where $\mathcal{L}$ is a polynomial, whose exact expression is not needed for the remaining steps in the proof of Theorem 1.2. Estimate (2.24) finally follows from (see (3.67))

$$ \int_{0}^{\sigma(t)} \|\nabla u\|_{L^3(\mathbb{R}^3)}^3 \le C_* C_0 \left(1 + \check{\phi}_{\Sigma}^{\frac{3}{2}} E_0^{\frac{1}{2}} \mathcal{A}_1(t)^{\frac{1}{2}}\right) + C_* E_0^{\frac{1}{4}} \left(\mathcal{A}_1(t)^{\frac{5}{4}} + C_0^{\frac{5}{4}}\right). $$

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

### 3.5. Proof of Lemma 2.4.

*Proof.* From(3.93), we have

$$ \begin{aligned} & \int_0^t \sigma^{\frac{1}{2}+\check{\alpha}} \langle\rangle^{1+r_0} \|\nabla \dot{u}\|_{L^3(\mathbb{R}^3)}^2 \\ & \le C_* \left( \int_0^t \sigma^{1+\check{\alpha}} \langle\rangle^{\frac{3}{2}(1+r_0)} \|\rho \ddot{u}\|_{L^2(\mathbb{R}^3)}^2 \right)^{1/2} \left( \int_0^t \sigma^{\check{\alpha}} \langle\rangle^{1+r_0} \|(-\Delta)^{-1} \nabla \mathcal{P}(\rho \ddot{u}), (-\Delta)^{-1} \mathrm{div}(\rho \ddot{u})\|_{L^2(\mathbb{R}^3)}^2 \right)^{1/2} \\ & \quad + C_* \int_0^t \sigma^{\frac{1}{2}+\check{\alpha}} \langle\rangle^{1+r_0} \left( \|\nabla u, G(\rho, c)\|_{L^6(\mathbb{R}^3)}^4 + \|\mathrm{div} u\|_{L^3(\mathbb{R}^3)}^2 + \check{\phi}_{\Sigma}^{\frac{2}{4}} \|\nabla \dot{u}\|_{L^2(\mathbb{R}^3)}^2 \right) \\ & \le C_* \mathcal{A}_3(t)^{1/2} \left( \int_0^t \sigma^{\check{\alpha}} \langle\rangle^{1+r_0} \left( \|\nabla \dot{u}\|_{L^2(\mathbb{R}^3)}^2 + \|\nabla u\|_{L^4(\mathbb{R}^3)}^4 + \|\mathrm{div} u\|_{L^2(\mathbb{R}^3)}^2 \right) \right)^{1/2} \\ & \quad + C_* \int_0^t \sigma^{\frac{1}{2}+\check{\alpha}} \langle\rangle^{1+r_0} \left( \|\nabla u, G(\rho, c)\|_{L^6(\mathbb{R}^3)}^4 + \|\mathrm{div} u\|_{L^3(\mathbb{R}^3)}^2 + \check{\phi}_{\Sigma}^{\frac{2}{4}} \|\nabla \dot{u}\|_{L^2(\mathbb{R}^3)}^2 \right). \end{aligned} \quad (3.94) $$

Using (3.86), we deduce

$$ \int_0^t \sigma^{\check{\alpha}} \langle\rangle^{1+r_0} \left( \|\nabla \dot{u}\|_{L^2(\mathbb{R}^3)}^2 + \|\nabla u\|_{L^4(\mathbb{R}^3)}^4 + \|\mathrm{div} u\|_{L^2(\mathbb{R}^3)}^2 \right) \\ \le C_* (\check{\mathcal{A}}_2(t) + C_0(1 + \check{\phi}_{\Sigma}^{\frac{6}{4}} \mathcal{A}_1(t)) + (C_0 + \mathcal{A}_1(t)) \mathcal{A}_1(t)^2), $$

and with the help of (3.88), the remaining terms can be estimated as follows:

$$ \begin{aligned} & \int_0^t \sigma^{\frac{1}{2}+\check{\alpha}} \langle\rangle^{1+r_0} (\|\nabla u, G(\rho, c)\|_{L^6(\mathbb{R}^3)}^4 + \|\mathrm{div} u\|_{L^3(\mathbb{R}^3)}^2 + \check{\phi}_{\Sigma}^{\frac{2}{4}} \|\nabla \dot{u}\|_{L^2(\mathbb{R}^3)}^2) \\ & \le \left( \int_0^t \langle\rangle^{1+r_0} \|\nabla u, G(\rho, c)\|_{L^6(\mathbb{R}^3)}^2 \right)^{1/2} \left( \int_0^t \sigma^{1+2\check{\alpha}} \langle\rangle^{1+r_0} (\|\nabla u, G(\rho, c)\|_{L^6(\mathbb{R}^3)}^6 + \|\mathrm{div} u\|_{L^2(\mathbb{R}^3)}^2) \right)^{1/2} + C_* \check{\phi}_{\Sigma}^{\frac{2}{4}} \check{\mathcal{A}}_2(t) \\ & \le \left( \int_0^t \langle\rangle^{1+r_0} \|\nabla u, G(\rho, c)\|_{L^6(\mathbb{R}^3)}^2 \right)^{1/2} \left( C_* C_0 (1 + \check{\phi}_{\Sigma}^{\frac{12}{4}} \mathcal{A}_1(t)^2) + C_* \mathcal{A}_2(t) (1 + \mathcal{A}_1(t) \check{\mathcal{A}}_2(t)) \right)^{1/2} + C_* \check{\phi}_{\Sigma}^{\frac{2}{4}} \check{\mathcal{A}}_2(t). \end{aligned} $$

The $L^2((0, t), L^6(\mathbb{R}^3))$-norm of $\langle\rangle^{(1+r_0)/2}(\nabla u, G(\rho, c))$ will be estimated in what follows.
To achieve this, we first deduce from (3.48)-(3.49) to obtain

$$ \frac{d}{dt} \int_{\mathbb{R}^3} H_5(\rho, c) + \int_{\mathbb{R}^3} \frac{1}{v(\rho, c)} |G(\rho, c)|^6 = - \int_{\mathbb{R}^3} \frac{1}{v(\rho, c)} FG(\rho, c) |G(\rho, c)|^4, $$

and from (3.46)-(3.47), we have

$$ (3.95) \qquad \tilde{\rho}(c) g(\rho, c) H_l(\rho, c) \le |G(\rho, c)|^{l+1} \le \frac{\rho + \tilde{\rho}(c)}{a_l(c)} g(\rho, c) H_l(\rho, c), \quad \text{where } g(\rho, c) = \frac{G(\rho, c)}{\rho - \tilde{\rho}(c)}. $$

We will use the following notations:

$$ (3.96) \qquad a_{l,*} = \inf_c a_l(c), \quad g^* = \sup_{\rho,c} g(\rho,c), \quad g_* = \inf_{\rho,c} g(\rho,c), \quad k_* = \frac{\rho_* g_*}{2\rho^* g^*} \quad \text{and} \quad k_l^* = \frac{v^*}{a_{l,*} v_*}. $$

It follows that

$$ \begin{aligned} \frac{d}{dt} \int_{\mathbb{R}^3} H_5(\rho, c) + \int_{\mathbb{R}^3} \tilde{\rho}(c) \frac{g(\rho, c)}{v(\rho, c)} H_5(\rho, c) & \le \left( \int_{\mathbb{R}^3} \frac{1}{v(\rho, c)} |F|^6 \right)^{1/6} \left( \int_{\mathbb{R}^3} \frac{\rho + \tilde{\rho}(c)}{a_5(c)} \frac{g(\rho, c)}{v(\rho, c)} H_5(\rho, c) \right)^{5/6} \\ & \le \frac{1}{v_*} \left( \frac{2\rho^* g^*}{a_{5,*}} \right)^{5/6} \|F\|_{L^6(\mathbb{R}^3)} \left( \int_{\mathbb{R}^3} H_5(\rho, c) \right)^{5/6}. \end{aligned} $$
Hence

$$ (3.97) \quad 3 \frac{d}{dt} \left( \int_{\mathbb{R}^3} H_5(\rho, c) \right)^{1/3} + \frac{\rho_* g_*}{v^*} \left( \int_{\mathbb{R}^3} H_5(\rho, c) \right)^{1/3} \le \frac{1}{v_*} \left( \frac{2\rho^* g^*}{a_{5,*}} \right)^{5/6} \|F\|_{L^6(\mathbb{R}^3)} \left( \int_{\mathbb{R}^3} H_5(\rho, c) \right)^{1/6}, $$

and Young's inequality yields

$$ 3 \frac{d}{dt} \left( \int_{\mathbb{R}^3} H_5(\rho, c) \right)^{1/3} + \frac{\rho_* g_*}{2v^*} \left( \int_{\mathbb{R}^3} H_5(\rho, c) \right)^{1/3} \le \frac{v^* v_*^{-2}}{2\rho_* g_*} \left( \frac{2\rho^* g^*}{a_{5,*}} \right)^{5/3} \|F\|_{L^6(\mathbb{R}^3)}^2. $$

