbb{R}^d)}^{8/d} + \left[ \int_0^T \|\nabla \delta \Phi(v)\|_{L^8(\mathbb{R}^d)}^{16} \right]^{1/(2d)} \\ + \|\sigma \partial_{tt} \delta \Phi(v)\|_{L^\infty((0,T), L^2(\mathbb{R}^d))}^{8/d} \\ + \|\nabla \delta \Phi(v)\|_{L^4((0,T) \times \mathbb{R}^d)}^{8/d} + \left[ \int_0^T \|\nabla \partial_t \delta \Phi(v), \partial_t \delta \tilde{F}(v)\|_{L^2(\mathbb{R}^d)}^2 \right]^{4/d}. $$

Owing to (3.54), (3.52), (3.53), (3.51) we only need to estimate the first term of the right hand side. And just by using (3.50), one has:

$$ \int_0^T \sigma^{4/d} \|\partial_t \delta \tilde{F}(v)\|_{L^4(\mathbb{R}^d)}^{8/d} \lesssim |\theta - \theta_0|^{8/d} \|\delta v\|_{X_T}^{8/d} $$

and one obtains an appropriate estimate for the underlying term.

Step 6: Piecewise Hölder norm estimate for $\nabla\delta\Phi(v)$. We begin by recalling that thanks to (3.10), one has the following:

$$ \int_0^T \sigma^r \|\nabla \delta \Phi(v)\|_{\mathscr{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)}^4 \lesssim (1 + \mathcal{P}(C)) \int_0^T \sigma^r \|\delta \tilde{F}(v)\|_{\mathscr{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} + \|\delta \tilde{F}(v), \nabla \delta \Phi(v), \partial_t \delta \Phi(v), \sqrt{\sigma} \nabla \partial_t \delta \Phi(v)\|_{L^\infty((0,T), L^2(\mathbb{R}^d))}^4 \\ + \left[ \int_0^T \|\delta \tilde{F}(v), \nabla \delta \Phi(v), \partial_t \delta \Phi(v), \nabla \partial_t \delta \Phi(v)\|_{L^2(\mathbb{R}^d)}^2 \right]^2 + \left[ \int_0^T \sigma^{4/d} \|\nabla \partial_t \delta \Phi(v)\|_{L^4(\mathbb{R}^d)}^{8/d} \right]^{d/2} $$

and we will only have to estimate the first term of the right hand side above. Thanks to (3.50), the latter can be estimated as follows:

$$ \int_0^T \sigma^r \|\delta \tilde{F}(v)\|_{\mathscr{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)}^4 \lesssim |\theta - \theta_0|^4 \|\delta v\|_{X_T}^4 $$

and this provides an appropriate estimate for the underlying term.

Gathering all of these estimates, one has:

$$ \|\delta\Phi(v)\|_{X_T} \le C|\theta - \theta_0|\|\delta v\|_{X_T} $$

with a constant $C$ that depends on the regularity of viscosity $\mu$, $\lambda$ and the initial density $\rho_0$, as well on $\mathcal{P}(C)$. Thus, there exists $\eta > 0$ small related to the norms of the viscosity and their upper an lower bounds, as well on $\mathcal{P}(C)$ such that if $|\theta - \theta_0| \le \eta$ then, the application $\Phi$ is a contraction on $B_{X_T(u_0)}(0, R_T)$, hereby admits an unique fixed point $u$ that solves (3.38). Since $\eta$ does not depend on $u_0$ nor $F$, it makes the condition (3.37) true, for all $(F, u_0) \in Y_T^\theta$ and one deduces that $\mathscr{E}$ is an open set of $[0, 1]$.
and bounded away from vacuum, one has that $u$ solves (3.53) if and only if $v := \rho_{0,\delta}u$ solves the following Cauchy problem:

$$ (3.54) \qquad \left\{ \begin{array}{l} v_t - \operatorname{div} \left( \frac{\tilde{\mu}}{\rho_{0,\delta}} \mathbb{D}v \right) - \nabla \left( \frac{\tilde{\lambda}}{\rho_{0,\delta}} \operatorname{div} v \right) = \operatorname{div} F_{\delta} - \tilde{\mu} \operatorname{div} \left( \rho_{0,\delta}^{-2} (\nabla \rho_{0,\delta} \otimes v + v \otimes \nabla \rho_{0,\delta}) \right) \\ v|_{t=0} \end{array} \right. = \tilde{\lambda} \nabla \left( \rho_{0,\delta}^{-2} v \cdot \nabla \rho_{0,\delta} \right), $$

In order to ease computations, we consider the following problem for $\mu'_0, \lambda'_0, b_0 = \mu'_0(x), \lambda'_0(x), b_0(x)$

$$ (3.55) \qquad \left\{ \begin{array}{l} v_t - \operatorname{div}(2\mu'_0 \mathbb{D}v + \lambda'_0 \operatorname{div} v I_d) = f + \operatorname{div}(b_0 \cdot v), \\ v|_{t=0} = v_0 \end{array} \right. $$

and we have the following:

**Theorem 3.3.** Let $0 < T < \infty$ and $f \in L^2((0,T) \times \mathbb{R}^d)$, $0 < \mu_* \le \mu'_0$, $\lambda'_0 \in W^{1,\infty}(\mathbb{R}^d)$, $b_0 \in W^{1,\infty}(\mathbb{R}^d)$, and finally $v_0 \in H^1(\mathbb{R}^d)$. Then there exists a unique solution $v \in \mathcal{C}([0,T], H^1(\mathbb{R}^d))$ with $v_t, \nabla^2 v \in L^2((0,T), L^2(\mathbb{R}^d))$ for the system (3.55). Moreover, there exist constants $\Lambda, K > 0$ that depend on the norms of $\mu'_0, \lambda'_0, b_0$ and the lower and upper bounds $\mu_*, \mu^*$ such that

$$ \sup_{t \in [0,T]} e^{-2\Lambda t} \left\{ \|v(t)\|_{H^1(\mathbb{R}^d)}^2 + \int_0^t \|v_t, \nabla v, \nabla^2 v\|_{L^2(\mathbb{R}^d)}^2 \right\} \le K \left( \|v_0\|_{H^1(\mathbb{R}^d)}^2 + \int_0^T \|f(s)\|_{L^2(\mathbb{R}^d)}^2 ds \right). $$

The proof of Theorem 3.3 is the purpose of the Step below, and it implies immediately:

**Lemma 3.4.** For all $\delta > 0$, there exists a unique classical solution $u^\delta$ of the Cauchy problem (3.53). The sequence $(u^\delta)_\delta$ converges to $u \in X_T$ as $\delta \to 0$, the unique solution of the Cauchy problem (3.52).

All these discussions establish that $\mathcal{E}$ is a non-empty set. In the following, we present the proof of Theorem 3.3 and Lemma 3.4.

**Detailed proof of Theorem 3.3 and Lemma 3.4.** We start with the proof of Theorem 3.3.

**Proof of Theorem 3.3.** The proof of Theorem 3.3 is based on a homotopy argument. So, we define

$$ \mu_0^{\theta} = (1 - \theta)\tilde{\mu} + \theta\mu_0' \quad \text{and} \quad \lambda_0^{\theta} = (1 - \theta)\tilde{\lambda} + \theta\lambda_0', $$

and we consider the equation (3.55) where the viscosity $\mu'_0, \lambda'_0$ are replaced by $\mu_0^{\theta}, \lambda_0^{\theta}$ respectively. Also, we define

$$ X_T' := \{v \in \mathcal{C}([0,T], H^1(\mathbb{R}^d)): v_t, \nabla^2 v \in L^2((0,T) \times \mathbb{R}^d)\} $$

and the map:

$$ \Pi_\theta: \begin{array}{l} X_T' \to L^2((0,T) \times \mathbb{R}^d) \times H^1(\mathbb{R}^d) \\ v \mapsto (v_t - \operatorname{div}(2\mu_0^{\theta} \mathbb{D}v + \lambda_0^{\theta} \operatorname{div} v I_d) - \operatorname{div}(b_0 \cdot v), v|_{t=0}) \end{array} $$

and finally we consider the set:

$$ \mathcal{E}' := \{\theta \in [0,1]: \Pi_\theta \text{ is one to one}\}. $$

Obviously, the map $\Pi_\theta$ is well-defined and continuous, we will only have to show that $1 \in \mathcal{E}'$ to conclude the proof of Theorem 3.3. To obtain $0 \in \mathcal{E}'$, we observe that for $\theta = 0$, the equation (3.55)$_\theta$ reduces to:

$$ (3.56) \qquad v_t - \tilde{\mu}\Delta v - (\tilde{\mu} + \tilde{\lambda})\nabla \operatorname{div} v = f + \operatorname{div}(b_0 \cdot v). $$

The existence of a unique solution $v$ of (3.56) is established in Appendix B, resulting in $0 \in \mathcal{E}'$. The remainder of the proof of Theorem 3.3 consists in first deriving a priori estimates for the solution $v \in X'_T$ of (3.55). Then, we prove that $\mathcal{E}'$ is at the same time open and closed.
We turn to the proof of the fact that $\mathcal{E}$ is a closed set of $[0, 1]$. So let $(\theta_n)_n$ a sequence of element of $\mathcal{E}$ that converges to some $\theta_0 \in [0, 1]$. Our aim is to prove that $\theta_0 \in \mathcal{E}$. In other words, the following Cauchy problem

$$ (3.55) \qquad \begin{cases} \rho_0 u_t - \operatorname{div}(2\mu^{\theta_n} \mathcal{D}u + \lambda^{\theta_n} \operatorname{div} u I_d) = \operatorname{div} F + (\theta_0 - \theta_n) \operatorname{div}\{2(\mu - \tilde{\mu})\mathcal{D}u + (\lambda - \tilde{\lambda}) \operatorname{div} u I_d\}, \\ u|_{t=0} = u_0 \end{cases} $$

admits an unique solution in $X_T$ for all $(F, u_0) \in Y_T^{\theta_0}$. By following the procedure outlined in steps 1 to 6, one can choose a sufficiently large $n$ to solve the aforementioned problem, as the norms of $\mu^\theta$ and $\lambda^\theta$ are uniformly bounded with respect to $\theta$. Hereby, $\mathcal{E}$ is a closed set of $[0, 1]$.

In conclusion, $\mathcal{E}$ is at same time a non-empty, open and closed set of $[0, 1]$, since $[0, 1]$ is a connected set, $\mathcal{E} = [0, 1]$, $1 \in \mathcal{E}$ and **Theorem 3.1** finally holds. $\square$

## 4. STUDY OF THE FULL NONLINEAR SYSTEM

4.1. **Preliminary.** This section is devoted to the solvability of the full nonlinear two-fluid model in Lagrangian coordinates (2.7). The initial data $(c_0, \rho_0, u_0)$ is such that the compatibility condition holds true, that is:

$$ (4.1) \qquad \operatorname{div}(2\mu(\rho_0, c_0)\mathcal{D}u_0 + (\lambda(\rho_0, c_0)\operatorname{div} u_0 - P(\rho_0, c_0))I_d) \in L^2(\mathbb{R}^d). $$

Our method relies on the fixed point theorem, thus we will begin by setting tools in order to use this theorem. We recall that such solution $u$ verifies:

$$ (4.2) \qquad \rho_0 u_t - \operatorname{div}\{2\mu(J_u^{-1}\rho_0, c_0)\mathcal{D}u + \lambda(J_u^{-1}\rho_0, c_0)\} \operatorname{div} u I_d = \operatorname{div} I(u, u) $$

where

$$ I(u, u) = I_1(u, u) + I_2(u, u) + I_3(u) $$

and

$$ \left\{ \begin{aligned} I_1(v, w) &= 2\mu(J_v^{-1}\rho_0, c_0)\{\operatorname{Adj}(D\mathcal{X}_v)\mathcal{D}_{A_v}w - \mathcal{D}w\}, \\ I_2(v, w) &= \lambda(J_v^{-1}\rho_0, c_0)\{\operatorname{Adj}(D\mathcal{X}_v)\operatorname{div}_{A_v}w - \operatorname{div}w\}, \\ I_3(v) &= \operatorname{Adj}(D\mathcal{X}_v)\{P(J_v^{-1}\rho_0, c_0) - \tilde{P}\}I_d. \end{aligned} \right. $$

Regarding the previous section, we will apply the Banach fixed point theorem in a closed subset of the vector space $X_T$ as defined in (3.3) for some small time $T$. Thus, we consider, for some $v \in X_T$ with $v|_{t=0} = u_0$ the linear Cauchy problem:

$$ (4.3) \qquad \begin{cases} \rho_0 u_t - \operatorname{div}\{2\mu(J_v^{-1}\rho_0, c_0)\mathcal{D}u + \lambda(J_v^{-1}\rho_0, c_0)\} \operatorname{div} u I_d = \operatorname{div} I(v, v), \\ u|_{t=0} = u_0. \end{cases} $$

The fact that $v$ is chosen such that $v|_{t=0} = u_0$ and the initial data verifies (4.1) ensure the validity of the last condition in the definition of $Y_T := Y_T(\mu(\rho_0, c_0), \lambda(\rho_0, c_0))$. The first part of the following section will consist in proving that the regularity of $v \in X_T$ is sufficient to ensure that $I(v, v)$ belongs to $Y_T$. From there, we will be able to deduce the existence of a solution for the Cauchy problem (4.3) using **Theorem 3.1**.

Let us recall some estimates for the flow of a vector field whose gradient is piecewise Hölder continuous on both sides of a hyper-surface $\gamma$. The proof follows the same steps as in [3, Lemma A.3]:

**Lemma 4.1.** Let $v$ be a vector field with $\nabla v \in L^1((0, T), L^\infty(\mathbb{R}^d))$. Suppose that

$$ c = \int_0^T \|\nabla v\|_{L^\infty(\mathbb{R}^d)} < 1. $$

Then, there exists a constant $k = k(c)$ such that the following estimates hold true:

$$ \left\{ \begin{array}{ll} \|I_d - \operatorname{Adj}(D\mathcal{X}_v(t)); I_d - A_v(t)\|_{L^\infty(\mathbb{R}^d)} & \le k \|\nabla v\|_{L^1((0,t), L^\infty(\mathbb{R}^d))}, \\ \|J_v^{\pm 1}(t) - 1\|_{L^\infty(\mathbb{R}^d)} & \le k \|\nabla v\|_{L^1((0,t), L^\infty(\mathbb{R}^d))}. \end{array} \right. $$

Moreover, one has the following:

$$ \left\{ \begin{array}{ll} \|\operatorname{Adj}(D\mathcal{X}_v(t)) D_{A_v(t)} w - D w\|_{L^p(\mathbb{R}^d)} & \le k \|\nabla v\|_{L^1((0,t), L^\infty(\mathbb{R}^d))} \|D w\|_{L^p(\mathbb{R}^d)}, \\ \|\operatorname{Adj}(D\mathcal{X}_v(t)) \operatorname{div}_{A_v(t)} w - \operatorname{div} w\|_{L^p(\mathbb{R}^d)} & \le k \|\nabla v\|_{L^1((0,t), L^\infty(\mathbb{R}^d))} \|D w\|_{L^p(\mathbb{R}^d)}. \end{array} \right. $$

In the spirit of the above results, and owing to the fact that $\mathcal{E}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)$ is an algebra, one has the following estimates.
**A priori estimates.** We multiply (3.55) by $v$ and integrate in time and space in order to obtain:

$$ 
\begin{aligned}
\frac{1}{2} \|v(t)\|_{L^2(\mathbb{R}^d)}^2 + \int_0^t \int_{\mathbb{R}^d} 2\mu_0' |\mathbb{D}v|^2 + \int_0^t \int_{\mathbb{R}^d} \lambda_0' |\text{div } v|^2 &= \frac{1}{2} \|v_0\|_{L^2(\mathbb{R}^d)}^2 + \int_0^t \int_{\mathbb{R}^d} fv - \int_0^t \int_{\mathbb{R}^d} b_0 \cdot v \nabla v \\
&\le \eta \int_0^t \|\nabla v\|_{L^2(\mathbb{R}^d)}^2 + \frac{1}{2} \|v_0\|_{L^2(\mathbb{R}^d)}^2 + \int_0^t \|f\|_{L^2(\mathbb{R}^d)}^2 + C_\eta \int_0^t (1 + \|b_0\|_{L^\infty(\mathbb{R}^d)}^2) \|v\|_{L^2(\mathbb{R}^d)}^2
\end{aligned}
$$

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

$$ (3.57) \quad \|v(t)\|_{L^2(\mathbb{R}^d)}^2 + \int_0^t \|\nabla v\|_{L^2(\mathbb{R}^d)}^2 \le C_* \|v_0\|_{L^2(\mathbb{R}^d)}^2 + C_* \int_0^t \|f\|_{L^2(\mathbb{R}^d)}^2 + C_* \int_0^t (1 + \|b_0\|_{L^\infty(\mathbb{R}^d)}^2) \|v\|_{L^2(\mathbb{R}^d)}^2, $$

where $C_*$ depends only on the bounds of the viscosity.

Next, we use $\Delta v$ as a test function in (3.55). The first term is:

$$ \int_{\mathbb{R}^d} v_t \Delta v = - \int_{\mathbb{R}^d} \nabla v_t \cdot \nabla v = - \frac{1}{2} \frac{d}{dt} \int_{\mathbb{R}^d} |\nabla v|^2 $$

and the second one is:

$$ 
\begin{aligned}
-2 \int_{\mathbb{R}^d} \partial_k (\mu_0' \mathbb{D}^{jk} v) \Delta v^j & - \int_{\mathbb{R}^d} \partial_j (\lambda_0' \text{div } v) \Delta v^j = - \int_{\mathbb{R}^d} \mu_0' |\Delta v|^2 - \int_{\mathbb{R}^d} \mu_0' \partial_j \text{div } v \Delta v^j - \int_{\mathbb{R}^d} \partial_k \mu_0' \mathbb{D}^{jk} v \Delta v^j \\
& - \int_{\mathbb{R}^d} \lambda_0' |\nabla \text{div } v|^2 - \int_{\mathbb{R}^d} \text{div } v \nabla \lambda_0' \cdot \nabla \text{div } v \\
& = - \int_{\mathbb{R}^d} \mu_0' |\Delta v|^2 - \int_{\mathbb{R}^d} (\mu_0' + \lambda_0') |\nabla \text{div } v|^2 + \int_{\mathbb{R}^d} \text{div } v \partial_j \mu_0' \Delta v^j \\
& - \int_{\mathbb{R}^d} \text{div } v \nabla \mu_0' \cdot \nabla \text{div } v - \int_{\mathbb{R}^d} \partial_k \mu_0' \mathbb{D}^{jk} v \Delta v^j - \int_{\mathbb{R}^d} \text{div } v \nabla \lambda_0' \cdot \nabla \text{div } v.
\end{aligned}
$$

Gathering all of these computations, we have:

$$ 
\begin{aligned}
\frac{1}{2} \frac{d}{dt} \|\nabla v\|_{L^2(\mathbb{R}^d)}^2 + \int_{\mathbb{R}^d} \mu_0' |\Delta v|^2 + \int_{\mathbb{R}^d} (\mu_0' + \lambda_0') |\nabla \text{div } v|^2 &= \int_{\mathbb{R}^d} f^j \Delta v^j + \int_{\mathbb{R}^d} \text{div}(b_0 \cdot v) \Delta v^j \\
&\quad + \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \text{div } v \partial_j \mu_0' \Delta v^j - \int_{\mathbb{R}^d} \text{div } v \nabla \mu_0' \cdot \nabla \text{div } v - \int_{\mathbb{R}^d} \partial_k \mu_0' \mathbb{D}^{jk} v \Delta v^j - \int_{\mathbb{R}^d} \text{div } v \nabla \lambda_0' \cdot \nabla \text{div } v.
\end{aligned}
$$

Thus, Hölder's and Young's inequalities yield:

$$ (3.58) \quad \|\nabla v(t)\|_{L^2(\mathbb{R}^d)}^2 + \int_0^t \|\Delta v\|_{L^2(\mathbb{R}^d)}^2 \le C_* \|\nabla v_0\|_{L^2(\mathbb{R}^d)}^2 + C_* \int_0^t \|f\|_{L^2(\mathbb{R}^d)}^2 + C_* \int_0^t \|\nabla v\|_{L^2(\mathbb{R}^d)}^2 \|\nabla \mu_0', \nabla \lambda_0', b_0, \nabla b_0\|_{L^\infty(\mathbb{R}^d)}^2. $$

Finally, we gather (3.57) and (3.58), and we make use of Grönwall's lemma and (3.55) to obtain:

$$ \sup_{t \in [0, T]} e^{-\Lambda t} \left\{ \|v(t)\|_{H^1(\mathbb{R}^d)}^2 + \int_0^t \|v_t, \nabla v, \nabla^2 v\|_{L^2(\mathbb{R}^d)}^2 \right\} \le C_* \|v_0\|_{H^1(\mathbb{R}^d)}^2 + C_* \int_0^T \|f\|_{L^2(\mathbb{R}^d)}^2 $$

with

$$ \Lambda = C_* (1 + \|\nabla \mu_0', \nabla \lambda_0', b_0, \nabla b_0\|_{L^\infty(\mathbb{R}^d)}^2). $$

This ends the computations of the a priori estimates and we turn to the homotopy argument.

**Closing of the homotopy argument.** In this section, we will show that $\mathcal{E}' = [0, 1]$. We have already showed that $\mathcal{E}'$ is not empty. Also, we may notice that if $\Pi_\theta$ is invertible then, from the previous a priori estimates, $\Pi_\theta^{-1}$ is linear continuous with a norm that depends only on $\Lambda$ and time $T$. In particular, we have:

$$ K_T := \sup_{\theta \in \mathcal{E}'} \|\Pi_\theta^{-1}\|_{\mathcal{L}} < \infty. $$

Let us prove that $\mathcal{E}'$ is an open set. Let $\theta_0 \in \mathcal{E}'$, then, $\Pi_{\theta_0}$ is a one to one map and one can write, for all $\theta \in [0, 1]$,

$$ \Pi_{\theta} = \Pi_{\theta_0} + (\theta - \theta_0)(\Pi_1 - \Pi_0) = \Pi_{\theta_0} (I + (\theta - \theta_0)\Pi_{\theta_0}^{-1}(\Pi_1 - \Pi_0)). $$

Thus, all $\theta$ such that $2|\theta - \theta_0| \|\Pi_{\theta_0}^{-1}(\Pi_1 - \Pi_0)\|_{\mathcal{L}(X'_T)} < 1$ belongs to $\mathcal{E}'$, therefore $\mathcal{E}'$ is an open set of $[0, 1]$. We turn to the last part. Let $(\theta_n)_n$ be a sequence of element of $\mathcal{E}'$ that converges to some $\theta \in [0, 1]$. As above, one can write, for some $n_0$

$$ \Pi_{\theta} = \Pi_{\theta_{n_0}} (I + (\theta - \theta_{n_0})\Pi_{\theta_{n_0}}^{-1}(\Pi_1 - \Pi_0)). $$
**Lemma 4.2.** Let $v$ be a vector field fulfilling $\nabla v \in L^1((0,T), \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d))$. Suppose that

$$ (4.4) \qquad c = \int_0^T \|\nabla v\|_{\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} < 1 $$

Then there exists a constant $k = k(c)$ such that the following estimates holds true:

$$ \left\{ \begin{array}{l} \|I_d - \operatorname{Adj}(D\mathcal{X}_v(t)); I_d - A_v(t)\|_{\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \le k \|\nabla v\|_{L^1((0,t), \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d))}, \\ \|J_v^{\pm 1}(t) - 1\|_{\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \le k \|\nabla v\|_{L^1((0,t), \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d))}. \end{array} \right. $$

Moreover, one has the following:

$$ \left\{ \begin{array}{l} \|\operatorname{Adj}(D\mathcal{X}_v(t)) D_{A_v(t)} w - D w\|_{\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \le k \|\nabla v\|_{L^1((0,t), \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d))} \|Dw\|_{\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)}, \\ \|\operatorname{Adj}(D\mathcal{X}_v(t)) \operatorname{div}_{A_v(t)} w - \operatorname{div} w\|_{\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \le k \|\nabla v\|_{L^1((0,t), \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d))} \|Dw\|_{\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)}. \end{array} \right. $$

Notice that we will need to estimate the first and the second time derivative of $I(v, v)$ as it is required in the definition of $Y_T$. For this purpose, from (2.6), $\operatorname{Adj}(D\mathcal{X}_v) \cdot D\mathcal{X}_v = \det(D\mathcal{X}_v)I_d$ and $\partial_t D\mathcal{X}_v = Dv$ one has the following expressions:

**Lemma 4.3.** The following expressions hold true:

$$ (4.5) \qquad \partial_t J_v^{-1}(t, y) = -J_v^{-1}(t, y) \operatorname{div}_{A_v}(v), $$

$$ (4.6) \qquad \partial_t A_v = -A_v \cdot Dv \cdot A_v, $$

$$ (4.7) \qquad \partial_t \operatorname{Adj}(D\mathcal{X}_v) = (Dv: \operatorname{Adj}(D\mathcal{X}_v))A_v - \operatorname{Adj}(D\mathcal{X}_v) \cdot Dv \cdot A_v. $$

Next, taking the time derivative of the above quantities, one has:

$$ (4.8) \qquad J_v \partial_{tt} J_v^{-1} = (\operatorname{div}_{A_v}(v))^2 - Dv_t: A_v + Dv: (A_v \cdot Dv \cdot A_v), $$

$$ (4.9) \qquad \partial_{tt} A_v = 2A_v \cdot (Dv \cdot A_v)^2 - A_v \cdot Dv_t \cdot A_v, $$

$$ \partial_{tt} \operatorname{Adj}(D\mathcal{X}_v) = (Dv_t: \operatorname{Adj}(D\mathcal{X}_v))A_v + J_v(Dv: A_v)^2 A_v - (Dv: (\operatorname{Adj}(D\mathcal{X}_v) \cdot Dv \cdot A_v))A_v $$

$$ (4.10) \qquad -2(Dv: \operatorname{Adj}(D\mathcal{X}_v))A_v \cdot Dv \cdot A_v + 2J_v A_v \cdot (Dv \cdot A_v)^2 - \operatorname{Adj}(D\mathcal{X}_v) \cdot Dv_t \cdot A_v. $$

**4.2. Implementation of the fixed point theorem.** In this section, we will set up tools for applying the fixed point theorem. Basically we will justify the well-posedness of a contracting application whose fixed point is the solution of the nonlinear system (4.2). For this purpose, for $v$ in $X_{T,u_0} := \{u \in X_T: u|_{t=0} = u_0\}$, we will begin by proving that the source $I(v, v)$ of (4.3) is regular enough in order to obtain a solution of the system (4.3) using **Theorem 3.1**. In order to ease the reading, for $T > 0$, and a Banach space $\mathcal{X}$, the space $L^p((0,T), \mathcal{X})$ will be denoted $L_T^p\mathcal{X}$. Also, we will only present the main steps, technical computations are carried out in **Section 4.3**.

First of all, let us point out that since $\sigma^{r/4}\nabla v \in L_T^4 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)$, for some $0 < r < 2$, one has:

$$ c := \int_0^T \|\nabla v\|_{\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \lesssim T^{\frac{3-r}{4}} \|v\|_{X_T}. $$

Suppose that $\|v\|_{X_T} \le R$ for some $R > 0$ to be fixed later. One can choose $T$ small in such a way that $2c < 1$. Thus, the constants involved in **Lemma 4.1** and **Lemma 4.2** are assumed to be fixed. The following bounds for $I(v, v)$ will be proved in **Section 4.3**:

$$ (4.11) \qquad \sup_{[0,T]} \|I(v,v)\|_{L^2(\mathbb{R}^d)} + \|I(v,v)\|_{L_T^\infty L^6(\mathbb{R}^d)} \lesssim \|P(\rho_0, c_0) - \tilde{P}\|_{L^2 \cap L^6(\mathbb{R}^d)} + T^{\frac{3-r}{4}} R(1+R), $$

$$ (4.12) \qquad \|I(v,v)\|_{L_T^{16} L^8(\mathbb{R}^d)} \lesssim T^{1/8} \|P(\rho_0, c_0) - \tilde{P}\|_{L^8(\mathbb{R}^d)} + T^{\frac{3-r}{4}} R(T^{1/8} + R), $$

$$ (4.13) \qquad \|\sigma^{r/4} I(v,v)\|_{L_T^4 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \lesssim T^{1/4} \|P(\rho_0, c_0) - \tilde{P}\|_{\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} + T^{\frac{3-r}{4}} R(T^{1/4} + R). $$

On the other hand, the following estimates for $\partial_t I(v, v)$ will be proved in **Section 4.3**:

$$ (4.14) \qquad \|\partial_t I(v,v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim T^{1/2} R(T^{\frac{3-r}{4}} R + 1) + T^{\frac{3-r}{4}} R^2 (RT^{1/2} + 1) + RT^{1/2}, $$
Since $K_T < \infty$, one can choose large $n_0$ such that

$$2|\theta - \theta_{n_0}|K_T \|\Pi_1 - \Pi_0\|_{\mathcal{L}} < 1$$

thus $\Pi_\theta$ is one to one as composition of two one to one operators. Therefore $\theta \in \mathcal{E}'$ and consequently $\mathcal{E}'$ is a closed set of $[0, 1]$. One then concludes that $1 \in \mathcal{E}'$ and **Theorem 3.3** then follows. □

**Proof of Lemma 3.4.** One deduces from **Theorem 3.3**, (3.54) and then (3.53) that the Cauchy problem (3.53) admits a unique solution $u^\delta \in \mathcal{C}([0, T], H^1(\mathbb{R}^d))$ with $u_t^\delta, \nabla^2 u^\delta \in L^2((0, T) \times \mathbb{R}^d)$. Moreover, by differentiating (3.53) with respect to space, and noting that $\rho_0^\delta$ and $u_0^\delta$ are smooth, we obtain that for all $s \ge 2$, $u^\delta \in \mathcal{C}([0, T], H^s(\mathbb{R}^d))$, and $u_t, \nabla^2 u^\delta \in L^2((0, T), H^{s-1}(\mathbb{R}^d))$. In the same manner, we differentiate (3.53) with respect to time, in order to obtain the equation satisfied by $u_t^\delta$. Of course, $F_t^\delta$ is regular enough and the regularity $u_t^\delta \in L^2((0, T), H^{s-1}(\mathbb{R}^d))$ is adequate for the differentiating to make sense. Thus, applying again **Theorem 3.3** one deduces from the previous discussion that $u_t^\delta \in \mathcal{C}([0, T], H^s(\mathbb{R}^d))$ and $u_{tt}^\delta, \nabla^2 u_t^\delta \in L^2((0, T), H^{s-1}(\mathbb{R}^d))$. This closes the existence of classical solution of the Cauchy problem (3.53).

The sequence $(u^\delta)_\delta$ verifies the estimates for $u, u_t$, (in (3.7)) $u_{tt}$ (in (3.9)) and the $L^p$ estimates in **Theorem 3.1** for $u$ and $u_t$, uniformly with respect to the regularisation parameter $\delta > 0$ with the exception of the piecewise Hölder estimate. Hence, one can let $\delta$ go to zero in order to obtain a solution of the Cauchy problem (3.52). Since $(u^\delta)_\delta$ is bounded in $L^\infty((0, T), H^1(\mathbb{R}^d))$ and its time derivative $(u_t^\delta)_\delta$ is bounded in $L^2((0, T), H^1(\mathbb{R}^d))$, it then turns out that $u \in \mathcal{C}([0, T], H^1(\mathbb{R}^d))$. On the other hand, given that $u_t \in L^\infty((0, T), L^2(\mathbb{R}^d))$ and $\sigma u_{tt} \in L^2((0, T), L^2(\mathbb{R}^d))$ it immediately follows $u_t \in \mathcal{C}((0, T], L^2(\mathbb{R}^d))$. It only remains the continuity of $u_t$ at the initial time. To achieve this, we first observe that for all $\tau > 0$:

$$\frac{1}{2} \int_{\mathbb{R}^d} \rho_{0,\delta} |u_t^\delta(\tau)|^2 + \int_0^\tau \tilde{\mu} \int_{\mathbb{R}^d} |\nabla u_t^\delta|_{L^2(\mathbb{R}^d)}^2 + (\tilde{\mu} + \tilde{\lambda}) \int_0^\tau \int_{\mathbb{R}^d} (\mathrm{div} \, u_t^\delta)^2 = \frac{1}{2} \int_{\mathbb{R}^d} \rho_{0,\delta} |u_t^\delta|_{t=0}^2 - \int_0^\tau \int_{\mathbb{R}^d} \partial_t F_\delta^{jk} \partial_k (u^\delta)_t^j,$$

and hence:

$$\frac{1}{2} \int_{\mathbb{R}^d} \rho_{0,\delta} |u_t^\delta(\tau)|^2 \le \frac{1}{2} \int_{\mathbb{R}^d} \rho_{0,\delta} |u_t^\delta|_{t=0}|^2 + C_* \int_0^\tau \|\partial_t F\|_{L^2(\mathbb{R}^d)}^2.$$

Using the compact embedding of $H^1(\mathbb{R}^d)$ into $L^2_{\mathrm{loc}}(\mathbb{R}^d)$ and the Aubin-Lions Lemma, we obtain $u_t^\delta(\tau) \to u_t(\tau)$ a.e in $\mathbb{R}^d$. Consequently, applying Fatou's Lemma yields:

$$\frac{1}{2} \int_{\mathbb{R}^d} \rho_0 |u_t(\tau)|^2 \le \frac{1}{2} \int_{\mathbb{R}^d} \rho_0 |u_t|_{t=0}|^2 + C_* \int_0^\tau \|\partial_t F\|_{L^2(\mathbb{R}^d)}^2,$$

from which we deduce:

$$\limsup_{\tau \to 0} \|\sqrt{\rho_0} u_t(\tau)\|_{L^2(\mathbb{R}^d)} \le \|\sqrt{\rho_0} u_{t|t=0}\|_{L^2(\mathbb{R}^d)}.$$

Since

$$\|\sqrt{\rho_0} u_{t|t=0}\|_{L^2(\mathbb{R}^d)} \le \liminf_{\tau \to 0} \|\sqrt{\rho_0} u_t(\tau)\|_{L^2(\mathbb{R}^d)},$$

we infer that $\sqrt{\rho_0} u_t(\tau)$ converges to $\sqrt{\rho_0} u_{t|t=0}$ strongly in $L^2(\mathbb{R}^d)$ and consequently we obtain the continuity of $u_t$ at the initial time given that $\rho_0$ is bounded away from vacuum.

This solution is unique: the difference $\delta u$ of two solutions $u_1$ and $u_2$ of (3.52) verifies

$$\begin{cases} \rho_0 \delta u_t - \tilde{\mu} \Delta \delta u - (\tilde{\mu} + \tilde{\lambda}) \nabla \delta u = 0, \\ \delta u|_{t=0} = 0 \end{cases}$$

and testing the above equation with $\delta$ leads to $\delta u = 0$.

From (3.52), it is obvious to derive the following expression for $\nabla u$:

$$\nabla u = -\frac{1}{\tilde{\mu}}(-\Delta)^{-1}\nabla [P(\rho_0 u_t) - P \operatorname{div} F] - \frac{1}{2\tilde{\mu} + \tilde{\lambda}}(-\Delta)^{-1}\nabla [Q(\rho_0 u_t) - Q \operatorname{div} F].$$

where $P$ is the Leray projector onto the space of divergence-free vector fields and $Q$ is the projector onto the space of curl-free vector fields. Finally, we can easily show that (3.11) also holds for $\nabla u$. □

This concludes the proof that $\mathcal{E}$ is a nonempty set. The only remaining task is to prove that $\mathcal{E}$ is both an open and closed subset of $[0, 1]$. This will be addressed in the following step.
### 3.2.3. Remainder of the proof of Theorem 3.1.
We now proceed to the final step in the proof of Theorem 3.1, which is to show that $\mathcal{E}$ is both an open and closed subset of $[0, 1]$.

*Proof.* We begin by proving that $\mathcal{E}$ is an open subset of $[0, 1]$. So let $\theta_0 \in \mathcal{E}$, the goal is to find some $\eta > 0$ such that

$$ (3.59) \qquad (\theta_0 - \eta, \theta_0 + \eta) \cap [0, 1] \subset \mathcal{E}. $$

For this purpose, let $\theta \in [0, 1]$, $(F, u_0) \in Y_T^\theta$ and consider the Cauchy problem (3.50) which can be written as:

$$ (3.60) \qquad \begin{cases} \rho_0 u_t - \operatorname{div}(2\mu^{\theta_0} \mathbb{D}u + \lambda^{\theta_0} \operatorname{div} u I_d) = \operatorname{div} F + (\theta - \theta_0) \operatorname{div}\{2(\mu - \tilde{\mu})\mathbb{D}u + (\lambda - \tilde{\lambda}) \operatorname{div} u I_d\}, \\ u|_{t=0} = u_0. \end{cases} $$

We will apply a fixed-point argument to solve the above Cauchy problem with a small $\eta$ that does not depend on $F$ nor $u_0$. In fact, the proof shows that the constant $\eta$ only depends on the norm of the viscosity and initial density as required in (3.2), as well on the interface characteristics. The solution will be constructed in $X_{T,u_0}$ which is defined as:

$$ X_{T,u_0} := \{v \in X_T : v|_{t=0} = u_0\}, $$

a complete subset of the Banach space $X_T$. So for $v \in X_{T,u_0}$, we consider the following equation:

$$ (3.61) \qquad \begin{cases} \rho_0 u_t - \operatorname{div}(2\mu^{\theta_0} \mathbb{D}u + \lambda^{\theta_0} \operatorname{div} u I_d) = \operatorname{div} F + (\theta - \theta_0) \operatorname{div}\{2(\mu - \tilde{\mu})\mathbb{D}v + (\lambda - \tilde{\lambda}) \operatorname{div} v I_d\}, \\ u|_{t=0} = u_0. \end{cases} $$

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

$$ (3.62) \qquad F(v) := F + (\theta - \theta_0)\{2(\mu - \tilde{\mu})\mathbb{D}v + (\lambda - \tilde{\lambda}) \operatorname{div} v I_d\} $$

verifies $(F(v), u_0) \in Y_T^{\theta_0}$. The last condition in the definition of $Y_T^{\theta_0}$ (see (3.3)) is valid since $v|_{t=0} = u_0$. Indeed, we have

$$ \begin{align} & \operatorname{div} \left( 2\mu_0^{\theta_0} \mathbb{D}u_0 + \lambda_0^{\theta_0} \operatorname{div} u_0 I_d + F(v)|_{t=0} \right) \nonumber \\ &= \operatorname{div} \left( 2(\mu_0^{\theta_0} + (\theta - \theta_0)(\mu_0 - \tilde{\mu}))\mathbb{D}u_0 + (\lambda_0^{\theta_0} + (\theta - \theta_0)(\lambda_0 - \tilde{\lambda})) \operatorname{div} u_0 I_d + F|_{t=0} \right) \nonumber \\ & \qquad - \operatorname{div} \left( 2\mu_0^{\theta_0} \mathbb{D}u_0 + \lambda_0^{\theta_0} \operatorname{div} u_0 I_d + F|_{t=0} \right) \in L^2(\mathbb{R}^d) \quad \text{since } (F, u_0) \in Y_T^\theta. \tag{3.63} \end{align} $$

Now, owing to the fact that $\theta_0 \in \mathcal{E}$, the underlying equation admits a unique solution $\Phi(v) \in X_{T,u_0}$ that verifies all the estimates in Theorem 3.1, where $F$ is replaced by $F(v)$.

In the following we are looking for some small $\theta - \theta_0$ and some $R_T$ which depends on the initial data such that the condition $\|v\|_{X_T} \le R_T$ implies the same for $\Phi(v)$. As a standard notation, from now on $A \lesssim B$ means $A \le CB$ with a constant $C$ depending on the bounds of $\mu, \lambda, \rho_0$, interface characteristics, and on all of the norms of $\mu, \lambda$ as required in (3.2).

**Step 1: Estimates for $\Phi(v)$ and $\partial_t \Phi(v)$**. Since $\Phi(v)$ solves the Cauchy problem (3.61), $(F(v), u_0) \in Y_T^{\theta_0}$, and $\theta_0 \in \mathcal{E}$, one has from the a priori estimate the following estimate for $\Phi(v)$:

$$ (3.64) \qquad \begin{aligned} & \sup_{[0,T]} \|\Phi(v), \partial_t \Phi(v), \nabla \Phi(v)\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|\nabla \Phi(v), \nabla \partial_t \Phi(v), \partial_t \Phi(v)\|_{L^2(\mathbb{R}^d)}^2 \lesssim \|u_0\|_{H^1(\mathbb{R}^d)}^2 + \|\partial_t \Phi(v)\|_{t=0} \|v\|_{L^2(\mathbb{R}^d)}^2 \\ & \qquad + \sup_{[0,T]} \|F(v)\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|\partial_t F(v)\|_{L^2(\mathbb{R}^d)}^2 \mathrm{d}t + \int_0^T \|F(v)\|_{L^2 \cap L^6(\mathbb{R}^d)}^2 \mathrm{d}t. \end{aligned} $$

On the one hand, from (3.62), we infer that:

$$ (3.65) \qquad \begin{aligned} & \sup_{[0,T]} \|F(v) - F\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|F(v) - F\|_{L^2(\mathbb{R}^d)}^2 \lesssim |\theta - \theta_0|^2 \left\{ \sup_{[0,T]} \|\nabla v\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|\nabla v\|_{L^2(\mathbb{R}^d)}^2 \mathrm{d}t \right\} \\ & \qquad \lesssim |\theta - \theta_0|^2 \|v\|_{X_T}^2 \end{aligned} $$

and

$$ \int_0^T \|F(v) - F\|_{L^6(\mathbb{R}^d)}^2 \lesssim |\theta - \theta_0|^2 \|v\|_{X_T}^2. $$

On the other hand, from (3.62) one can express the time derivative of $F(v)$ as:

$$ (3.66) \qquad \partial_t F(v) = F_t + (\theta - \theta_0)\{2(\mu - \tilde{\mu})\mathbb{D}v_t + (\lambda - \tilde{\lambda}) \operatorname{div} v_t I_d + 2\mu_t \mathbb{D}v + \lambda_t \operatorname{div} v I_d\} $$
$L_T^{8/d} L^4(\mathbb{R}^d)$ norm estimate for $\sqrt{\sigma}\nabla u_t$.

$$ (4.20) \quad \int_0^T \sigma^{4/d} \|\nabla u_t\|_{L^4(\mathbb{R}^d)}^{8/d} \lesssim \int_0^T \sigma^{4/d} \|\partial_t I(v, v)\|_{L^4(\mathbb{R}^d)}^{8/d} + T^{1-\frac{1}{d}} R^{8/d} \|\nabla u\|_{L^1 T^{16} L^8(\mathbb{R}^d)}^{8/d} + \|\sigma u_{tt}\|_{L^\infty T^2 L^2(\mathbb{R}^d)}^{8/d} \\ \qquad + \left[ \int_0^T \|\nabla u_t, \partial_t I(v, v)\|_{L^2(\mathbb{R}^d)}^2 \right]^{4/d} + T^{4/d} R^{8/d} \|\nabla u\|_{L^\infty T^4 L^4(\mathbb{R}^d)}^{8/d}. $$

The norms of $I(v, v)$ are given in (4.15) and (4.14) whereas the norms of $u$ involved are controlled in (4.17), (4.18) and (4.19).

$L_T^4 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)$ estimate for $\sigma^{r/4}\nabla u$.

$$ \int_0^T \sigma^r \|\nabla u\|_{\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)}^4 \lesssim \int_0^T \sigma^r \|I(v,v)\|_{\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)}^4 + \|I(v,v), \nabla u, u_t, \sqrt{\sigma}\nabla u_t\|_{L^\infty T^2 L^2(\mathbb{R}^d)}^4 \\ + \left[ \int_0^T \|I(v,v), \nabla u, u_t, \nabla u_t\|_{L^2(\mathbb{R}^d)}^2 \right]^2 + \left[ \int_0^T \sigma^{4/d} \|\nabla u_t\|_{L^4(\mathbb{R}^d)}^{8/d} \right]^{d/2}. $$

The norms of $I(v, v)$ are given in (4.11) and (4.13) whereas the norms of $u$ involved are controlled in (4.17), (4.18) and (4.20).

By putting together all these computations, one can choose $R$ related to the norm of the initial data and some small $T^*$ such that for all $0 < T \le T_*$,

$$ \|u\|_{X_T} \le R. $$

Moreover, for all $0 < T \le T_*$, the map:

$$ \Phi_T: X_{T,u_0}(R) \to X_{T,u_0}(R) \\ v \mapsto \Phi_T(v) := u $$

where $u$ is the unique solution of the Cauchy problem (4.3), and $X_{T,u_0}(R) := \{u \in X_{T,u_0} : \|u\|_{X_T} \le R\}$, is well-defined. It is obvious that $X_{T,u_0}$ is a complete metric space as a closed subset of the Banach space $X_T$. So every closed subset of $X_{T,u_0}$ is complete with respect to the induced distance. In the following we will show that for some small $T \le T_*$, the maps $\Phi_T$ is a contraction over $X_{T,u_0}(R)$ thus admits an unique fixed point, that solves our nonlinear equations (4.2).

Let us fix some $0 < T \le T_*$ and let $v_1, v_2 \in X_{T,u_0}(R)$, $u_i = \Phi_T(v_i)$ and denote $\delta u := u_2 - u_1$, $\delta v := v_2 - v_1$. By taking differences in the equations solved by $u_1$ and $u_2$, we have that $\delta u$ solves the Cauchy problem:

$$ \left\{ \begin{array}{l} \rho_0 \partial_t \delta u - \text{div}\{2\mu(J_{v_2}^{-1} \rho_0, c_0) \mathbb{D}\delta u + \lambda(J_{v_2}^{-1} \rho_0, c_0) \text{div}\delta u\} = \text{div}(\delta \mathcal{I}_1 + \delta \mathcal{I}_2) + \text{div}(I(v_2, v_2) - I(v_1, v_1)) =: \text{div}(\delta \mathcal{I}(v)), \\ \delta u|_{t=0} = 0 \end{array} \right. $$

where

$$ \delta \mathcal{I}_1 = 2\{(\mu(J_{v_2}^{-1}\rho_0, c_0) - \mu(J_{v_1}^{-1}\rho_0, c_0))\mathbb{D}u_1 \quad \text{and} \quad \delta \mathcal{I}_2 = \{\lambda(J_{v_2}^{-1}\rho_0, c_0) - \lambda(J_{v_1}^{-1}\rho_0, c_0)\}\text{div}u_1. $$

Moreover, $\delta u|_{t=0} = 0$ and $\partial_t \delta u|_{t=0} = 0$.

In order to estimate various norm of $\delta \mathcal{I}(v)$ in $Y_T$ and then deduce the norm of $\delta u$ in $X_T$ we will use the following Lemma:

**Lemma 4.4.** Let $v_1, v_2$ be two vectors fields verifying the condition (4.4). Then, there exists a constant $k = k(c)$ such that the following estimates hold true:

$$ \left\{ \begin{array}{l} \|A_{v_2}(t) - A_{v_1}(t)\|_{L^p(\mathbb{R}^d)} \le k \|\nabla \delta v\|_{L^1_t L^p(\mathbb{R}^d)}, \\ \|\text{Adj}(D\mathcal{X}_{v_2}(t)) - \text{Adj}(D\mathcal{X}_{v_1}(t))\|_{L^p(\mathbb{R}^d)} \le k \|\nabla \delta v\|_{L^1_t L^p(\mathbb{R}^d)}, \\ \|J_{v_2}^{\pm 1}(t) - J_{v_1}^{\pm 1}(t)\|_{L^p(\mathbb{R}^d)} \le k \|\nabla \delta v\|_{L^1_t L^p(\mathbb{R}^d)} \end{array} \right. $$

and

$$ \left\{ \begin{array}{l} \|A_{v_2}(t) - A_{v_1}(t)\|_{\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \le k \|\nabla \delta v\|_{L^1_t \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)}, \\ \|\text{Adj}(D\mathcal{X}_{v_2}(t)) - \text{Adj}(D\mathcal{X}_{v_1}(t))\|_{\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \le k \|\nabla \delta v\|_{L^1_t \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)}, \\ \|J_{v_2}^{\pm 1}(t) - J_{v_1}^{\pm 1}(t)\|_{\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \le k \|\nabla \delta v\|_{L^1_t \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)}. \end{array} \right. $$
and consequently, it holds:

$$ (3.67) \quad \int_0^T \|\partial_t F(v) - F_t\|_{L^2(\mathbb{R}^d)}^2 \lesssim |\theta - \theta_0|^2 \left\{ \|\nabla v\|_{L^4((0,T)\times\mathbb{R}^d)}^2 + \int_0^T \|\nabla v_t\|_{L^2(\mathbb{R}^d)}^2 \right\} \lesssim |\theta - \theta_0|^2 \|v\|_{X_T}^2. $$

Gathering all of these estimates, one has the following:

$$ (3.68) \quad \sup_{[0,T]} \|\Phi(v), \partial_t \Phi(v), \nabla \Phi(v)\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|\nabla \Phi(v), \nabla \partial_t \Phi(v), \partial_t \Phi(v)\|_{L^2(\mathbb{R}^d)}^2 \lesssim \|(F, u_0)\|_{Y_T^\theta}^2 + |\theta - \theta_0|^2 \|v\|_{X_T}^2. $$

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

$$ (3.69) \quad \|\nabla \Phi(v)\|_{L^{\infty}((0,T),L^6(\mathbb{R}^d))} \lesssim \|F(v)\|_{L^{\infty}((0,T),L^6(\mathbb{R}^d))} + \sup_{[0,T]} \|F(v), \partial_t \Phi(v), \nabla \Phi(v)\|_{L^2(\mathbb{R}^d)}. $$

Owing to the fact that the estimate (3.68) provides a bound for $\partial_t \Phi(v)$ in $L^\infty((0,T), L^2(\mathbb{R}^d))$ and for $\Phi(v)$ in $L^\infty((0,T), H^1(\mathbb{R}^d))$ and owing to the estimate (3.65), we only need to estimate the $L^\infty((0,T), L^6(\mathbb{R}^d))$ norm of $F(v)$. In fact, this can be easily obtained from (3.62) as:

$$ \|F(v) - F\|_{L^{\infty}((0,T),L^6(\mathbb{R}^d))} \lesssim |\theta - \theta_0| \|\nabla v\|_{L^{\infty}((0,T),L^6(\mathbb{R}^d))} \lesssim |\theta - \theta_0| \|v\|_{X_T}. $$

Gathering all of these estimates, one has:

$$ (3.70) \quad \|\nabla \Phi(v)\|_{L^{\infty}((0,T),L^6(\mathbb{R}^d))} \lesssim \|(F, u_0)\|_{Y_T^\theta} + |\theta - \theta_0| \|v\|_{X_T}. $$

We turn to the $L^{16}((0,T), L^8(\mathbb{R}^d))$ estimate for $\nabla \Phi(v)$. For this purpose, we recall from (3.8) that:

$$ \int_0^T \|\nabla \Phi(v)\|_{L^8(\mathbb{R}^d)}^{16} \lesssim \int_0^T \|F(v)\|_{L^8(\mathbb{R}^d)}^{16} + \|F(v), \partial_t \Phi(v), \nabla \Phi(v)\|_{L^2 \cap L^\infty((0,T), L^2(\mathbb{R}^d))}^{16} + \|\nabla \partial_t \Phi(v)\|_{L^2((0,T) \times \mathbb{R}^d)}^{16}. $$

We have already obtained bound for the last two terms above in (3.68) and (3.65) and we have only to estimate the first one. From (3.62), we have:

$$ \int_0^T \|F(v) - F\|_{L^8(\mathbb{R}^d)}^{16} \lesssim |\theta - \theta_0|^{16} \int_0^T \|\nabla v\|_{L^8(\mathbb{R}^d)}^{16} \lesssim |\theta - \theta_0|^p \|v\|_{X_T}^{16} $$

and hence:

$$ (3.71) \quad \int_0^T \|\nabla \Phi(v)\|_{L^8(\mathbb{R}^d)}^{16} \lesssim \|(F, u_0)\|_{Y_T^\theta}^p + |\theta - \theta_0|^{16} \|v\|_{X_T}^{16}. $$

**Step 2: Estimates for $\partial_{tt}\Phi(v)$ and $L^4$-norm estimate for $\nabla\partial_t\Phi(v)$.** As in the previous step, one first recalls that $\Phi(v)$ verifies:

$$ \int_0^T \|\sqrt{\sigma} \partial_{tt} \Phi(v), \sigma \nabla \partial_{tt} \Phi(v)\|_{L^2(\mathbb{R}^d)}^2 + \|\sqrt{\sigma} \partial_t \nabla \Phi(v), \sigma \partial_{tt} \Phi(v)\|_{L^\infty((0,T), L^2(\mathbb{R}^d))}^2 \\ \lesssim \int_0^{\sigma(T)} \|\partial_t \nabla \Phi(v)\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|\partial_t F(v), \sigma \partial_{tt} F(v)\|_{L^2(\mathbb{R}^d)}^2 + \|\nabla \Phi(v)\|_{L^\infty((0,T), L^4(\mathbb{R}^d))}^2. $$

All terms that appear in the right hand side above are controlled in (3.68), (3.70), (3.65) except the $L^2((0,T) \times \mathbb{R}^d)$ norm of $\sigma \partial_{tt} F(v)$ which is:

$$ \int_0^T \sigma^2 \|\partial_{tt} F(v)\|_{L^2(\mathbb{R}^d)}^2 \lesssim \int_0^T \sigma^2 \|F_{tt}\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \sigma^2 \|\partial_{tt} F(v) - F_{tt}\|_{L^2(\mathbb{R}^d)}^2. $$

In order to link the last term to the norm of $v$, we first express $\partial_{tt} F(v) - F_{tt}$, from (3.66) as

$$ \partial_{tt} F(v) - F_{tt} = (\theta - \theta_0)\{2(\mu - \tilde{\mu})\mathbb{D}v_{tt} + (\lambda - \tilde{\lambda})\operatorname{div}v_{tt}I_d + 4\mu_t\mathbb{D}v_t + 2\lambda_t\operatorname{div}v_tI_d + 2\mu_{tt}\mathbb{D}v + \lambda_{tt}\operatorname{div}vI_d\} $$

and then

$$ \int_0^T \sigma^2 \|\partial_{tt} F(v) - F_{tt}\|_{L^2(\mathbb{R}^d)}^2 \lesssim |\theta - \theta_0|^2 \|v\|_{X_T}^2 $$

from which, we deduce that

$$ (3.72) \quad \int_0^T \|\sqrt{\sigma} \partial_{tt} \Phi(v), \sigma \nabla \partial_{tt} \Phi(v)\|_{L^2(\mathbb{R}^d)}^2 + \|\sqrt{\sigma} \partial_t \nabla \Phi(v), \sigma \partial_{tt} \Phi(v)\|_{L^\infty((0,T), L^2(\mathbb{R}^d))}^2 \\ \lesssim \|(F, u_0)\|_{Y_T^\theta}^2 + |\theta - \theta_0|^2 \|v\|_{X_T}^2. $$
The proof of the above Lemma follows the same ideas as in [3, Lemma A.4] and it is left as an exercise for the reader. We turn to the computation of the norm of $\delta\mathcal{I}(v)$.

First, by combining estimates obtained in Section 4.3, Section 4.3 and in Section 4.3, we have:

$$ \|\delta\mathcal{I}(v)\|_{L_T^\infty(L^2 \cap L^6)(\mathbb{R}^d)} + \|\sigma^{r/4}\delta\mathcal{I}(v)\|_{L_T^4\mathcal{C}_{\rho w, \gamma}^{\alpha}(\mathbb{R}^d)} + \|\delta\mathcal{I}(v)\|_{L_T^{16}L^8(\mathbb{R}^d)} \lesssim T^{\frac{3-r}{4}}R\|\delta v\|_{X_T} + T^{\frac{3-r}{4}}(R^2 + T^{\frac{1+r}{4}})\|\delta v\|_{X_T}. $$

On the other hand, the estimates we will obtain in Section 4.3, Section 4.3 and in Section 4.3, yield the following:

$$ \|\partial_t \delta \mathcal{I}(v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim T^{\frac{3-r}{4}} R(1 + T^{1/2}) \|\delta v\|_{X_T} + T^{1/2}(1 + R) \|\delta v\|_{X_T} + T^{\frac{3-r}{2}} R^2 \|\delta v\|_{X_T} $$

and

$$ \|\sqrt{\sigma} \partial_t \delta \mathcal{I}(v)\|_{L_T^{8/d} L^4(\mathbb{R}^d)} \lesssim T^{\frac{3-r}{4}} R \|\delta v\|_{X_T} (1 + T^{d/8}) + T^{\frac{3-r}{4} + \frac{d-1}{8}} R^2 \|\delta v\|_{X_T} + T^{\frac{d-1}{8}} R \|\delta v\|_{X_T} \\ + T^{\frac{3-r}{2}} R^2 \|\delta v\|_{X_T} + T^{d/8} \|\delta v\|_{X_T}. $$

As well, from Section 4.3, one has:

$$ \|\sigma \partial_{tt} \delta \mathcal{I}_1, \sigma \partial_{tt} \delta \mathcal{I}_2\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \left[ T^{\frac{3-r}{4}} + T^{\frac{1}{2}-\frac{d}{8}} + T^{\frac{10-2r-d}{8}} R + T^{1/2} R \right] R \|\delta v\|_{X_T}. $$

Finally, the estimates we will obtain in Section 4.3, Section 4.3, Section 4.3 and in Section 4.3, yield the following:

$$ \|\sigma \partial_{tt}(I(v_1, v_1) - I(v_2, v_2))\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim T^{\frac{3-r}{4}} R \|\delta v\|_{X_T} + T^{\frac{3-r}{2}} R^2 \|\delta v\|_{X_T} + T^{1/2} R^2 \|\delta v\|_{X_T} (1 + R^2) \\ + T^{\frac{10-2r-d}{8}} R^2 (1 + R) \|\delta v\|_{X_T} + T^{\frac{16-4r-d}{8}} R^2 (1 + R) \|\delta v\|_{X_T} + T^{\frac{1}{2}-\frac{d}{8}} R \|\delta v\|_{X_T}. $$

This gives an appropriate estimate for the norm of $(\delta\mathcal{I}(v), 0)$ in $Y_T$, which we use to estimate the norm of $\delta u$ in the following lines.

### Estimates for $u$ and $u_t$.

$$ \sup_{[0,T]} \|\delta u, \partial_t \delta u, \nabla \delta u\|_{L^2(\mathbb{R}^d)}^2 + \|\nabla \delta u\|_{L_T^\infty L^6(\mathbb{R}^d)}^2 + \int_0^T \|\nabla \delta u, \nabla \partial_t \delta u, \partial_t \delta u\|_{L^2(\mathbb{R}^d)}^2 \\ \lesssim \left( \| (\delta \mathcal{I}(v), 0) \|_{Y_T}^2 + T R^2 \| \delta \mathcal{I}(v) \|_{L_T^\infty(L^2 \cap L^6)(\mathbb{R}^d)}^2 \right) $$

with the norms of $\delta\mathcal{I}(v)$ given in the previous discussion.

### Estimates for $u_{tt}$.

$$ \int_0^T \|\sqrt{\sigma} \partial_{tt} \delta u, \sigma \nabla \partial_{tt} \delta u\|_{L^2(\mathbb{R}^d)}^2 + \|\sqrt{\sigma} \nabla \partial_t \delta u, \sigma \partial_{tt} \delta u\|_{L_T^\infty L^2(\mathbb{R}^d)}^2 \lesssim \left[ \int_0^{\sigma(T)} \|\nabla \partial_t \delta u\|_{L^2(\mathbb{R}^d)}^2 \right. \\ \left. + \int_0^T \|\partial_t \delta \mathcal{I}(v), \sigma \partial_{tt} \delta \mathcal{I}(v)\|_{L^2(\mathbb{R}^d)}^2 + \left[ T R^2 + (T^{\frac{d-1}{4}} R^4 + R^2) T^{1-\frac{d}{4}} \right] \|\nabla \delta u\|_{L_T^\infty L^4(\mathbb{R}^d)}^2 \right]. $$

### $L_T^{16} L^8(\mathbb{R}^d)$ estimates for $\nabla u$.

$$ \|\nabla \delta u\|_{L_T^{16} L^8(\mathbb{R}^d)} \lesssim \|\delta \mathcal{I}(v)\|_{L_T^{16} L^8(\mathbb{R}^d)} + \|\delta \mathcal{I}(v), \partial_t \delta u, \nabla \delta u\|_{L_T^2 \cap L_T^\infty L^2(\mathbb{R}^d)} + \|\nabla \partial_t \delta u\|_{L_T^2 L^2(\mathbb{R}^d)}. $$

### $L_T^{8/d} L^4(\mathbb{R}^d)$ estimate for $\nabla u_t$.

$$ \int_0^T \sigma^{4/d} \|\nabla \partial_t \delta u\|_{L^4(\mathbb{R}^d)}^{8/d} \lesssim \int_0^T \sigma^{4/d} \|\partial_t \delta \mathcal{I}(v)\|_{L^4(\mathbb{R}^d)}^{8/d} + T^{1-\frac{1}{d}} R^{8/d} \|\nabla \delta u\|_{L^{16}((0,T),L^8(\mathbb{R}^d))}^{8/d} + \|\sigma \partial_{tt} \delta u\|_{L_T^\infty L^2(\mathbb{R}^d)}^{8/d} \\ + \left[ \int_0^T \|\nabla \partial_t \delta u, \partial_t \delta \mathcal{I}(v)\|_{L^2(\mathbb{R}^d)}^2 \right]^{4/d} + T^{4/d} R^{8/d} \|\nabla \delta u\|_{L_T^\infty L^4(\mathbb{R}^d)}^{8/d}. $$
This ends the computations of the higher-order Hoff estimates. Now, we may turn to the $L^{8/d}((0,T), L^4(\mathbb{R}^d))$ norm estimate for $\sigma^4 \nabla \partial_t \Phi(v)$. Thanks to (3.10), Hölder's and Young's inequality, we have:

$$ \int_0^T \sigma^2 \|\nabla \partial_t \Phi(v)\|_{L^4(\mathbb{R}^d)}^{8/d} \lesssim \int_0^T \sigma^2 \|\partial_t F(v)\|_{L^4(\mathbb{R}^d)}^{8/d} + \left[ \int_0^T \|\nabla \Phi(v)\|_{L^8(\mathbb{R}^d)}^{16} \right]^{1/(2d)} + \|\sigma \partial_t \Phi(v)\|_{L^\infty((0,T), L^2(\mathbb{R}^d))}^{8/d} \\ + \|\nabla \Phi(v)\|_{L^4((0,T) \times \mathbb{R}^d)}^{8/d} + \left[ \int_0^T \|\nabla \partial_t \Phi(v), \partial_t F(v)\|_{L^2(\mathbb{R}^d)}^2 \right]^{4/d} . $$

The estimates (3.72) and (3.71), (3.70) and (3.67) control all of the terms that appear in the left hand side above, except the $L^{8/d}((0,T), L^4(\mathbb{R}^d))$-norm of $\sigma^4 \partial_t F(v)$. Starting from (3.66), one has:

$$ \int_0^T \sigma^2 \|\partial_t F(v) - F_t\|_{L^4(\mathbb{R}^d)}^{8/d} \lesssim |\theta - \theta_0|^{8/d} \left\{ \int_0^T \sigma^2 \|\nabla v_t\|_{L^4(\mathbb{R}^d)}^{8/d} + \left[ \int_0^T \|\nabla v\|_{L^8(\mathbb{R}^d)}^{16} \right]^{1/(2d)} \right\} \\ \lesssim |\theta - \theta_0|^{8/d} \|v\|_{X_T}^{8/d}. $$

Gathering all of the estimates one obtains the following:

$$ \int_0^T \sigma^2 \|\partial_t \nabla \Phi(v)\|_{L^4(\mathbb{R}^d)}^{8/d} \lesssim \|(F, u_0)\|_{Y_T^\theta}^{8/d} + |\theta - \theta_0|^{8/d} \|v\|_{X_T}^{8/d}. $$

We now turn to the last step:

**Step 3: Piecewise Hölder-norm estimate for $\nabla \Phi(v)$.** As above, thanks to (3.11), the following Hölder estimate holds for $\nabla \Phi(v)$:

$$ \int_0^T \sigma^r \|\nabla \Phi(v)\|_{\mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)}^4 \lesssim \int_0^T \sigma^r \|F(v)\|_{\mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)}^4 + \left[ \int_0^T \sigma^{4/d} \|\partial_t \nabla \Phi(v)\|_{L^4(\mathbb{R}^d)}^{8/d} \right]^{d/2} \\ + \|F(v), \nabla \Phi(v), \partial_t \Phi(v), \sqrt{\sigma} \nabla \partial_t \Phi(v)\|_{L^\infty((0,T), L^2(\mathbb{R}^d))}^4 \\ + \left[ \int_0^T \|F(v), \nabla \Phi(v), \partial_t \Phi(v), \partial_t \nabla \Phi(v)\|_{L^2(\mathbb{R}^d)}^2 \right]^2 . $$

According to the above computations, one only needs to estimate the first term of the right hand side above and from (3.62) one has:

$$ \int_0^T \sigma^r \|F(v) - F\|_{\mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)}^4 \lesssim |\theta - \theta_0|^4 \|v\|_{X_T}^4 . $$

Gathering all of these estimates, one has the following:

$$ (3.73) \qquad \|\Phi(v)\|_{X_T} \le C(R_0 + |\theta - \theta_0|\|v\|_{X_T}) $$

where $R_0$ depends on the norm of $(F, u_0) \in Y_T^\theta$ and the constant $C$ that depends not only on the bounds of the density and viscosity, on the interface characteristics, and all of the norm of the viscosity $\mu$, $\lambda$ and initial density $\rho_0$ as required in (3.2). So, choosing large $R_T$, for instance $R_T = 2CR_0$ and $|\theta - \theta_0| \le \min\{1, 1/(2C)\}$, one obtains $\|\Phi(v)\|_{X_T} \le R_T$ as soon as $\|v\|_{X_T} \le R_T$. We thus define a map $\Phi$ from the close ball $B_{X_{T,u_0}}(0, R_T)$ of $X_{T,u_0}$ to itself.

**Step 4: Construction of the approximate sequence and proof of its convergence.** We construct the sequence of approximate solutions $(u^n)_{n \in \mathbb{N}}$, inductively, as follows:

$$ u_{n+1} = \Phi(u_n), \quad n \ge 0, $$

with

$$ (3.74) \qquad \begin{cases} \rho_0 \partial_t u_{n+1} - \operatorname{div}(2\mu^{\theta_0} \mathbb{D} u_{n+1} + \lambda^{\theta_0} \operatorname{div} u_{n+1} I_d) = \operatorname{div} F + (\theta - \theta_0) \operatorname{div} (2(\mu - \tilde{\mu}) \mathbb{D} u_n \\ + (\lambda - \tilde{\lambda}) \operatorname{div} u_n I_d), \\ u_{n+1|_{t=0}} = u_0. \end{cases} $$

It follows from the conclusion of Step 3 that the sequence $(u_n)_n$ is uniformly bounded in $X_T$:

$$ (3.75) \qquad \sup_{n \in \mathbb{N}} \|u_n\|_{X_T} \le R_T. $$
$L_T^4 \mathscr{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)$ estimate for $\nabla u$.

$$
\int_0^T \sigma^r \|\nabla \delta u\|_{\mathscr{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)}^4 \lesssim \int_0^T \sigma^r \|\delta \mathcal{I}(v)\|_{\mathscr{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)}^4 + \|\delta \mathcal{I}(v), \nabla \delta u, \partial_t \delta u, \sqrt{\sigma} \nabla \partial_t \delta u\|_{L_T^\infty L^2(\mathbb{R}^d)}^4 \\
+ \left[ \int_0^T \|\delta \mathcal{I}(v), \nabla \delta u, \partial_t \delta u, \nabla \partial_t \delta u\|_{L^2(\mathbb{R}^d)}^2 \right]^2 + \left[ \int_0^T \sigma^{4/d} \|\nabla \partial_t \delta u\|_{L^4(\mathbb{R}^d)}^{8/d} \right]^{d/2}.
$$

Plugging all of these computations, one obtains the fact that, for some sufficiently small $T$,

$$
\|\Phi_T(u_2) - \Phi_T(u_1)\|_{X_T} \le \frac{1}{2} \|v_2 - v_1\|_{X_T},
$$

thus $\Phi_T$ is a contraction over the complete set $X_{T,u_0}$, hence admits an unique fixed point $u$. By construction this fixed point solves the system (2.7) and by change of variable in Eulerian coordinates, $u$ solves the system (1.5).