Upon multiplying by the time weight $\kappa_1 = \langle\rangle^{1+r_0}$ and integrating in time, we obtain:

$$ \kappa_1(t) \left( \int_{\mathbb{R}^3} H_5(\rho, c)(t) \right)^{1/3} + \frac{\rho_* g_*}{6v^*} \int_0^t \kappa_1 \left( \int_{\mathbb{R}^3} H_5(\rho, c) \right)^{1/3} \le \left( \int_{\mathbb{R}^3} H_5(\rho_0, c_0) \right)^{1/3} + \int_0^t \kappa'_1 \left( \int_{\mathbb{R}^3} H_5(\rho, c) \right)^{1/3} \\ + \frac{v^* v_*^{-2}}{6\rho_* g_*} \left( \frac{2\rho^* g^*}{a_{5,*}} \right)^{5/3} \int_0^t \kappa_1 \|F\|_{L^6(\mathbb{R}^3)}^2. $$

From (3.64)-(3.52) we infer

$$ \left( \int_{\mathbb{R}^3} H_5(\rho_0, c_0) \right)^{1/3} + \int_0^t \kappa'_1 \left( \int_{\mathbb{R}^3} H_5(\rho, c) \right)^{1/3} \le C_* C_0, $$

and (3.95), yields

$$ \int_0^t \kappa_1 \left( \int_{\mathbb{R}^3} H_5(\rho, c) \right)^{1/3} \ge \left( \frac{a_{5,*}}{2\rho^* g^*} \right)^{1/3} \int_0^t \kappa_1 \|G(\rho, c)\|_{L^6(\mathbb{R}^3)}^2. $$

It then follows that

$$ k_*^2 \int_0^t \kappa_1 \|G(\rho, c)\|_{L^6(\mathbb{R}^3)}^2 \le C_* C_0 + k_*^{*2} \int_0^t \kappa_1 \|F\|_{L^6(\mathbb{R}^3)}^2. $$

Once again, we combine (3.63)-(A.8) to deduce that

$$ \begin{aligned} k_*^2 \int_0^t \kappa_1 \left( \|G(\rho, c)\|_{L^6(\mathbb{R}^3)}^2 + \mu_*^2 \|\nabla u\|_{L^6(\mathbb{R}^3)}^2 \right) &\le C_* C_0 + C_* \int_0^t \kappa_1 \|(-\Delta)^{-1} \nabla P(\rho \dot{u})\|_{L^6(\mathbb{R}^3)}^2 \\ &\quad + C(k_*^2 + k_5^{*2}) \int_0^t \kappa_1 \|[\mathcal{H}, \mu(\rho, c) - \tilde{\mu}] \mathbb{D}u, [\mathcal{H}', \mu(\rho, c) - \tilde{\mu}] \mathbb{D}u\|_{L^6(\mathbb{R}^3)}^2 \\ &\le C_* C_0 + C_* \int_0^t \kappa_1 \left( \|\sqrt{\rho} \dot{u}\|_{L^2(\mathbb{R}^3)}^2 + \tilde{\phi}_\Sigma^2 \|\nabla u\|_{L^{6/(1+\alpha)}(\mathbb{R}^3)}^2 \right) \\ &\quad + C(k_*^2 + k_5^{*2}) \sup_{[0,t]} \left\| \left[ \mu(\rho, c) \right] \right\|_{L^\infty(\Sigma)} \int_0^t \kappa_1 \|\nabla u\|_{L^6(\mathbb{R}^3)}^2. \end{aligned} $$

Under the smallness assumption on the shear-viscosity jump

$$ \frac{C}{\mu_*^2} \left( 1 + \frac{k_5^{*2}}{k_*^2} \right) \sup_{[0,t]} \left\| \left[ \mu(\rho, c) \right] \right\|_{L^\infty(\Sigma)}^2 \le \frac{1}{2}, $$

the last term can be absorbed into the LHS, and Young's inequality gives

$$ \int_0^t \kappa_1 \|G(\rho, c), \nabla u\|_{L^6(\mathbb{R}^3)}^2 \le C_* (C_0(1 + \tilde{\phi}_\Sigma^{\frac{2}{\alpha}}) + \mathcal{A}_1(t)). $$

This completes the estimate of the last term in (3.94).

Using an interpolation inequality, we have:

$$ \int_0^t \|\dot{u}\|_{L^{6/(2-\alpha)}(\mathbb{R}^3)} \le C_* \left( \int_0^t \kappa_1 \|\dot{u}\|_{L^2(\mathbb{R}^3)}^2 \right)^{\frac{1-\alpha}{4}} \left( \int_0^t \sigma \kappa_1 \|\nabla \dot{u}\|_{L^2(\mathbb{R}^3)}^2 \right)^{\frac{1+\alpha}{4}} \left( \int_0^t \kappa_1^{-1} \sigma^{-\frac{1+\alpha}{2}} \right)^{\frac{1}{2}}, $$

and
◇ if $\alpha \in (0, 1/2)$,

$$ 
\begin{aligned}
\int_0^t \|\dot{u}\|_{L^{3/(1-\alpha)}(\mathbb{R}^3)} &\le C_* \int_0^t \|\rho \dot{u}\|_{L^2(\mathbb{R}^3)^{\frac{1-2\alpha}{2}}} \|\nabla \dot{u}\|_{L^2(\mathbb{R}^3)^{\frac{1+2\alpha}{2}}} \\
&\le C_* \left( \int_0^t \kappa_1 \|\rho \dot{u}\|_{L^2(\mathbb{R}^3)}^2 \right)^{\frac{1-2\alpha}{4}} \left( \int_0^t \sigma \kappa_1 \|\nabla \dot{u}\|_{L^2(\mathbb{R}^3)}^2 \right)^{\frac{1+2\alpha}{4}} \left( \int_0^t \kappa_1^{-1} \sigma^{-\frac{1+2\alpha}{2}} \right)^{1/2};
\end{aligned}
$$

◇ If $\alpha \in [1/2, 1)$, we have $\check{\alpha} \in (0, 3/2 - \alpha)$ and whence

$$ 
\begin{aligned}
\int_0^t \|\dot{u}\|_{L^{3/(1-\alpha)}(\mathbb{R}^3)} &\le C_* \int_0^t \|\nabla \dot{u}\|_{L^2(\mathbb{R}^3)^{2(1-\alpha)}} \|\nabla \dot{u}\|_{L^3(\mathbb{R}^3)^{2\alpha-1}} \\
&\le C_* \left( \int_0^t \sigma^{\check{\alpha}} \kappa_1 \|\nabla \dot{u}\|_{L^2(\mathbb{R}^3)}^2 \right)^{1-\alpha} \left( \int_0^t \sigma^{\frac{1}{2}+\check{\alpha}} \kappa_1 \|\nabla \dot{u}\|_{L^3(\mathbb{R}^3)}^2 \right)^{\alpha-\frac{1}{2}} \left( \int_0^t \kappa_1^{-1} \sigma^{\frac{1}{2}-(\alpha+\check{\alpha})} \right)^{\frac{1}{2}},
\end{aligned}
$$

The final step is devoted to obtaining an $L^1((0, t), L^p(\mathbb{R}^3))$ estimate for $\nabla u$ and $G(\rho, c)$, with $p \in [6/\alpha, \infty)$. To achieve this, we follow the computations that lead to (3.97) and we obtain

$$ p \frac{d}{dt} \left[ \int_{\mathbb{R}^3} H_{p-1}(\rho, c) \right]^{\frac{1}{p}} + \frac{\rho_* g_*}{v^*} \left[ \int_{\mathbb{R}^3} H_{p-1}(\rho, c) \right]^{\frac{1}{p}} \le \frac{1}{v_*} \left( \frac{2\rho^* g^*}{a_{p,*}} \right)^{\frac{1}{p'}} \|F\|_{L^p(\mathbb{R}^3)}, $$

and time integration, together with (3.95), yields:

$$ k_* \int_0^t \|G(\rho, c)\|_{L^p(\mathbb{R}^3)} \le C_* C_0 + k_p^* \int_0^t \|F\|_{L^p(\mathbb{R}^3)}. $$

It follows that (using again (3.63)-(A.8)):

$$ 
\begin{aligned}
k_* \int_0^t \|G(\rho, c), \mu_* \nabla u\|_{L^p(\mathbb{R}^3)} &\le C_* C_0 + C_* \int_0^t \left( \|\rho \dot{u}\|_{L^{3p/(3+p)}(\mathbb{R}^3)} + \check{\phi}_\Sigma \|\nabla u\|_{L^{6p/(1+\rho\alpha)}(\mathbb{R}^3)} \right) \\
&\quad + C(k_* + k_p^*) \sup_{[0,t]} \|\llbracket \mu(\rho, c) \rrbracket\|_{L^\infty(\Sigma)} \int_0^t \|\nabla u\|_{L^p(\mathbb{R}^3)}.
\end{aligned}
$$

Under the smallness assumption on the shear-viscosity jump

$$ \frac{C}{\mu_*} \left(1 + \frac{k_p^*}{k_*}\right) \sup_{[0,t]} \|\llbracket \mu(\rho, c) \rrbracket\|_{L^\infty(\Sigma)} \le \frac{1}{2}, $$

the last term can be absorbed into the LHS, and Young's inequality gives

$$ 
\begin{aligned}
\int_0^t \|G(\rho, c), \nabla u\|_{L^p(\mathbb{R}^3)} &\le C_* C_0 + C_* \int_0^t \left( \|\rho \dot{u}\|_{L^{3p/(3+p)}(\mathbb{R}^3)} + \check{\phi}_\Sigma^{\frac{3}{a}} \left(1-\frac{2}{p}\right) \|\nabla u\|_{L^2(\mathbb{R}^3)} \right) \\
&\le C_* C_0 \left( 1 + \check{\phi}_\Sigma^{\frac{3}{a}} \left(1-\frac{2}{p}\right) \right) + C_* \left( \int_0^t \kappa_1 \|\dot{u}\|_{L^2(\mathbb{R}^3)}^2 \right)^{\frac{1}{4}+\frac{3}{2p}} \left( \int_0^t \sigma \kappa_1 \|\nabla \dot{u}\|_{L^2(\mathbb{R}^3)}^2 \right)^{\frac{1}{4}-\frac{3}{2p}} \left( \int_0^t \kappa_1^{-1} \sigma^{-\frac{1}{2}+\frac{3}{p}} \right)^{\frac{1}{2}}.
\end{aligned}
$$

This completes the proof of Lemma 2.4. □

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

### 4. PROOF OF THE MAIN RESULT

This section is devoted to the final step in the proof of the main result, Theorem 1.2. It is divided into three steps. First, (see Section 4.1) we construct an approximate sequence $(c^n, \rho^n, u^n)_{n \in \mathbb{N}}$ for the Cauchy problem (2.1). This sequence comes with the a priori estimates summarized in Section 2. In Section 4.2, we close these estimates using a bootstrap argument.

4.1. **Construction of approximate solutions.** Usually, the approximate sequence $(c^n, \rho^n, u^n)$ is constructed as the solution of the Cauchy problem (1.1) with initial data $(c_0^n, \rho_0^n, u_0^n)$ obtained by mollifying $(c_0, \rho_0, u_0)$ with a smooth kernel. This approach is unsuited here, because smoothing the initial data would remove the density discontinuity. Furthermore, classical arguments à la Feireisl-Novotný-Petzeltová, or à la P.-L. Lions fail in the density-dependent viscosity case; in particular, the well-known effective flux and vorticity degenerate completely. Motivated by this, we establish in [44] a local-in-time existence of weak solutions to the system (2.1) within a functional framework tailored to our setting. In particular, beyond the initial data assumptions (1.12)-(1.13)-(1.18), it
is enough to require $(\rho \dot{u})|_{t=0} \in L^2(\mathbb{R}^3)$ to construct the approximate sequence $(c^n, \rho^n, u^n)$ to (2.1). In other terms, we shall consider the Cauchy problem:

$$ (4.1) \qquad \begin{cases} \partial_t c^n + u^n \cdot \nabla c^n = 0, \\ \partial_t \rho^n + \text{div}(\rho^n u^n) = 0, \\ \partial_t (\rho^n u^n) + \text{div}(\rho^n u^n \otimes u^n) + \nabla P(\rho^n, c^n) = \text{div}(2\mu(\rho^n, c^n)\mathbb{D}u^n) + \nabla(\lambda(\rho^n, c^n)\text{div}u^n), \\ (c^n, \rho^n, u^n)|_{t=0} = (c_0, \rho_0, u_0^n), \end{cases} $$

with $u_0^n \in H^1(\mathbb{R}^3) \cap L^{p_0}(\mathbb{R}^3)$, fulfilling the so-called compatibility condition

$$ (\rho^n \dot{u}^n)|_{t=0} = \text{div}(2\mu(\rho_0, c_0)\mathbb{D}u_0^n + (\lambda(\rho_0, c_0)\text{div}u_0^n - P(\rho_0, c_0) + \tilde{P})I_3) \in L^2(\mathbb{R}^3). $$

We construct the sequence of initial velocity $(u_0^n)_n$ as follows.

**Lemma 4.1.** There exists a sequence $(u_0^n)_n$ of initial velocities such that

$$ \text{div}(2\mu(\rho_0, c_0)\mathbb{D}u_0^n) + \nabla(\lambda(\rho_0, c_0)\text{div}u_0^n) - \nabla P(\rho_0, c_0) \in L^2(\mathbb{R}^3); $$

and

$$ (4.2) \qquad \|u_0^n - u_0\|_{H^1(\mathbb{R}^3)}^2 + \|u_0^n - u_0\|_{L^{p_0}(\mathbb{R}^3)}^2 \xrightarrow{n \to \infty} 0. $$

We provide the proof of Lemma 4.1 below. With the sequence $(u_0^n)_n$ in place, we construct $(c^n, \rho^n, u^n)_n$ by applying the Theorem A.1. This theorem refines our recent work in [44], where local-in-time well-posedness for (4.1) was obtained assuming

$$ \|\mu(\rho_0, c_0) - \tilde{\mu}\|_{\text{pw}, \Sigma_0(\mathbb{R}^3)} \ll 1. $$

Indeed, here we only require smallness of the shear viscosity jump. In particular, when the shear viscosity is continuous, the condition holds trivially. Thus, our result aligns with the well-posedness theory for smooth data.

**Proof of Lemma 4.1.** Following Section 3.5 of [43], we consider a sequence of nonnegative and smooth mollifier $(\omega_n)_{n \in \mathbb{N}}$, and we denote by $\Pi_0$ the stress tensor at the initial time

$$ \Pi_0 = 2\mu(\rho_0, c_0)\mathbb{D}u_0 + (\lambda(\rho_0, c_0)\text{div}u_0 - P(\rho_0, c_0) + \tilde{P})I_3. $$

We now consider the elliptic equation

$$ (4.3) \qquad -\text{div}(2\mu(\rho_0, c_0)\mathbb{D}u_0^n + (\lambda(\rho_0, c_0)\text{div}u_0^n - P(\rho_0, c_0) + \tilde{P})I_3) + d_n u_0^n = -\text{div}(\omega_n * \Pi_0), $$

where $(d_n)_{n \in \mathbb{N}}$ is a sequence of positive real numbers is a sequence of positive real numbers to be specified later.

For each $n \in \mathbb{N}$, the existence and uniqueness of $u_0^n \in H^1(\mathbb{R}^3)$ follow from the Lax-Milgram theorem. We are now left to prove (4.2). We add $\text{div}(\Pi_0)$ to both sides of (4.3)

$$ (4.4) \qquad -\text{div}(2\mu(\rho_0, c_0)\mathbb{D}(u_0^n - u_0)) - \nabla(\lambda(\rho_0, c_0)\text{div}(u_0^n - u_0)) + d_n u_0^n = -\text{div}(\omega_n * \Pi_0 - \Pi_0), $$

and basic energy estimate yields

$$ \int_{\mathbb{R}^3} (2\mu(\rho_0, c_0)|\mathbb{D}(u_0^n - u_0)|^2 + \lambda(\rho_0, c_0)|\text{div}(u_0^n - u_0)|^2)(x)dx + d_n \int_{\mathbb{R}^3} |u_0^n|^2(x)dx \\ = \int_{\mathbb{R}^3} (\omega_n * \Pi_0 - \Pi_0)(x)\mathbb{D}(u_0^n - u_0)(x)dx + d_n \int_{\mathbb{R}^3} u_0^n(x)u_0(x)dx. $$

Applying Young's and Korn's inequalities, we get

$$ (4.5) \qquad \int_{\mathbb{R}^3} [\mu(\rho_0, c_0)|\mathbb{D}(u_0^n - u_0)|^2 + \lambda(\rho_0, c_0)(\text{div}(u_0^n - u_0))^2](x)dx + \frac{d_n}{2}\|u_0^n\|_{L^2(\mathbb{R}^3)}^2 \\ \leq \frac{1}{\mu_*}\|\omega_n * \Pi_0 - \Pi_0\|_{L^2(\mathbb{R}^3)}^2 + \frac{d_n}{2}\|u_0^n\|_{L^2(\mathbb{R}^3)}^2, $$

and therefore

$$ \lim_{n \to \infty} \|\nabla u_0^n - \nabla u_0\|_{L^2(\mathbb{R}^3)} = 0 \quad \text{and} \quad \limsup_{\delta \to 0} \|u_0^n\|_{L^2(\mathbb{R}^3)} \leq \|u_0\|_{L^2(\mathbb{R}^3)}, $$

as soon as $(d_n)_n$ satisfies

$$ (4.6) \qquad d_n \xrightarrow{n \to \infty} 0 \quad \text{and} \quad d_n^{-1} \|\omega_n * \Pi_0 - \Pi_0\|_{L^2(\mathbb{R}^3)} \xrightarrow{n \to \infty} 0. $$
As a consequence, $u_0^{n \xrightarrow{n \to \infty}} u_0$ strongly in $H^1(\mathbb{R}^3)$. It now remains to prove $\|u_0^n - u_0\|_{L^{p_0}(\mathbb{R}^3)} \xrightarrow{n \to \infty} 0$.
To achieve this, we set $v_0^n = u_0^n - u_0$, and we rewrite (4.4) as

$$ -\tilde{\mu}\Delta v_0^n - (\tilde{\mu} + \tilde{\lambda})\nabla \operatorname{div} v_0^n + d_n v_0^n = -d_n u_0 - \operatorname{div}(\omega_n * \Pi_0 - \Pi_0) \\ + \operatorname{div}(2(\mu(\rho_0, c_0) - \tilde{\mu})\mathbb{D}v_0^n) + \nabla((\lambda(\rho_0, c_0) - \tilde{\lambda})\operatorname{div} v_0^n) =: h_0^n. $$