The proof of uniqueness of solution of the equations (1.5) follows the same lines as [4]. It only requires that the
density and viscosity be far away from vacuum and that the velocity gradient verifies the following:

$$
\nabla u \in L^1((0, T), L^\infty(\mathbb{R}^d)), \quad \text{and} \quad \sqrt{\sigma} \nabla u \in L^2((0, T), L^\infty(\mathbb{R}^d)).
$$

In particular, no small contrast on dynamic viscosity is required.

4.3. **Estimates.** In this section, we provide the full computations of the norm of $I(v, v)$ and $\delta I(v)$ used in the preceding section.

Step 1: Lebesgue and Hölder norm estimates for $I(v, v)$. We begin this estimate by noticing that, from **Lemma 4.1**, we have the following estimates for the first two terms of $I(v, v)$:

$$
\| I_1(v, v); I_2(v, v) \|_{L_T^m L^q(\mathbb{R}^d)} \lesssim \| \nabla v \|_{L_T^1 L^\infty(\mathbb{R}^d)} \| \nabla v \|_{L_T^m L^q(\mathbb{R}^d)}
$$

for all $2 \le m, q \le \infty$. Regarding the Hölder estimates, one has from **Lemma 4.2** the following:

$$
\| \sigma^{r/4} (I_1(v, v); I_2(v, v)) \|_{L_T^4 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \lesssim \| \nabla v \|_{L_T^1 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \| \sigma^{r/4} \nabla v \|_{L_T^4 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)}.
$$

It now remains the norm of $I_3(v)$. We may first write:

$$
I_3(v) = \operatorname{Adj}(D\mathcal{X}_v)\{P(J_v^{-1}\rho_0, c_0) - P(\rho_0, c_0) + P(\rho_0, c_0) - \tilde{P}\}
$$

and thus, deduce from **Lemma 4.1**, the following:

$$
\| I_3(v) \|_{L_T^m L^q(\mathbb{R}^d)} \lesssim T^{1/m} \| \nabla v \|_{L_T^1 L^q(\mathbb{R}^d)} + T^{1/m} \| P(\rho_0, c_0) - \tilde{P} \|_{L^q(\mathbb{R}^d)}
$$

and finally, from **Lemma 4.2**, one has the following Hölder estimate for $I_3(v)$:

$$
\| \sigma^{r/4} I_3(v) \|_{L_T^4 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \lesssim T^{1/4} \| \nabla v \|_{L_T^1 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} + T^{1/4} \| P(\rho_0, c_0) - \tilde{P} \|_{\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)}.
$$

We gather all of these estimates in order to obtain:

$$
\| I(v, v) \|_{L_T^m L^q(\mathbb{R}^d)} \lesssim \| \nabla v \|_{L_T^1 L^\infty(\mathbb{R}^d)} \| \nabla v \|_{L_T^m L^q(\mathbb{R}^d)} + T^{1/m} \| \nabla v \|_{L_T^1 L^q(\mathbb{R}^d)} + T^{1/m} \| P(\rho_0, c_0) - \tilde{P} \|_{L^q(\mathbb{R}^d)}
$$

and

$$
\| \sigma^{r/4} I_3(v) \|_{L_T^4 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \lesssim \| \nabla v \|_{L_T^1 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \| \sigma^{r/4} \nabla v \|_{L_T^4 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} + T^{1/4} \| \nabla v \|_{L_T^1 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \\
\qquad + T^{1/4} \| P(\rho_0, c_0) - \tilde{P} \|_{\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)}.
$$

We turn to the norm of $\partial_t I(v, v)$ required in $Y_T$.

Step 2: Lebesgue norm estimates for $\partial_t I(v, v)$. In this step, we will estimate the $L_T^2 L^2(\mathbb{R}^d)$ of $\partial_t I(v, v)$ and the $L_T^{8/d} L^4(\mathbb{R}^d)$ norm of $\sqrt{\sigma} \partial_t I(v, v)$. Let us begin by computing the time derivative of the third term $I_3(v)$. We have:

$$
(4.21) \qquad \partial_t I_3(v) = \partial_t \operatorname{Adj}(D\mathcal{X}_v)\{P(J_v^{-1}\rho_0, c_0) - \tilde{P}\} + \rho_0 \partial_\rho P(J_v^{-1}\rho_0, c_0) \operatorname{Adj}(D\mathcal{X}_v)\partial_t J_v^{-1}
$$

and thus one obtains from (4.5) and (4.7) the following:

$$
(4.22) \qquad \| \partial_t I_3(v) \|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \| \nabla v \|_{L_T^2 L^2(\mathbb{R}^d)} ( \| \nabla v \|_{L_T^1 L^\infty(\mathbb{R}^d)} + 1 ) .
$$

On the other hand, the following estimate holds true for $\partial_t I_3(v)$:

$$
\| \sqrt{\sigma} \partial_t I_3(v) \|_{L_T^{8/d} L^4(\mathbb{R}^d)} \lesssim \| \nabla v \|_{L_T^{8/d} L^4(\mathbb{R}^d)} ( \| \nabla v \|_{L_T^1 L^\infty(\mathbb{R}^d)} + 1 ) .
$$
In the following lines, we will prove that $(u_n)_n$ converges strongly in $L^\infty((0,T), L^2(\mathbb{R}^d)) \cap L^2((0,T), H^1(\mathbb{R}^d))$ to some $u \in X_{T,u_0}$ which satisfies (3.60).

To achieve this, we take the differences in (3.74) and obtain:

$$ \begin{cases} \rho_0 \partial_t w_{n+1} - \operatorname{div} (2\mu^{\theta_0} \mathbb{D}(w_{n+1}) + \lambda^{\theta_0} \operatorname{div}(w_{n+1}) I_d) = (\theta - \theta_0) \operatorname{div} (2(\mu - \tilde{\mu})\mathbb{D}(w_n) + (\lambda - \tilde{\lambda}) \operatorname{div}(w_n) I_d), \\ w_{n+1}|_{t=0} = 0, \end{cases} $$

where $w_{k+1} = u_{k+1} - u_k$, with $k \ge 1$. By multiplying the equation above by $w_{n+1}$, integrating in space, and applying Korn's inequality (see (3.13)), along with Young's and Hölder's inequalities, we obtain:

$$ \frac{1}{2} \frac{d}{dt} \|\sqrt{\rho_0} w_{n+1}\|_{L^2(\mathbb{R}^d)}^2 + \frac{\mu_*}{2} \|\nabla w_{n+1}\|_{L^2(\mathbb{R}^d)}^2 \le C |\theta - \theta_0|^2 \|\nabla w_n\|_{L^2(\mathbb{R}^d)}^2. $$

Time integration, along with the condition $w_{n+1}|_{t=0} = 0$ yields:

$$ \sup_{[0,T]} \|w_{n+1}\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|\nabla w_{n+1}\|_{L^2(\mathbb{R}^d)}^2 \le C |\theta - \theta_0|^2 \left( \sup_{[0,T]} \|w_n\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|\nabla w_n\|_{L^2(\mathbb{R}^d)}^2 \right), $$

and therefore:

$$ \begin{aligned} \sup_{[0,T]} \|u_{n+1} - u_n\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|\nabla(u_{n+1} - u_n)\|_{L^2(\mathbb{R}^d)}^2 \le & C^n |\theta - \theta_0|^{2n} \left( \sup_{[0,T]} \|w_1\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|\nabla w_1\|_{L^2(\mathbb{R}^d)}^2 \right) \\ \le & 2C^n |\theta - \theta_0|^{2n} R_T^2. \end{aligned} $$

By choosing

$$ (3.76) \qquad |\theta - \theta_0|^2 < \frac{1}{2C}, $$

we end up with the strong convergence of $(u_n)_n$ to a limit $u$ in $L^\infty((0,T), L^2(\mathbb{R}^d)) \cap L^2((0,T), H^1(\mathbb{R}^d))$. It is a simple matter to check from (3.74) that $u$ is a weak solution of

$$ (3.77) \qquad \begin{cases} \rho_0 \partial_t u - \operatorname{div}(2\mu^{\theta_0} \mathbb{D}u + \lambda^{\theta_0} \operatorname{div} u I_d) = \operatorname{div} F + (\theta - \theta_0) \operatorname{div} (2(\mu - \tilde{\mu})\mathbb{D}u + (\lambda - \tilde{\lambda}) \operatorname{div} u I_d), \\ u|_{t=0} = u_0, \end{cases} $$

and consequently, of (3.50). Then, by virtue of lower semi-continuity of norms, we deduce from the uniform bound (3.75):

$$ \|w\|_{X_T} \le R_T. $$

Thus $u$ has all the regularity required in (3.4), and it only remains to establish time continuity:

$$ u \in \mathcal{C}([0, T], H^1(\mathbb{R}^d)), \text{ and } \partial_t u \in \mathcal{C}([0, T], L^2(\mathbb{R}^d)). $$

The first one follows from the fact that $\partial_t u \in L^2((0,T), H^1(\mathbb{R}^d))$. Moreover, since $\sqrt{\sigma}u_{tt} \in L^2((0,T) \times \mathbb{R}^d)$, it transpires that $u_t \in \mathcal{C}((0,T], L^2(\mathbb{R}^d))$. Therefore, it only remains to prove the continuity of $u_t$ at the initial time. To achieve this, we observe that the sequence $(u_n)_n$ satisfies the following estimate (see (3.17)):

$$ \begin{aligned} \frac{1}{2} \int_{\mathbb{R}^d} \rho_0 |\partial_t u_n(\tau)|^2 &\le \frac{1}{2} \int_{\mathbb{R}^d} \rho_0 |\partial_t u_n|_{t=0}^2 |\tau|^2 + C \left( \int_0^\tau \|F_t\|_{L^2(\mathbb{R}^d)}^2 + \int_0^\tau \|\mu_t^\theta, \lambda_t^\theta\|_{L^4(\mathbb{R}^d)}^2 \|\nabla u_n\|_{L^4(\mathbb{R}^d)}^2 \right) \\ &\le \frac{1}{2} \int_{\mathbb{R}^d} \rho_0 |\partial_t u_n|_{t=0}^2 |\tau|^2 + \tau C R_T^2 + C \int_0^\tau \|F_t\|_{L^2(\mathbb{R}^d)}^2. \end{aligned} $$

For all $n \ge 1$, since $u_{n+1}|_{t=0} = u_n|_{t=0} = u_0$, it follows from (3.77)-(3.63) that:

$$ \rho_0 \partial_t u_{n+1}|_{t=0} = \operatorname{div} (2\mu_0^\theta \mathbb{D}u_0 + \lambda_0^\theta \operatorname{div} u_0 + F|_{t=0}) = \rho_0 u_{t|t=0}. $$

Next, using the compact embedding $H^1(\mathbb{R}^d)$ into $L^2_{\text{loc}}(\mathbb{R}^d)$ and Aubins-Lions theorem, we obtain $w_t^n(\tau) \to u_t(\tau)$ a.e in $\mathbb{R}^d$. Therefore, Fatou's Lemma yields:

$$ \frac{1}{2} \int_{\mathbb{R}^d} \rho_0 |u_t(\tau)|^2 \le \frac{1}{2} \int_{\mathbb{R}^d} \rho_0 |u_{t|t=0}|^2 + \tau C R_T^2 + C \int_0^\tau \|F_t\|_{L^2(\mathbb{R}^d)}^2 $$

from which we deduce

$$ \limsup_{\tau \to 0} \|\sqrt{\rho_0} u_t(\tau)\|_{L^2(\mathbb{R}^d)} \le \|\sqrt{\rho_0} u_{t|t=0}\|_{L^2(\mathbb{R}^d)}. $$

This yields the continuity of $u_t$ at the initial time, and thereby completing the proof that $u \in X_{T,u_0}$.
We turn to the estimate of the norms of $\partial_t I_1(v, v)$ and $\partial_t I_2(v, v)$. First, we begin by writing:

$$ (4.23) \qquad \begin{aligned} \partial_t I_1(v, v) ={}& 2\rho_0\partial_\rho\mu(J_v^{-1}\rho_0, c_0)\partial_t J_v^{-1}\{\operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v - \mathbb{D}v\} \\ & + 2\mu(J_v^{-1}\rho_0, c_0)\{\operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v_t - \mathbb{D}v_t\} \\ & + 2\mu(J_v^{-1}\rho_0, c_0)\partial_t \operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v \\ & + 2\mu(J_v^{-1}\rho_0, c_0)\operatorname{Adj}(D\mathcal{X}_v)(Dv \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v). \end{aligned} $$

Using Lemma 4.1 and the expression of the time derivative of $J_v^{-1}$ (4.5), the following bounds hold for the first term in the right hand side above:

$$ \|2\rho_0\partial_\rho\mu(J_v^{-1}\rho_0, c_0)\partial_t J_v^{-1}\{\operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v - \mathbb{D}v\}\|_{L^2_T L^2(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L^1_T L^\infty(\mathbb{R}^d)} \|\nabla v\|_{L^4_T L^4(\mathbb{R}^d)}^2 $$

and

$$ \|\sqrt{\sigma}\rho_0\partial_\rho\mu(J_v^{-1}\rho_0, c_0)\partial_t J_v^{-1}\{\operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v - \mathbb{D}v\}\|_{L^{s/d}_T L^4(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L^1_T L^\infty(\mathbb{R}^d)} \|\nabla v\|_{L^{16/d}_T L^8(\mathbb{R}^d)}^2. $$

The same argument helps us to control the second term of the right hand side of (4.23) as follows:

$$ \|2\mu(J_v^{-1}\rho_0, c_0)\{\operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v_t - \mathbb{D}v_t\}\|_{L^2_T L^2(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L^1_T L^\infty(\mathbb{R}^d)} \|\nabla v_t\|_{L^2_T L^2(\mathbb{R}^d)} $$

and

$$ \|\sqrt{\sigma}\mu(J_v^{-1}\rho_0, c_0)\{\operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v_t - \mathbb{D}v_t\}\|_{L^{s/d}_T L^4(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L^1_T L^\infty(\mathbb{R}^d)} \|\sqrt{\sigma}\nabla v_t\|_{L^{s/d}_T L^4(\mathbb{R}^d)}. $$

Next, from the expression of the time derivative of $A_v$, (4.6), one has the following:

$$ Dv \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v = -(Dv \cdot A_v)^2 + (A_v^T \cdot \nabla v)^2 $$

thus, the others terms of (4.23) are controlled as follows:

$$ \|\mu(J_v^{-1}\rho_0, c_0)\partial_t \operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v; \mu(J_v^{-1}\rho_0, c_0)\operatorname{Adj}(D\mathcal{X}_v)\{Dv \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v\}\|_{L^2_T L^2(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L^4_T L^4(\mathbb{R}^d)}^2 $$

and

$$ \|\sqrt{\sigma}\mu(J_v^{-1}\rho_0, c_0)\partial_t \operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v; \sqrt{\sigma}\mu(J_v^{-1}\rho_0, c_0)\operatorname{Adj}(D\mathcal{X}_v)\{Dv \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v\}\|_{L^{s/d}_T L^4(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L^{16/d}_T L^8(\mathbb{R}^d)}^2. $$

Gathering all of these computations and noticing that the estimate for $\partial_t I_1(v, v)$ and $\sqrt{\sigma}I_1(v, v)$ also holds for $\partial_t I_2(v, v)$ and $\sqrt{\sigma}I_2(v, v)$ respectively, one has:

$$ \|\partial_t I(v, v)\|_{L^2_T L^2(\mathbb{R}^d)} \lesssim \left( \|\nabla v\|_{L^2_T L^2(\mathbb{R}^d)} + \|\nabla v\|_{L^4_T L^4(\mathbb{R}^d)}^2 \right) \left( \|\nabla v\|_{L^1_T L^\infty(\mathbb{R}^d)} + 1 \right) + \|\nabla v\|_{L^1_T L^\infty(\mathbb{R}^d)} \|\nabla v_t\|_{L^2_T L^2(\mathbb{R}^d)} $$

and

$$ \|\sqrt{\sigma}\partial_t I(v, v)\|_{L^{8/d}_T L^4(\mathbb{R}^d)} \lesssim \left( \|\nabla v\|_{L^{8/d}_T L^4(\mathbb{R}^d)} + \|\nabla v\|_{L^{16/d}_T L^8(\mathbb{R}^d)}^2 \right) \left( \|\nabla v\|_{L^1_T L^\infty(\mathbb{R}^d)} + 1 \right) + \|\nabla v\|_{L^{16/d}_T L^8(\mathbb{R}^d)} \|\sqrt{\sigma}\nabla v_t\|_{L^{8/d}_T L^4(\mathbb{R}^d)}. $$

Step 3: $L_T^2 L^2(\mathbb{R}^d)$ norm estimate for $\sigma\partial_{tt}I(v, v)$. As above, we may begin by computing the second time derivative of $I(v, v)$. First, from (4.21), the following expression for $\partial_{tt}I_3(v)$ holds:

$$ \partial_{tt}I_3(v, v) = \partial_{tt}I_{3,1} + \partial_{tt}I_{3,2} $$

where

$$ (4.24) \qquad \partial_{tt}I_{3,1} = \partial_{tt}\operatorname{Adj}(D\mathcal{X}_v)\{P(J_v^{-1}\rho_0, c_0) - \tilde{P}\} + \rho_0\partial_\rho P(J_v^{-1}\rho_0, c_0)\partial_t J_v^{-1}\partial_t\operatorname{Adj}(D\mathcal{X}_v) $$

and

$$ (4.25) \qquad \begin{aligned} \partial_{tt}I_{3,2} ={}& \rho_0^2\partial_\rho^2 P(J_v^{-1}\rho_0, c_0)(\partial_t J_v^{-1})^2 \operatorname{Adj}(D\mathcal{X}_v) + \rho_0\partial_\rho P(J_v^{-1}\rho_0, c_0)\partial_t J_v^{-1}\partial_t \operatorname{Adj}(D\mathcal{X}_v) \\ & + \rho_0\partial_\rho P(J_v^{-1}\rho_0, c_0)\partial_{tt}J_v^{-1}\operatorname{Adj}(D\mathcal{X}_v). \end{aligned} $$

Using the expression of the time derivative of $J_v^{-1}$, (4.5) and of $\operatorname{Adj}(D\mathcal{X}_v)$, (4.7), the last term of (4.24) and the first two terms of the right hand side of (4.25) can be estimated as follows:

$$ \|\sigma\rho_0\partial_\rho P(J_v^{-1}\rho_0, c_0)\partial_t J_v^{-1}\partial_t \operatorname{Adj}(D\mathcal{X}_v), \sigma\rho_0^2\partial_\rho^2 P(J_v^{-1}\rho_0, c_0)(\partial_t J_v^{-1})^2 \operatorname{Adj}(D\mathcal{X}_v)\|_{L^{\frac{8}{d}}T L^2(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L^4_T L^4(\mathbb{R}^d)}^2. $$

Owing to the expression of the second time derivative of $J_v^{-1}$ (4.8) and $\operatorname{Adj}(D\mathcal{X}_v)$ (4.10), the remainder terms of $\partial_{tt}I_3(v)$ are estimated as follows:

$$ \|\sigma\partial_{tt}\operatorname{Adj}(D\mathcal{X}_v)\{P(J_v^{-1}\rho_0, c_0) - \tilde{P}\}, \sigma\rho_0\partial_\rho P(J_v^{-1}\rho_0, c_0)\partial_{tt}J_v^{-1}\operatorname{Adj}(D\mathcal{X}_v)\|_{L^2_T L^2(\mathbb{R}^d)} \lesssim \|\sigma\nabla v_t\|_{L^2_T L^2(\mathbb{R}^d)} + \|\nabla v\|_{L^4_T L^4(\mathbb{R}^d)}^2. $$
Given that the constant $C$ appearing in (3.73) and (3.76) does not depend on $u_0$ nor $F$, it makes the condition (3.59) true, for all $(F, u_0) \in Y_T^\theta$ and one deduces that $\mathcal{E}$ is an open set of $[0, 1]$.

We turn to the proof of the fact that $\mathcal{E}$ is a closed set of $[0, 1]$. Let $(\theta_n)_n$ a sequence of element of $\mathcal{E}$ that converges to some $\theta_0 \in [0, 1]$. Our aim is to prove that $\theta_0 \in \mathcal{E}$. In other words, the following Cauchy problem

$$ (3.78) \qquad \begin{cases} \rho_0 u_t - \operatorname{div}(2\mu^{\theta_n} \mathbb{D}u + \lambda^{\theta_n} \operatorname{div} u I_d) = \operatorname{div} F + (\theta_0 - \theta_n) \operatorname{div}\{2(\mu - \tilde{\mu})\mathbb{D}u + (\lambda - \tilde{\lambda}) \operatorname{div} u I_d\}, \\ u|_{t=0} = u_0 \end{cases} $$

admits a unique solution in $X_T$ for all $(F, u_0) \in Y_T^{\theta_0}$. By following the computations in Step 1-Step 4, and given that the norms of $\mu^\theta$ and $\lambda^\theta$ are uniformly bounded with respect to $\theta$, we can choose a sufficiently large $n$ independently of $F, u_0$. Hereby, $\mathcal{E}$ is a closed set of $[0, 1]$.

In conclusion, $\mathcal{E}$ is at same time a non-empty, open and closed set of $[0, 1]$, since $[0, 1]$ is a connected set, $\mathcal{E} = [0, 1]$, $1 \in \mathcal{E}$ and Theorem 3.1 finally holds. $\square$

## 4. STUDY OF THE FULL NONLINEAR SYSTEM (2.8)

### 4.1. Preliminary.
This section is devoted to the solvability of the full nonlinear two-fluid model in Lagrangian coordinates (2.8). The initial data $(c_0, \rho_0, u_0)$ is such that the compatibility condition holds true, that is:

$$ (4.1) \qquad \operatorname{div}(2\mu(\rho_0, c_0)\mathbb{D}u_0 + (\lambda(\rho_0, c_0)\operatorname{div} u_0 - P(\rho_0, c_0))I_d) \in L^2(\mathbb{R}^d). $$

Our method relies on a fixed-point argument, thus we will begin by setting some tools. We recall that such solution $u$ verifies (for simplicity, we write $u$ instead of $\bar{u}$):

$$ (4.2) \qquad \rho_0 u_t - \operatorname{div}\{2\mu(J_u^{-1}\rho_0, c_0)\mathbb{D}u + \lambda(J_u^{-1}\rho_0, c_0)\} \operatorname{div} u I_d = \operatorname{div} I(u, u) $$

where

$$ I(u, u) = I_1(u, u) + I_2(u, u) + I_3(u) $$

and

$$ (4.3) \qquad \begin{cases} I_1(v, w) = 2\mu(J_v^{-1}\rho_0, c_0)\{\operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}w - \mathbb{D}w\}, \\ I_2(v, w) = \lambda(J_v^{-1}\rho_0, c_0)\{\operatorname{Adj}(D\mathcal{X}_v)\operatorname{div}_{A_v}w - \operatorname{div} w\}, \\ I_3(v) = \operatorname{Adj}(D\mathcal{X}_v)\{P(J_v^{-1}\rho_0, c_0) - \tilde{P}\}I_d. \end{cases} $$

According to the previous section, we will construct $u$ in a closed subset of the vector space $X_T$ as defined in (3.4) for some small time $T$. For some $v \in X_T$ with $v|_{t=0} = u_0$, we consider the linear Cauchy problem:

$$ (4.4) \qquad \begin{cases} \rho_0 u_t - \operatorname{div}\{2\mu(J_v^{-1}\rho_0, c_0)\mathbb{D}u + \lambda(J_v^{-1}\rho_0, c_0)\} \operatorname{div} u I_d = \operatorname{div} I(v, v), \\ u|_{t=0} = u_0. \end{cases} $$

The validity of the last condition in the definition of $Y_T := Y_T(\mu(\rho_0, c_0), \lambda(\rho_0, c_0))$ (see (3.3)) results from the fact that $v|_{t=0} = u_0$, $(J_v^{-1})|_{t=0} = 1$, $I(v, v)|_{t=0} = (P(\rho_0, c_0) - \tilde{P})I_d$ and (4.1). In the following section, we set up tools for the construction of the solution to (4.2).

### 4.2. Implementation of the fixed-point theorem.
In this section, we will justify the well-posedness of a contracting map whose fixed-point is the solution of the nonlinear system (4.2). To this end, we fix some $v$ in $X_{T,u_0} = \{u \in X_T: u|_{t=0} = u_0\}$. The first part of the following section will consist in proving that the regularity of $v \in X_T$ is sufficient to ensure that $I(v, v)$ belongs to $Y_T$. From there, we will be able to deduce the existence of a solution for the Cauchy problem (4.4) using Theorem 3.1. This leads to the following result.

**Proposition 4.1.** There are $T^* > 0$ and $R > 0$ such that for all $T \in (0, T_*)$ the map:

$$ (4.5) \qquad \begin{gathered} \Phi_T: X_{T,u_0}(R) \to X_{T,u_0}(R) \\ v \mapsto \Phi_T(v) := u, \end{gathered} $$

where $u$ is the unique solution of the Cauchy problem (4.4), and $X_{T,u_0}(R) := \{u \in X_{T,u_0}: \|u\|_{X_T} \le R\}$, is well-defined.

The final step of this section is dedicated to proving that for some small $T \in (0, T_*)$, the map $\Phi_T$ admits a fixed point, see Proposition 4.2 below.
Gathering all of these estimates, one has:

$$ (4.26) \qquad \|\sigma \partial_{tt} I_3(v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\sigma \nabla v_t\|_{L_T^2 L^2(\mathbb{R}^d)} + \|\nabla v\|_{L_T^4 L^4(\mathbb{R}^d)}^2, $$

and we turn to the $L_T^2 L^2(\mathbb{R}^d)$ norm estimate of $\sigma \partial_{tt} I_1(v, v)$. From (4.23), one can express:

$$ \partial_{tt} I_1(v, v) = \sum_{k=0}^{4} \partial_{tt} I_{1,k}(v, v) $$

where:

$$ \begin{aligned} \partial_{tt} I_{1,1}(v, v) ={}& 2(\rho_0)^2 \partial_{\rho}^2 \mu(J_v^{-1} \rho_0, c_0) (\partial_t J_v^{-1})^2 \{\operatorname{Adj}(D\mathcal{X}_v) \mathbb{D}_{A_v} v - \mathbb{D}v\} \\ & + 2\rho_0 \partial_{\rho} \mu(J_v^{-1} \rho_0, c_0) \partial_{tt} J_v^{-1} \{\operatorname{Adj}(D\mathcal{X}_v) \mathbb{D}_{A_v} v - \mathbb{D}v\} \\ & + 2\rho_0 \partial_{\rho} \mu(J_v^{-1} \rho_0, c_0) \partial_t J_v^{-1} \{\operatorname{Adj}(D\mathcal{X}_v) \mathbb{D}_{A_v} v_t - \mathbb{D}v_t\} \\ & + 2\rho_0 \partial_{\rho} \mu(J_v^{-1} \rho_0, c_0) \partial_t J_v^{-1} \partial_t \operatorname{Adj}(D\mathcal{X}_v) \mathbb{D}_{A_v} v \\ & + 2\rho_0 \partial_{\rho} \mu(J_v^{-1} \rho_0, c_0) \partial_t J_v^{-1} \operatorname{Adj}(D\mathcal{X}_v) \{\mathbb{D}v \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v\}, \end{aligned} $$

$$ \begin{aligned} \partial_{tt} I_{1,2}(v, v) ={}& 2\rho_0 \partial_{\rho} \mu(J_v^{-1} \rho_0, c_0) \partial_t J_v^{-1} \{\operatorname{Adj}(D\mathcal{X}_v) \mathbb{D}_{A_v} v_t - \mathbb{D}v_t\} \\ & + 2\mu(J_v^{-1} \rho_0, c_0) \{\operatorname{Adj}(D\mathcal{X}_v) \mathbb{D}_{A_v} v_{tt} - \mathbb{D}v_{tt}\} \\ & + 2\mu(J_v^{-1} \rho_0, c_0) \partial_t \operatorname{Adj}(D\mathcal{X}_v) \mathbb{D}_{A_v} v_t \\ & + 2\mu(J_v^{-1} \rho_0, c_0) \operatorname{Adj}(D\mathcal{X}_v) \{\mathbb{D}v_t \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v_t\}, \end{aligned} $$

$$ \begin{aligned} \partial_{tt} I_{1,3}(v, v) ={}& 2\rho_0 \partial_{\rho} \mu(J_v^{-1} \rho_0, c_0) \partial_t J_v^{-1} \partial_t \operatorname{Adj}(D\mathcal{X}_v) \mathbb{D}_{A_v} v + 2\mu(J_v^{-1} \rho_0, c_0) \partial_{tt} \operatorname{Adj}(D\mathcal{X}_v) \mathbb{D}_{A_v} v \\ & + 2\mu(J_v^{-1} \rho_0, c_0) \partial_t \operatorname{Adj}(D\mathcal{X}_v) \{\mathbb{D}_{A_v} v_t + \mathbb{D}v \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v\} \end{aligned} $$

and finally,

$$ \begin{aligned} \partial_{tt} I_{1,4}(v, v) ={}& 2\rho_0 \partial_{\rho} \mu(J_v^{-1} \rho_0, c_0) \partial_t J_v^{-1} \operatorname{Adj}(D\mathcal{X}_v) (\mathbb{D}v \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v) \\ & + 2\mu(J_v^{-1} \rho_0, c_0) \partial_t \operatorname{Adj}(D\mathcal{X}_v) (\mathbb{D}v \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v) \\ & + 2\mu(J_v^{-1} \rho_0, c_0) \operatorname{Adj}(D\mathcal{X}_v) (\mathbb{D}v_t \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v_t + \mathbb{D}v \cdot \partial_{tt} A_v + \partial_{tt} A_v^T \cdot \nabla v). \end{aligned} $$

Combining (4.5), (4.7), (4.6), (4.8) and (4.10) one has:

$$ \begin{aligned} \|\sigma \partial_{tt} I_1(v, v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \left( \|\nabla v\|_{L_T^6 L^6(\mathbb{R}^d)}^3 + \|\sqrt{\sigma} \nabla v_t\|_{L_T^{s/d} L^4(\mathbb{R}^d)} \|\nabla v\|_{L_T^{s/(4-d)} L^4(\mathbb{R}^d)} + \|\sigma \nabla v_{tt}\|_{L_T^2 L^2(\mathbb{R}^d)} \right) \\ + \|\sqrt{\sigma} \nabla v_t\|_{L_T^{s/d} L^4(\mathbb{R}^d)} \|\nabla v\|_{L_T^{s/(4-d)} L^4(\mathbb{R}^d)} + \|\nabla v\|_{L_T^6 L^6(\mathbb{R}^d)}^3. \end{aligned} $$

Finally, the same estimate holds true for the $L_T^2 L^2(\mathbb{R}^d)$ norm of $\sigma \partial_{tt} I_2(v, v)$ and gathering (4.26) one has the $L_T^2 L^2(\mathbb{R}^d)$ norm estimate for $\sigma \partial_{tt} I(v, v)$. We turn to the estimation of the norms of the viscosity.

Step 4: Lebesgue and Hölder norm estimate for viscosity. This step makes more sense to the estimate of the lower, upper bound and the Hölder regularity of the viscosity $\mu$ and $\lambda$. With the help of Lemma 4.1, one has the following estimate for $\mu$

$$ \|\mu(J_v^{-1} \rho_0, c_0) - \tilde{\mu}\|_{L_T^\infty L^\infty(\mathbb{R}^d)} \le \|\mu(\rho_0, c_0) - \tilde{\mu}\|_{L^\infty(\mathbb{R}^d)} + \|\partial_\rho \mu\|_{L^\infty} \|\nabla v\|_{L_T^1 L^\infty(\mathbb{R}^d)} $$

and the same holds for $\lambda$. Also, one has the following Hölder estimate:

$$ \|\mu(J_v^{-1} \rho_0, c_0)\|_{L_T^\infty \mathcal{C}_{\rho w, \gamma}^{\alpha}(\mathbb{R}^d)} \le \|\mu(\rho_0, c_0)\|_{\mathcal{C}_{\rho w, \gamma}^{\alpha}(\mathbb{R}^d)} + \|\partial_{\rho} \mu\|_{L^\infty} \|\nabla v\|_{L_T^1 \mathcal{C}_{\rho w, \gamma}^{\alpha}(\mathbb{R}^d)}. $$

The same estimate also holds for $\lambda$. We now turn to the estimate for the time derivative of the viscosity.

Step 5: Estimates for time derivative of the viscosity. In this section we will estimate the norm of the time derivative of $\mu(J_v^{-1} \rho_0, c_0)$ and $\lambda(J_v^{-1} \rho_0, c_0)$ as required in (3.2). First, one can express:

$$ (4.27) \qquad \partial_t \mu(J_v^{-1} \rho_0, c_0) = -\rho_0 \partial_\rho \mu(J_v^{-1} \rho_0, c_0) J_v^{-1} \operatorname{div}_{A_v}(v), $$

and thus, it transpires that:

$$ \|\partial_t \mu(J_v^{-1} \rho_0, c_0)\|_{L_T^m L^q(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L_T^m L^q(\mathbb{R}^d)}, \quad \|\sqrt{\sigma} \partial_t \mu(J_v^{-1} \rho_0, c_0)\|_{L_T^2 L^\infty(\mathbb{R}^d)} \lesssim \|\sqrt{\sigma} \nabla v\|_{L_T^2 L^\infty(\mathbb{R}^d)}. $$

A similar estimates hold true for the time derivative of $\lambda(J_v^{-1} \rho_0, c_0)$. We turn to the estimates of the norm of $\partial_{tt} \mu$ and $\partial_{tt} \lambda$ as required in (3.2).
Proof of Proposition 4.1. To ease readability, for $T > 0$, and a normed space $\mathcal{X}$, the space $L^p((0, T), \mathcal{X})$ will be denoted $L_T^p\mathcal{X}$. Also, we will only present the main steps, technical computations are carried out in Section 4.3 below.

First of all, let us point out that since $\sigma^{r/4}\nabla v \in L_T^4 \mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)$, with $0 < r < 8/3$, one has:

$$c = \int_0^T \|\nabla v\|_{\mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} \le T^{\frac{3-r}{4}} \|v\|_{X_T}.$$

Suppose that $\|v\|_{X_T} \le R$ for some $R > 0$ to be fixed later. One chooses $T \le 1$ small in such a way that $2c < 1$. Thus, the constants involved in Lemma 4.4 and Lemma 4.5 below are assumed to be fixed. Furthermore, the constant in the notation $\lesssim$ in all subsequent steps, may depend on $c < 1/2$, the bounds of $\rho_0, \mu(\rho_0, c_0), \lambda(\rho_0, c_0)$, the piecewise Hölder norm of $\mu(\rho_0, c_0)$, $\lambda(\rho_0, c_0)$ and the interface regularity.

The following bounds for $I(v, v)$ will be proved in Section 4.3.1 (we freely use $T \le 1$):

$$(4.6) \quad \sup_{[0,T]} \|I(v,v)\|_{L^2(\mathbb{R}^d)} + \|I(v,v)\|_{L_T^\infty L^6(\mathbb{R}^d)} \lesssim \|P(\rho_0, c_0) - \tilde{P}\|_{L^2 \cap L^6(\mathbb{R}^d)} + T^{\frac{3-r}{4}} R(1+R),$$

$$(4.7) \quad \|I(v,v)\|_{L_T^{16} L^8(\mathbb{R}^d)} \lesssim T^{1/16} \|P(\rho_0, c_0) - \tilde{P}\|_{L^8(\mathbb{R}^d)} + T^{\frac{3-r}{4}} R(1+R),$$

$$(4.8) \quad \|\sigma^{r/4} I(v,v)\|_{L_T^4 \mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} \lesssim T^{1/4} \|P(\rho_0, c_0) - \tilde{P}\|_{\mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} + T^{\frac{3-r}{4}} R(1+R).$$

On the other hand, the following estimates for $\partial_t I(v, v)$ will be proved in Section 4.3.2 (we freely use $T^{\frac{3-r}{4}} R \le 1$):

$$(4.9) \quad \|\partial_t I(v,v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim T^{1/2} R(R+1) + T^{\frac{3-r}{4}} R^2,$$

$$(4.10) \quad \|\sigma^{\frac{d}{4}} \partial_t I(v,v)\|_{L_T^{s/d} L^4(\mathbb{R}^d)} \lesssim T^{\frac{d}{8}} R(R+1) + T^{\frac{3-r}{4}} R^2,$$

and finally, we will derive in Section 4.3.3 the following estimates for $\partial_{tt} I(v, v)$:

$$(4.11) \quad \|\sigma \partial_{tt} I(v,v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim T^{1/2} R(R+1) + T^{\frac{1-d}{8}} R^2 (1+T^{\frac{d}{8}} R) + T^{\frac{3-r}{4}} R^2.$$

We obtain that $I(v, v)$ satisfies all conditions required in the definition of $Y_T$. Now, we shall ensure that the viscosity $\mu(J_v^{-1}\rho_0, c_0)$ and $\lambda(J_v^{-1}\rho_0, c_0)$ have the regularities required in (3.2), and the dynamic viscosity $\mu(J_v^{-1}\rho_0, c_0)$ remains close to $\tilde{\mu}$. We prove in Section 4.3.4:

$$(4.12) \quad \left\{ \begin{array}{l} \|\mu(J_v^{-1}\rho_0, c_0) - \tilde{\mu}\|_{L_T^\infty \mathcal{C}_{pw,c_0}^\alpha(\mathbb{R}^d)} \le \|\mu(\rho_0, c_0) - \tilde{\mu}\|_{\mathcal{C}_{pw,c_0}^\alpha(\mathbb{R}^d)} + \|\partial_\rho \mu\|_{L^\infty} T^{\frac{3-r}{4}} R, \\ \|\llbracket \mu(J_v^{-1}\rho_0, c_0) \rrbracket\|_{L_T^\infty L^\infty(c_0)} \le \|\llbracket \mu(\rho_0, c_0) \rrbracket\|_{L^\infty(c_0)} + \|\partial_\rho \mu\|_{L^\infty} T^{\frac{3-r}{4}} R, \end{array} \right.$$

with similar bounds for $\lambda(J_v^{-1}\rho_0, c_0)$. Assuming (1.14)-(1.15), we choose $T$ small such that

$$\sup_{[0,T]} \left[ 1 + \|\lambda(J_v^{-1}\rho_0, c_0)\|_{\dot{\mathcal{C}}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} + (\mathfrak{P}_{c_0} + \ell_{\varphi_0}^{-\alpha}) \|\lambda(J_v^{-1}\rho_0, c_0)\|_{L^\infty(c_0)} \right] \sup_{[0,T]} \left\| \mu(J_v^{-1}\rho_0, c_0) - \tilde{\mu} \right\|_{\mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} \\ + (\mathfrak{P}_{c_0} + \ell_{\varphi_0}^{-\alpha}) \sup_{[0,T]} \left\| \|\mu(J_v^{-1}\rho_0, c_0)\|_{L^\infty(c_0)} \right\| \\ + \sup_{[0,T]} \left[ \|\|\mu(J_v^{-1}\rho_0, c_0)\|\|_{L^\infty(c_0)} \left\| 1 - \frac{\tilde{\mu}}{\langle \mu(J_v^{-1}\rho_0, c_0) \rangle} \right\|_{L^\infty(c_0)} \right] \le \frac{[\mu]}{2},$$

and

$$\mu_{*,0}/4 \le \mu(J_v^{-1}\rho_0, c_0) \quad \text{and} \quad \lambda(J_v^{-1}\rho_0, c_0), \mu(J_v^{-1}\rho_0, c_0) \le 4 \max \left\{ \sup_{\mathbb{R}^d} \mu(\rho_0, c_0), \sup_{\mathbb{R}^d} \lambda(\rho_0, c_0) \right\}.$$

We prove in Section 4.3.5 below, the following estimates for the time derivative of the viscosity:

$$(4.13) \quad \|\partial_t \mu(J_v^{-1}\rho_0, c_0), \partial_t \lambda(J_v^{-1}\rho_0, c_0)\|_{L_T^\infty L^4(\mathbb{R}^d)} \lesssim R, \quad \|\partial_t \mu(J_v^{-1}\rho_0, c_0), \partial_t \lambda(J_v^{-1}\rho_0, c_0)\|_{L_T^{16} L^8(\mathbb{R}^d)} \lesssim R,$$

and

$$(4.14) \quad \|\partial_t \mu(J_v^{-1}\rho_0, c_0), \partial_t \lambda(J_v^{-1}\rho_0, c_0)\|_{L_T^1 L^\infty(\mathbb{R}^d)} + \|\sqrt{\sigma} \partial_t \mu(J_v^{-1}\rho_0, c_0), \sqrt{\sigma} \partial_t \lambda(J_v^{-1}\rho_0, c_0)\|_{L_T^2 L^\infty(\mathbb{R}^d)} \lesssim T^{\frac{3-r}{4}} R.$$

Finally, the $L_T^2 L^4(\mathbb{R}^d)$-norm of the second time derivative of the viscosity is obtained in Section 4.3.6, from which we deduce:

$$(4.15) \quad \|\sigma \partial_{tt} \mu(J_v^{-1}\rho_0, c_0), \sigma \partial_{tt} \lambda(J_v^{-1}\rho_0, c_0)\|_{L_T^2 L^4(\mathbb{R}^d)} \lesssim T^{\frac{1-d}{8}} R(1+R).$$
Step 6: Estimates for the second time derivative of the viscosity. This step is devoted to the $L_T^1 L^2(\mathbb{R}^d)$ norm estimate for $\partial_{tt}\mu(J_v^{-1}\rho_0, c_0)$, $\partial_{tt}\lambda(J_v^{-1}\rho_0, c_0)$ and the $L_T^{8/d} L^4(\mathbb{R}^d)$ norm estimate for $\sqrt{\sigma}\partial_{tt}\mu(J_v^{-1}\rho_0, c_0)$, $\sqrt{\sigma}\partial_{tt}\lambda(J_v^{-1}\rho_0, c_0)$. It is obvious from (4.27), that the second time derivative of $\mu(J_v^{-1}\rho_0, c_0)$ reads:

$$ \begin{aligned} \partial_{tt}\mu(J_v^{-1}\rho_0, c_0) ={}& \rho_0^2\partial_{\rho}^2\mu(J_v^{-1}\rho_0, c_0)(J_v^{-1}\operatorname{div}_{A_v}(v))^2 + \rho_0\partial_{\rho}\mu(J_v^{-1}\rho_0, c_0)J_v^{-1}(\operatorname{div}_{A_v}(v))^2 \\ & - \rho_0\partial_{\rho}\mu(J_v^{-1}\rho_0, c_0)J_v^{-1}\operatorname{div}_{A_v}v_t - \rho_0\partial_{\rho}\mu(J_v^{-1}\rho_0, c_0)J_v^{-1}\partial_t A_v^T : \nabla v, \end{aligned} $$

hence, the following estimates hold true:

$$ \|\partial_{tt}\mu(J_v^{-1}\rho_0, c_0)\|_{L_T^{1/2}L^2(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L_T^2L^4(\mathbb{R}^d)}^2 + \|\nabla v_t\|_{L_T^{1/2}L^2(\mathbb{R}^d)} $$

and

$$ \|\sqrt{\sigma}\partial_{tt}\mu(J_v^{-1}\rho_0, c_0)\|_{L_T^{8/d}L^4(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L_T^{16/d}L^8(\mathbb{R}^d)}^2 + \|\sqrt{\sigma}\nabla v_t\|_{L_T^{8/d}L^4(\mathbb{R}^d)} $$

A similar estimates hold true for the second time derivative of $\lambda(J_v^{-1}\rho_0, c_0)$.

Step 7: Lebesgue and Hölder estimates for $\delta\mathcal{I}_1$ and $\delta\mathcal{I}_2$. Obviously, from Lemma 4.4, the following estimates for the Lebesgue norms hold true for $\delta\mathcal{I}_1$:

$$ \|\delta\mathcal{I}_1\|_{L_T^mL^q(\mathbb{R}^d)} \lesssim \|\nabla\delta v\|_{L_T^1L^\infty(\mathbb{R}^d)} \|\nabla u_1\|_{L_T^mL^q(\mathbb{R}^d)} $$

for $2 \le m, q \le \infty$ and moreover, the following estimate for the Hölder norm holds true as well:

$$ \|\sigma^{r/4}\delta\mathcal{I}_1\|_{L_T^4\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \lesssim \|\sigma^{r/4}\nabla u_1\|_{L_T^4\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \|\nabla\delta v\|_{L_T^1\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} $$

A similar estimates hold true for $\delta\mathcal{I}_2$.

Step 8: Lebesgue norm estimate for $\partial_t\delta\mathcal{I}_1$ and $\partial_t\delta\mathcal{I}_2$. The purpose of this step is to estimate the $L_T^2 L^2(\mathbb{R}^d)$ norm of $\partial_t\delta\mathcal{I}_1$ and $\partial_t\delta\mathcal{I}_2$, the $L_T^{8/d} L^4(\mathbb{R}^d)$ norm of $\sqrt{\sigma}\partial_t\delta\mathcal{I}_1$ and $\sqrt{\sigma}\partial_t\delta\mathcal{I}_2$. First, one expresses the time derivative of $\delta\mathcal{I}_1$ as:

$$ \begin{aligned} \partial_t\delta\mathcal{I}_1 ={}& 2\rho_0\{\partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1}\}\partial_{\rho}\mu(J_{v_2}^{-1}\rho_0, c_0)\mathbb{D}u_1 + 2\rho_0\partial_t J_{v_1}^{-1}\{\partial_{\rho}\mu(J_{v_2}^{-1}\rho_0, c_0) - \partial_{\rho}\mu(J_{v_1}^{-1}\rho_0, c_0)\}\mathbb{D}u_1 \\ & + 2\{\mu(J_{v_2}^{-1}\rho_0, c_0) - \mu(J_{v_1}^{-1}\rho_0, c_0)\}\mathbb{D}\partial_t u_1. \end{aligned} \quad (4.28) $$

With the help of Lemma 4.4, one can estimate the last term above as:

$$ \|\{\mu(J_{v_2}^{-1}\rho_0, c_0) - \mu(J_{v_1}^{-1}\rho_0, c_0)\}\mathbb{D}\partial_t u_1\|_{L_T^2L^2(\mathbb{R}^d)} \lesssim \|\nabla\partial_t u_1\|_{L_T^2L^2(\mathbb{R}^d)} \|\nabla\delta v\|_{L_T^1L^\infty(\mathbb{R}^d)} $$

and

$$ \|\sqrt{\sigma}\{\mu(J_{v_2}^{-1}\rho_0, c_0) - \mu(J_{v_1}^{-1}\rho_0, c_0)\}\mathbb{D}\partial_t u_1\|_{L_T^{8/d}L^4(\mathbb{R}^d)} \lesssim \|\sqrt{\sigma}\nabla\partial_t u_1\|_{L_T^{8/d}L^4(\mathbb{R}^d)} \|\nabla\delta v\|_{L_T^1L^\infty(\mathbb{R}^d)} $$

whereas the penultimate term of (4.28) can be controlled as follows:

$$ \|\rho_0\partial_t J_{v_1}^{-1}\{\partial_{\rho}\mu(J_{v_2}^{-1}\rho_0, c_0) - \partial_{\rho}\mu(J_{v_1}^{-1}\rho_0, c_0)\}\mathbb{D}u_1\|_{L_T^2L^2(\mathbb{R}^d)} \lesssim \|\nabla v_1\|_{L_T^4L^4(\mathbb{R}^d)} \|\nabla u_1\|_{L_T^4L^4(\mathbb{R}^d)} \|\nabla\delta v\|_{L_T^1L^\infty(\mathbb{R}^d)} $$

and

$$ \|\sqrt{\sigma}\rho_0\partial_t J_{v_1}^{-1}\{\partial_{\rho}\mu(J_{v_2}^{-1}\rho_0, c_0) - \partial_{\rho}\mu(J_{v_1}^{-1}\rho_0, c_0)\}\mathbb{D}u_1\|_{L_T^{8/d}L^4(\mathbb{R}^d)} \lesssim \|\nabla v_1, \nabla u_1\|_{L_T^{16/d}L^8(\mathbb{R}^d)}^2 \|\nabla\delta v\|_{L_T^1L^\infty(\mathbb{R}^d)} $$

According to (4.5), we express:

$$ \begin{aligned} \partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1} ={}& -J_{v_2}^{-1}\operatorname{div}_{A_{v_2}}(v_2) + J_{v_1}^{-1}\operatorname{div}_{A_{v_1}}(v_1) \\ ={}& \{J_{v_1}^{-1} - J_{v_2}^{-1}\}\operatorname{div}_{A_{v_2}}(v_2) + J_{v_1}^{-1}\{\operatorname{div}_{A_{v_1}} - \operatorname{div}_{A_{v_2}}\}v_1 - J_{v_1}^{-1}\operatorname{div}_{A_{v_2}}(\delta v) \end{aligned} \quad (4.29) $$

and thus, the first term of the right hand side of (4.28) can be estimated, thanks to Lemma 4.4, as:

$$ \begin{aligned} \|\rho_0\{\partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1}\}\partial_{\rho}\mu(J_{v_2}^{-1}\rho_0, c_0)\mathbb{D}u_1\|_{L_T^2L^2(\mathbb{R}^d)} \lesssim{}& \|\nabla v_1, \nabla v_2\|_{L_T^4L^4(\mathbb{R}^d)} \|\nabla u_1\|_{L_T^4L^4(\mathbb{R}^d)} \|\nabla\delta v\|_{L_T^1L^\infty(\mathbb{R}^d)} \\ & + \|\nabla u_1\|_{L_T^4L^4(\mathbb{R}^d)} \|\nabla\delta v\|_{L_T^1L^\infty(\mathbb{R}^d)} \end{aligned} \quad (4.30) $$

and

$$ \begin{aligned} \|\sqrt{\sigma}\rho_0\{\partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1}\}\partial_{\rho}\mu(J_{v_2}^{-1}\rho_0, c_0)\mathbb{D}u_1\|_{L_T^{8/d}L^4(\mathbb{R}^d)} \lesssim{}& \|\nabla v_1, \nabla v_2\|_{L_T^{16/d}L^8(\mathbb{R}^d)} \|\nabla u_1\|_{L_T^{16/d}L^8(\mathbb{R}^d)} \|\nabla\delta v\|_{L_T^1L^\infty(\mathbb{R}^d)} \\ & + \|\nabla u_1\|_{L_T^{16/d}L^8(\mathbb{R}^d)} \|\nabla\delta v\|_{L_T^{16/d}L^8(\mathbb{R}^d)}. \end{aligned} \quad (4.31) $$

In sum, one has:

$$ \begin{aligned} \|\partial_t\delta\mathcal{I}_1\|_{L_T^2L^2(\mathbb{R}^d)} \lesssim{}& \|\nabla\partial_t u_1\|_{L_T^2L^2(\mathbb{R}^d)} \|\nabla\delta v\|_{L_T^1L^\infty(\mathbb{R}^d)} + \|\nabla v_1, \nabla v_2\|_{L_T^4L^4(\mathbb{R}^d)} \|\nabla u_1\|_{L_T^4L^4(\mathbb{R}^d)} \|\nabla\delta v\|_{L_T^1L^\infty(\mathbb{R}^d)} \\ & + \|\nabla u_1\|_{L_T^4L^4(\mathbb{R}^d)} \|\nabla\delta v\|_{L_T^4L^4(\mathbb{R}^d)}, \end{aligned} $$
Hence, the viscosities $\mu$ and $\lambda$ verify all the conditions stated in (3.2) and in particular **Theorem 3.1** provides us with a unique solution $u \in X_{T,u_0}$ of the Cauchy problem (4.4).

**Estimates for $u$ and $u_t$:**

$$ (4.16) \qquad \sup_{[0,T]} \|u, u_t, \nabla u\|_{L^2(\mathbb{R}^d)}^2 + \|\nabla u\|_{L^\infty_T L^6(\mathbb{R}^d)}^2 + \|\nabla u\|_{L^T L^8(\mathbb{R}^d)}^2 + \int_0^T \|\nabla u, \nabla u_t, u_t\|_{L^2(\mathbb{R}^d)}^2 \\ \lesssim \left( \| (I(v,v), u_0)\|_{Y_T}^2 + T R^2 \|I(v,v)\|_{L^\infty_T(L^2 \cap L^6)(\mathbb{R}^d)}^2 \right) \exp \left( C_* T^{\frac{3-r}{4}} R + C_* T R^2 \right) $$

with the estimates of $I(v,v)$ given in (4.6)-(4.11).

**Estimates for $u_{tt}$:**

$$ (4.17) \qquad \int_0^T \|\sqrt{\sigma} u_{tt}, \sigma \nabla u_{tt}\|_{L^2(\mathbb{R}^d)}^2 + \|\sqrt{\sigma} \nabla u_t, \sigma u_{tt}\|_{L^\infty_T L^2(\mathbb{R}^d)}^2 \lesssim \left[ \int_0^T \|\nabla u_t\|_{L^2(\mathbb{R}^d)}^2 \right. \\ \left. + \int_0^T \|\partial_t I(v,v), \sigma \partial_{tt} I(v,v)\|_{L^2(\mathbb{R}^d)}^2 + (T R^2 + T^{1-\frac{d}{4}} R^2 (1+R)^2) \|\nabla u\|_{L^\infty_T L^4(\mathbb{R}^d)}^2 \right] \times \exp \left( C_* T^{\frac{3-r}{4}} R \right). $$

The norms of $u$ involved in this estimate are controlled in (4.16), while those of $I(v,v)$ are given in (4.6)-(4.11). $L_T^{8/d} L^4(\mathbb{R}^d)$-norm estimate for $\sqrt{\sigma}\nabla u_t$.

$$ (4.18) \qquad \int_0^T \sigma^2 \|\nabla u_t\|_{L^4(\mathbb{R}^d)}^{8/d} \lesssim \int_0^T \sigma^2 \|\partial_t I(v,v)\|_{L^4(\mathbb{R}^d)}^{8/d} + T^{1-\frac{1}{d}} R^{8/d} \|\nabla u\|_{L_T^{16} L^8(\mathbb{R}^d)}^{8/d} + \|\sigma u_{tt}\|_{L_T^\infty L^2(\mathbb{R}^d)}^{8/d} \\ + \left[ \int_0^T \|\nabla u_t, \partial_t I(v,v)\|_{L^2(\mathbb{R}^d)}^2 \right]^{4/d} + T^{4/d} R^{8/d} \|\nabla u\|_{L_T^\infty L^4(\mathbb{R}^d)}^{8/d}. $$

The norms of $I(v,v)$ are given in (4.10) and (4.9), while the norms of $u$ are controlled in (4.16) and (4.17).

$L_T^4 \mathcal{C}_{pw,c_0}^\alpha(\mathbb{R}^d)$ estimate for $\sigma^{r/4}\nabla u$.

$$ \int_0^T \sigma^r \|\nabla u\|_{\mathcal{C}_{pw,c_0}^\alpha(\mathbb{R}^d)}^4 \lesssim \int_0^T \sigma^r \|I(v,v)\|_{\mathcal{C}_{pw,c_0}^\alpha(\mathbb{R}^d)}^4 + \|I(v,v), \nabla u, u_t, \sqrt{\sigma} \nabla u_t\|_{L_T^\infty L^2(\mathbb{R}^d)}^4 \\ + \left[ \int_0^T \|I(v,v), \nabla u, u_t, \nabla u_t\|_{L^2(\mathbb{R}^d)}^2 \right]^2 + \left[ \int_0^T \sigma^2 \|\nabla u_t\|_{L^4(\mathbb{R}^d)}^{8/d} \right]^{d/2}. $$

The norms of $I(v,v)$ are given in (4.6) and (4.8) whereas the norms of $u$ involved are controlled in (4.16), (4.17) and (4.18).

By combining all these computations, we choose $R$ related to the norm of the initial data and a small $T^*$ such that for all $0 < T \le T_*$,

$$ \|u\|_{X_T} \le R. $$

This concludes the proof of **Proposition 4.1**.

□

The final step of this section is dedicated to constructing the fixed point $u \in X_{T,u_0}$ of $\Phi_T$.

**Proposition 4.2.** There exists $T \in (0, T_*)$ and $u \in X_{T,u_0}$ such that

$$ u = \Phi_T(u). $$

This fixed point $u$ satisfies the nonlinear equations (4.2), thereby completing the proof of **Theorem 2.2**.

**Proof of Proposition 4.2.** To prove **Proposition 4.2**, we begin by considering the following sequence:

$$ u_{n+1} = \Phi_T(u_n), \quad n \ge 0, $$

with

$$ (4.19) \qquad \begin{cases} \rho_0 \partial_t u_{n+1} - \operatorname{div}\{2\mu(J_{u_n}^{-1}\rho_0, c_0) \mathbb{D}u_{n+1} + \lambda(J_{u_n}^{-1}\rho_0, c_0) \operatorname{div} u_{n+1} I_d\} = \operatorname{div}(I(u_n, u_n)), \\ u_{n+1}|_{t=0} = u_0. \end{cases} $$

From **Proposition 4.1**, it follows that the sequence $(u_n)_n$ is uniformly bounded in $X_T$:

$$ (4.20) \qquad \sup_{n \in \mathbb{N}} \|u_n\|_{X_T} \le R. $$
and

$$ \begin{aligned} \|\sqrt{\sigma} \partial_t \delta \mathcal{I}_1\|_{L_T^{s/d} L^4(\mathbb{R}^d)} &\lesssim \|\sqrt{\sigma} \nabla \partial_t u_1\|_{L_T^{s/d} L^4(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} + \|\nabla u_1\|_{L_T^{16/d} L^8(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^{16/d} L^8(\mathbb{R}^d)} \\ &\quad + \|\nabla v_1, \nabla v_2, \nabla u_1\|_{L_T^{16/d} L^8(\mathbb{R}^d)} \|\nabla u_1, \nabla v_1\|_{L_T^{16/d} L^8(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)}. \end{aligned} $$

A similar estimate holds for $\delta\mathcal{I}_2$. We turn to the norm of the second time derivative of $\delta\mathcal{I}_1$ and $\delta\mathcal{I}_2$.

Step 9: $L_T^2 L^2(\mathbb{R}^d)$ norm estimate for $\sigma \partial_{tt} \delta \mathcal{I}_1$ and $\sigma \partial_{tt} \delta \mathcal{I}_2$. We begin by computing $\partial_{tt} \delta \mathcal{I}_1$ from (4.28) as:

$$ \partial_{tt} \delta \mathcal{I}_1 = \partial_{tt} \delta \mathcal{I}_{1,3} + \partial_{tt} \delta \mathcal{I}_{1,4} + \partial_{tt} \delta \mathcal{I}_{1,5} $$

where

$$ \begin{aligned} \partial_{tt} \delta \mathcal{I}_{1,3} = {}& 2\rho_0\{\partial_{tt} J_{v_2}^{-1} - \partial_{tt} J_{v_1}^{-1}\} \partial_{\rho} \mu(J_{v_2}^{-1} \rho_0, c_0) \mathbb{D}u_1 + 2\rho_0^2\{\partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1}\} \partial_{\rho}^2 \mu(J_{v_2}^{-1} \rho_0, c_0) \partial_t J_{v_2}^{-1} \mathbb{D}u_1 \\ &+ 2\rho_0\{\partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1}\} \partial_{\rho} \mu(J_{v_2}^{-1} \rho_0, c_0) \mathbb{D}\partial_t u_1, \end{aligned} $$

$$ \begin{aligned} \partial_{tt} \delta \mathcal{I}_{1,4} = {}& 2\rho_0 \partial_{tt} J_{v_1}^{-1} \{\partial_{\rho} \mu(J_{v_2}^{-1} \rho_0, c_0) - \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0)\} \mathbb{D}u_1 + 2\rho_0^2 \partial_t J_{v_1}^{-1} \partial_t J_{v_2}^{-1} \{\partial_{\rho}^2 \mu(J_{v_2}^{-1} \rho_0, c_0) - \partial_{\rho}^2 \mu(J_{v_1}^{-1} \rho_0, c_0)\} \mathbb{D}u_1 \\ &+ 2\rho_0^2 \partial_t J_{v_1}^{-1} \partial_{\rho}^2 \mu(J_{v_2}^{-1} \rho_0, c_0) \{\partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1}\} \mathbb{D}u_1 + 2\rho_0 \partial_t J_{v_1}^{-1} \{\partial_{\rho} \mu(J_{v_2}^{-1} \rho_0, c_0) - \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0)\} \mathbb{D}\partial_t u_1 \end{aligned} $$

and

$$ \begin{aligned} \partial_{tt} \delta \mathcal{I}_{1,5} = {}& 2\rho_0 \partial_t J_{v_2}^{-1} \{\partial_{\rho} \mu(J_{v_2}^{-1} \rho_0, c_0) - \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0)\} \mathbb{D}\partial_t u_1 + 2\rho_0 \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \{\partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1}\} \mathbb{D}\partial_t u_1 \\ &+ 2\{\mu(J_{v_2}^{-1} \rho_0, c_0) - \mu(J_{v_1}^{-1} \rho_0, c_0)\} \mathbb{D}\partial_{tt} u_1. \end{aligned} $$

Thanks to Lemma 4.4, the expression (4.29) and (4.8), the following estimates for $\partial_{tt} \delta \mathcal{I}_{1,4}$ and $\partial_{tt} \delta \mathcal{I}_{1,5}$ hold true:

$$ \begin{aligned} \|\sigma \partial_{tt} \delta \mathcal{I}_{1,4}\|_{L_T^2 L^2(\mathbb{R}^d)} &\lesssim \|\delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \left[ \|\nabla u_1\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla v_1\|_{L_T^6 L^6(\mathbb{R}^d)} \left( \|\nabla v_1\|_{L_T^6 L^6(\mathbb{R}^d)} + \|\nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \right) \right. \\ &\qquad \left. + \|\sigma \nabla \partial_t u_1\|_{L_T^2 L^4(\mathbb{R}^d)} \|\nabla v_1\|_{L_T^\infty L^4(\mathbb{R}^d)} \right] + \|\nabla u_1\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla v_1\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^6 L^6(\mathbb{R}^d)}, \\ \|\sigma \partial_{tt} \delta \mathcal{I}_{1,5}\|_{L_T^2 L^2(\mathbb{R}^d)} &\lesssim \|\delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \left( \|\sigma \nabla \partial_t u_1\|_{L_T^2 L^4(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^\infty L^4(\mathbb{R}^d)} + \|\sigma \nabla \partial_{tt} u_1\|_{L_T^2 L^2(\mathbb{R}^d)} \right) \\ &\quad + \|\nabla \delta v\|_{L_T^\infty L^4(\mathbb{R}^d)} \|\sigma \nabla \partial_t u_1\|_{L_T^2 L^4(\mathbb{R}^d)}. \end{aligned} $$

It only remains the term $\partial_{tt} \delta \mathcal{I}_{1,3}$. In fact, its last two terms are controlled by:

$$ \begin{aligned} &\|2\sigma\rho_0\{\partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1}\} \partial_{\rho} \mu(J_{v_2}^{-1} \rho_0, c_0) \mathbb{D}\partial_t u_1, \quad 2\sigma\rho_0^2\{\partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1}\} \partial_{\rho}^2 \mu(J_{v_2}^{-1} \rho_0, c_0) \partial_t J_{v_2}^{-1} \mathbb{D}u_1\|_{L_T^2 L^2(\mathbb{R}^d)} \\ &\lesssim \|\delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \left( \|\sigma \nabla \partial_t u_1\|_{L_T^2 L^4(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^\infty L^4(\mathbb{R}^d)} + \|\nabla u_1\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \right) \\ &\quad + \|\sigma \nabla \partial_t u_1\|_{L_T^2 L^4(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^\infty L^4(\mathbb{R}^d)} + \|\nabla u_1\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^6 L^6(\mathbb{R}^d)}. \end{aligned} \quad (4.32) $$

For first term, we express, from (4.8):

$$ \begin{aligned} \partial_{tt} J_{v_2}^{-1} - \partial_{tt} J_{v_1}^{-1} = {}& \{J_{v_2}^{-1} - J_{v_1}^{-1}\} J_{v_2} \partial_{tt} J_{v_2}^{-1} + J_{v_1}^{-1} \{\mathrm{div}_{A_{v_2}} v_2 - \mathrm{div}_{A_{v_1}} v_1\} \{\mathrm{div}_{A_{v_2}} v_2 + \mathrm{div}_{A_{v_1}} v_1\} \\ & - J_{v_1}^{-1} \mathrm{D}\partial_t \delta v: A_{v_2} - J_{v_1}^{-1} \mathrm{D}\partial_t v_1: (A_{v_2} - A_{v_1}) + J_{v_1}^{-1} \mathrm{D}\delta v: (A_{v_2} \cdot Dv_2 \cdot A_{v_2}) \\ & + J_{v_1}^{-1} \mathrm{D}v_1: \{A_{v_2} - A_{v_1}\} \cdot Dv_2 \cdot A_{v_2}) + J_{v_1}^{-1} \mathrm{D}v_1: A_{v_1} \cdot D\delta v \cdot A_{v_2} \\ & + J_{v_1}^{-1} \mathrm{D}v_1: A_{v_1} \cdot Dv_1 \cdot \{A_{v_2} - A_{v_1}\}. \end{aligned} \quad (4.33) $$

It then turns out that the first term of $\partial_{tt} \delta \mathcal{I}_{1,3}$ is bounded as follows:

$$ \begin{aligned} &\|2\sigma\rho_0\{\partial_{tt} J_{v_2}^{-1} - \partial_{tt} J_{v_1}^{-1}\} \partial_{\rho} \mu(J_{v_2}^{-1} \rho_0, c_0) \mathbb{D}u_1\|_{L_T^2 L^2(\mathbb{R}^d)} \\ &\lesssim \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla u_1\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \\ &\quad + \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\sigma \nabla \partial_t u_1\|_{L_T^2 L^4(\mathbb{R}^d)} \|\mathbb{D}u_1\|_{L_T^\infty L^4(\mathbb{R}^d)} + \|\sigma \nabla \partial_t \delta v\|_{L_T^2 L^4(\mathbb{R}^d)} \|\nabla u_1\|_{L_T^\infty L^4(\mathbb{R}^d)} \\ &\quad + \|\nabla \delta v\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla u_1\|_{L_T^6 L^6(\mathbb{R}^d)} \\ &\quad + \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla u_1\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla v_1\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \end{aligned} \quad (4.34) $$

where we have used Lemma 4.4, (4.29). Finally we gather (4.32) and (4.34) in order to obtain the norm of the remainder term $\partial_{tt} \delta \mathcal{I}_{1,3}$.
We take the differences in (4.19) and obtain:

$$
(4.21) \qquad \begin{cases}
\rho_0 \partial_t w_{n+1} - \operatorname{div} (2\mu(J_{u_n}^{-1} \rho_0, c_0) \mathbb{D}w_{n+1} + \lambda(J_{u_n}^{-1} \rho_0, c_0) \operatorname{div} w_{n+1} I_d) \\
= \operatorname{div} (\delta \mathcal{I}) + \operatorname{div} (I(u_n, u_n) - I(u_{n-1}, u_{n-1})), \\
w_{n+1}|_{t=0} = 0,
\end{cases}
$$

where $w_{k+1} = u_{k+1} - u_k$, $k \ge 1$, and

$$
\delta \mathcal{I} = 2 \left( (\mu(J_{u_n}^{-1} \rho_0, c_0) - \mu(J_{u_{n-1}}^{-1} \rho_0, c_0)) \mathbb{D}u_{n-1} + (\lambda(J_{u_n}^{-1} \rho_0, c_0) - \lambda(J_{u_{n-1}}^{-1} \rho_0, c_0)) \operatorname{div} u_{n-1} I_d \right).
$$

An energy estimate in (4.21) yields:

$$
(4.22) \quad \sup_{[0,T]} \|w_{n+1}\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|\nabla w_{n+1}\|_{L^2(\mathbb{R}^d)}^2 \lesssim \int_0^T \|\delta \mathcal{I}, I(u_n, u_n) - I(u_{n-1}, u_{n-1})\|_{L^2(\mathbb{R}^d)}^2.
$$

To estimate the right-hand side above in terms of the norms of $u_n$ and $u_{n-1}$, we rely on the following result from [5, Lemma A.4].