From this, we deduce (recall that $\mathbb{P}$ denotes the Leray projector)

$$ \mathbb{P}v_0^n = Z_n * \mathbb{P}h_0^n, \quad \text{and} \quad (\mathbb{I} - \mathbb{P})v_0^n = \check{Z}_n * (\mathbb{I} - \mathbb{P})h_0^n; $$

where

$$ (\mathfrak{F}Z_n)(\xi) = \frac{1}{d_n + \tilde{\mu}|\xi|^2}, \quad \text{and} \quad (\mathfrak{F}\check{Z}_n)(\xi) = \frac{1}{d_n + (2\tilde{\mu} + \tilde{\lambda})|\xi|^2}. $$

For any $a > 0$, we have

$$ \frac{1}{1 + a|\xi|^2} = \int_0^\infty e^{-t(1+a|\xi|^2)} dt, $$

and whence

$$ \check{K}_a(x) := \mathfrak{F}^{-1}\left(\xi \mapsto \frac{1}{1 + a|\xi|^2}\right)(x) = \int_0^\infty e^{-t} \mathfrak{F}^{-1}\left(\xi \mapsto e^{-at|\xi|^2}\right) dt = a^{-3/2} \int_0^\infty e^{-t} K_t\left(\frac{x}{\sqrt{a}}\right) dt, $$

where $(K_t)_{t>0}$ denotes the heat kernel. As a result, for all $q \in [1, \infty]$ and $p \in [1, 3/2)$, we have

$$ \|\check{K}_a\|_{L^q(\mathbb{R}^3)} \le C a^{-\frac{3}{2}(1-1/q)}, \quad \text{and} \quad \|\nabla \check{K}_a\|_{L^p(\mathbb{R}^3)} \le C a^{-2+3/(2p)}. $$

Since $Z_n = d_n^{-1} \check{K}_{\tilde{\mu}/d_n}$, it is straightforward to derive the following bound (for some $p \in (1, p_0)$, and $1/p + 1/p = 1/2 + 1/p_0$):

$$ \begin{aligned} \|Z_n * \operatorname{div} \mathbb{P}(2(\mu(\rho_0, c_0) - \tilde{\mu})\mathbb{D}v_0^n)\|_{L^{p_0}(\mathbb{R}^3)} &+ \|Z_n * \mathbb{P}\nabla((\lambda(\rho_0, c_0) - \tilde{\lambda})\operatorname{div} v_0^n)\|_{L^{p_0}(\mathbb{R}^3)} \\ &\le C \|\nabla Z_n\|_{L^p(\mathbb{R}^3)} \|\nabla v_0^n\|_{L^2(\mathbb{R}^3)} \|\mu(\rho_0, c_0) - \tilde{\mu}, \lambda(\rho_0, c_0) - \tilde{\lambda}\|_{L^p(\mathbb{R}^3)} \\ &\le C_* d_n^{1-\frac{3}{2p}} \|\nabla v_0^n\|_{L^2(\mathbb{R}^3)} \|\mu(\rho_0, c_0) - \tilde{\mu}, \lambda(\rho_0, c_0) - \tilde{\lambda}\|_{L^p(\mathbb{R}^3)}. \end{aligned} $$

Hence, from (4.6)-(4.5), we obtain

$$ (4.7) \qquad \|Z_n * \operatorname{div} \mathbb{P}(2(\mu(\rho_0, c_0) - \tilde{\mu})\mathbb{D}v_0^n)\|_{L^{p_0}(\mathbb{R}^3)} + \|Z_n * \mathbb{P}\nabla((\lambda(\rho_0, c_0) - \tilde{\lambda})\operatorname{div} v_0^n)\|_{L^{p_0}(\mathbb{R}^3)} \xrightarrow{n \to \infty} 0. $$

Next, for $R > 0$ we write

$$ \begin{aligned} d_n Z_n * \mathbb{P} u_0 &= \check{K}_{\tilde{\mu}/d_n} * \mathbb{P} u_0 \\ &= \check{K}_{\tilde{\mu}/d_n} * \mathbb{P}(\mathbb{1}_{B_R} u_0) + \check{K}_{\tilde{\mu}/d_n} * \mathbb{P}(\mathbb{1}_{B_R^c} u_0), \end{aligned} $$

and it follows that

$$ \|d_n Z_n * \mathbb{P} u_0\|_{L^{p_0}(\mathbb{R}^3)} \le C_* \left( d_n^{\frac{3}{2}(1/p-1/p_0)} R^{3(1/p-1/p_0)} \|u_0\|_{L^{p_0}(\mathbb{R}^3)} + \|\mathbb{1}_{B_R^c} u_0\|_{L^{p_0}(\mathbb{R}^3)} \right). $$

Choosing $R = d_n^{-1/4}$, it is immediate that the first term converges to zero as $n \to \infty$; the same holds for the second term by the dominated convergence theorem. In the following will now prove

$$ (4.8) \qquad \|Z_n * \operatorname{div} \mathbb{P}(\omega_n * \Pi_0 - \Pi_0)\|_{L^{p_0}(\mathbb{R}^3)} \xrightarrow{n \to \infty} 0. $$

Following the computations that led to (4.7), we obtain (4.8) provided that

$$ d_n^{-\frac{1}{2}} \|\omega_n * \Pi_0 - \Pi_0\|_{L^{p_0}(\mathbb{R}^3)} \xrightarrow{n \to \infty} 0. $$

However, this would require assuming $\Pi_0 \in L^{p_0}(\mathbb{R}^3)$, and hence $\nabla u_0 \in L^{p_0}(\mathbb{R}^3)$ which we aim to avoid. To this end, we express

$$ \begin{aligned} \Pi_0 &= 2(\mu(\rho_0, c_0) - \tilde{\mu})\mathbb{D}u_0 + (\lambda(\rho_0, c_0) - \tilde{\lambda})\operatorname{div} u_0 + \tilde{\mu}\Delta u_0 + (2\tilde{\mu} + \tilde{\lambda})\nabla \operatorname{div} u_0 - P(\rho_0, c_0) + \tilde{P} \\ &=: \Pi_{0,1} + \tilde{\mu}\Delta u_0 + (2\tilde{\mu} + \tilde{\lambda})\nabla \operatorname{div} u_0, \end{aligned} $$

and assuming that

$$ d_n^{-\frac{1}{2}} \|\omega_n * \Pi_{0,1} - \Pi_{0,1}\|_{L^{p_0}(\mathbb{R}^3)} \xrightarrow{n \to 0} 0, \quad \text{we readily obtain} \quad \|Z_n * \operatorname{div} \mathbb{P}(\omega_n * \Pi_{0,1} - \Pi_{0,1})\|_{L^{p_0}(\mathbb{R}^3)} \xrightarrow{n \to \infty} 0. $$
Then we write the remaining term as

$$-\tilde{\mu}\mathcal{F}^{-1}\left(\xi \mapsto \frac{|\xi|^2}{d_n + \tilde{\mu}|\xi|^2} \mathcal{F}\mathcal{P}(\omega_n * u_0 - u_0)\right) - (2\tilde{\mu} + \tilde{\lambda})\mathcal{F}^{-1}\left(\xi \mapsto \frac{\xi}{d_n + \tilde{\mu}|\xi|^2} \xi \cdot \mathcal{F}\mathcal{P}(\omega_n * u_0 - u_0)\right),$$

and a Fourier multiplier theorem (see e.g. Theorem 6.2.7 of [22]) yields

$$\left\| \tilde{\mu} \mathcal{F}^{-1} \left( \xi \mapsto \frac{|\xi|^2}{d_n + \tilde{\mu}|\xi|^2} \mathcal{F} \mathcal{P} (\omega_n * u_0 - u_0) \right) + (2\tilde{\mu} + \tilde{\lambda}) \mathcal{F}^{-1} \left( \xi \mapsto \frac{\xi}{d_n + \tilde{\mu}|\xi|^2} \xi \cdot \mathcal{F} \mathcal{P} (\omega_n * u_0 - u_0) \right) \right\|_{L^{p_0}(\mathbb{R}^3)} \\ \le C_* \|\omega_n * u_0 - u_0\|_{L^{p_0}(\mathbb{R}^3)} \xrightarrow{n \to \infty} 0.$$