**Lemma 4.3.** Let $v_1, v_2$ be two vectors fields verifying

$$
c = \max_j \int_0^T \|Dv_j\|_{L^\infty(\mathbb{R}^d)} < 1.
$$

There exists a constant $k = k(c)$ such that the following estimates hold true: for all $p \in [2, \infty]$

$$
\left\{
\begin{array}{l}
\|A_{v_2}(t) - A_{v_1}(t)\|_{L^p(\mathbb{R}^d)} \\
\|\operatorname{Adj}(D\mathcal{X}_{v_2}(t)) - \operatorname{Adj}(D\mathcal{X}_{v_1}(t))\|_{L^p(\mathbb{R}^d)} \\
\|J_{v_2}^{\pm 1}(t) - J_{v_1}^{\pm 1}(t)\|_{L^p(\mathbb{R}^d)}
\end{array}
\right.
\leq
\begin{array}{l}
k \|D(v_2 - v_1)\|_{L^1 t L^p(\mathbb{R}^d)}, \\
k \|D(v_2 - v_1)\|_{L^1 t L^p(\mathbb{R}^d)}, \\
k \|D(v_2 - v_1)\|_{L^1 t L^p(\mathbb{R}^d)}.
\end{array}
$$

Lemma 4.3 immediately implies:

$$
\begin{align*}
\int_0^T \|\delta \mathcal{I}(t)\|_{L^2(\mathbb{R}^d)}^2 dt &\lesssim \int_0^T \|J_{u_n}^{-1}(t) - J_{u_{n-1}}^{-1}(t)\|_{L^2(\mathbb{R}^d)}^2 \|Du_{n-1}(t)\|_{L^\infty(\mathbb{R}^d)}^2 dt \\
&\lesssim \left( \int_0^T t \|Du_{n-1}(t)\|_{L^\infty(\mathbb{R}^d)}^2 dt \right) \left( \int_0^T \|Dw_n\|_{L^2(\mathbb{R}^d)}^2 \right) \\
&\lesssim T^{\frac{3-r}{2}} R^2 \int_0^T \|Dw_n\|_{L^2(\mathbb{R}^d)}^2 dt.
\end{align*}
$$

Similarly, by expressing (see the definitions of $I_1$ and $I_2$ in (4.3))

$$
I_3(u_n) - I_3(u_{n-1}) = \{\operatorname{Adj}(D\mathcal{X}_{u_n}) - \operatorname{Adj}(D\mathcal{X}_{u_{n-1}})\} (P(J_{u_n}^{-1}\rho_0, c_0) - \tilde{P}) \\
+ \operatorname{Adj}(D\mathcal{X}_{u_{n-1}})\{P(J_{u_n}^{-1}\rho_0, c_0) - P(J_{u_{n-1}}^{-1}\rho_0, c_0)\},
$$

and

$$
\begin{align*}
I_1(u_n, u_n) - I_1(u_{n-1}, u_{n-1}) &= 2\{\mu(J_{u_n}^{-1}\rho_0, c_0) - \mu(J_{u_{n-1}}^{-1}\rho_0, c_0)\}\{\operatorname{Adj}(D\mathcal{X}_{u_n})\mathbb{D}_{A_{u_n}}u_n - \mathbb{D}u_n\} \\
&\quad + 2\mu(J_{u_{n-1}}^{-1}\rho_0, c_0)\{\operatorname{Adj}(D\mathcal{X}_{u_n})\mathbb{D}_{A_{u_n}}w_n - \mathbb{D}w_n\} \\
&\quad + 2\mu(J_{u_{n-1}}^{-1}\rho_0, c_0)\{\operatorname{Adj}(D\mathcal{X}_{u_n}) - \operatorname{Adj}(D\mathcal{X}_{u_{n-1}})\}\mathbb{D}_{A_{u_n}}u_{n-1} \\
&\quad + 2\mu(J_{u_{n-1}}^{-1}\rho_0, c_0)\operatorname{Adj}(D\mathcal{X}_{u_{n-1}})\{\mathbb{D}_{A_{u_n}} - \mathbb{D}_{A_{u_{n-1}}}\}u_{n-1},
\end{align*}
$$

we infer

$$
\int_0^T \|I_3(u_n) - I_3(u_{n-1})\|_{L^2(\mathbb{R}^d)}^2 \lesssim T^2 \int_0^T \|\nabla w_n\|_{L^2(\mathbb{R}^d)}^2,
$$

and

$$
\int_0^T \| I_1(u_n, u_n) - I_1(u_{n-1}, u_{n-1}) \|_{L^2(\mathbb{R}^d)}^2 \lesssim T^{\frac{3-r}{2}} R^2 \int_0^T \| \nabla w_n \|_{L^2(\mathbb{R}^d)}^2.
$$

A similar estimate also holds for $I_2(u_n, u_n) - I_2(u_{n-1}, u_{n-1})$. It therefore follows from (4.22) that:

$$
\sup_{[0,T]} \|w_{n+1}\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|\nabla w_{n+1}\|_{L^2(\mathbb{R}^d)}^2 \lesssim CT^{\frac{3-r}{2}} (1+R^2) \left( \sup_{[0,T]} \|w_n\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|\nabla w_n\|_{L^2(\mathbb{R}^d)}^2 \right).
$$
Step 10: Lebesgue and Hölder norm estimates for $\delta I_3(v) := I_3(v_2) - I_3(v_1)$. We begin by writing:

$$I_3(v_2) - I_3(v_1) = \{\operatorname{Adj}(D\mathcal{X}_{v_2}) - \operatorname{Adj}(D\mathcal{X}_{v_1})\} (P(J_{v_2}^{-1}\rho_0, c_0) - \tilde{P}) \\ + \operatorname{Adj}(D\mathcal{X}_{v_1})\{P(J_{v_2}^{-1}\rho_0, c_0) - P(J_{v_1}^{-1}\rho_0, c_0)\}$$

and thus, from Lemma 4.4, the following estimates hold true:

$$\|\delta I_3(v)\|_{L_T^m L^q(\mathbb{R}^d)} \lesssim T^{1/m} \|\nabla \delta v\|_{L_T^1 L^q(\mathbb{R}^d)}, \\ \|\sigma^{r/4} \delta I_3(v)\|_{L_T^4 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \lesssim T^{1/4} \|\nabla \delta v\|_{L_T^1 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)}.$$

We turn to the Lebesgue norms estimates for the time derivative of $\delta I_3(v)$.

Step 11: Lebesgue norms estimate for $\partial_t \delta I_3(v)$. We recall that the aim is to estimate the $L_T^2 L^2(\mathbb{R}^d)$ norm of $\partial_t \delta I_3(v)$ and the $L_T^{8/d} L^4(\mathbb{R}^d)$ norm of $\sqrt{\sigma} \partial_t \delta I_3(v)$. So we begin by the following expression of $\partial_t \delta I_3(v)$:

$$\begin{align} \partial_t \delta I_3(v) ={}& \{\partial_t \operatorname{Adj}(D\mathcal{X}_{v_2}) - \partial_t \operatorname{Adj}(D\mathcal{X}_{v_1})\} (P(J_{v_2}^{-1}\rho_0, c_0) - \tilde{P}) \nonumber \\ & + \{\operatorname{Adj}(D\mathcal{X}_{v_2}) - \operatorname{Adj}(D\mathcal{X}_{v_1})\}\rho_0 \partial_\rho P(J_{v_2}^{-1}\rho_0, c_0) \partial_t J_{v_2}^{-1} \nonumber \\ & + \partial_t \operatorname{Adj}(D\mathcal{X}_{v_1})\{P(J_{v_2}^{-1}\rho_0, c_0) - P(J_{v_1}^{-1}\rho_0, c_0)\} \nonumber \\ & + \rho_0 \partial_\rho P(J_{v_2}^{-1}\rho_0, c_0) \operatorname{Adj}(D\mathcal{X}_{v_1})\{\partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1}\} \nonumber \\ & + \rho_0 \partial_t J_{v_1}^{-1} \operatorname{Adj}(D\mathcal{X}_{v_1})\{\partial_\rho P(J_{v_2}^{-1}\rho_0, c_0) - \partial_\rho P(J_{v_1}^{-1}\rho_0, c_0)\} \tag{4.35} \\ ={}& \sum_{k=1}^5 \partial_t \delta I_{3,k}. \nonumber \end{align}$$

From Lemma 4.4 and identities (4.5) and (4.7), the $L_T^2 L^2(\mathbb{R}^d)$ norm of the second, third and last terms of the right hand side above are estimated as:

$$\|\partial_t \delta I_{3,2}, \partial_t \delta I_{3,3}, \partial_t \delta I_{3,5}\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla v_1, \nabla v_2\|_{L_T^2 L^2(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)}$$

whereas their $L_T^{8/d} L^4(\mathbb{R}^d)$ norms are estimated as:

$$\|\partial_t \delta I_{3,2}, \partial_t \delta I_{3,3}, \partial_t \delta I_{3,5}\|_{L_T^{8/d} L^4(\mathbb{R}^d)} \lesssim \|\nabla v_1, \nabla v_2\|_{L_T^{8/d} L^4(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)}.$$

On the other hand, the following estimates hold for the remainder terms:

$$\|\partial_t \delta I_{3,1}, \partial_t \delta I_{3,4}\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla \delta v\|_{L_T^2 L^2(\mathbb{R}^d)} + \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^2 L^2(\mathbb{R}^d)}, \\ \|\sqrt{\sigma} \partial_t \delta I_{3,1}, \sqrt{\sigma} \partial_t \delta I_{3,4}\|_{L_T^{8/d} L^4(\mathbb{R}^d)} \lesssim \|\nabla \delta v\|_{L_T^{8/d} L^4(\mathbb{R}^d)} + \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^{8/d} L^4(\mathbb{R}^d)}$$

where we have used (4.29), Lemma 4.4 and the fact that, owing to (4.7) and (4.6), we have:

$$\begin{align} \partial_t \operatorname{Adj}(D\mathcal{X}_{v_2}) - \partial_t \operatorname{Adj}(D\mathcal{X}_{v_1}) ={}& (Dv_2 - Dv_1) : \operatorname{Adj}(D\mathcal{X}_{v_2}) A_{v_2} + Dv_1 : \{\operatorname{Adj}(D\mathcal{X}_{v_2}) - \operatorname{Adj}(D\mathcal{X}_{v_1})\} A_{v_2} \nonumber \\ & + Dv_1 : \operatorname{Adj}(Dv_1)\{A_{v_2} - A_{v_1}\} - \{\operatorname{Adj}(D\mathcal{X}_{v_2}) - \operatorname{Adj}(D\mathcal{X}_{v_1})\} \cdot Dv_2 \cdot A_{v_2} \nonumber \\ & - \operatorname{Adj}(D\mathcal{X}_{v_1}) \cdot \{Dv_2 - Dv_1\} \cdot A_{v_2} - \operatorname{Adj}(D\mathcal{X}_{v_1}) \cdot Dv_1 \cdot \{A_{v_2} - A_{v_1}\} \tag{4.36} \\ \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} ={}& \|\nabla \delta v\|_{L_T^2 L^2(\mathbb{R}^d)} + \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^2 L^2(\mathbb{R}^d)} \nonumber \end{align}$$

and

$$\partial_t A_{v_2} - \partial_t A_{v_1} = \{A_{v_1} - A_{v_2}\} \cdot D_{v_2} \cdot A_{v_2} + A_{v_1} \cdot (Dv_1 - Dv_2) \cdot A_{v_2} + A_{v_1} \cdot Dv_1 \cdot (A_{v_1} - A_{v_2}).$$

Summing-up, we have:

$$\|\partial_t \delta I_3(v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla v_1, \nabla v_2\|_{L_T^2 L^2(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} + \|\nabla \delta v\|_{L_T^2 L^2(\mathbb{R}^d)}$$

along with

$$\|\sqrt{\sigma} \partial_t \delta I_3(v)\|_{L_T^{8/d} L^4(\mathbb{R}^d)} \lesssim \|\nabla v_1, \nabla v_2\|_{L_T^{8/d} L^4(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} + \|\nabla \delta v\|_{L_T^{8/d} L^4(\mathbb{R}^d)}.$$

We turn to the $L_T^2 L^2(\mathbb{R}^d)$ norm estimate for $\sigma \partial_t \delta I_3(v)$.
We then choose $T < T_*$ small such that

$$CT^{\frac{3-r}{2}}(1 + R^2) \le \frac{1}{2},$$

which leads to

$$\sup_{[0,T]} \|w_{n+1}\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|\nabla w_{n+1}\|_{L^2(\mathbb{R}^d)}^2 \le \frac{1}{2} \left( \sup_{[0,T]} \|w_n\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|\nabla w_n\|_{L^2(\mathbb{R}^d)}^2 \right),$$

and hence

$$\begin{aligned}
\sup_{[0,T]} \|w_{n+1}\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|\nabla w_{n+1}\|_{L^2(\mathbb{R}^d)}^2 \le \frac{1}{2^n} \left( \sup_{[0,T]} \|w_1\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|\nabla(w_1)\|_{L^2(\mathbb{R}^d)}^2 \right) \\
\le \frac{R^2}{2^{n-1}}.
\end{aligned}$$

This implies the strong convergence of the sequence $(u_n)_n$ in $L^\infty((0,T), L^2(\mathbb{R}^d)) \cap L^2((0,T), H^1(\mathbb{R}^d))$ to some limit $u$. Furthermore, by virtue of lower semi-continuity of norms, we deduce from the uniform bound (4.20):

$$\|u\|_{X_T} \le R.$$

This enables us to apply Lemma 4.3 with $v_1 = u_n$ and $v_2 = u$:

$$\left\{
\begin{array}{ll}
\|A_{u_n} - A_u\|_{L_T^\infty L^2(\mathbb{R}^d)} & \le k \|D(u_n - u)\|_{L_T^1 L^2(\mathbb{R}^d)}, \\
\|\operatorname{Adj}(D\mathcal{X}_{u_n}) - \operatorname{Adj}(D\mathcal{X}_u)\|_{L_T^\infty L^2(\mathbb{R}^d)} & \le k \|D(u_n - u)\|_{L_T^1 L^2(\mathbb{R}^d)}, \\
\|J_{u_n}^{\pm 1} - J_u^{\pm 1}\|_{L_T^\infty L^2(\mathbb{R}^d)} & \le k \|D(u_n - u)\|_{L_T^1 L^2(\mathbb{R}^d)},
\end{array}
\right.$$

from which we deduce the strong convergence in $L_T^\infty L^2(\mathbb{R}^d)$ of the sequences $(I_d - A_{u_n})_n$, $(I_d - \operatorname{Adj}(D\mathcal{X}_{u_n}))_n$ and $(J_{u_n}^{\pm 1} - 1)_n$ to $I_d - A_u$, $I_d - \operatorname{Adj}(D\mathcal{X}_u)$, and $J_u^{\pm 1} - 1$, respectively. In fact, these strong convergences hold in $L^\infty((0,T), L^p(\mathbb{R}^d))$ for all $p \in [2, \infty)$, given that the sequences are uniformly bounded in $L^\infty((0,T) \times \mathbb{R}^d)$. Additionally, the bound in (4.20) indicates that the sequence $(\nabla u_n)_n$ is bounded in $L^{16}((0,T), L^8(\mathbb{R}^d))$, which implies that $(\nabla u_n)_n$ converges strongly to $\nabla u$ in $L^p((0,T) \times \mathbb{R}^d)$ for all $p \in [2, 8)$. We are then able to pass to the limit in (4.19) and conclude that the limit $u$ is a weak solution of (4.2). The remaining step is to establish the following time continuity:

$$u \in \mathcal{C}([0, T], H^1(\mathbb{R}^d)), \text{ and } u_t \in \mathcal{C}([0, T], L^2(\mathbb{R}^d)).$$

This property can be proved by following the arguments in Step 4 of Section 3.2.3 and using (4.9). This completes the proof of Proposition 4.2. $\square$

4.3. **Estimates.** In this section, we derive the estimates for $I(v, v)$, $\mu(\rho_0 J_v^{-1}, c_0)$ and $\lambda(\rho_0 J_v^{-1}, c_0)$ used in the proof of Proposition 4.1 above. We first recall some estimates for the flow of a Lipschitz vector field. We refer to [5, Lemma A.3] for the proof.

**Lemma 4.4.** Let $v$ be a vector field with $\nabla v \in L^1((0,T), L^\infty(\mathbb{R}^d))$. Suppose that

$$c = \int_0^T \|\nabla v\|_{L^\infty(\mathbb{R}^d)} < 1.$$

There exists a constant $k = k(c)$ such that the following estimates hold true:

$$\left\{
\begin{array}{ll}
\|I_d - \operatorname{Adj}(D\mathcal{X}_v(t)); I_d - A_v(t)\|_{L^\infty(\mathbb{R}^d)} & \le k \|\nabla v\|_{L^1((0,t), L^\infty(\mathbb{R}^d))}, \\
\|J_v^{\pm 1}(t) - 1\|_{L^\infty(\mathbb{R}^d)} & \le k \|\nabla v\|_{L^1((0,t), L^\infty(\mathbb{R}^d))}.
\end{array}
\right.$$

Moreover, one has the following:

$$\left\{
\begin{array}{ll}
\|\operatorname{Adj}(D\mathcal{X}_v(t)) D_{A_v(t)} w - D w\|_{L^p(\mathbb{R}^d)} & \le k \|\nabla v\|_{L^1((0,t), L^\infty(\mathbb{R}^d))} \|D w\|_{L^p(\mathbb{R}^d)}, \\
\|\operatorname{Adj}(D\mathcal{X}_v(t)) \operatorname{div}_{A_v(t)} w - \operatorname{div} w I_d\|_{L^p(\mathbb{R}^d)} & \le k \|\nabla v\|_{L^1((0,t), L^\infty(\mathbb{R}^d))} \|D w\|_{L^p(\mathbb{R}^d)}.
\end{array}
\right.$$

In the spirit of the above result, and owing to the fact that $\mathcal{C}_{pw,c_0}^\alpha(\mathbb{R}^d)$ is an algebra, one has the following estimates.
Step 12: $L_T^2 L^2(\mathbb{R}^d)$ norm estimate for $\sigma \partial_{tt} \delta I_3(v)$. We begin by computing the second time derivative of $\delta I_3(v)$ from (4.35) as:

$$ \partial_{tt} \delta I_3(v) = \partial_{tt} \delta I_{3,1}(v) + \partial_{tt} \delta I_{3,2}(v) + \partial_{tt} \delta I_{3,3}(v) + \partial_{tt} \delta I_{3,4}(v) + \partial_{tt} \delta I_{3,5}(v) $$

where

$$ \partial_{tt} \delta I_{3,1}(v) = \{ \partial_{tt} \operatorname{Adj}(D\mathcal{X}_{v_2}) - \partial_{tt} \operatorname{Adj}(D\mathcal{X}_{v_1}) \} (P(J_{v_2}^{-1} \rho_0, c_0) - \tilde{P}) \\ + \{ \partial_t \operatorname{Adj}(D\mathcal{X}_{v_2}) - \partial_t \operatorname{Adj}(D\mathcal{X}_{v_1}) \} \rho_0 \partial_{\rho} P(J_{v_2}^{-1} \rho_0, c_0) \partial_t J_{v_2}^{-1}, $$

$$ \partial_{tt} \delta I_{3,2}(v) = \{ \partial_t \operatorname{Adj}(D\mathcal{X}_{v_2}) - \partial_t \operatorname{Adj}(D\mathcal{X}_{v_1}) \} \rho_0 \partial_{\rho} P(J_{v_2}^{-1} \rho_0, c_0) \partial_t J_{v_2}^{-1} \\ + \{ \operatorname{Adj}(D\mathcal{X}_{v_2}) - \operatorname{Adj}(D\mathcal{X}_{v_1}) \} \rho_0^2 \partial_{\rho}^2 P(J_{v_2}^{-1} \rho_0, c_0) (\partial_t J_{v_2}^{-1})^2 \\ + \{ \operatorname{Adj}(D\mathcal{X}_{v_2}) - \operatorname{Adj}(D\mathcal{X}_{v_1}) \} \rho_0 \partial_{\rho} P(J_{v_2}^{-1} \rho_0, c_0) \partial_{tt} J_{v_2}^{-1}, $$

$$ \partial_{tt} \delta I_{3,3}(v) = \partial_{tt} \operatorname{Adj}(D\mathcal{X}_{v_1}) \{ P(J_{v_2}^{-1} \rho_0, c_0) - P(J_{v_1}^{-1} \rho_0) \} \\ + \rho_0 \partial_t J_{v_1}^{-1} \partial_t \operatorname{Adj}(D\mathcal{X}_{v_1}) \{ \partial_{\rho} P(J_{v_2}^{-1} \rho_0, c_0) - \partial_{\rho} P(J_{v_1}^{-1} \rho_0, c_0) \} \\ + \rho_0 \partial_{\rho} P(J_{v_2}^{-1} \rho_0, c_0) \partial_t \operatorname{Adj}(D\mathcal{X}_{v_1}) \{ \partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1} \}, $$

$$ \partial_{tt} \delta I_{3,4}(v) = \rho_0^2 \partial_{\rho}^2 P(J_{v_2}^{-1} \rho_0, c_0) \partial_t J_{v_2}^{-1} \operatorname{Adj}(D\mathcal{X}_{v_1}) \{ \partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1} \} \\ + \rho_0 \partial_{\rho} P(J_{v_2}^{-1} \rho_0, c_0) \partial_t \operatorname{Adj}(D\mathcal{X}_{v_1}) \{ \partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1} \} \\ + \rho_0 \partial_{\rho} P(J_{v_2}^{-1} \rho_0, c_0) \operatorname{Adj}(D\mathcal{X}_{v_1}) \{ \partial_{tt} J_{v_2}^{-1} - \partial_{tt} J_{v_1}^{-1} \} $$

and finally,

$$ \partial_{tt} \delta I_{3,5}(v) = \rho_0 \partial_{tt} J_{v_1}^{-1} \operatorname{Adj}(D\mathcal{X}_{v_1}) \{ \partial_{\rho} P(J_{v_2}^{-1} \rho_0, c_0) - \partial_{\rho} P(J_{v_1}^{-1} \rho_0, c_0) \} \\ + \rho_0 \partial_t J_{v_1}^{-1} \partial_t \operatorname{Adj}(D\mathcal{X}_{v_1}) \{ \partial_{\rho} P(J_{v_2}^{-1} \rho_0, c_0) - \partial_{\rho} P(J_{v_1}^{-1} \rho_0, c_0) \} \\ + \rho_0^2 \partial_t J_{v_1}^{-1} \partial_t J_{v_2}^{-1} \operatorname{Adj}(D\mathcal{X}_{v_1}) \{ \partial_{\rho}^2 P(J_{v_2}^{-1} \rho_0, c_0) - \partial_{\rho}^2 P(J_{v_1}^{-1} \rho_0, c_0) \} \\ + \rho_0 \partial_t J_{v_1}^{-1} \partial_{\rho}^2 P(J_{v_1}^{-1} \rho_0, c_0) \operatorname{Adj}(D\mathcal{X}_{v_1}) \{ \partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1} \}. $$

We have the following estimates:

$$ \|\sigma \partial_{tt} \delta I_{3,5}(v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla \delta v\|_{L_1^1 L^\infty(\mathbb{R}^d)} (\|\sigma \nabla \partial_t v_1\|_{L_T^2 L^2(\mathbb{R}^d)} + \|\nabla v_1\|_{L_4^4 L^4(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^4 L^4(\mathbb{R}^d)}) \\ + \|\nabla v_1\|_{L_4^4 L^4(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^4 L^4(\mathbb{R}^d)}, $$

$$ \|\sigma \partial_{tt} \delta I_{3,4}(v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla \delta v\|_{L_1^1 L^\infty(\mathbb{R}^d)} (\|\sigma \nabla \partial_t v_2, \sigma \nabla \partial_t v_1\|_{L_T^2 L^2(\mathbb{R}^d)} + \|\nabla v_2, \nabla v_1\|_{L_4^4 L^4(\mathbb{R}^d)} \|\nabla v_2, \nabla v_1\|_{L_T^4 L^4(\mathbb{R}^d)}) \\ + \|\nabla v_1, \nabla v_2\|_{L_4^4 L^4(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^4 L^4(\mathbb{R}^d)} + \|\sigma \nabla \partial_t \delta v\|_{L_T^2 L^2(\mathbb{R}^d)}, $$

$$ \|\sigma \partial_{tt} \delta I_{3,3}(v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla \delta v\|_{L_1^1 L^\infty(\mathbb{R}^d)} (\|\nabla \partial_t v_1\|_{L_T^2 L^2(\mathbb{R}^d)} + \|\nabla v_1\|_{L_4^4 L^4(\mathbb{R}^d)} \|\nabla v_2, \nabla v_1\|_{L_T^4 L^4(\mathbb{R}^d)}) \\ + \|\nabla \delta v\|_{L_T^4 L^4(\mathbb{R}^d)} \|\nabla v_1\|_{L_4^4 L^4(\mathbb{R}^d)}, $$

$$ \|\sigma \partial_{tt} \delta I_{3,2}(v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla \delta v\|_{L_1^1 L^\infty(\mathbb{R}^d)} (\|\sigma \nabla \partial_t v_2\|_{L_T^2 L^2(\mathbb{R}^d)} + \|\nabla v_2\|_{L_4^4 L^4(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^4 L^4(\mathbb{R}^d)}) \\ + \|\nabla \delta v\|_{L_T^4 L^4(\mathbb{R}^d)} \|\nabla v_2\|_{L_4^4 L^4(\mathbb{R}^d)}. $$

It only remains the estimate for $\partial_{tt} \delta I_{3,1}(v)$. The following inequality holds for its last term:

$$ (4.37) \quad \|\sigma\{\partial_t \operatorname{Adj}(D\mathcal{X}_{v_2}) - \partial_t \operatorname{Adj}(D\mathcal{X}_{v_1})\} \rho_0 \partial_{\rho} P(J_{v_2}^{-1} \rho_0, c_0) \partial_t J_{v_2}^{-1}\|_{L_T^2 L^2(\mathbb{R}^d)} \\ \lesssim \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_4^4 L^4(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^4 L^4(\mathbb{R}^d)} + \|\nabla \delta v\|_{L_T^4 L^4(\mathbb{R}^d)} \|\nabla v_2\|_{L_4^4 L^4(\mathbb{R}^d)} $$
**Lemma 4.5.** Let $v$ be a vector field fulfilling $\nabla v \in L^1((0,T), \mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d))$. Suppose that

$$ (4.23) \qquad c = \int_0^T \|\nabla v\|_{\mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} < 1. $$

There exists a constant $k = k(c)$ such that the following estimates holds true:

$$ \left\{ \begin{array}{l} \|I_d - \operatorname{Adj}(D\mathcal{X}_v(t)); I_d - A_v(t)\|_{\mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} \le k\|\nabla v\|_{L^1((0,t), \mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d))}, \\ \|J_v^{\pm 1}(t) - 1\|_{\mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} \le k\|\nabla v\|_{L^1((0,t), \mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d))}. \end{array} \right. $$

Moreover, one has the following:

$$ \left\{ \begin{array}{l} \|\operatorname{Adj}(D\mathcal{X}_v(t))\mathbb{D}_{A_v(t)}w - \mathbb{D}w\|_{\mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} \le k\|\nabla v\|_{L^1((0,t), \mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d))}\|\mathbb{D}w\|_{\mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)}, \\ \|\operatorname{Adj}(D\mathcal{X}_v(t))\operatorname{div}_{A_v(t)}w - \operatorname{div} w I_d\|_{\mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} \le k\|\nabla v\|_{L^1((0,t), \mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d))}\|\mathbb{D}w\|_{\mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)}. \end{array} \right. $$

Notice that we will need to estimate the first and the second time derivative of $I(v, v)$ as it is required in the definition of $Y_T$. From (2.6) we have $\operatorname{Adj}(D\mathcal{X}_v) \cdot D\mathcal{X}_v = \operatorname{det}(D\mathcal{X}_v)I_d$ and $\partial_t D\mathcal{X}_v = Dv$ and consequently:

**Lemma 4.6.** The following expressions hold true:

$$ (4.24) \qquad \partial_t J_v^{-1}(t, y) = -J_v^{-1}(t, y) \operatorname{div}_{A_v}(v), $$

$$ (4.25) \qquad \partial_t A_v = -A_v \cdot Dv \cdot A_v, $$

$$ (4.26) \qquad \partial_t \operatorname{Adj}(D\mathcal{X}_v) = (Dv: \operatorname{Adj}(D\mathcal{X}_v))A_v - \operatorname{Adj}(D\mathcal{X}_v) \cdot Dv \cdot A_v. $$

Next, taking the time derivative of the above quantities, one has:

$$ (4.27) \qquad J_v \partial_t t J_v^{-1} = (\operatorname{div}_{A_v}(v))^2 - Dv_t: A_v + Dv: (A_v \cdot Dv \cdot A_v), $$

$$ (4.28) \qquad \partial_{tt} A_v = 2A_v \cdot (Dv \cdot A_v)^2 - A_v \cdot Dv_t \cdot A_v, $$

$$ \partial_{tt} \operatorname{Adj}(D\mathcal{X}_v) = (Dv_t: \operatorname{Adj}(D\mathcal{X}_v))A_v + J_v(Dv: A_v)^2 A_v - (Dv: (\operatorname{Adj}(D\mathcal{X}_v) \cdot Dv \cdot A_v))A_v $$

$$ (4.29) \qquad -2(Dv: \operatorname{Adj}(D\mathcal{X}_v))A_v \cdot Dv \cdot A_v + 2J_v A_v \cdot (Dv \cdot A_v)^2 - \operatorname{Adj}(D\mathcal{X}_v) \cdot Dv_t \cdot A_v. $$

We now proceed to the estimates for $I(v, v)$, $\mu(\rho_0 J_v^{-1}, c_0)$ and $\lambda(\rho_0 J_v^{-1}, c_0)$.