Putting everything together, we obtain

$$\|\mathbb{P}(u_0^n - u_0)\|_{L^{p_0}(\mathbb{R}^3)} \xrightarrow{n \to \infty} 0.$$

Following the same computations, we also get

$$\|(\mathbb{1} - \mathbb{P})(u_0^n - u_0)\|_{L^{p_0}(\mathbb{R}^3)} \xrightarrow{n \to \infty} 0.$$

This completes the proof of Lemma 4.1. In addition, from (4.4) one readily deduces the strong convergence

$$\mathrm{div}(2\mu(\rho_0, c_0)\mathbb{D}u_0^n) + \nabla(\lambda(\rho_0, c_0)\mathrm{div}u_0^n) - \nabla P(\rho_0, c_0) \xrightarrow{n \to \infty} \mathrm{div}(\Pi_0), \\ \text{in } \dot{H}^{-\hat{\alpha}}(\mathbb{R}^3), \text{ as soon as } \mathrm{div}(\Pi_0) \in \dot{H}^{-\hat{\alpha}}(\mathbb{R}^3). \quad \square$$

**4.2. The bootstrap argument and final steps.** This section is devoted to closing all the a priori estimates derived in Section 2. Since the initial density is bounded away from zero and, in view of the assumptions on the shear viscosity coefficient $\mu = \mu(\rho, c)$, we have the following bounds

$$0 < \rho_{0,*} = \inf_{\mathbb{R}^3} \rho_0 \le \sup_{\mathbb{R}^3} \rho_0 = \rho_0^* < \infty;$$

$$0 < \mu_{0,*} := \inf_{\mathbb{R}^3} \mu(\rho_0, c_0) \le \sup_{\mathbb{R}^3} \mu(\rho_0, c_0) = \mu_0^* < \infty.$$

Let us define

$$L_0 = |f(\rho_0, c_0)|_{\dot{\mathcal{P}}_{\mathrm{pw}, \Sigma_0}(\mathbb{R}^3)} + \|\Sigma_0\|_{\mathrm{Lip}} + |\Sigma_0|_{\dot{\mathcal{P}}^{1+\alpha}} + |\nabla\varphi_0|_{\mathrm{inf}}^{-1} + |\Sigma_0|_{\mathrm{inf}}^{-1},$$

and let us consider the constants $\varepsilon_1$, and $\varepsilon_2$ given in Theorem A.1 below, for $\alpha' = \alpha/2$. It is obvious that there exists $\delta_0 = \delta_0(\mu_{0,*}, \mu_0^*, L_0) > 0$ such that if

$$\|\llbracket\mu(\rho_0, c_0)\rrbracket\|_{L^\infty(\Sigma_0)} \le \delta_0,$$

then

$$\frac{1}{\mu_{0,*}} \|\llbracket\mu(\rho_0, c_0)\rrbracket\|_{L^\infty(\Sigma_0)} \le \frac{\varepsilon_1}{10}, \quad \text{and} \quad \Psi(\Sigma_0, \mu(\rho_0, c_0), v(\rho_0, c_0)) \le \frac{\varepsilon_2}{10}.$$

As a consequence, Theorem A.1 ensures the existence of a unique solution $(c^n, \rho^n, u^n)$ to (4.1), defined on a maximal time interval $[0, T_n)$. On this time interval, the solution has enough regularity such that the computations performed in the Section 2-3 make sense. We now consider the functionals $\vartheta, \check{\vartheta}, \delta, \mathcal{A}_1, \mathcal{A}_2$ and $\mathcal{A}_3$, with $(c, \rho, u)$ replaced by $(c^n, \rho^n, u^n)$. Let $T \in (0, T_n)$ be such that, for every $t \in [0, T]$ we have

$$\begin{cases} 0 < \frac{\rho_{0,*}}{10} \le \inf_{x \in \mathbb{R}^3} \rho^n(t, x) \le \sup_{x \in \mathbb{R}^3} \rho^n(t, x) \le 10\rho_0^*, \\ \vartheta(t) \le L_1, \quad \check{\vartheta}(t) \le L_2, \quad \delta(t) \le 10\delta_0, \quad \mathcal{A}_1(t) \le L_3; \end{cases}$$

where $L_1, L_2$, and $L_3$ will be fixed later. On the time interval $[0, T]$ the density and the viscosity are lower and upper bounded as in (2.7). In particular, whenever $\delta_0, E_0$ are below thresholds depending only on the upper and lower bounds of density (and viscosity), Lemma 2.3 ensures that

$$\mathcal{A}_1(t) \le C_* \left( C_0 \left( 1 + \check{\phi}_{\Sigma_n}^{\frac{6}{\alpha}} E_0 \right) + \|\nabla u_0\|_{L^2(\mathbb{R}^3)}^2 + E_0^{\frac{1}{4}} \left( \mathcal{A}_1(t)^{\frac{5}{4}} + C_0^{\frac{5}{4}} \right) \right),$$

so that we can choose

$$L_3 = 5C_* \left( C_0 \left( 1 + \check{\phi}_{\Sigma_n}^{\frac{6}{\alpha}} E_0 \right) + \|\nabla u_0\|_{L^2(\mathbb{R}^3)}^2 \right)$$

and deduce that on the time interval $[0, T]$, we have

$$\mathcal{A}_1(t) \le \frac{3}{5} L_3.$$
Recalling the expression of $\check{\phi}_{\Sigma_n}$ from (2.10), it follows that

$$ \check{\phi}_{\Sigma_n} \le C_* L_2^{\frac{1}{2}} (1 + \delta_0 K_0 e^{C' L_1}), $$

and we shall assume that

$$ C_* L_2^{\frac{1}{2}} (1 + \delta_0 K_0 e^{C' L_1}) (C_0 + \|\nabla u_0\|_{L^2(\mathbb{R}^3)}^2)^\beta \le C, \quad \text{for some } \beta \in (0, \alpha). $$

This implies that

$$ \check{\phi}_{\Sigma_n}^{\frac{6}{\alpha}} (C_0 + \|\nabla u_0\|_{L^2(\mathbb{R}^3)}^2) \le C, $$

and Lemma 2.4 provides us with

$$ \int_0^t \left( \|\nabla u^n, G(\rho^n, c^n)\|_{L^p(\mathbb{R}^3)} + \|\dot{u}^n\|_{L^{6/(2-\alpha)}(\mathbb{R}^3)} \right) \le C_* L_3^{1/2} (1+L_3), \quad \text{and} \quad \int_0^t \|\rho^n \dot{u}^n\|_{L^{3/(1-\alpha)}(\mathbb{R}^3)} \le C_* \check{L}_3, $$

where $\check{L}_3$ depends nonlinearly on $\rho_{*,0}, \rho_0^*, \check{C}_0$.

Let us denote by $\rho^n$ the density $\rho^n$ along the flow map $\mathcal{X}^n$ of $u^n$. It is straightforward to deduce from the mass equation (4.1) that:

$$ \partial_t \log(\rho^n / \tilde{\rho}(c_0)) + \frac{1}{v(\rho^n, c_0)} \frac{P(\rho^n, c_0) - \tilde{P}}{\log(\rho^n / \tilde{\rho}(c_0))} \log(\rho^n / \tilde{\rho}(c_0)) = - \frac{1}{v(\rho^n, c_0)} F^n \circ \mathcal{X}^n, $$

and hence

$$ \log(\rho^n / \tilde{\rho}(c_0)) = e^{-\int_0^t g^n(t') dt'} \log(\rho_0 / \tilde{\rho}(c_0)) - \int_0^t \frac{e^{-\int_{t'}^t g^n(t'') dt''}}{v(\rho^n(t'), c_0)} (F^n \circ \mathcal{X}^n)(t') dt', $$

where

$$ g^n = \frac{1}{v(\rho^n, c_0)} \frac{P(\rho^n, c_0) - \tilde{P}}{\log(\rho^n / \tilde{\rho}(c_0))}. $$

It follows that

$$ \sup_{x \in \mathbb{R}^3} \rho^n(t, x) \le \rho_0^* e^{C_* \int_0^t \|F^n\|_{L^\infty(\mathbb{R}^3)}} \text{, and } \inf_{x \in \mathbb{R}^3} \rho^n(t, x) \ge \rho_{0,*} e^{-C_* \int_0^t \|F^n\|_{L^\infty(\mathbb{R}^3)}} $$

where the $L^1((0, t), L^\infty(\mathbb{R}^3))$ norm of $F^n$ is estimated as

$$ \int_0^t \|F^n\|_{L^\infty(\mathbb{R}^3)} \le C_* \int_0^t \left( \|\rho^n \dot{u}^n\|_{L^2(\mathbb{R}^3)} + \|\rho^n \dot{u}^n\|_{L^{6/(2-\alpha)}(\mathbb{R}^3)} + \phi_{\Sigma_n} \|\nabla u^n\|_{L^p(\mathbb{R}^3)} \right) \\ + C_* \sup_{[0, t]} \|\llbracket \mu(\rho^n, c^n) \rrbracket\|_{L^\infty(\Sigma_n)} \int_0^t \|\nabla u^n\|_{C_{\text{pw}, \Sigma_n}^{\alpha/2}(\mathbb{R}^3)}.$$

From the expression of $\phi_{\Sigma_n}$ given (2.10), we obtain

$$ \int_0^t \|F^n\|_{L^\infty(\mathbb{R}^3)} \le C_* L_3^{1/2} (1 + K_0 e^{C' L_1} L_2^{1/2}) (1 + L_3) + C_* \delta_0 L_1^{1/2} (L_1^{1/2} + L_2^{1/2}). $$

We assume that $C_0, \|\nabla u_0\|_{L^2(\mathbb{R}^3)}$ and $\delta_0$ are small so that the RHS above is less than $2\log(3)$. This implies

$$ \sup_{x \in \mathbb{R}^3} \rho^n(t, x) \le 9\rho_0^*, \quad \text{and} \quad \inf_{x \in \mathbb{R}^3} \rho^n(t, x) \ge \frac{\rho_{0,*}}{9}. $$

Under the assumption that

$$ C_* \delta_0^{1/2} (\delta_0^{1/2} e^{C' L_1} + L_2^{1/2}) \le 1, $$

(2.11) holds and the bounds (2.12)–(2.13) follow. In particular, one can choose

$$ L_2 = 3C_* (\check{\theta}(0) + \check{L}_3) e^{C_* K_0 (L_1 + L_3^{1/2} (1+L_3))} e^{C L_1}. $$

What remains is to close the estimates for $\theta$ and $\delta$. Recall that

$$ \theta(t) \le e^{C_* \int_0^t \|w^n\|_{L^\infty(\mathbb{R}^3)}} \left\{ C_* |f(\rho_0, c_0)|_{C_{\text{pw}, \Sigma_0}^{\alpha/2}(\mathbb{R}^3)} + C_* L_3^{1/2} (1+L_3)(1+L_2) + C_* \delta_0^{1/2} (1+\delta_0^{1/2} L_2) K_0 e^{C' L_1} L_3^{1/2} (1+L_3) \right. \\ \left. + C_* \delta_0^{1/2} L_2^{1/2} (L_2 + \delta_0^{1/2} L_1 (1+K_0 e^{C' L_1})) + C_* \int_0^t \left[ \ell_{\Sigma_n}^{-\frac{\alpha}{2}} \|\llbracket G(\rho^n, c^n) \rrbracket\|_{L^\infty \cap L^p(\Sigma_n)} + \|\llbracket G(\rho^n, c^n) \rrbracket\|_{\dot{e}^{\alpha/2}(\Sigma_n)} \right] \right\}, $$
with

$$ (4.9) \qquad \sup_{[0,t]} \|f(\rho^n, c^n)\|_{L^\infty(\mathbb{R}^3)} + \int_0^t \|w^n\|_{L^\infty(\mathbb{R}^3)} \\ \le C_* \left( \|f(\rho_0, c_0)\|_{L^\infty(\mathbb{R}^3)} + (1+L_2^{1/2}(1+K_0 e^{C'L_1}))L_3^{1/2}(1+L_3) + \delta_0 L_1^{1/2}(L_1^{1/2} + L_2^{1/2}) \right). $$

From the rough bounds (2.14)–(3.18) for the pressure jumps, we derive

$$ (4.10) \qquad \int_0^t \left[ \ell_{\Sigma_n}^{-\frac{\alpha}{2}} \|\!\| [G(\rho^n, c^n)] \|\!\|_{L^\infty \cap L^p(\Sigma_n)} + \|\!\| [G(\rho^n, c^n)] \|\!\|_{\dot{\mathcal{P}}^{\alpha/2}(\Sigma_n)} \right] \\ \le C_* e^{C'L_1} \left( L_2^{1/2} K_0 \|f(\rho_0, c_0)\|_{L^\infty(\mathbb{R}^3)} + |f(\rho_0, c_0)|_{\dot{\mathcal{P}}^{\alpha/2}_{\text{pw},\Sigma_0}(\mathbb{R}^3)} + L_3^{1/2}(1+L_3)(1+L_2+K_0 e^{C'L_1}) \right. \\ \left. + (1+L_2^{1/2}(1+K_0 e^{C'L_1}))L_3^{1/2}(1+L_3) + \delta_0^{1/2} L_1^{1/2} L_2^{1/2} (L_1^{1/2} + K_0 \delta_0^{1/2} (L_1 e^{C'L_1} + L_2^{1/2})) \right). $$

One can first choose

$$ L_1 = \left[ 10^5 C_* \left( (\dot{\vartheta}(0) + \check{L}_3)^{1/2} K_0 \|f(\rho_0, c_0)\|_{L^\infty(\mathbb{R}^3)} + |f(\rho_0, c_0)|_{\dot{\mathcal{P}}^{\alpha/2}_{\text{pw},\Sigma_0}(\mathbb{R}^3)} + L_3^{1/4} \right) \right] e^{C_* \left( (1+L_3^{1/2}) + \|f(\rho_0, c_0)\|_{L^\infty(\mathbb{R}^3)} \right)}, $$

and then $L_3$ and $\delta_0$ small enough so that

$$ \vartheta(t) \le (10^{-4} L_1) e^{C_* K_0 (L_1^{1/2} + L_3^{1/2} (1+L_3))} e^{cL_1}. $$

Therefore, under the smallness assumption on $\|f(\rho_0, c_0)\|_{\mathcal{C}_{\text{pw},\Sigma_0}^{\alpha/2}(\mathbb{R}^3)}$ and $L_3$, we have

$$ \vartheta(t) \le \frac{9}{10} L_1. $$

Observe that when the constitutive coefficients satisfy (1.6) or (1.9), we obtain sharper bounds (see Lemma 2.2) for the pressure jump, which we shall use instead of (4.10). This sharper bounds allow us to require smallness only on $\delta_0$ and $L_3$, rather than $\|f(\rho_0, c_0)\|_{\mathcal{C}_{\text{pw},\Sigma_0}^{\alpha/2}(\mathbb{R}^3)}$. Furthermore, one readily proves that

$$ \delta(t) \le 9\delta_0, $$

assuming that $\|\!\| [f(\rho_0, c_0)] \|\!\|_{\mathcal{C}_{\text{pw},\Sigma_0}^{\alpha/2}(\Sigma_0)}$ (and also $\|\!\| [\mu(c_0)] \|\!\|_{\mathcal{C}_{\text{pw},\Sigma_0}^{\alpha/2}(\Sigma_0)} = \text{cnst}$, if (1.6) holds) is small enough. If neither (1.6) nor (1.9) holds, we should make use of the rough bound (4.9) and deduce the above conclusion assuming that $L_1$ and $L_3$ are small. A continuation argument yields $T = T_n$.

To complete Theorem 1.2, we shall show that $T_n = \infty$ using the continuation criterion (see item 3 of Theorem A.1). The convergence of the sequence $(c^n, \rho^n, u^n)_n$ to a solution $(c, \rho, u)$ of (1.1) follows by arguments analogous those of [43, Section 3.2]. The solution $(c, \rho, u)$ has the regularity stated in Theorem 1.2. In particular the velocity $u$ is Lipschitz, which allows a change of variables to Lagrangian coordinates and, by adapting Proposition 3.1 of the above reference, yields the uniqueness of $(c, \rho, u)$. Let us observe that we will need $\sigma \nabla u \in L^2_{\text{loc}}([0, \infty), L^\infty(\mathbb{R}^3))$. This improved time integrability follows by repeating the computations leading to (2.13). This proves Theorem 1.2.

## ACKNOWLEDGMENT

This work was carried out during my PhD studies; and I thank my supervisors Cosmin Burtea and David Gérard-Varet, for fruitful discussions and careful reading of this manuscript.

I was funded by the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 945332.