4.3.1. Lebesgue and Hölder norm estimates for $I(v, v)$. We first recall

$$ (4.30) \qquad I(v, v) = I_1(v, v) + I_2(v, v) + I_3(v) $$

where

$$ (4.31) \qquad \left\{ \begin{array}{l} I_1(v, v) = 2\mu(J_v^{-1}\rho_0, c_0)\{\operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v - \mathbb{D}v\}, \\ I_2(v, v) = \lambda(J_v^{-1}\rho_0, c_0)\{\operatorname{Adj}(D\mathcal{X}_v)\operatorname{div}_{A_v}v - \operatorname{div} v\}, \\ I_3(v) = \operatorname{Adj}(D\mathcal{X}_v)\{P(J_v^{-1}\rho_0, c_0) - \tilde{P}\}I_d. \end{array} \right. $$

From Lemma 4.4, we have the following estimates for the first two terms:

$$ \|I_1(v, v); I_2(v, v)\|_{L^m_t L^q_s(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L^1_t L^\infty(\mathbb{R}^d)} \|\nabla v\|_{L^m_t L^q_s(\mathbb{R}^d)} $$

for all $2 \le m, q \le \infty$. Regarding the Hölder estimates, one has from Lemma 4.5 the following:

$$ \|\sigma^{r/4}(I_1(v, v); I_2(v, v))\|_{L^4_T \mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L^1_T \mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} \|\sigma^{r/4} \nabla v\|_{L^4_T \mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)}. $$

It now remains the norm of $I_3(v)$. We may first write:

$$ I_3(v) = \operatorname{Adj}(D\mathcal{X}_v)\{P(J_v^{-1}\rho_0, c_0) - P(\rho_0, c_0) + P(\rho_0, c_0) - \tilde{P}\} $$

and thus, we deduce from Lemma 4.4, the following:

$$ \|I_3(v)\|_{L^m_t L^q_s(\mathbb{R}^d)} \lesssim T^{1/m} \|\nabla v\|_{L^1_t L^q_s(\mathbb{R}^d)} + T^{1/m} \|P(\rho_0, c_0) - \tilde{P}\|_{L^q(\mathbb{R}^d)} $$

and finally, from Lemma 4.5, we have the following Hölder estimate for $I_3(v)$:

$$ \|\sigma^{r/4} I_3(v)\|_{L^4_T \mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} \lesssim T^{1/4} \|\nabla v\|_{L^1_T \mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} + T^{1/4} \|P(\rho_0, c_0) - \tilde{P}\|_{\mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)}. $$

We gather all of these estimates in order to obtain:

$$ \|I(v, v)\|_{L^m_t L^q_s(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L^1_t L^\infty(\mathbb{R}^d)} \|\nabla v\|_{L^m_t L^q_s(\mathbb{R}^d)} + T^{1/m} \|\nabla v\|_{L^1_t L^q_s(\mathbb{R}^d)} + T^{1/m} \|P(\rho_0, c_0) - \tilde{P}\|_{L^q(\mathbb{R}^d)} $$
and in order to provide $L_T^2 L^2(\mathbb{R}^d)$ estimate for the first term, we take differences in (4.10) and we obtain:

$$
\begin{align*}
\partial_{tt} \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_2}) - \partial_{tt} \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1}) &= D\partial_t \delta v: \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_2} A_{v_2} + D\partial_t v_1: \{\operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_2}) - \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1})\} \cdot A_{v_2} \\
&\quad + D\partial_t v_1: \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1}) \cdot \{A_{v_2} - A_{v_1}\} - 2D\delta v: \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_2}) A_{v_2} \cdot Dv_2 \cdot A_{v_2} \\
&\quad - 2Dv_1: \{\operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_2}) - \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1})\} A_{v_2} \cdot Dv_2 \cdot A_{v_2} - 2Dv_1: \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1}) \{A_{v_2} - A_{v_1}\} \cdot Dv_2 \cdot A_{v_2} \\
&\quad - 2Dv_1: \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1}) A_{v_1} \cdot \delta Dv \cdot A_{v_2} - 2Dv_1: \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1}) A_{v_1} \cdot Dv_1 \cdot \{A_{v_2} - A_{v_1}\} \\
&\quad + \{J_{v_2} - J_{v_1}\}(Dv_2: A_{v_2})^2 A_{v_2} + J_{v_1} \{(Dv_2: A_{v_2})^2 - (Dv_1: A_{v_1})^2\} \\
&\quad + J_{v_1}(Dv_1: A_{v_1})^2 \{A_{v_2} - A_{v_1}\} + 2\{J_{v_2} - J_{v_1}\} A_{v_2} \cdot (Dv_2 \cdot A_{v_2})^2 \\
&\quad + 2J_{v_1} \{A_{v_2} - A_{v_1}\} \cdot (Dv_2 \cdot A_{v_2})^2 + 2J_{v_1} A_{v_1} \cdot \{(Dv_2 \cdot A_{v_2})^2 - (Dv_1 \cdot A_{v_1})^2\} \\
&\quad - D\delta v: (\operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_2}) \cdot Dv_2 \cdot A_{v_2}) A_{v_2} - Dv_1: \{\operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_2} - \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1}) \cdot Dv_2 \cdot A_{v_2}) A_{v_2} \\
&\quad - Dv_1: (\operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1} \cdot D\delta v \cdot A_{v_2}) A_{v_2} - Dv_1: (\operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1} \cdot Dv_1 \cdot \{A_{v_2} - A_{v_1}\}) A_{v_2} \\
&\quad - Dv_1: (\operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1} \cdot Dv_1 \cdot A_{v_1}) \{A_{v_2} - A_{v_1}\} - \{\operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_2}) - \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1})\} \cdot D\partial_t v_2 \cdot A_{v_2} \\
&\quad - \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1}) \cdot D\partial_t \delta v \cdot A_{v_2} - \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1}) \cdot D\partial_t v_1 \cdot \{A_{v_2} - A_{v_1}\}.
\end{align*}
$$

It then follows that

$$
\begin{align}
& \|\sigma\{\partial_{tt} \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_2}) - \partial_{tt} \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1})\}(P(J_{v_2}^{-1}\rho_0, c_0) - \tilde{P})\|_{L_T^2 L^2(\mathbb{R}^d)} \nonumber \\
& \lesssim \|\sigma \nabla \partial_t \delta v\|_{L_T^2 L^2(\mathbb{R}^d)} + \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\sigma(\nabla \partial_t v_1, \nabla \partial_t v_2)\|_{L_T^2 L^2(\mathbb{R}^d)} \nonumber \\
& \quad + \|\nabla \delta v\|_{L_T^4 L^4(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^4 L^4(\mathbb{R}^d)} + \|\nabla \delta v\|_{L_T^4 L^\infty(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^4 L^4(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^4 L^4(\mathbb{R}^d)}. \tag{4.38}
\end{align}
$$

We finally gather (4.37) and (4.38) in order to obtain the $L_T^2 L^2(\mathbb{R}^d)$ norm estimate for $\sigma \partial_{tt} \delta I_{3,1}(v)$. We turn to the computations of the norms of $\delta I_1(v, v)$ and $\delta I_2(v, v)$.

Step 13: Lebesgue estimates for $\delta I_1(v, v)$ and $\delta I_2(v, v)$. As above, we begin computations by writing:

$$
\begin{align}
I_1(v_2, v_2) - I_1(v_1, v_1) &= 2\{\mu(J_{v_2}^{-1}\rho_0, c_0) - \mu(J_{v_1}^{-1}\rho_0, c_0)\}\{\operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_2})\mathbb{D}_{A_{v_2}}v_2 - \mathbb{D}v_2\} \nonumber \\
&\quad + 2\mu(J_{v_1}^{-1}\rho_0, c_0)\{\operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_2})\mathbb{D}_{A_{v_2}}\delta v - \mathbb{D}\delta v\} \nonumber \\
&\quad + 2\mu(J_{v_1}^{-1}\rho_0, c_0)\{\operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_2}) - \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1})\}\mathbb{D}_{A_{v_2}}v_1 \nonumber \\
&\quad + 2\mu(J_{v_1}^{-1}\rho_0, c_0)\operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1})\{\mathbb{D}_{A_{v_2}} - \mathbb{D}_{A_{v_1}}\}v_1, \tag{4.39}
\end{align}
$$

thus, it holds that:

$$
\|\delta I_1(v, v)\|_{L_T^m L^q(\mathbb{R}^d)} \lesssim \|\nabla \delta v\|_{L_T^{1/4} L^{\infty}(\mathbb{R}^d)} \left( \|\nabla v_2\|_{L_T^1 L^{\infty}(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^m L^q(\mathbb{R}^d)} + \|\nabla v_1\|_{L_T^m L^q(\mathbb{R}^d)} \right) \\
+ \|\nabla v_2\|_{L_T^1 L^{\infty}(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^m L^q(\mathbb{R}^d)}
$$

and

$$
\|\sigma^{r/4} \delta I_1(v, v)\|_{L_T^4 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \lesssim \|\nabla \delta v\|_{L_T^1 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \left( \|\nabla v_2\|_{L_T^1 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \|\sigma^{r/4} \nabla v_2\|_{L_T^4 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} + \|\sigma^{r/4} \nabla v_1\|_{L_T^4 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \right) \\
+ \|\nabla v_2\|_{L_T^1 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \|\sigma^{r/4} \nabla \delta v\|_{L_T^4 \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)}.
$$

Step 14: Estimate for the time derivative of $\delta I_1(v, v)$. In the following lines we will estimate the $L_T^2 L^2(\mathbb{R}^d)$ norm of $\partial_t \delta I_1(v, v)$ and the $L_T^{8/d} L^4(\mathbb{R}^d)$ norm of $\sqrt{\sigma} \partial_t \delta I_1(v, v)$. We begin by writing:

$$
(4.40) \qquad \partial_t \delta I_1(v, v) = \partial_t \delta I_{1,6}(v, v) + \partial_t \delta I_{1,7}(v, v) + \partial_t \delta I_{1,8}(v, v) + \partial_t \delta I_{1,9}(v, v)
$$

according to the expression (4.39). We will only estimate the first three terms since the last one is similar to the penultimate one. We express:

$$
\begin{align*}
\partial_t \delta I_{1,8}(v, v) &= \rho_0 \partial_t J_{v_1}^{-1} \partial_\rho \mu(J_{v_1}^{-1} \rho_0, c_0) \{\operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_2}) - \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1})\} \mathbb{D}_{A_{v_2}} v_1 \\
&\quad + \mu(J_{v_1}^{-1} \rho_0, c_0) \{\partial_t \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_2}) - \partial_t \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1})\} \mathbb{D}_{A_{v_2}} v_1 \\
&\quad + \mu(J_{v_1}^{-1} \rho_0, c_0) \{\operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_2}) - \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_1})\} \partial_t \mathbb{D}_{A_{v_2}} v_1, \\
\partial_t \delta I_{1,7}(v, v) &= \rho_0 \partial_t J_{v_1}^{-1} \partial_\rho \mu(J_{v_1}^{-1} \rho_0, c_0) \{\operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \delta v - \mathbb{D} \delta v\} \\
&\quad + \mu(J_{v_1}^{-1} \rho_0, c_0) \partial_t \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \delta v \\
&\quad + \mu(J_{v_1}^{-1} \rho_0, c_0) \{\operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \partial_t \delta v - \mathbb{D} \partial_t \delta v\} \\
&\quad + \mu(J_{v_1}^{-1} \rho_0, c_0) \operatorname{Adj}(\mathcal{D}\mathcal{X}_{v_2}) \{\mathbb{D} \delta v \partial_t A_{v_2} + \partial_t A_{v_2}^T \nabla \delta v\},
\end{align*}
$$
and

$$ \|\sigma^{r/4} I(v, v)\|_{L_T^4 \mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L_T^1 \mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} \|\sigma^{r/4} \nabla v\|_{L_T^4 \mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} + T^{1/4} \|\nabla v\|_{L_T^1 \mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} \\ + T^{1/4} \|P(\rho_0, c_0) - \tilde{P}\|_{\mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)}. $$

This proves (4.6), (4.7), and (4.8), and we now move on to estimating the time derivative of $I(v, v)$.

4.3.2. Lebesgue norm estimates for $\partial_t I(v, v)$. In this step, we will estimate the $L_T^2 L^2(\mathbb{R}^d)$ of $\partial_t I(v, v)$ and the $L_T^{8/d} L^4(\mathbb{R}^d)$ norm of $\sigma^{\frac{d}{4}} \partial_t I(v, v)$. Let us begin by computing the time derivative of the third term $I_3(v)$. We have:

$$ (4.32) \qquad \partial_t I_3(v) = \partial_t \operatorname{Adj}(D\mathcal{X}_v)\{P(J_v^{-1}\rho_0, c_0) - \tilde{P}\} + \rho_0 \partial_\rho P(J_v^{-1}\rho_0, c_0) \operatorname{Adj}(D\mathcal{X}_v)\partial_t J_v^{-1} $$

and thus one obtains from (4.24) and (4.26) the following:

$$ (4.33) \qquad \|\partial_t I_3(v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L_T^2 L^2(\mathbb{R}^d)} (\|\nabla v\|_{L_T^1 L^\infty(\mathbb{R}^d)} + 1). $$

On the other hand, the following estimate holds true for $\partial_t I_3(v)$:

$$ \|\sigma^{\frac{d}{4}} \partial_t I_3(v)\|_{L_T^{8/d} L^4(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L_T^{8/d} L^4(\mathbb{R}^d)} (\|\nabla v\|_{L_T^1 L^\infty(\mathbb{R}^d)} + 1). $$

We turn to the estimate of the norms of $\partial_t I_1(v, v)$ and $\partial_t I_2(v, v)$. First, we begin by writing:

$$ (4.34) \qquad \begin{aligned} \partial_t I_1(v, v) = & 2\rho_0 \partial_\rho \mu(J_v^{-1}\rho_0, c_0) \partial_t J_v^{-1} \{\operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v - \mathbb{D}v\} \\ & + 2\mu(J_v^{-1}\rho_0, c_0) \{\operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v_t - \mathbb{D}v_t\} \\ & + 2\mu(J_v^{-1}\rho_0, c_0) \partial_t \operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v \\ & + 2\mu(J_v^{-1}\rho_0, c_0) \operatorname{Adj}(D\mathcal{X}_v) (Dv \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v). \end{aligned} $$

Using Lemma 4.4 and the expression of the time derivative of $J_v^{-1}$ (4.24), the following bounds hold for the first term of the expression above:

$$ \|2\rho_0 \partial_\rho \mu(J_v^{-1}\rho_0, c_0) \partial_t J_v^{-1} \{\operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v - \mathbb{D}v\} \|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v\|_{L_T^4 L^4(\mathbb{R}^d)}^2 $$

and

$$ \|\sigma^{\frac{d}{4}} \rho_0 \partial_\rho \mu(J_v^{-1}\rho_0, c_0) \partial_t J_v^{-1} \{\operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v - \mathbb{D}v\} \|_{L_T^{8/d} L^4(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v\|_{L_T^{16/d} L^8(\mathbb{R}^d)}^2. $$

The same argument helps us to control the second term of the right hand side of (4.34) as follows:

$$ \|2\mu(J_v^{-1}\rho_0, c_0)\{\operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v_t - \mathbb{D}v_t\}\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v_t\|_{L_T^2 L^2(\mathbb{R}^d)} $$

and

$$ \|\sigma^{\frac{d}{4}} \mu(J_v^{-1}\rho_0, c_0)\{\operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v_t - \mathbb{D}v_t\}\|_{L_T^{8/d} L^4(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\sigma^{\frac{d}{4}} \nabla v_t\|_{L_T^{8/d} L^4(\mathbb{R}^d)} $$

Next, from the expression of the time derivative of $A_v$, (4.25), we have:

$$ Dv \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v = -(Dv \cdot A_v)^2 - (A_v^T \cdot \nabla v)^2 $$

thus, the others terms of (4.34) are controlled as follows:

$$ \|\mu(J_v^{-1}\rho_0, c_0) \partial_t \operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v; \mu(J_v^{-1}\rho_0, c_0) \operatorname{Adj}(D\mathcal{X}_v)\{Dv \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v\} \|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L_T^4 L^4(\mathbb{R}^d)}^2 $$

and

$$ \|\sigma^{\frac{d}{4}} \mu(J_v^{-1}\rho_0, c_0) \partial_t \operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v; \sigma^{\frac{d}{4}} \mu(J_v^{-1}\rho_0, c_0) \operatorname{Adj}(D\mathcal{X}_v)\{Dv \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v\} \|_{L_T^{8/d} L^4(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L_T^{16/d} L^8(\mathbb{R}^d)}^2 $$

Gathering all of these computations and noticing that the estimate for $\partial_t I_1(v, v)$ and $\sigma^{\frac{d}{4}} I_1(v, v)$ also holds for $\partial_t I_2(v, v)$ and $\sigma^{\frac{d}{4}} I_2(v, v)$ respectively, one has:

$$ \|\partial_t I(v, v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim (\|\nabla v\|_{L_T^2 L^2(\mathbb{R}^d)} + \|\nabla v\|_{L_T^4 L^4(\mathbb{R}^d)}^2) (\|\nabla v\|_{L_T^1 L^\infty(\mathbb{R}^d)} + 1) + \|\nabla v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v_t\|_{L_T^2 L^2(\mathbb{R}^d)} $$

and

$$ \|\sigma^{\frac{d}{4}} \partial_t I(v, v)\|_{L_T^{8/d} L^4(\mathbb{R}^d)} \lesssim (\|\nabla v\|_{L_T^{8/d} L^4(\mathbb{R}^d)} + \|\nabla v\|_{L_T^{16/d} L^8(\mathbb{R}^d)}^2) (\|\nabla v\|_{L_T^1 L^\infty(\mathbb{R}^d)} + 1) \\ + \|\nabla v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\sigma^{\frac{d}{4}} \nabla v_t\|_{L_T^{8/d} L^4(\mathbb{R}^d)} $$
$$
\begin{equation} \tag{4.41}
\begin{aligned}
\partial_t \delta I_{1,6}(v,v) ={}& \rho_0 \{\partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1}\} \partial_\rho \mu(J_{v_2}^{-1} \rho_0, c_0) \{\operatorname{Adj}(D\mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} v_2 - \mathbb{D}v_2\} \\
& + \rho_0 \partial_t J_{v_1}^{-1} \{\partial_\rho \mu(J_{v_2}^{-1} \rho_0, c_0) - \partial_\rho \mu(J_{v_1}^{-1} \rho_0, c_0)\} \{\operatorname{Adj}(D\mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} v_2 - \mathbb{D}v_2\} \\
& + \{\mu(J_{v_2}^{-1} \rho_0, c_0) - \mu(J_{v_1}^{-1} \rho_0, c_0)\} \{\operatorname{Adj}(D\mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \partial_t v_2 - \mathbb{D}\partial_t v_2\} \\
& + \{\mu(J_{v_2}^{-1} \rho_0, c_0) - \mu(J_{v_1}^{-1} \rho_0, c_0)\} \partial_t \operatorname{Adj}(D\mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} v_2 \\
& + \{\mu(J_{v_2}^{-1} \rho_0, c_0) - \mu(J_{v_1}^{-1} \rho_0, c_0)\} \operatorname{Adj}(D\mathcal{X}_{v_2}) \{Dv_2 \partial_t A_{v_2} + \partial_t A_{v_2}^T \nabla v_2\}.
\end{aligned}
\end{equation}
$$

The same arguments as in the preceding steps help us to obtain the following:

$$
\|\partial_t \delta I_{1,8}(v, v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} (\|\nabla \partial_t v_1\|_{L_T^2 L^2(\mathbb{R}^d)} + \|\nabla v_1\|_{L_T^4 L^4(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^4 L^4(\mathbb{R}^d)}) \\
+ \|\nabla \delta v\|_{L_T^4 L^4(\mathbb{R}^d)} \|\nabla v_1\|_{L_T^4 L^4(\mathbb{R}^d)},
$$

$$
\|\sqrt{\sigma} \partial_t \delta I_{1,8}(v, v)\|_{L_T^{8/d} L^4(\mathbb{R}^d)} \lesssim \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \left( \|\sqrt{\sigma} \nabla \partial_t v_1\|_{L_T^{8/d} L^4(\mathbb{R}^d)} + \|\nabla v_1\|_{L_T^{16/d} L^8(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^{16/d} L^8(\mathbb{R}^d)} \right) \\
+ \|\nabla \delta v\|_{L_T^{16/d} L^8(\mathbb{R}^d)} \|\nabla v_1\|_{L_T^{16/d} L^8(\mathbb{R}^d)},
$$

$$
\|\partial_t \delta I_{1,7}(v, v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla v_2\|_{L_T^1 L^\infty(\mathbb{R}^d)} (\|\nabla \partial_t \delta v\|_{L_T^2 L^2(\mathbb{R}^d)} + \|\nabla \delta v\|_{L_T^4 L^4(\mathbb{R}^d)} \|\nabla v_1\|_{L_T^4 L^4(\mathbb{R}^d)}) \\
+ \|\nabla \delta v\|_{L_T^4 L^4(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^4 L^4(\mathbb{R}^d)},
$$

$$
\|\sqrt{\sigma} \partial_t \delta I_{1,7}(v, v)\|_{L_T^{8/d} L^4(\mathbb{R}^d)} \lesssim \|\nabla v_2\|_{L_T^1 L^\infty(\mathbb{R}^d)} \left( \|\sqrt{\sigma} \nabla \partial_t \delta v\|_{L_T^{8/d} L^4(\mathbb{R}^d)} + \|\nabla \delta v\|_{L_T^{16/d} L^8(\mathbb{R}^d)} \|\nabla v_1\|_{L_T^{16/d} L^8(\mathbb{R}^d)} \right) \\
+ \|\nabla \delta v\|_{L_T^{16/d} L^8(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^{16/d} L^8(\mathbb{R}^d)},
$$

$$
\|\partial_t \delta I_{1,6}(v, v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^1 L^\infty(\mathbb{R}^d)} (\|\nabla \partial_t v_2\|_{L_T^2 L^2(\mathbb{R}^d)} + \|\nabla v_2\|_{L_T^4 L^4(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^4 L^4(\mathbb{R}^d)}) \\
+ \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^2 L^4(\mathbb{R}^d)} + \|\nabla v_2\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^4 L^4(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^4 L^4(\mathbb{R}^d)},
$$

$$
\|\sqrt{\sigma} \partial_t \delta I_{1,6}(v, v)\|_{L_T^{8/d} L^4(\mathbb{R}^d)} \lesssim \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^1 L^\infty(\mathbb{R}^d)} \left( \|\sqrt{\sigma} \nabla \partial_t v_2\|_{L_T^{8/d} L^4(\mathbb{R}^d)} \right. \\
\left. \qquad + \|\nabla v_2\|_{L_T^{16/d} L^8(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^{16/d} L^8(\mathbb{R}^d)} \right) + \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^{26/d} L^8(\mathbb{R}^d)} \\
\left. + \|\nabla v_2\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^{16/d} L^8(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^{16/d} L^8(\mathbb{R}^d)} \right).
$$

Step 15: $L_T^2 L^2(\mathbb{R}^d)$ norm estimate for $\sigma \partial_{tt} \delta I_{1,6}(v, v)$. According to (4.41), we can write:

$$
(4.42) \quad \partial_{tt} \delta I_{1,6}(v, v) = \partial_{tt} \delta I_{1,6,1}(v, v) + \partial_{tt} \delta I_{1,6,2}(v, v) + \partial_{tt} \delta I_{1,6,3}(v, v) + \partial_{tt} \delta I_{1,6,4}(v, v) + \partial_{tt} \delta I_{1,6,5}(v, v)
$$

where

$$
\begin{aligned}
\partial_{tt} \delta I_{1,6,1}(v, v) ={}& \rho_0 \{\partial_{tt} J_{v_2}^{-1} - \partial_{tt} J_{v_1}^{-1}\} \partial_\rho \mu(J_{v_2}^{-1} \rho_0, c_0) \{\operatorname{Adj}(D\mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} v_2 - \mathbb{D}v_2\} \\
& + \rho_0^2 \{\partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1}\} \partial_\rho^2 \mu(J_{v_2}^{-1} \rho_0, c_0) \partial_t J_{v_2}^{-1} \{\operatorname{Adj}(D\mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} v_2 - \mathbb{D}v_2\} \\
& + \rho_0 \{\partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1}\} \partial_\rho \mu(J_{v_2}^{-1} \rho_0, c_0) \{\operatorname{Adj}(D\mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \partial_t v_2 - \mathbb{D}\partial_t v_2\} \\
& + \rho_0 \{\partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1}\} \partial_\rho \mu(J_{v_2}^{-1} \rho_0, c_0) \partial_t \operatorname{Adj}(D\mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} v_2 \\
& + \rho_0 \{\partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1}\} \partial_\rho \mu(J_{v_2}^{-1} \rho_0, c_0) \operatorname{Adj}(D\mathcal{X}_{v_2}) \{Dv_2 \cdot \partial_t A_{v_2} + \partial_t A_{v_2}^T \cdot \nabla v_2\}
\end{aligned}
$$

with the following $L_T^2 L^2(\mathbb{R}^d)$ norm estimate:

$$
\|\sigma \partial_{tt} \delta I_{1,6,1}(v, v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla v_2\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \\
\times (\|\nabla v_1, \nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} + \|\nabla \delta v\|_{L_T^6 L^6(\mathbb{R}^d)}) \\
+ \|\nabla v_2\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v_2, \nabla v_1\|_{L_T^\infty L^4(\mathbb{R}^d)} \|\sigma \nabla \partial_t v_1, \sigma \nabla \partial_t v_2\|_{L_T^2 L^4(\mathbb{R}^d)} \\
+ \|\nabla v_2\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^4 L^4(\mathbb{R}^d)} \|\sigma \nabla \partial_t \delta v\|_{L_T^2 L^4(\mathbb{R}^d)} \\
+ \|\nabla v_2\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\sigma \nabla \partial_t v_2\|_{L_T^2 L^4(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^\infty L^4(\mathbb{R}^d)}.
$$
This leads to (4.9) and (4.10).

4.3.3. $L_T^2 L^2(\mathbb{R}^d)$ norm estimate for $\sigma \partial_{tt} I(v, v)$. As above, we may begin by computing the second time derivative of $I(v, v)$. First, from (4.32), we have the following expression for $\partial_{tt} I_3(v)$:

$$ \partial_{tt} I_3(v, v) = \partial_{tt} I_{3,1} + \partial_{tt} I_{3,2} $$

where

$$ (4.35) \qquad \partial_{tt} I_{3,1} = \partial_{tt} \operatorname{Adj}(D\mathcal{X}_v)\{P(J_v^{-1}\rho_0, c_0) - \tilde{P}\} + \rho_0 \partial_{\rho} P(J_v^{-1}\rho_0, c_0) \partial_t J_v^{-1} \partial_t \operatorname{Adj}(D\mathcal{X}_v) $$

and

$$ (4.36) \qquad \begin{aligned} \partial_{tt} I_{3,2} = & \rho_0^2 \partial_{\rho}^2 P(J_v^{-1}\rho_0, c_0)(\partial_t J_v^{-1})^2 \operatorname{Adj}(D\mathcal{X}_v) + \rho_0 \partial_{\rho} P(J_v^{-1}\rho_0, c_0) \partial_t J_v^{-1} \partial_t \operatorname{Adj}(D\mathcal{X}_v) \\ & + \rho_0 \partial_{\rho} P(J_v^{-1}\rho_0, c_0) \partial_{tt} J_v^{-1} \operatorname{Adj}(D\mathcal{X}_v). \end{aligned} $$

Using the expression of the time derivative of $J_v^{-1}$, (4.24) and of $\operatorname{Adj}(D\mathcal{X}_v)$, (4.26), the last term of (4.35) and the first two terms of the right hand side of (4.36) can be estimated as follows:

$$ \|\sigma \rho_0 \partial_{\rho} P(J_v^{-1} \rho_0, c_0) \partial_t J_v^{-1} \partial_t \operatorname{Adj}(D\mathcal{X}_v) - \tilde{P}\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L_T^4 L^4(\mathbb{R}^d)}^2 $$

Owing to the expression of the second time derivative of $J_v^{-1}$ (4.27) and $\operatorname{Adj}(D\mathcal{X}_v)$ (4.29), the remainder terms of $\partial_{tt} I_3(v)$ are estimated as follows:

$$ \|\sigma \partial_{tt} \operatorname{Adj}(D\mathcal{X}_v)\{P(J_v^{-1}\rho_0, c_0) - \tilde{P}\}, \sigma \rho_0 \partial_{\rho} P(J_v^{-1}\rho_0, c_0) \partial_{tt} J_v^{-1} \operatorname{Adj}(D\mathcal{X}_v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\sigma \nabla v_t\|_{L_T^2 L^2(\mathbb{R}^d)} + \|\nabla v\|_{L_T^4 L^4(\mathbb{R}^d)}^2 $$

Gathering all of these estimates, one has:

$$ (4.37) \qquad \|\sigma \partial_{tt} I_3(v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\sigma \nabla v_t\|_{L_T^2 L^2(\mathbb{R}^d)} + \|\nabla v\|_{L_T^4 L^4(\mathbb{R}^d)}^2 $$

and we turn to the $L_T^2 L^2(\mathbb{R}^d)$ norm estimate of $\sigma \partial_{tt} I_1(v, v)$. From (4.34), we express:

$$ \partial_{tt} I_1(v, v) = \sum_{k=0}^{4} \partial_{tt} I_{1,k}(v, v) $$

where:

$$ \begin{aligned} \partial_{tt} I_{1,1}(v, v) = & 2(\rho_0)^2 \partial_{\rho}^2 \mu(J_v^{-1}\rho_0, c_0)(\partial_t J_v^{-1})^2 \{\operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v - \mathbb{D}v\} \\ & + 2\rho_0 \partial_{\rho} \mu(J_v^{-1}\rho_0, c_0) \partial_{tt} J_v^{-1} \{\operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v - \mathbb{D}v\} \\ & + 2\rho_0 \partial_{\rho} \mu(J_v^{-1}\rho_0, c_0) \partial_t J_v^{-1} \{\operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v_t - \mathbb{D}v_t\} \\ & + 2\rho_0 \partial_{\rho} \mu(J_v^{-1}\rho_0, c_0) \partial_t J_v^{-1} \partial_t \operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v \\ & + 2\rho_0 \partial_{\rho} \mu(J_v^{-1}\rho_0, c_0) \partial_t J_v^{-1} \operatorname{Adj}(D\mathcal{X}_v)\{Dv \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v\}, \end{aligned} $$

$$ \begin{aligned} \partial_{tt} I_{1,2}(v, v) = & 2\rho_0 \partial_{\rho} \mu(J_v^{-1}\rho_0, c_0) \partial_t J_v^{-1} \{\operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v_t - \mathbb{D}v_t\} \\ & + 2\mu(J_v^{-1}\rho_0, c_0) \{\operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v_{tt} - \mathbb{D}v_{tt}\} \\ & + 2\mu(J_v^{-1}\rho_0, c_0) \partial_t \operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v_t \\ & + 2\mu(J_v^{-1}\rho_0, c_0) \operatorname{Adj}(D\mathcal{X}_v)\{\mathbb{D}v_t \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v_t\}, \end{aligned} $$

$$ \begin{aligned} \partial_{tt} I_{1,3}(v, v) = & 2\rho_0 \partial_{\rho} \mu(J_v^{-1}\rho_0, c_0) \partial_t J_v^{-1} \partial_t \operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v + 2\mu(J_v^{-1}\rho_0, c_0) \partial_{tt} \operatorname{Adj}(D\mathcal{X}_v)\mathbb{D}_{A_v}v \\ & + 2\mu(J_v^{-1}\rho_0, c_0) \partial_t \operatorname{Adj}(D\mathcal{X}_v)\{\mathbb{D}_{A_v}v_t + \mathbb{D}v \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v\} \end{aligned} $$

and finally,

$$ \begin{aligned} \partial_{tt} I_{1,4}(v, v) = & 2\rho_0 \partial_{\rho} \mu(J_v^{-1}\rho_0, c_0) \partial_t J_v^{-1} \operatorname{Adj}(D\mathcal{X}_v) (Dv \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v) \\ & + 2\mu(J_v^{-1}\rho_0, c_0) \partial_t \operatorname{Adj}(D\mathcal{X}_v) (Dv \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v) \\ & + 2\mu(J_v^{-1}\rho_0, c_0) \operatorname{Adj}(D\mathcal{X}_v) (Dv_t \cdot \partial_t A_v + \partial_t A_v^T \cdot \nabla v_t + \mathbb{D}v \cdot \partial_{tt} A_v + \partial_{tt} A_v^T \cdot \nabla v) \end{aligned} $$

Combining (4.24), (4.25), (4.26), (4.27) and (4.29) one has:

$$ \begin{aligned} \|\sigma \partial_{tt} I_1(v, v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim & \|\nabla v\|_{L_T^1 L^{\infty}(\mathbb{R}^d)} \left( \|\nabla v\|_{L_T^6 L^6(\mathbb{R}^d)}^3 + \|\sigma^{\frac{d}{4}} \nabla v_t\|_{L_T^{8/d} L^4(\mathbb{R}^d)} \|\nabla v\|_{L_T^{8/(4-d)} L^4(\mathbb{R}^d)} + \|\sigma \nabla v_{tt}\|_{L_T^2 L^2(\mathbb{R}^d)} \right) \\ & + \|\sigma^{\frac{d}{4}} \nabla v_t\|_{L_T^{8/d} L^4(\mathbb{R}^d)} \|\nabla v\|_{L_T^{8/(4-d)} L^4(\mathbb{R}^d)} + \|\nabla v\|_{L_T^6 L^6(\mathbb{R}^d)}^3 \end{aligned} $$
Next, the second term of (4.42) is:

$$
\begin{align*}
\partial_{tt} \delta I_{1,6,2}(v, v) ={}& \rho_0 \partial_{tt} J_{v_1}^{-1} \{ \partial_{\rho} \mu(J_{v_2}^{-1} \rho_0, c_0) - \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \} \{ \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} v_2 - \mathbb{D} v_2 \} \\
& + \rho_0^2 \partial_t J_{v_1}^{-1} \partial_t J_{v_2}^{-1} \{ \partial_{\rho}^2 \mu(J_{v_2}^{-1} \rho_0, c_0) - \partial_{\rho}^2 \mu(J_{v_1}^{-1} \rho_0, c_0) \} \{ \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} v_2 - \mathbb{D} v_2 \} \\
& + \rho_0^2 \partial_t J_{v_1}^{-1} \partial_{\rho}^2 \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1} \} \{ \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} v_2 - \mathbb{D} v_2 \} \\
& + \rho_0 \partial_t J_{v_1}^{-1} \{ \partial_{\rho} \mu(J_{v_2}^{-1} \rho_0, c_0) - \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \} \{ \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \partial_t v_2 - \mathbb{D} \partial_t v_2 \} \\
& + \rho_0 \partial_t J_{v_1}^{-1} \{ \partial_{\rho} \mu(J_{v_2}^{-1} \rho_0, c_0) - \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \} \partial_t \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} v_2 \\
& + \rho_0 \partial_t J_{v_1}^{-1} \{ \partial_{\rho} \mu(J_{v_2}^{-1} \rho_0, c_0) - \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \} \operatorname{Adj}(D \mathcal{X}_{v_2}) \{ D v_2 \cdot \partial_t A_{v_2} + \partial_t A_{v_2}^T \nabla \partial_t v_2 \}
\end{align*}
$$

and the following estimate holds true:

$$
\begin{aligned}
\| \sigma \partial_{tt} \delta I_{1,6,2}(v, v) \|_{L_T^2 L^2(\mathbb{R}^d)} & \lesssim \| \nabla v_2 \|_{L_T^1 L^\infty(\mathbb{R}^d)} \| \nabla v_1 \|_{L_T^6 L^6(\mathbb{R}^d)} \| \nabla v_2 \|_{L_T^6 L^6(\mathbb{R}^d)} \\
& \quad \times \left( \| \nabla v_1, \nabla v_2 \|_{L_T^6 L^6(\mathbb{R}^d)} \| \nabla \delta v \|_{L_T^1 L^\infty(\mathbb{R}^d)} + \| \nabla \delta v \|_{L_T^6 L^6(\mathbb{R}^d)} \right) \\
& \quad + \| \nabla v_2 \|_{L_T^1 L^\infty(\mathbb{R}^d)} \| \nabla v_1, \nabla v_2 \|_{L_T^\infty L^4(\mathbb{R}^d)} \| \sigma \nabla \partial_t v_1, \sigma \nabla \partial_t v_2 \|_{L_T^2 L^4(\mathbb{R}^d)} \| \nabla \delta v \|_{L_T^1 L^\infty(\mathbb{R}^d)} \\
& \quad + \| \nabla \delta v \|_{L_T^1 L^\infty(\mathbb{R}^d)} \| \nabla v_1 \|_{L_T^6 L^6(\mathbb{R}^d)} \| \nabla v_2 \|_{L_T^6 L^6(\mathbb{R}^d)}^2.
\end{aligned}
$$

As well, the third term of the expression (4.42) is:

$$
\begin{align*}
\partial_{tt} \delta I_{1,6,3}(v, v) ={}& \rho_0 \partial_t J_{v_2}^{-1} \{ \partial_{\rho} \mu(J_{v_2}^{-1} \rho_0, c_0) - \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \} \{ \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \partial_t v_2 - \mathbb{D} \partial_t v_2 \} \\
& + \rho_0 \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1} \} \{ \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \partial_t v_2 - \mathbb{D} \partial_t v_2 \} \\
& + \{ \mu(J_{v_2}^{-1} \rho_0, c_0) - \mu(J_{v_1}^{-1} \rho_0, c_0) \} \{ \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \partial_t v_2 - \mathbb{D} \partial_t v_2 \} \\
& + \{ \mu(J_{v_2}^{-1} \rho_0, c_0) - \mu(J_{v_1}^{-1} \rho_0, c_0) \} \partial_t \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \partial_t v_2 \\
& + \{ \mu(J_{v_2}^{-1} \rho_0, c_0) - \mu(J_{v_1}^{-1} \rho_0, c_0) \} \operatorname{Adj}(D \mathcal{X}_{v_2}) \{ D v_2 \cdot \partial_t A_{v_2} + \partial_t A_{v_2}^T \nabla \partial_t v_2 \}
\end{align*}
$$

with the following norm estimate:

$$
\| \sigma \partial_{tt} \delta I_{1,6,3}(v, v) \|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \| \sigma \nabla \partial_t v_2 \|_{L_T^2 L^4(\mathbb{R}^d)} \| \nabla v_2 \|_{L_T^1 L^\infty(\mathbb{R}^d)} \left( \| \nabla v_1, \nabla v_2 \|_{L_T^\infty L^4(\mathbb{R}^d)} \| \nabla \delta v \|_{L_T^1 L^\infty(\mathbb{R}^d)} \right. \\
\left. + \| \nabla \delta v \|_{L_T^\infty L^4(\mathbb{R}^d)} \right) + \| \nabla \delta v \|_{L_T^1 L^\infty(\mathbb{R}^d)} \left( \| \nabla v_2 \|_{L_T^1 L^\infty(\mathbb{R}^d)} \| \sigma \nabla \partial_t v_2 \|_{L_T^2 L^2(\mathbb{R}^d)} \right. \\
\left. + \| \nabla v_2 \|_{L_T^\infty L^4(\mathbb{R}^d)} \| \sigma \nabla \partial_t v_2 \|_{L_T^2 L^4(\mathbb{R}^d)} \right).
$$

Finally, the fourth term of (4.42) is given by:

$$
\begin{align*}
\partial_{tt} \delta I_{1,6,4}(v, v) ={}& \rho_0 \partial_t J_{v_2}^{-1} \{ \partial_{\rho} \mu(J_{v_2}^{-1} \rho_0, c_0) - \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \} \partial_t \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} v_2 \\
& + \rho_0 \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \partial_t J_{v_2}^{-1} - \partial_t J_{v_1}^{-1} \} \partial_t \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} v_2 \\
& + \{ \mu(J_{v_2}^{-1} \rho_0, c_0) - \mu(J_{v_1}^{-1} \rho_0, c_0) \} \partial_{tt} \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} v_2 \\
& + \{ \mu(J_{v_2}^{-1} \rho_0, c_0) - \mu(J_{v_1}^{-1} \rho_0, c_0) \} \partial_t \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \partial_t v_2 \\
& + \{ \mu(J_{v_2}^{-1} \rho_0, c_0) - \mu(J_{v_1}^{-1} \rho_0, c_0) \} \partial_t \operatorname{Adj}(D \mathcal{X}_{v_2}) \{ D v_2 \cdot \partial_t A_{v_2} + \partial_t A_{v_2}^T \nabla \partial_t v_2 \}
\end{align*}
$$

with the following norm estimate:

$$
\| \sigma \partial_{tt} \delta I_{1,6,4}(v, v) \|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \| \nabla v_2 \|_{L_T^6 L^6(\mathbb{R}^d)} \left( \| \nabla v_2, \nabla v_1 \|_{L_T^6 L^6(\mathbb{R}^d)} \| \nabla \delta v \|_{L_T^1 L^\infty(\mathbb{R}^d)} + \| \nabla \delta v \|_{L_T^6 L^6(\mathbb{R}^d)} \right) \\
+ \| \nabla \delta v \|_{L_T^1 L^\infty(\mathbb{R}^d)} \| \nabla v_2 \|_{L_T^\infty L^4(\mathbb{R}^d)} \| \sigma \nabla \partial_t v_2 \|_{L_T^2 L^4(\mathbb{R}^d)}.
$$

A similar estimate holds true for the $L_T^2 L^2(\mathbb{R}^d)$ norm of $\sigma \partial_{tt} \delta I_{1,6,5}(v, v)$, the last term of (4.42).

Step 16: $L_T^2 L^2(\mathbb{R}^d)$ norm estimate for $\sigma \partial_{tt} \delta I_{1,7}(v, v)$. The second time derivative of $\delta I_{1,7}(v, v)$ reads:

$$
(4.43) \qquad \partial_{tt} \delta I_{1,7}(v, v) = \partial_{tt} \delta I_{1,7,1}(v, v) + \partial_{tt} \delta I_{1,7,2}(v, v) + \partial_{tt} \delta I_{1,7,3}(v, v) + \partial_{tt} \delta I_{1,7,4}(v, v)
$$
Finally, the same estimate holds true for the $L_T^2 L^2(\mathbb{R}^d)$ norm of $\sigma \partial_{tt} I_2(v, v)$ and gathering (4.37) we obtain (4.11). We turn to the estimation of the norms of the viscosity.

### 4.3.4. Lebesgue and Hölder norm estimate for viscosity.
This step makes more sense to the estimate of the lower, upper bound and the Hölder regularity of the viscosity $\mu$ and $\lambda$. With the help of Lemma 4.4, we have the following estimates for $\mu$

$$ \left\{ \begin{array}{ll} \|\mu(J_v^{-1}\rho_0, c_0) - \tilde{\mu}\|_{L_T^\infty L^\infty(\mathbb{R}^d)} & \le \|\mu(\rho_0, c_0) - \tilde{\mu}\|_{L^\infty(\mathbb{R}^d)} + \|\partial_\rho \mu\|_{L^\infty} \|\nabla v\|_{L_T^1 L^\infty(\mathbb{R}^d)}, \\ \|\llbracket \mu(J_v^{-1}\rho_0, c_0) \rrbracket\|_{L_T^\infty c_0} & \le \|\llbracket \mu(\rho_0, c_0) \rrbracket\|_{L_T^\infty c_0} + \|\partial_\rho \mu\|_{L^\infty} \|\nabla v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \end{array} \right. $$

and a similar estimates hold for $\lambda(J_v^{-1}\rho_0, c_0)$. Also, one has the following Hölder estimate:

$$ \|\mu(J_v^{-1}\rho_0, c_0)\|_{L_T^\infty \mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} \le \|\mu(\rho_0, c_0)\|_{\mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)} + \|\partial_\rho \mu\|_{L^\infty} \|\nabla v\|_{L_T^1 \mathcal{C}_{pw,c_0}^{\alpha}(\mathbb{R}^d)}. $$

The same estimate also holds for $\lambda(J_v^{-1}\rho_0, c_0)$ and we obtain (4.12). We now turn to the estimate for the time derivative of the viscosity.

### 4.3.5. Estimates for time derivative of the viscosity.
We estimate the norm of the time derivative of $\mu(J_v^{-1}\rho_0, c_0)$ and $\lambda(J_v^{-1}\rho_0, c_0)$ as required in (3.2). First, we express:

$$ (4.38) \qquad \partial_t \mu(J_v^{-1} \rho_0, c_0) = -\rho_0 \partial_\rho \mu(J_v^{-1} \rho_0, c_0) J_v^{-1} \mathrm{div}_{A_v}(v), $$

and it transpires that:

$$ \|\partial_t \mu(J_v^{-1} \rho_0, c_0)\|_{L_T^m L^q(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L_T^m L^q(\mathbb{R}^d)}, \quad \|\sqrt{\sigma} \partial_t \mu(J_v^{-1} \rho_0, c_0)\|_{L_T^2 L^\infty(\mathbb{R}^d)} \lesssim \|\sqrt{\sigma} \nabla v\|_{L_T^2 L^\infty(\mathbb{R}^d)}. $$

A similar estimate holds true for the time derivative of $\lambda(J_v^{-1}\rho_0, c_0)$ and implicitly (4.13)-(4.14) follow. We turn to the estimates of the norm of $\partial_{tt}\mu$ and $\partial_{tt}\lambda$ as required in (3.2).

### 4.3.6. Estimates for the second time derivative of the viscosity.
From (4.38), it is obvious that the second time derivative of $\mu(J_v^{-1}\rho_0, c_0)$ reads:

$$ \partial_{tt}\mu(J_v^{-1}\rho_0, c_0) = \rho_0^2 \partial_\rho^2 \mu(J_v^{-1}\rho_0, c_0) (J_v^{-1} \mathrm{div}_{A_v}(v))^2 + \rho_0 \partial_\rho \mu(J_v^{-1}\rho_0, c_0) J_v^{-1} (\mathrm{div}_{A_v}(v))^2 \\ - \rho_0 \partial_\rho \mu(J_v^{-1}\rho_0, c_0) J_v^{-1} \mathrm{div}_{A_v} v_t - \rho_0 \partial_\rho \mu(J_v^{-1}\rho_0, c_0) J_v^{-1} \partial_t A_v^T : \nabla v, $$

hence:

$$ \|\sigma \partial_{tt} \mu(J_v^{-1} \rho_0, c_0)\|_{L_T^2 L^4(\mathbb{R}^d)} \lesssim \|\nabla v\|_{L_T^4 L^8(\mathbb{R}^d)}^2 + \|\sigma \nabla v_t\|_{L_T^2 L^4(\mathbb{R}^d)}. $$

A similar estimate holds true for the second time derivative of $\lambda(J_v^{-1}\rho_0, c_0)$. This leads to (4.15) and concludes the estimates used in Proposition 4.1.

### Acknowledgment.
I would like to express my gratitude to my PhD advisors, Cosmin Burtea and David Gérard-Varet, for their insightful discussions and thorough review of this work. I also sincerely thank the referees for their careful reading and valuable comments and suggestions on the manuscript.

### Funding.
This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 945332. 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).

## APPENDIX A. QUANTITATIVE HÖLDER ESTIMATES

This section contains two results concerning the regularity of even-order Riesz transform of discontinuous functions. The first one is a recent contribution by Gancedo and García-Juárez [11]. Roughly speaking, they established that these operators are continuous on the space of piecewise Hölder continuous functions, provided that the discontinuity surfaces enjoy certain regularities. The second result, obtained in Hoff [14] for $d = 2$, concerns the extension of discontinuous functions.

The quantitative Hölder estimates for even-order Riesz operator obtained by Gancedo and García-Juárez reads as follows:
where the first term is

$$
\begin{align*}
\partial_{tt} \delta I_{1,7,1}(v,v) ={}& \rho_0 \partial_{tt} J_{v_1}^{-1} \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \delta v - \mathbb{D} \delta v \} \\
& + (\rho_0 \partial_t J_{v_1}^{-1})^2 \partial_{\rho}^2 \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \delta v - \mathbb{D} \delta v \} \\
& + \rho_0 \partial_t J_{v_1}^{-1} \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \partial_t \delta v - \mathbb{D} \partial_t \delta v \} \\
& + \rho_0 \partial_t J_{v_1}^{-1} \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \partial_t \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \delta v \\
& + \rho_0 \partial_t J_{v_1}^{-1} \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \operatorname{Adj}(D \mathcal{X}_{v_2}) \{ D \delta v \cdot \partial_t A_{v_2} + \partial_t A_{v_2}^T \cdot \nabla \delta v \}
\end{align*}
$$

with the following norm estimate:

$$
\|\sigma \partial_{tt} \delta I_{1,7,1}(v, v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla v_1\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^6 L^6(\mathbb{R}^d)} \left( \|\nabla v_1\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^1 L^\infty(\mathbb{R}^d)} + \|\nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \right) \\
+ \|\nabla v_2\|_{L_T^1 L^\infty(\mathbb{R}^d)} \left( \|\nabla v_1\|_{L_T^\infty L^4(\mathbb{R}^d)} \|\sigma \nabla \partial_t \delta v\|_{L_T^2 L^4(\mathbb{R}^d)} + \|\sigma \nabla \partial_t v_1\|_{L_T^2 L^4(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^\infty L^4(\mathbb{R}^d)} \right).
$$

Next, the second term of (4.43) is:

$$
\begin{align*}
\partial_{tt} \delta I_{1,7,2}(v,v) ={}& \rho_0 \partial_t J_{v_1}^{-1} \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \partial_t \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \delta v \\
& + \mu(J_{v_1}^{-1} \rho_0, c_0) \partial_{tt} \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \delta v \\
& + \mu(J_{v_1}^{-1} \rho_0, c_0) \partial_t \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \partial_t \delta v \\
& + \mu(J_{v_1}^{-1} \rho_0, c_0) \partial_t \operatorname{Adj}(D \mathcal{X}_{v_2}) \{ D \delta v \cdot \partial_t A_{v_2} + \partial_t A_{v_2}^T \cdot \nabla \delta v \}
\end{align*}
$$

and its $L_T^2 L^2(\mathbb{R}^d)$ norm can be estimated as follows:

$$
\|\sigma \partial_{tt} \delta I_{1,7,2}(v, v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^6 L^6(\mathbb{R}^d)} \\
+ \|\nabla v_2\|_{L_T^\infty L^4(\mathbb{R}^d)} \|\sigma \nabla \partial_t \delta v\|_{L_T^2 L^4(\mathbb{R}^d)} + \|\sigma \nabla \partial_t v_2\|_{L_T^2 L^4(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^\infty L^4(\mathbb{R}^d)}.
$$

As well, the third term of the expression (4.43) is:

$$
\begin{align*}
\partial_{tt} \delta I_{1,7,3}(v,v) ={}& \rho_0 \partial_t J_{v_1}^{-1} \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \partial_t \delta v - \mathbb{D} \partial_t \delta v \} \\
& + \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \partial_{tt} \delta v - \mathbb{D} \partial_{tt} \delta v \} \\
& + \mu(J_{v_1}^{-1} \rho_0, c_0) \partial_t \operatorname{Adj}(D \mathcal{X}_{v_2}) \mathbb{D}_{A_{v_2}} \partial_t \delta v \\
& + \mu(J_{v_1}^{-1} \rho_0, c_0) \operatorname{Adj}(D \mathcal{X}_{v_2}) \{ D \partial_t \delta v \cdot \partial_t A_{v_2} + \partial_t A_{v_2}^T \cdot \nabla \partial_t \delta v \}
\end{align*}
$$

thus, the following estimate holds true:

$$
\|\sigma \partial_{tt} \delta I_{1,7,3}(v, v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \left( \|\nabla v_1\|_{L_T^\infty L^4(\mathbb{R}^d)} \|\sigma \nabla \partial_t \delta v\|_{L_T^2 L^4(\mathbb{R}^d)} + \|\sigma \nabla \partial_{tt} \delta v\|_{L_T^2 L^2(\mathbb{R}^d)} \right) \\
+ \|\nabla v_2\|_{L_T^\infty L^4(\mathbb{R}^d)} \|\sigma \nabla \partial_t \delta v\|_{L_T^2 L^4(\mathbb{R}^d)}.
$$

Finally the last term of (4.43) is:

$$
\begin{align*}
\partial_{tt} \delta I_{1,7,4}(v,v) ={}& \rho_0 \partial_t J_{v_1}^{-1} \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \operatorname{Adj}(D \mathcal{X}_{v_2}) \{ D \delta v \partial_t A_{v_2} + \partial_t A_{v_2}^T \nabla \delta v \} \\
& + \mu(J_{v_1}^{-1} \rho_0, c_0) \partial_t \operatorname{Adj}(D \mathcal{X}_{v_2}) \{ D \delta v \partial_t A_{v_2} + \partial_t A_{v_2}^T \nabla \delta v \} \\
& + \mu(J_{v_1}^{-1} \rho_0, c_0) \operatorname{Adj}(D \mathcal{X}_{v_2}) \{ D \partial_t \delta v \partial_t A_{v_2} + \partial_t A_{v_2}^T \nabla \partial_t \delta v \} \\
& + \mu(J_{v_1}^{-1} \rho_0, c_0) \operatorname{Adj}(D \mathcal{X}_{v_2}) \{ D \delta v \partial_{tt} A_{v_2} + \partial_{tt} A_{v_2}^T \nabla \delta v \}
\end{align*}
$$

with the following estimate:

$$
\|\sigma \partial_{tt} \delta I_{1,7,4}(v, v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla v_1, \nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla \delta v\|_{L_T^6 L^6(\mathbb{R}^d)} \\
+ \|\sigma \nabla \partial_t \delta v\|_{L_T^2 L^4(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^\infty L^4(\mathbb{R}^d)} + \|\nabla \delta v\|_{L_T^\infty L^4(\mathbb{R}^d)} \|\sigma \nabla \partial_t v_2\|_{L_T^2 L^4(\mathbb{R}^d)}.
$$

Step 17: $L_T^2 L^2(\mathbb{R}^d)$ norm estimate for $\sigma \partial_{tt} \delta I_{1,8}(v, v)$. From (4.40), We express:

$$
(4.44) \qquad \partial_{tt} \delta I_{1,8}(v, v) = \partial_{tt} \delta I_{1,8,1}(v, v) + \partial_{tt} \delta I_{1,8,2}(v, v) + \partial_{tt} \delta I_{1,8,3}(v, v)
$$
**Lemma A.1 ([11, 12]).** Assume the hypotheses in Section 1.1.1 hold for $\mathcal{C}$ and consider a singular integral operator $\mathcal{T}$:

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

where the kernel $K$ is:

$$ K(x) = \frac{P_{2l}(x)}{|x|^{d+2l}}. $$

Here, $P_{2l}$ is a homogeneous polynomial in $\mathbb{R}^d$ of degree $2l$, $l \in \mathbb{N} \setminus \{0\}$.

* $L^\infty(\mathbb{R}^d)$-Estimate. Given $\alpha' \in (0,1)$ and $q > d/\alpha'$, there exists a constant $C = C(d, q, \alpha')$ such that for all $g \in \mathcal{C}_{pw,C}^{\alpha'}(\mathbb{R}^d)$ and $w \in \mathcal{C}_{pw,C}^{\alpha'}(\mathbb{R}^d) \cap L^q(\mathbb{R}^d)$:

$$ (A.2) \qquad \|\mathcal{T}(gw)\|_{L^\infty(\mathbb{R}^d)} \le C \left( \|g\|_{\dot{\mathcal{C}}_{pw,C}^{\alpha'}(\mathbb{R}^d)} + \ell_\varphi^{-\frac{d}{q}} \|g\|_{L^\infty(\mathbb{R}^d)} \right) \|w\|_{L^q(\mathbb{R}^d)} + C \|g\|_{L^\infty(\mathbb{R}^d)} \|w\|_{\mathcal{C}_{pw,C}^{\alpha'}(\mathbb{R}^d)} $$

* $\dot{\mathcal{C}}_{pw,C}^{\alpha'}(\mathbb{R}^d)$-Estimate. There exists a constant $C = C(d, \alpha)$ such that for all $w \in \mathcal{C}_{pw,C}^{\alpha'}(\mathbb{R}^d)$:

$$ (A.3) \qquad \|\mathcal{T}(w)\|_{\dot{\mathcal{C}}_{pw,C}^{\alpha'}(\mathbb{R}^d)} \le C \|w\|_{\dot{\mathcal{C}}_{pw,C}^{\alpha'}(\mathbb{R}^d)} + C(1+|C|) \|w\|_{L^\infty(\mathbb{R}^d)} \mathfrak{P}(\|C\|_{Lip} + |C|_*) |C|_{\dot{\mathcal{C}}_{1+\alpha}} $$

where $\mathfrak{P}$ is a polynomial depending on $P_{2l}$.

For $d=2$, Lemma A.1 summarizes the computations in Step 4 and Step 5 of [12]. It should be noted that the proof uses a geometric lemma derived in [1] for $d=2$. This extra cancellation property of even-order Riesz operators is also valid in higher dimensions. We refer to [25, Section 3], and more precisely to the estimate of the term in equation (12). Let us now move on to the second result, which concerns the extension of piecewise Hölder continuous functions.

**Lemma A.2.** Assume the hypotheses in Section 1.1.1 hold for $\mathcal{C}$ and let $q \in [1, \infty]$. For $g \in L^q(\mathbb{R}^d) \cap \mathcal{C}_{pw,C}^{\alpha'}(\mathbb{R}^d)$ there exists $g^e \in \mathcal{C}^{\alpha'}(\mathbb{R}^d)$ such that $g^e = g$ on $D$ (or $D^c$) and:

$$ (A.4) \qquad \left\{ \begin{array}{ll} \|g^e\|_{L^\infty(\mathbb{R}^d)} & \le C\|g\|_{L^\infty(\mathbb{R}^d)}, \\ \|g-g^e\|_{L^q(\mathbb{R}^d)} & \le C\ell_\varphi^{1/q}\|\llbracket g \rrbracket\|_{L^q(\mathcal{C})}, \\ \|g^e\|_{\dot{\mathcal{C}}_{pw,C}^{\alpha'}} + \|g-g^e\|_{\dot{\mathcal{C}}_{pw,C}^{\alpha'}} & \le C\|g\|_{\dot{\mathcal{C}}_{pw,C}^{\alpha'}} + C\|\llbracket g \rrbracket\|_{L^\infty(\mathcal{C})} \ell_\varphi^{-\alpha}. \end{array} \right. $$

The constant $C$ above is independent from $q$, $\alpha$ and $\mathcal{C}$.

For $d=2$, the lemma above does not differ from Lemma 5.2 in [14], except for the dependence on the interface regularity. The proof for $d=3$ follows almost the same lines.

**Proof of Lemma A.2.** Given that $\mathcal{C} \in \mathcal{C}^{1+\alpha}$, for each $x = (x', x_d) \in \mathcal{C}^2$ there exist a neighborhood $W'$ of $x'$ in $\mathbb{R}^{d-1}$ and a function $z: W' \to (x_d - \ell, x_d + \ell) \in \mathcal{C}^{1+\alpha}$ such that:

$$ \mathcal{C} \cap W = \{(y', z(y')): y' \in W'\}, \quad \text{where} \quad W = W' \times (x_d - \ell, x_d + \ell). $$

For all $y \in \mathbb{R}^d$, we have:

$$ |\nabla\varphi(y)| \ge |\nabla\varphi(x)| - |\nabla\varphi(x) - \nabla\varphi(y)| \ge |\nabla\varphi(x)| - |x - y|^\alpha \|\nabla\varphi\|_{\dot{\mathcal{C}}^\alpha}. $$

Therefore, for $y$ such that:

$$ |x - y|^\alpha \|\nabla\varphi\|_{\dot{\mathcal{C}}^\alpha} \le \frac{1}{2} \|\nabla\varphi\|_{\inf}, $$

it follows that

$$ |\nabla\varphi(y)| \ge \frac{1}{2} \|\nabla\varphi\|_{\inf}. $$

In consequence, we can consider the local chart $(W', z)$ provided by the implicit function theorem, and $W'$ can be taken as the $(d-1)$-dimensional ball centered at $x'$ with radius $\ell = C_d \ell_\varphi$. Furthermore, we might have:

$$ (A.5) \qquad \|\nabla z\|_{L^\infty(W')} \le 2. $$

<footnote n="2">²For $y = (y_1, \dots, y_d) \in \mathbb{R}^d$ we use the notation $y = (y', y_d)$, where $y' = (y_1, \dots, y_{d-1})$.</footnote>
where the first term is

$$
\begin{align*}
\partial_{tt} \delta I_{1,8,1}(v, v) ={}& \rho_0 \partial_{tt} J_{v_1}^{-1} \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \operatorname{Adj}(D\mathcal{X}_{v_2}) - \operatorname{Adj}(D\mathcal{X}_{v_1}) \} \mathbb{D}_{A_{v_2}} v_1 \\
& + (\rho_0 \partial_t J_{v_1}^{-1})^2 \partial_{\rho}^2 \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \operatorname{Adj}(D\mathcal{X}_{v_2}) - \operatorname{Adj}(D\mathcal{X}_{v_1}) \} \mathbb{D}_{A_{v_2}} v_1 \\
& + \rho_0 \partial_t J_{v_1}^{-1} \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \partial_t \operatorname{Adj}(D\mathcal{X}_{v_2}) - \partial_t \operatorname{Adj}(D\mathcal{X}_{v_1}) \} \mathbb{D}_{A_{v_2}} v_1 \\
& + \rho_0 \partial_t J_{v_1}^{-1} \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \operatorname{Adj}(D\mathcal{X}_{v_2}) - \operatorname{Adj}(D\mathcal{X}_{v_1}) \} \mathbb{D}_{A_{v_2}} \partial_t v_1 \\
& + \rho_0 \partial_t J_{v_1}^{-1} \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \operatorname{Adj}(D\mathcal{X}_{v_2}) - \operatorname{Adj}(D\mathcal{X}_{v_1}) \} \{ Dv_1 \cdot \partial_t A_{v_2} + \partial_t A_{v_2}^T \cdot \nabla v_1 \}
\end{align*}
$$

with the following norm estimate:

$$
\|\sigma\partial_{tt}\delta I_{1,8,1}(v, v)\|_{L_T^2 L^2(\mathbb{R}^d)} \lesssim \|\nabla v_1\|_{L_T^\infty L^4(\mathbb{R}^d)} \|\nabla\delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\sigma\nabla\partial_t v_1\|_{L_T^2 L^4(\mathbb{R}^d)} \\
+ \|\nabla v_1\|_{L_T^6 L^6(\mathbb{R}^d)}^2 \left( \|\nabla\delta v\|_{L_T^6 L^6(\mathbb{R}^d)} + \|\nabla\delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \right).
$$

On the other hand, the second term of (4.44) is given by:

$$
\begin{align*}
\partial_{tt} \delta I_{1,8,2}(v, v) ={}& \rho_0 \partial_t J_{v_1}^{-1} \partial_{\rho} \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \partial_t \operatorname{Adj}(D\mathcal{X}_{v_2}) - \partial_t \operatorname{Adj}(D\mathcal{X}_{v_1}) \} \mathbb{D}_{A_{v_2}} v_1 \\
& + \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \partial_{tt} \operatorname{Adj}(D\mathcal{X}_{v_2}) - \partial_{tt} \operatorname{Adj}(D\mathcal{X}_{v_1}) \} \mathbb{D}_{A_{v_2}} v_1 \\
& + \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \partial_t \operatorname{Adj}(D\mathcal{X}_{v_2}) - \partial_t \operatorname{Adj}(D\mathcal{X}_{v_1}) \} \mathbb{D}_{A_{v_2}} \partial_t v_1 \\
& + \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \partial_t \operatorname{Adj}(D\mathcal{X}_{v_2}) - \partial_t \operatorname{Adj}(D\mathcal{X}_{v_1}) \} \{ Dv_1 \cdot \partial_t A_{v_2} + \partial_t A_{v_2}^T \cdot \nabla v_1 \}
\end{align*}
$$

with the following norm estimate:

$$
\begin{align*}
\|\sigma\partial_{tt}\delta I_{1,8,2}(v, v)\|_{L^2} &\lesssim \|\nabla v_1\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \\
&\qquad \left( \|\nabla\delta v\|_{L_T^6 L^6(\mathbb{R}^d)} + \|\nabla\delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \right) \\
&\qquad + \|\nabla v_1\|_{L_T^\infty L^4(\mathbb{R}^d)} \left( \|\sigma\nabla\partial_t\delta v\|_{L_T^2 L^4(\mathbb{R}^d)} + \|\nabla\delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\sigma\nabla\partial_t v_1, \sigma\nabla\partial_t v_2\|_{L_T^2 L^4(\mathbb{R}^d)} \right) \\
&\qquad + \|\sigma\nabla\partial_t v_1\|_{L_T^2 L^4(\mathbb{R}^d)} \left( \|\nabla\delta v\|_{L_T^\infty L^4(\mathbb{R}^d)} + \|\nabla\delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^\infty L^4(\mathbb{R}^d)} \right).
\end{align*}
$$

Finally, the last term of (4.44) is given by:

$$
\begin{align*}
\partial_{tt} \delta I_{1,8,3}(v, v) ={}& \rho_0 \partial_t J_{v_1}^{-1} \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \operatorname{Adj}(D\mathcal{X}_{v_2}) - \operatorname{Adj}(D\mathcal{X}_{v_1}) \} \partial_t \mathbb{D}_{A_{v_2}} v_1 \\
& + \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \partial_t \operatorname{Adj}(D\mathcal{X}_{v_2}) - \partial_t \operatorname{Adj}(D\mathcal{X}_{v_1}) \} \partial_t \mathbb{D}_{A_{v_2}} v_1 \\
& + \mu(J_{v_1}^{-1} \rho_0, c_0) \{ \operatorname{Adj}(D\mathcal{X}_{v_2}) - \operatorname{Adj}(D\mathcal{X}_{v_1}) \} \partial_{tt} \mathbb{D}_{A_{v_2}} v_1
\end{align*}
$$

with

$$
2 \partial_t \mathbb{D}_{A_{v_2}} v_1 = 2 \mathbb{D}_{A_{v_2}} \partial_t v_1 + Dv_1 \partial_t A_{v_2} + \partial_t A_{v_2}^T \cdot \nabla v_1
$$

and

$$
2 \partial_{tt} \mathbb{D}_{A_{v_2}} v_1 = 2 \mathbb{D}_{A_{v_2}} \partial_{tt} v_1 + 2 D \partial_t v_1 \partial_t A_{v_2} + 2 \partial_t A_{v_2}^T \cdot \nabla \partial_t v_1 + D v_1 \partial_{tt} A_{v_2} + \partial_{tt} A_{v_2}^T \cdot \nabla v_1.
$$

In consequence, one has:

$$
\begin{align*}
\|\sigma\partial_{tt}\delta I_{1,8,3}(v, v)\|_{L_T^2 L^2(\mathbb{R}^d)} &\lesssim \|\nabla\delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \\
&\qquad \left( \|\nabla v_1, \nabla v_2\|_{L_T^\infty L^4(\mathbb{R}^d)} \|\nabla\partial_t v_1, \nabla\partial_t v_2\|_{L_T^2 L^4(\mathbb{R}^d)} \right. \\
&\qquad \left. + \|\nabla v_1\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla v_1, \nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \right) \\
&\qquad + \|\nabla\partial_t v_1\|_{L_T^2 L^4(\mathbb{R}^d)} \|\nabla\delta v\|_{L_T^\infty L^4(\mathbb{R}^d)} \\
&\qquad + \|\nabla v_1\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla v_2\|_{L_T^6 L^6(\mathbb{R}^d)} \|\nabla\delta v\|_{L_T^6 L^6(\mathbb{R}^d)} \\
&\qquad + \|\nabla\delta v\|_{L_T^1 L^\infty(\mathbb{R}^d)} \|\sigma\nabla\partial_{tt} v_1\|_{L_T^2 L^2(\mathbb{R}^d)}.
\end{align*}
$$

We gather all of these estimates in order to obtain a bound for the $L_T^2 L^2(\mathbb{R}^d)$ norm of $\sigma\partial_{tt} I_{1,8}(v, v)$. A similar estimate holds true for the $L_T^2 L^2(\mathbb{R}^d)$ norm of $\sigma\partial_{tt} I_{1,9}(v, v)$.

**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.
We now cover $\mathcal{C}$ with finite open bounded sets $W_j$, $j = 2, \dots, N$ and we set $W_0 = D$ and $W_1 = \mathbb{R}^d \setminus \bar{D}$. For each $j = 2, 3, \dots, N$, we define the projection:

$$ \Xi_j : W_j \to W_j \cap \mathcal{C} \\ x = (x', x_d) \mapsto \Xi_j(x) = (x', z_j(x')) $$

and the local extension $g_j^e$:

$$ (A.6) \qquad g_j^e(x) = \begin{cases} g(x) & \text{if } x \in W_j \cap \bar{D}, \\ g(x) + \llbracket g(\Xi_j(x)) \rrbracket & \text{if } x \in W_j \cap (\mathbb{R}^d \setminus \bar{D}). \end{cases} $$

Clearly, $g_j^e$ is $\alpha$-Hölder continuous on $W_j \cap \bar{D}$ and $W_j \cap (\mathbb{R}^d \setminus D)$. However, it is even more than that: $g_j^e$ is $\alpha$-Hölder in the whole $W_j$ with:

$$ (A.7) \qquad \begin{cases} \|g_j^e - g\|_{L^\infty(W_j)} \le \|\llbracket g \rrbracket\|_{L^\infty(W_j \cap \mathcal{C})}, \\ \|g_j^e\|_{\mathcal{C}^\alpha(W_j)} \le C \|g\|_{\mathcal{C}^\alpha_{\text{pw},C}(W_j)}. \end{cases} $$

Deriving (A.7)$_1$, and obtaining the estimate of the piecewise Hölder norm for $g_j^e$ is straightforward from (A.6). Next, we consider $x \in W_j \cap \bar{D}$ and $y \in W_j \cap (\mathbb{R}^d \setminus \bar{D})$ and we express:

$$ (A.8) \qquad \begin{aligned} |g_j^e(x) - g_j^e(y)| &= |g(x) - g^+(y', z_j(y')) + g^-(y', z_j(y')) - g(y)| \\ &\le \|g\|_{\mathcal{C}^\alpha_{\text{pw},C}(W_j)} (|x' - y'|^\alpha + |x_d - z_j(y')|^\alpha + |y_d - z_j(y')|^\alpha). \end{aligned} $$

Recalling $y_d \le z_j(y')$ and $z_j(x') \le x_d$, we have:

* If $z_j(y') \le x_d$, then $y_d \le z_j(y') \le x_d$ and hence:

$$ |x_d - z_j(y')| + |y_d - z_j(y')| \le |x_d - y_d|. $$

* If $z_j(y') > x_d$, then $z_j(x') \le x_d \le z_j(y')$ and using (A.5) we find:

$$ |x_d - z_j(x')| + |x_d - z_j(y')| \le |z_j(x') - z_j(y')| \le 2|x' - y'|. $$

Whence:

$$ |y_d - z_j(y')| \le |y_d - x_d| + |x_d - z_j(y')| \le 3|x - y|. $$

Summing up, (A.8) becomes:

$$ |g_j^e(x) - g_j^e(y)| \le C \|g\|_{\mathcal{C}^\alpha_{\text{pw},C}(W_j)} |x - y|^\alpha $$

and (A.7)$_2$ follows.