## APPENDIX A. LOCAL-IN-TIME WELL-POSEDNESS

This section aims to outline how the smallness condition on the shear viscosity fluctuation in Theorem 1.1 and Corollary 1.5 of [44] can be relaxed to a smallness condition on the viscosity jump, as stated in Theorem A.1 below.
**Theorem A.1.** Consider the system (1.1) with initial data $(c_0, \rho_0, u_0)$ satisfying (1.12)-(1.13)-(1.18)-(2.2), and let $\alpha' \in (0, \alpha)$. There exist positive constants $\varepsilon_1$, and $\varepsilon_2$ depending only on $\alpha$ and $\alpha'$ such that if

$$ (A.1) \qquad \frac{1}{\inf_{\mathbb{R}^3} \mu(\rho_0, c_0)} \|\|\mu(\rho_0, c_0)\|\|_{L^\infty(\Sigma_0)} \le \varepsilon_1, $$

and<fnref n="2" />

$$ \begin{aligned} \Psi(\Sigma_0, \mu(\rho_0, c_0), v(\rho_0, c_0)) := & \frac{1}{\inf_{\mathbb{R}^3} \mu(\rho_0, c_0)} \left( \left\|\|\mu(\rho_0, c_0)\|_{\dot{\epsilon}^{\alpha'}(\Sigma_0)} + (\ell_{\Sigma_0}^{-\alpha'} + \mathfrak{P}\Sigma_0 |\Sigma_0|_{\dot{\epsilon}^{1+\alpha'}}) \|\|\mu(\rho_0, c_0)\|\|_{L^\infty(\Sigma_0)} \right. \right. \\ & \left. + \left( \frac{1}{\inf_{\mathbb{R}^3} \mu(\rho_0, c_0)} \|\|\mu(\rho_0, c_0)\|\|_{L^\infty(\Sigma_0)} \right)^{\frac{\alpha}{\alpha'}} \left\{ \frac{1}{\inf_{\mathbb{R}^3} \mu(\rho_0, c_0)} \left[ |\mu(\rho_0, c_0)|_{\dot{\epsilon}_{pw,\Sigma_0}^{\alpha}(\mathbb{R}^3)} \right. \right. \\ & \left. + \|\|\mu(\rho_0, c_0)\|\|_{L^\infty(\Sigma_0)} \left( \frac{1}{\inf_{\mathbb{R}^3} \mu(\rho_0, c_0)} |\mu(\rho_0, c_0), v(\rho_0, c_0)|_{\dot{\epsilon}_{pw,\Sigma_0}^{\alpha}(\mathbb{R}^3)} + \mathfrak{P}\Sigma_0 |\Sigma_0|_{\dot{\epsilon}^{1+\alpha}} + \ell_{\Sigma_0}^{-\alpha} \right) \right] \right. \\ & \left. + \left( \frac{1}{\inf_{\mathbb{R}^3} \mu(\rho_0, c_0)} \right)^{\frac{\alpha}{\alpha-\alpha'}} \left[ |\mu(\rho_0, c_0)|_{\dot{\epsilon}_{pw,\Sigma_0}^{\alpha-\alpha'}(\mathbb{R}^3)} + \|\|\mu(\rho_0, c_0)\|\|_{L^\infty(\Sigma_0)} \right. \right. \\ & \left. \left. \times \left( \frac{1}{\inf_{\mathbb{R}^3} \mu(\rho_0, c_0)} |\mu(\rho_0, c_0), v(\rho_0, c_0)|_{\dot{\epsilon}_{pw,\Sigma_0}^{\alpha}(\mathbb{R}^3)} + \mathfrak{P}\Sigma_0 |\Sigma_0|_{\dot{\epsilon}^{1+\alpha}} + \ell_{\Sigma_0}^{-\alpha+\alpha'} \right) \right]^{\frac{\alpha}{\alpha-\alpha'}} \right\} \right) \\ < & \varepsilon_2, \end{aligned} \tag{A.2} $$

then there exist a time $T > 0$ and a unique solution $(c, \rho, u)$ satisfying the following properties.

(1) **Energy bounds:**

$$ \begin{aligned} & u \in C([0, T], H^1(\mathbb{R}^3)), \quad \dot{u} \in C([0, T], L^2(\mathbb{R}^3)); \\ & \sqrt{\sigma} \nabla \dot{u}, \sigma \ddot{u} \in L^\infty((0, T), L^2(\mathbb{R}^3)); \\ & \nabla \dot{u}, \sqrt{\sigma} \ddot{u}, \sigma \nabla \dot{u} \in L^2((0, T) \times \mathbb{R}^3). \end{aligned} $$

(2) **Piecewise Hölder bounds:** There exists $r_\alpha \in (0, 8/3)$ such that

$$ \sigma^{r_\alpha/4} \nabla u \in L^4((0, T); L^\infty(\mathbb{R}^3)). $$

As a consequence, the velocity field $u$ admits a flow map $\mathcal{X}$ that transports the initial domain $D_0$ to $D(t) = \mathcal{X}(t)D_0$, and $c(t) = \mathbb{1}_{D(t)}$, $t \in [0, T]$. Moreover, we have: for all $t \in [0, T]$,

$$ \rho(t) \in C_{pw, \partial D(t)}^{\alpha}(\mathbb{R}^3), \quad \text{and} \quad \int_0^T \sigma^{r_\alpha} \|\nabla u(t')\|_{\dot{\epsilon}_{pw, \partial D(t')}^{\alpha}(\mathbb{R}^3)}^4 dt' < \infty. $$

(3) **Continuation criterion:** Let $T^*$ be the lifespan of $(c, \rho, u)$. If $T^* < \infty$, then we have either

$$ \begin{aligned} & \limsup_{t \to T^*} \left( \left\| \rho(t), \frac{1}{\rho(t)}, \frac{1}{\mu(\rho(t), c(t))} \right\|_{L^\infty(\mathbb{R}^3)} + \|\rho(t)\|_{\dot{\epsilon}_{pw, \partial D(t)}^{\alpha}(\mathbb{R}^3)} + \|\nabla u(t), \dot{u}(t)\|_{L^2(\mathbb{R}^3)} \right) = \infty, \\ \text{or} \\ & \limsup_{t \to T^*} \left( \|\partial D(t)\|_{Lip} + |\partial D(t)|_{\dot{\epsilon}^{1+\alpha}} + \frac{1}{|\partial D(t)|_{inf}} \right) = \infty, \\ \text{or} \\ & \limsup_{t \to T^*} \left( \frac{1}{\inf_{\mathbb{R}^3} \mu(\rho(t), c(t))} \|\|\mu(\rho(t), c(t))\|\|_{L^\infty(\partial D(t))} \right) \ge 10\varepsilon_1, \\ \text{or} \\ & \limsup_{t \to T^*} \Psi(\partial D(t), \mu(\rho(t), c(t)), v(\rho(t), c(t))) \ge 10\varepsilon_2. \end{aligned} $$

As pointed out in Remark 1.2 of [44], the smallness assumption of the shear viscosity fluctuation is imposed to derive an estimate for the $L^p(\mathbb{R}^3) \cap C_{pw, \Sigma_0}^{\alpha}(\mathbb{R}^3)$-norm (with $p \in (2, 16]$) for the gradient of the solution $v = v(x) \in \mathbb{R}^3$ of the following (stationary) elliptic equation:

$$ (A.3) \qquad - \operatorname{div}(2\mu \mathbb{D} v + \lambda \operatorname{div} v I_3) = \operatorname{div} G. $$

<footnote n="2">²Recall that $\mathfrak{P}\Sigma$ is a polynomial in $|\Sigma|$, $\|\Sigma\|_{Lip}$, and $|\Sigma|_{inf}^{-1}$.</footnote>
Here $\mu = \mu(x)$; $\lambda = \lambda(x) \in C_{pw,\Sigma_0}^{\alpha}(\mathbb{R}^3)$, and $G = (G^{jk}(x))_{1\le j,k\le 3} \in L^p(\mathbb{R}^3) \cap C_{pw,\Sigma_0}^{\alpha}(\mathbb{R}^3)$ are given. The argument was to perturb the shear viscosity around the constant state $\tilde{\mu}$

$$ -\tilde{\mu}\Delta\nu - \nabla((\tilde{\mu} + \lambda)\operatorname{div}\nu) = \operatorname{div}G + \operatorname{div}(2(\mu - \tilde{\mu})\mathbb{D}\nu). $$

From this we deduce

$$ (A.4) \qquad \begin{cases} -\tilde{\mu}\Delta\mathbb{P}\nu = \operatorname{div}\mathbb{P}G + \operatorname{div}\mathbb{P}(2(\mu - \tilde{\mu})\mathbb{D}\nu), \\ -\Delta(2\tilde{\mu} + \lambda)\operatorname{div}\nu = \operatorname{div}\operatorname{div}G + \operatorname{div}\operatorname{div}(2(\mu - \tilde{\mu})\mathbb{D}\nu), \end{cases} $$

and hence

$$ \begin{align} \tilde{\mu}\nabla\nu &= \tilde{\mu}\nabla\mathbb{P}\nu + \tilde{\mu}\nabla(\mathbb{1} - \mathbb{P})\nu \nonumber \\ &= \nabla(-\Delta)^{-1}\operatorname{div}\mathbb{P}G - \tilde{\mu}\nabla(-\Delta)^{-1}\nabla\operatorname{div}\nu + \nabla(-\Delta)^{-1}\operatorname{div}\mathbb{P}(2(\mu - \tilde{\mu})\mathbb{D}\nu) \nonumber \\ &= \nabla(-\Delta)^{-1}\operatorname{div}\mathbb{P}G - \nabla(-\Delta)^{-1}\nabla\left(\frac{\tilde{\mu}}{2\tilde{\mu} + \lambda}(-\Delta)^{-1}\operatorname{div}\operatorname{div}G\right) \nonumber \\ &\quad + \nabla(-\Delta)^{-1}\operatorname{div}\mathbb{P}(2(\mu - \tilde{\mu})\mathbb{D}\nu) - \nabla(-\Delta)^{-1}\nabla\left(\frac{\tilde{\mu}}{2\tilde{\mu} + \lambda}(-\Delta)^{-1}\operatorname{div}\operatorname{div}(2(\mu - \tilde{\mu})\mathbb{D}\nu)\right). \tag{A.5} \end{align} $$

Since the linear operators $\partial_j(-\Delta)^{-1}\partial_k$ are continuous on $L^p(\mathbb{R}^3)$, with $p \in (1, \infty)$, it follows that there exists a constant $C(p) > 0$ such that

$$ \tilde{\mu}\|\nabla\nu\|_{L^p(\mathbb{R}^3)} \le C(p)\|G\|_{L^p(\mathbb{R}^3)} + C(p)\|\mu - \tilde{\mu}\|_{L^\infty(\mathbb{R}^3)}\|\nabla\nu\|_{L^p(\mathbb{R}^3)}. $$

Then we assume that the shear viscosity fluctuation is small

$$ (A.6) \qquad C(p) \|\mu - \tilde{\mu}\|_{L^{\infty}(\mathbb{R}^3)} < \tilde{\mu}, $$

in order to absorb the last term on the LHS, and whence derive an estimate for the $L^p(\mathbb{R}^3)$-norm of $\nabla\nu$. By a similar (yet more technical) perturbation argument, we derived piecewise Hölder bound for the velocity gradient.