The global extension $g^e$ reads:

$$ (A.9) \qquad g^e(x) = g(x) [\psi_0(x) + \psi_1(x)] + \sum_{j=2}^{N} g_j^e(x) \psi_j(x), $$

where $(\psi_j)_j$ is a partition of unity subordinate to the cover $(W_j)_j$ of $\mathbb{R}^d$. Given that $\psi_j$ is compactly supported in $W_j$, $g\psi_0$, $g\psi_1$ and $g_j^e\psi_j$, $j \ge 2$, extend over $\mathbb{R}^d$ into $\mathcal{C}^\alpha(\mathbb{R}^d)$ functions. Therefore, $g^e$ itself belongs to $\mathcal{C}^\alpha(\mathbb{R}^d)$, and (A.4)$_{1,2}$ immediately holds for $q = \infty$ from (A.7). From (A.9), we observe:

$$ (A.10) \qquad g^e(x) = g(x) + \sum_{j=2}^{N} (g_j^e(x) - g(x)) \psi_j(x), $$

and whence (recall (A.6) and the local finiteness of the cover):

$$ g^e(x) = g(x) \quad \text{for } x \in \bar{D} \quad \text{and} \quad \|g^e - g\|_{\mathcal{C}^\alpha_{\text{pw},C}(\mathbb{R}^d)} \le C \|g\|_{\mathcal{C}^\alpha_{\text{pw},C}(\mathbb{R}^d)} + C \|\llbracket g \rrbracket\|_{L^\infty(\mathcal{C})} \sup_{j \ge 2} \|\psi_j\|_{\mathcal{C}^\alpha(\mathbb{R}^d)}. $$
## 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 [8] regarding quantitative Hölder estimates for even singular integral operators on patches. This is the key element for their recent result [9] concerning the global well-posedness for the 2d Navier-Stokes free boundary problem for incompressible fluids. **Proposition A.2** summarizes their computations in step 4 and step 5. Finally, the last result we state in this section is from [11] 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 $2l$ ($l \ge 1$), given by:

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

where the kernel $K$ is given by:

$$ K(x) = \frac{P_{2l}(x)}{|x|^{d+2l}} $$

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 a hyper-surface

$$ (A.2) \qquad \partial D := \mathcal{C} = \{ \gamma(s) \in \mathbb{R}^d : s \in \mathbb{T}^{d-1} \} \in \mathcal{C}^{1+\alpha} $$

and verifies the following non self-intersection condition:

$$ (A.3) \qquad \exists c_{\gamma} > 0, \quad |\forall (s, s') \in \mathbb{T}^{d-1}: \quad |\gamma(s) - \gamma(s')| \ge c_{\gamma}^{-1}|s - s'|. $$

We finally consider $\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)$, the space of piecewise Hölder continuous function defined in **Notation 1.1**. The first result by Gancedo and García-Juárez [8] reads as follows.

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

$$ (A.4) \qquad \|T(1_D)\|_{\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \le C(1 + |\partial D|)P(c_{\gamma} + \|\nabla\gamma\|_{L^{\infty}})(1 + \|\nabla\gamma\|_{\mathcal{C}^{\alpha}}). $$

The second result from [9, Step 4 & Step 5] reads as follows.

**Proposition A.2.** Assume that the hypotheses in **Theorem A.1** hold true for singular integral operator $T$ and interface $C$ and let $1 \le p < \infty$. There exists a constant $C = C(d, \alpha, p)$ such that for all $g \in L^p(\mathbb{R}^d) \cap \mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)$, we have:

$$ \|T(g)\|_{\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} \le C \left[ \eta^{-\frac{d}{p}} \|g\|_{L^p(\mathbb{R}^d)} + \|g\|_{\mathcal{C}_{pw,\gamma}^{\alpha}(\mathbb{R}^d)} + \mathcal{P}(C) \|g\|_{L^{\infty}(\mathbb{R}^d)} \right], $$

where

$$ \mathcal{P}(\partial D) = (1 + |\partial D|)P(c_{\gamma} + \|\nabla\gamma\|_{L^{\infty}}) (1 + \|\nabla\gamma\|_{\mathcal{C}^{\alpha}}) \quad \text{and} \quad \eta^{\alpha} := \min \left\{ 1, \frac{1}{16c_{\gamma}\|\nabla\gamma\|_{\mathcal{C}^{\alpha}}} \right\}. $$

## APPENDIX B. STUDY OF LINEAR NON HOMOGENEOUS HEAT EQUATION

This section is devoted to the solvability of the following system

$$ (B.1) \qquad v_t - \tilde{\mu}\Delta u - (\tilde{\mu} + \tilde{\lambda})\nabla \operatorname{div} v = f + \operatorname{div}(b_0 : v) $$

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

$$ \left\{ \begin{array}{l} \partial_t \mathcal{A}v - k\mathcal{A}v = \mathcal{A}f + \mathcal{A}\operatorname{div}(b_0 : v), \\ \mathcal{A}v|_{t=0} = \mathcal{A}v_0 \end{array} \right. $$

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

$$ (B.2) \qquad \left\{ \begin{array}{l} \partial_t \mathcal{A}v - k\mathcal{A}v = \mathcal{A}f + \mathcal{A}\operatorname{div}(b_0 : w), \\ \mathcal{A}v|_{t=0} = \mathcal{A}v_0 \end{array} \right. $$
Difference in (A.9) yields:

$$
\begin{align*}
2(g^e(x) - g^e(y)) &= (g(x) - g(y))(\psi_0(x) + \psi_1(x) + \psi_0(y) + \psi_1(y)) \\
&\quad + (g(x) + g(y)) \sum_{j=0}^{j=1} (\psi_j(x) - \psi_j(y)) + \sum_{j=2}^{N} (g_j^e(x) - g_j^e(y))(\psi_j(x) + \psi_j(y)) \\
&\quad + \sum_{j=2}^{N} (g_j^e(x) + g_j^e(y))(\psi_j(x) - \psi_j(y)) \\
&= (g(x) - g(y))(\psi_0(x) + \psi_1(x) + \psi_0(y) + \psi_1(y)) \\
&\quad + \sum_{j=2}^{N} (g_j^e(x) - g_j^e(y))(\psi_j(x) + \psi_j(y)) + \sum_{j=2}^{N} ((g_j^e - g)(x) + (g_j^e - g)(y))(\psi_j(x) - \psi_j(y)) \\
(A.11) \qquad &\quad + (g(x) + g(y)) \sum_{j=0}^{N} (\psi_j(x) - \psi_j(y)),
\end{align*}
$$

where we notice that the last term vanishes. This leads to

$$
\|g^e\|_{\mathcal{H}^\alpha(\mathbb{R}^d)} \le C \|g\|_{\mathcal{H}_{pw,C}^\alpha(\mathbb{R}^d)} + C \|[g]\|_{L^\infty(C)} \sup_j \|\psi_j\|_{\mathcal{H}^\alpha}
$$

and (A.4)$_3$ follows since we have:

$$
\sup_j \|\psi_j\|_{\mathcal{H}^\alpha(\mathbb{R}^d)} \le C \ell^{-\alpha}.
$$

We may observe that when $x \in D$ and $y \in \mathbb{R}^d \setminus \bar{D}$, it follows that $\psi_1(x) = 0$ and $\psi_0(y) = 0$. Next, we choose $\tilde{x} \in C$ (which implies $\psi_0(\tilde{x}) = \psi_1(\tilde{x}) = 0$, since $\psi_0$ and $\psi_1$ vanish on $\mathbb{R}^d \setminus D$ and $\bar{D}$, respectively) such that

$$
|x - \tilde{x}|, |y - \tilde{x}| \le |x - y|.
$$

Consequently, the first term of the right hand side of (A.11) shall be expressed as follows:

$$
\begin{align*}
(g(x) - g(y))(\psi_0(x) + \psi_1(x) + \psi_0(y) + \psi_1(y))
  &= (g(x) - g(y))(\psi_0(x) + \psi_1(y)) \\
  &= (g(x) - g^+(\tilde{x}) + g^-(\tilde{x}) - g(y))(\psi_0(x) + \psi_0(y)) \\
  &\quad + (g^+(\tilde{x}) - g^-(\tilde{x}))(\psi_0(x) - \psi_0(\tilde{x}) + \psi_1(y) - \psi_1(\tilde{x})).
\end{align*}
$$

We now proceed with the proof of (A.4)$_2$ for $q \in [1, \infty)$. From (A.10) and Hölder's inequality, we deduce:

$$
\begin{align*}
\|g^e - g\|_{L^q(\mathbb{R}^d)}^q &= \int_{D^c} \left| \sum_{j=1}^N (g_j^e(x) - g(x)) \psi_j(x) \right|^q dx \\
&\le \int_{D^c} \left( \sum_{j=1}^N \psi_j(x) \right)^{q-1} \left( \sum_{j=1}^N |g_j^e(x) - g(x)|^q \psi_j(x) \right) dx \\
&\le \sum_{j=1}^N \int_{D^c \cap W_j} |g_j^e(x) - g(x)|^q \psi_j(x) dx \\
&\le 2\ell \sum_{j=1}^N \int_{W_j'} \|[g(\Xi_j(x'))]\|_q^q dx' \quad \text{(recall (A.6))} \\
&\le C\ell \sum_{j=1}^N \int_{W_j \cap C} \|[g]\|_q^q d\sigma
\end{align*}
$$

and (A.4)$_2$ follows. This concludes the proof of Lemma A.2.

APPENDIX B. STUDY OF LINEAR NON HOMOGENEOUS HEAT EQUATION

This section addresses the well-posedness of the following system of PDE:

$$
(B.1) \qquad \begin{cases} v_t - \tilde{\mu}\Delta v - (\tilde{\mu} + \tilde{\lambda})\nabla \text{div } v = f + \text{div}(b_0 \cdot v), \\ v|_{t=0} = v_0. \end{cases}
$$
is just a linear non homogeneous heat equation, so this equation is uniquely solvable and the solution is given by the Duhamel formula:

$$ (B.3) \qquad \mathcal{A}v = \mathcal{T}(kt)\mathcal{A}v_0 + \int_0^t \mathcal{T}(k(t-s))\mathcal{A}f(s)ds + \int_0^t \mathcal{T}(k(t-s))\mathcal{A}\operatorname{div}(b_0:w)(s)ds. $$

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

$$ (B.4) \qquad \sup_{[0,T]} \|v\|_{H^1(\mathbb{R}^d)} \le \|v_0\|_{H^1(\mathbb{R}^d)} + \int_0^t \|\mathcal{T}(k(t-s))\mathcal{A}[f(s) + \operatorname{div}(b_0:w)](s)\|_{H^1(\mathbb{R}^d)} ds. $$

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

$$ (B.5) \qquad \int_0^T \|\partial_t v, \nabla^2 v\|_{L^2(\mathbb{R}^d)}^2 \le \|v_0\|_{H^1(\mathbb{R}^d)}^2 + \int_0^T \left\| \int_0^t \Delta \mathcal{T}(k(t-s)) \mathcal{A}[f + \operatorname{div}(b_0:w)](s) ds \right\|_{L^2(\mathbb{R}^d)}^2 dt. $$

In consequence, for all $w \in \mathcal{C}([0, T], H^1(\mathbb{R}^d))$ we can define $\Phi(w) = \mathcal{P}\Phi(w) + \mathcal{Q}\Phi(w)$ the unique solution of the Cauchy problem (B.2), given by (B.3) and come with the estimate (B.4) and (B.5). This defines an application $\Phi: w \in \mathcal{C}([0, T], H^1(\mathbb{R}^d)) \mapsto \Phi(w) \in \mathcal{C}([0, T], H^1(\mathbb{R}^d))$ the unique solution of the Cauchy problem:

$$ (B.6) \qquad \begin{cases} v_t - \tilde{\mu}\Delta u - (\tilde{\mu} + \tilde{\lambda})\nabla \operatorname{div} v = f + \operatorname{div}(b_0 : w), \\ v|_{t=0} = v_0 \end{cases} $$

whose Helmholtz decomposition verifies (B.3). In the following we will set some tools in order to apply the Banach fixed point theorem. Let us begin by the following:

**Lemma B.1.** There exists a constant $C$ that depends on $b_0$, such that:

$$ \forall \varphi \in H^1(\mathbb{R}^d), \quad \forall t > 0 \quad \|\mathcal{T}(kt) \operatorname{div}(b_0 : \varphi)\|_{H^1(\mathbb{R}^d)} \le C(kt)^{-\frac{1}{2}} \|\varphi\|_{H^1(\mathbb{R}^d)}. $$

*Proof.* First, let us begin by writing:

$$ \mathcal{T}(kt) \operatorname{div}(b_0 : \varphi) = G_{kt} * \operatorname{div}(b_0 : \varphi) = \nabla G_{kt} * (b_0 : \varphi) $$

where $G_t$ is the heat kernel. On the first hand, Young's inequality yields:

$$ \|\mathcal{T}(kt) \operatorname{div}(b_0 : \varphi)\|_{L^2(\mathbb{R}^d)} \le \|\nabla G_{kt}\|_{L^1(\mathbb{R}^d)} \|b_0\|_{L^\infty(\mathbb{R}^d)} \|\varphi\|_{L^2(\mathbb{R}^d)} $$

where,

$$ \|\nabla G_t\|_{L^1(\mathbb{R}^d)} = \frac{(4\pi t)^{-\frac{d}{2}}}{2t} (2\sqrt{t})^{d+1} |\partial B(0, 1)| \int_0^\infty r^d e^{-r^2} dr. $$

On the other hand, one has:

$$ \begin{aligned} \|\nabla \mathcal{T}(kt) \operatorname{div}(b_0 : \varphi)\|_{L^2(\mathbb{R}^d)} &\le \|\nabla G_{kt}\|_{L^1(\mathbb{R}^d)} \|\nabla(b_0 : \varphi)\|_{L^2(\mathbb{R}^d)} \\ &\le \|\nabla G_{kt}\|_{L^1(\mathbb{R}^d)} \|b_0\|_{W^{1,\infty}(\mathbb{R}^d)} \|\varphi\|_{H^1(\mathbb{R}^d)}. \end{aligned} $$

This concludes the proof of **Lemma B.1**.

□

We turn to the stability of the (B.3). So let us set

$$ R_T = \sup_{[0,T]} \left\| \mathcal{T}(kt) \mathcal{A}v_0 + \int_0^t \mathcal{T}(k(t-s)) \mathcal{A}f(s) ds \right\|_{H^1(\mathbb{R}^d)}. $$

**Lemma B.2.** There exists an increasing function $g: [0, \infty) \to [0, \infty)$ and a positive constant $K$, such that the condition

$$ (B.7) \qquad \forall t > 0, \quad \|w(t)\|_{H^1(\mathbb{R}^d)} \le R_T K g(t) $$

implies the same for $\Phi(w)$ for all $w \in \mathcal{C}([0, T], H^1(\mathbb{R}^d))$.

*Proof.* One deduces easily from (B.3) the following:

$$ \|\Phi(w)(t)\|_{H^1(\mathbb{R}^d)} \le R_T + \int_0^t \|\mathcal{T}(k(t-s))\mathcal{A}\operatorname{div}(b_0:w)(s)\|_{H^1(\mathbb{R}^d)} ds. $$

Next, gathering **Lemma B.1** and the assumption (B.7), one has:

$$ \|\Phi(w)(t)\|_{H^1(\mathbb{R}^d)} \le R_T \left( 1 + Ck^{-\frac{1}{2}} K \int_0^t \frac{g(s)}{\sqrt{t-s}} ds \right). $$
The main result is presented in the following theorem.

**Theorem B.1.** Let $T \in (0, \infty)$, $v_0 \in H^1(\mathbb{R}^d)$, $b_0 \in W^{1,\infty}(\mathbb{R}^d)$ and $f \in L^2((0,T) \times \mathbb{R}^d)$. Then, the Cauchy problem (B.1) admits a unique solution $v \in \mathcal{C}([0,T], H^1(\mathbb{R}^d))$. Moreover $v_t, \nabla^2 v \in L^2((0,T) \times \mathbb{R}^d)$.

**B.1. Preliminary.** We define

$$A_k = \begin{cases} \mathcal{P} & \text{if } k = \tilde{\mu}, \\ \mathcal{Q} & \text{if } k = 2\tilde{\mu} + \tilde{\lambda} \end{cases}$$

where $\mathcal{P}$ (resp. $\mathcal{Q}$) is the projector onto the space of divergence-free (resp. curl-free) vector fields. Applying $A_k$ to (B.1), we infer that the divergence-free and curl-free part of any regular$^3$ solution $v$ of (B.1) verify:

$$(B.2) \qquad \left\{ \begin{aligned} \partial_t A_k v - k \Delta A_k v &= A_k f + A_k \operatorname{div}(b_0 \cdot v), \\ A_k v|_{t=0} &= A_k v_0. \end{aligned} \right.$$

Conversely, since $\mathcal{P} + \mathcal{Q} = I$, any regular vector field $v$ that fulfills (B.2) is a solution of (B.1). We shall now consider the following Cauchy problem

$$(B.3) \qquad \left\{ \begin{aligned} \partial_t A_k v - k \Delta A_k v &= A_k f + A_k \operatorname{div}(b_0 : w), \\ A_k v|_{t=0} &= A_k v_0, \end{aligned} \right.$$

where $w \in \mathcal{C}([0,T], H^1(\mathbb{R}^d))$ is a given vector field. The solution of this linear non-homogeneous heat equation is given by the Duhamel formula:

$$(B.4) \qquad A_k v(t) = \mathcal{T}(kt) A_k v_0 + \int_0^t \mathcal{T}(k(t-s)) A_k f(s) ds + \int_0^t \mathcal{T}(k(t-s)) A_k \operatorname{div}(b_0 : w)(s) ds,$$

where $(\mathcal{T}(t))_t$ is the heat semi-group. It follows $v \in \mathcal{C}([0,T], H^1(\mathbb{R}^d))$ with the following estimate:

$$(B.5) \qquad \sup_{[0,T]} \|v\|_{H^1(\mathbb{R}^d)} \le \|v_0\|_{H^1(\mathbb{R}^d)} + \sum_k \int_0^T \|\mathcal{T}(k(t-s)) A_k [f(s) + \operatorname{div}(b_0 : w)](s)\|_{H^1(\mathbb{R}^d)} ds.$$

Additionally, based on [6, Lemma 2.2], we infer that $\partial_t v, \nabla^2 v \in L^2((0,T) \times \mathbb{R}^d)$ and that there exists a constant $C_k = C_k(k) > 0$ such that we have:

$$\int_0^T \|\partial_t A_k v, \nabla^2 A_k v\|_{L^2(\mathbb{R}^d)}^2 \le C_k \|A_k v_0\|_{H^1(\mathbb{R}^d)}^2 + C_k \int_0^T \left\| \int_0^t \Delta \mathcal{T}(k(t-s)) A_k [f + \operatorname{div}(b_0 : w)](s) ds \right\|_{L^2(\mathbb{R}^d)}^2 dt.$$

This therefore defines a map

$$\Phi: \mathcal{C}([0,T], H^1(\mathbb{R}^d)) \rightarrow \mathcal{C}([0,T], H^1(\mathbb{R}^d)) \\ w \mapsto \Phi(w)$$

where $\Phi(w)$ is the unique solution of (B.3), which is given by (B.4) and fulfills (B.5). In the following we will set some tools in order to apply the Banach's fixed-point theorem.

**B.2. Tools.** We begin by the following Lemma:

**Lemma B.2.** There exists a constant $C = C(\|b\|_{W^{1,\infty}(\mathbb{R}^d)})$ such that:

$$\forall \varphi \in H^1(\mathbb{R}^d), \quad \forall t > 0 \quad \| \mathcal{T}(kt) A_k \operatorname{div}(b_0 : \varphi) \|_{H^1(\mathbb{R}^d)} \le C(kt)^{-\frac{1}{2}} \| \varphi \|_{H^1(\mathbb{R}^d)}.$$

*Proof.* We first write:

$$\mathcal{T}(kt) A_k \operatorname{div}(b_0 : \varphi) = G_{kt} * A_k \operatorname{div}(b_0 : \varphi) = \nabla G_{kt} * A_k(b_0 : \varphi)$$

where $G_t$ is the heat kernel. Thanks to the continuity of $A_k$ on $L^2(\mathbb{R}^d)$ and Young's inequality we obtain:

$$\| \mathcal{T}(kt) A_k \operatorname{div}(b_0 : \varphi) \|_{L^2(\mathbb{R}^d)} \le \| \nabla G_{kt} \|_{L^1(\mathbb{R}^d)} \| b_0 \|_{L^\infty(\mathbb{R}^d)} \| \varphi \|_{L^2(\mathbb{R}^d)}$$

where

$$\| \nabla G_t \|_{L^1(\mathbb{R}^d)} = \frac{(4\pi t)^{-\frac{d}{2}}}{2t} (2\sqrt{t})^{d+1} |\partial B(0, 1)| \int_0^\infty r^d e^{-r^2} dr.$$

<footnote n="3">³It is sufficient that $\partial_t v \in L^2((0,T) \times \mathbb{R}^d)$ and $v \in L^2((0,T), H^2(\mathbb{R}^d))$.</footnote>
For the condition (B.7) to be satisfied for $\Phi(w)$, it is sufficient that

$$ (B.8) \qquad 1 + Ck^{-\frac{1}{2}}K \int_{0}^{t} \frac{g(s)}{\sqrt{t-s}}ds \le Kg(t). $$

Let us look for some $\alpha > 0$ and $K > 0$ such that $g$ takes the form $g(t) = e^{\alpha t}$. By change of variable, one has:

$$ e^{-\alpha t} \int_{0}^{t} \frac{e^{\alpha s}}{\sqrt{t-s}} ds = \frac{1}{\sqrt{\alpha}} \int_{0}^{\alpha t} \frac{e^{-s}}{\sqrt{s}} ds \le \frac{1}{\sqrt{\alpha}} \int_{0}^{\infty} \frac{e^{-s}}{\sqrt{s}} ds = \sqrt{\frac{2}{\alpha}} \int_{0}^{\infty} e^{-\frac{r^2}{2}} ds = \sqrt{\frac{\pi}{\alpha}} $$

hence,

$$ 1 + Ck^{-\frac{1}{2}}K \int_{0}^{t} \frac{g(s)}{\sqrt{t-s}}ds \le 1 + Ck^{-\frac{1}{2}}K \sqrt{\frac{\pi}{\alpha}}g(t). $$

Thus, one can choose $K$ to be

$$ K = \frac{1}{4C} \sqrt{\frac{\alpha k}{\pi}} $$

and $\alpha$ large such that $K \ge 2$. In consequence, the condition (B.8) becomes, since $g(t) > 1$,

$$ 1 + Ck^{-\frac{1}{2}}K \int_{0}^{t} \frac{g(s)}{\sqrt{t-s}}ds \le 1 + \frac{g(t)}{4} \le Kg(t), $$

and this concludes the proof of Lemma B.2. □

**Lemma B.3.** There exist $\alpha > 0$ and a positive constant $0 < \kappa < 1$ such that for all $v, w \in \mathscr{C}([0, T], H^1(\mathbb{R}^d))$, one has:

$$ \sup_{[0,T]} e^{-\alpha t} \|\Phi(v)(t) - \Phi(w)(t)\|_{H^1(\mathbb{R}^d)} \le \kappa \sup_{[0,T]} e^{-\alpha t} \|v(t) - w(t)\|_{H^1(\mathbb{R}^d)}. $$

*Proof.* Starting from (B.3) and combining Lemma B.1, one has:

$$ \begin{aligned} \|\Phi(v)(t) - \Phi(w)(t)\|_{H^1(\mathbb{R}^d)} &= \left\| \int_0^t \mathcal{T}(k(t-s))\mathcal{A}\operatorname{div}(b_0 : (v-w))(s)ds \right\|_{H^1(\mathbb{R}^d)} \\ &\le Ck^{-\frac{1}{2}} \int_0^t \frac{1}{\sqrt{t-s}} \|v(s)-w(s)\|_{H^1(\mathbb{R}^d)} ds \\ &\le Ck^{-\frac{1}{2}} \sup_{s \in [0,T]} e^{-\alpha s} \|v(s)-w(s)\|_{H^1(\mathbb{R}^d)} \int_0^t \frac{e^{\alpha s}}{\sqrt{t-s}} ds. \end{aligned} $$

It then turns out that,

$$ \sup_{[0,T]} e^{-\alpha t} \|\Phi(v)(t) - \Phi(w)(t)\|_{H^1(\mathbb{R}^d)} \le Ck^{-\frac{1}{2}} \sqrt{\frac{\pi}{\alpha}} \sup_{t \in [0,T]} e^{-\alpha t} \|v(t) - w(t)\|_{H^1(\mathbb{R}^d)} $$

and Lemma B.3 follows by choosing $\alpha$ large enough. □

We turn to the application of the Banach fixed point theorem in the following vector space

$$ E := \left\{ v \in \mathscr{C}([0, T], H^1(\mathbb{R}^d)) : \sup_{t \in [0, T]} e^{-\alpha t} \|v(t)\|_{H^1(\mathbb{R}^d)} < \infty \right\} $$

endowed with the norm:

$$ \forall v \in E, \quad \|v\|_E = \sup_{t \in [0,T]} e^{-\alpha t} \|v(t)\|_{H^1(\mathbb{R}^d)}. $$

It is obvious that $(E, \|\cdot\|_E)$ is a Banach space and, in particular the closed ball $E_T = B(0, KR_T)$ is complete with respect to the norm $\|\cdot\|_E$. On the first hand, Lemma B.2 ensures that $\Phi$ maps $E_T$ into itself and the Lemma B.3 ensures that $\Phi$ is a contracting application. Hence, by Banach fixed point theorem, $\Phi$ admits an unique fixed point $v$ in $E_T$ that solves the equation (B.1). Moreover, since $v$ verifies

$$ \mathcal{A}v = \mathcal{T}(kt)\mathcal{A}v_0 + \int_{0}^{t} \mathcal{T}(k(t-s))\mathcal{A}f(s)ds + \int_{0}^{t} \mathcal{T}(k(t-s))\mathcal{A}\operatorname{div}(b_0 : v)(s)ds $$

one deduces using [4, Lemma 2.2] that:

$$ \int_{0}^{t} \|v_t, \nabla^2 v\|_{L^2(\mathbb{R}^d)} \le \|v_0\|_{H^1(\mathbb{R}^d)}^2 + \int_{0}^{t} \|f, \operatorname{div}(b_0 : v)\|_{L^2(\mathbb{R}^d)}^2. $$
Similarly, we obtain:

$$
\begin{aligned}
\|\nabla T(kt) \mathcal{A}_k \operatorname{div}(b_0 : \varphi)\|_{L^2(\mathbb{R}^d)} &\le \|\nabla G_{kt}\|_{L^1(\mathbb{R}^d)} \|\nabla(b_0 : \varphi)\|_{L^2(\mathbb{R}^d)} \\
&\le \|\nabla G_{kt}\|_{L^1(\mathbb{R}^d)} \|b_0\|_{W^{1,\infty}(\mathbb{R}^d)} \|\varphi\|_{H^1(\mathbb{R}^d)}.
\end{aligned}
$$

This concludes the proof of Lemma B.2.

<figure><img src="image_3.png" /></figure>

We now turn to the stability of equations (B.4) and to do this, we set

$$
R_{k,T} = \sup_{t \in [0,T]} \left\| \mathcal{T}(kt) \mathcal{A}v_0 + \int_0^t \mathcal{T}(k(t-s)) \mathcal{A}_k f(s) ds \right\|_{H^1(\mathbb{R}^d)} \quad \text{and} \quad R_T = R_{\bar{\mu},T} + R_{2\bar{\mu}+\bar{\lambda},T}.
$$

**Lemma B.3.** There exist constants $K > 0$ and $\beta_0 > 0$ such that for all $\beta \ge \beta_0$ the condition

$$
(B.6) \qquad \forall t \in [0, T], \quad \|w(t)\|_{H^1(\mathbb{R}^d)} \le e^{\beta t} K R_T
$$

implies the same for $\Phi(w)$, for all $w \in \mathcal{C}([0, T], H^1(\mathbb{R}^d))$.

*Proof.* From (B.4), Lemma B.2 and assumption (B.6) on $w$, we have:

$$
\begin{aligned}
\| \mathcal{A}_k \Phi(w)(t) \|_{H^1(\mathbb{R}^d)} &\le R_{k,T} + \int_0^t \| \mathcal{T}(k(t-s)) \mathcal{A}_k \operatorname{div}(b_0 : w)(s) \|_{H^1(\mathbb{R}^d)} ds \\
&\le R_{k,T} + Ck^{-\frac{1}{2}} K R_T \int_0^t \frac{e^{\beta s}}{\sqrt{t-s}} ds.
\end{aligned}
$$

For the condition (B.6) to be satisfied for $\Phi(w)$, it is sufficient that

$$
(B.7) \qquad 1 + 2C \tilde{\mu}^{-\frac{1}{2}} K \int_{0}^{t} \frac{e^{\beta s}}{\sqrt{t-s}} ds \le K e^{\beta t}.
$$

Owing to a change of variable, we have:

$$
e^{-\beta t} \int_0^t \frac{e^{\beta s}}{\sqrt{t-s}} ds = \frac{1}{\sqrt{\beta}} \int_0^{\beta t} \frac{e^{-s}}{\sqrt{s}} ds \le \frac{1}{\sqrt{\beta}} \int_0^{\infty} \frac{e^{-s}}{\sqrt{s}} ds = \sqrt{\frac{2}{\beta}} \int_0^{\infty} e^{-\frac{r^2}{2}} ds = \sqrt{\frac{\pi}{\beta}}
$$

hence,

$$
1 + 2C\tilde{\mu}^{-\frac{1}{2}} K \int_{0}^{t} \frac{e^{\beta s}}{\sqrt{t-s}} ds \le 1 + 2CKe^{\beta t} \sqrt{\frac{\pi}{\tilde{\mu}\beta}}.
$$

We choose $K$ as

$$
K = \frac{1}{4C} \sqrt{\frac{\beta_0 \tilde{\mu}}{\pi}}
$$

with $\beta_0$ large such that $K \ge 2$, and the condition (B.7) becomes

$$
1 + Ck^{-\frac{1}{2}} K \int_{0}^{t} \frac{e^{\beta s}}{\sqrt{t-s}} ds \le 1 + \frac{e^{\beta t}}{2} \sqrt{\frac{\beta_0}{\beta}} \le K e^{\beta t}.
$$

This concludes the proof of Lemma B.3.

<figure><img src="image_4.png" /></figure>

**Lemma B.4.** There exists $\beta'_0 > 0$ such that for all $\beta \ge \beta'_0$ and for all $v$, $w \in \mathcal{C}([0, T], H^1(\mathbb{R}^d))$, one has:

$$
(B.8) \qquad \sup_{t \in [0,T]} e^{-\beta t} \| \Phi(v)(t) - \Phi(w)(t) \|_{H^1(\mathbb{R}^d)} \le \frac{1}{2} \sup_{t \in [0,T]} e^{-\beta t} \| v(t) - w(t) \|_{H^1(\mathbb{R}^d)}.
$$

*Proof.* Starting from (B.4) and Lemma B.2, one has:

$$
\begin{aligned}
\| \mathcal{A}_k \Phi(v)(t) - \mathcal{A}_k \Phi(w)(t) \|_{H^1(\mathbb{R}^d)} &= \left\| \int_0^t \mathcal{T}(k(t-s)) \mathcal{A}_k \operatorname{div}(b_0 : (v-w))(s) ds \right\|_{H^1(\mathbb{R}^d)} \\
&\le Ck^{-\frac{1}{2}} \int_0^t \frac{1}{\sqrt{t-s}} \|v(s) - w(s)\|_{H^1(\mathbb{R}^d)} ds \\
&\le Ck^{-\frac{1}{2}} \sup_{s \in [0,T]} e^{-\beta s} \|v(s) - w(s)\|_{H^1(\mathbb{R}^d)} \int_0^t \frac{e^{\beta s}}{\sqrt{t-s}} ds.
\end{aligned}
$$

It then turns out that,

$$
\sup_{t \in [0, T]} e^{-\beta t} \| \mathcal{A}_k \Phi(v)(t) - \mathcal{A}_k \Phi(w)(t) \|_{H^1(\mathbb{R}^d)} \le Ck^{-\frac{1}{2}} \sqrt{\frac{\pi}{\beta}} \sup_{t \in [0, T]} e^{-\beta t} \| v(t) - w(t) \|_{H^1(\mathbb{R}^d)}
$$

and Lemma B.4 follows by choosing $\beta'_0 > 0$ large enough.

<figure><img src="image_5.png" /></figure>