To pass from the smallness condition of the viscosity fluctuation to that of its jump across the discontinuity surface, we first derive the following expression from (A.4)$_1$ (see the computations that led to (2.6)):

$$ \begin{align} \nabla\nu &= \nabla\mathbb{P}\nu - \nabla(-\Delta)^{-1}\nabla\operatorname{div}\nu \nonumber \\ &= \frac{1}{\mu}\nabla(-\Delta)^{-1}\operatorname{div}\mathbb{P}G - \nabla(-\Delta)^{-1}\nabla\left(\frac{1}{2\mu + \lambda}(-\Delta)^{-1}\operatorname{div}\operatorname{div}G\right) \nonumber \\ &\quad + \frac{1}{\mu}[\nabla(-\Delta)^{-1}\operatorname{div}\mathbb{P}, \mu - \tilde{\mu}](2\mathbb{D}\nu) - \nabla(-\Delta)^{-1}\nabla\left(\frac{1}{2\mu + \lambda}(-\Delta)^{-1}\operatorname{div}\operatorname{div}(\mu - \tilde{\mu})(2\mathbb{D}\nu)\right). \tag{A.7} \end{align} $$

We estimate the $L^p(\mathbb{R}^3)$ and piecewise Hölder norms of the last two terms above using the following lemma, whose proof is deferred to the end of this section.

**Lemma A.2.** Let $\mathcal{T}$ be a singular integral operator of Calderón-Zygmung type of even-order, associated with a homogeneous kernel $K$. Assume that the assumptions and notations in Definition 1.1 hold for $\Sigma$. Let $p \in (1, \infty)$, $w, g \in C_{pw,\Sigma}^{\alpha}(\mathbb{R}^3)$, and $g \in L^p(\mathbb{R}^3)$.

(1) $L^p(\mathbb{R}^3)$- estimate. Let $\alpha' \in (0, \alpha] \cap (0, \frac{3}{p})$ ($p'$ denotes the Hölder conjugate exponent of $p$), and assume that $g \in L^r(\mathbb{R}^3)$, with $\frac{1}{r} = \frac{\alpha'}{3} + \frac{1}{p} < 1$. We have

$$ (A.8) \qquad \| [\mathcal{T}, w] g \|_{L^p(\mathbb{R}^3)} \le C |w|_{\dot{C}_{pw,\Sigma}^{\alpha'}}(\mathbb{R}^3) \|g\|_{L^r(\mathbb{R}^3)} + C \| [w] \|_{L^\infty(\Sigma)} (\ell_{\Sigma}^{-\alpha'} \|g\|_{L^r(\mathbb{R}^3)} + \|g\|_{L^p(\mathbb{R}^3)}). $$

(2) $L^\infty(\mathbb{R}^3)$- estimate. Let $\alpha' \in (0, \alpha]$, $r \in (\frac{3}{\alpha'}, \infty]$, and assume that $g \in L^r(\mathbb{R}^3)$. We have

$$ \begin{align} \| [\mathcal{T}, w] g \|_{L^\infty(\mathbb{R}^3)} &\le C \left( |w|_{\dot{C}_{pw,\Sigma}^{\alpha'}}(\mathbb{R}^3) + \ell_{\Sigma}^{-\alpha'} \| [w] \|_{L^\infty(\Sigma)} \right) \| g \|_{L^r(\mathbb{R}^3)} \nonumber \\ &\quad + C \left( \| [w] \|_{L^\infty(\Sigma)} \| g \|_{C_{pw,\Sigma}^{\alpha'}(\mathbb{R}^3)} + \ell_{\Sigma}^{-\alpha'} \| w \|_{L^\infty(\mathbb{R}^3)} \| g \|_{L^p(\mathbb{R}^3)} \right). \tag{A.9} \end{align} $$
(3) $\dot{\mathcal{E}}_{pw,\Sigma}^{\beta}(\mathbb{R}^3)$ - **estimate**. Let $\beta \in (0, \alpha]$ and $\alpha' \in (0, \min\{\alpha, \beta\}]$. Then, we have

$$
(A.10) \qquad \begin{aligned}
|\mathcal{T}, w|g|_{\dot{\mathcal{E}}_{pw,\Sigma}^{\beta}(\mathbb{R}^3)} &\le C(|w|_{\dot{\mathcal{E}}_{pw,\Sigma}^{\beta}(\mathbb{R}^3)} + (\ell_{\Sigma}^{-\beta} + \mathfrak{P}_{\Sigma}|\Sigma|^{\beta} \dot{\mathcal{E}}_{1+\beta}) \|[[w]]\|_{L^{\infty}(\Sigma)}) \|\mathcal{T}g, g\|_{L^{\infty}(\mathbb{R}^3)} \\
&\quad + C(|w|_{\dot{\mathcal{E}}_{pw,\Sigma}^{\beta-\alpha'}(\mathbb{R}^3)} + \ell_{\Sigma}^{-\beta+\alpha'} \|[[w]]\|_{L^{\infty}(\Sigma)}) |g|_{\dot{\mathcal{E}}_{pw,\Sigma}^{\alpha'}(\mathbb{R}^3)} + C \|[[w]]\|_{L^{\infty}(\Sigma)} |g|_{\dot{\mathcal{E}}_{pw,\Sigma}^{\beta}(\mathbb{R}^3)}.
\end{aligned}
$$

Above, $\mathfrak{P}_{\Sigma}$ is a polynomial in $|\Sigma|$, $\|\Sigma\|_{Lip}$, and $|\Sigma|_{inf}^{-1}$.

The constant C is independent of g, w and Σ.

The proof of Lemma A.2 combines, in a nontrivial manner, the extension theorem for piecewise Hölder regular functions (see [24]), and the continuity of Calderón-Zygmung operators on $L^q(\mathbb{R}^3) \cap \mathcal{C}_{pw,\Sigma}^{\beta}(\mathbb{R}^3)$, with $q \in (1, \infty)$ (see [16, 17]).

We now apply it to the last terms of (A.7).

**Estimate of the** $L^p(\mathbb{R}^3)$ **norm.** We consider $p \in (2, \infty)$, and $r \in [2, p)$ such that $\frac{1}{r} = \frac{\alpha'}{3} + \frac{1}{p}$, for some $\alpha' \in (0, \alpha]$. By applying (A.8) to the last terms in (A.7), we obtain

$$
(A.11) \qquad \begin{aligned}
\left\| \frac{1}{\mu} [\nabla(-\Delta)^{-1} \operatorname{div} \mathbb{P}, \mu - \tilde{\mu}] (2\mathbb{D}v) ; \nabla(-\Delta)^{-1} \nabla \left( \frac{1}{2\mu + \lambda} [(-\Delta)^{-1} \operatorname{div} \operatorname{div}, \mu - \tilde{\mu}] (2\mathbb{D}v) \right) \right\|_{L^p(\mathbb{R}^3)} \\
\le \frac{C}{\mu_*} \|[[\mu]]\|_{L^\infty(\Sigma_0)} \|\nabla v\|_{L^p(\mathbb{R}^3)} + C_* \|\nabla v\|_{L^r(\mathbb{R}^3)} \left( |\mu|_{\dot{\mathcal{E}}_{pw,\Sigma_0}^{\alpha'}(\mathbb{R}^3)} + \|[[\mu]]\|_{L^\infty(\Sigma_0)} \ell_{\Sigma_0}^{-\alpha'} \right),
\end{aligned}
$$

and it follows that

$$
\begin{equation} \tag{A.12}
\begin{aligned}
\|\nabla v\|_{L^p(\mathbb{R}^3)} &\le \frac{C}{\mu_*} \|[[\mu]]\|_{L^\infty(\Sigma_0)} \|\nabla v\|_{L^p(\mathbb{R}^3)} \\
&\quad + C_* \left( \|F\|_{L^p(\mathbb{R}^3)} + \|\nabla v\|_{L^r(\mathbb{R}^3)} \left( |\mu|_{\dot{\mathcal{E}}_{pw,\Sigma_0}^{\alpha'}(\mathbb{R}^3)} + \|[[\mu]]\|_{L^\infty(\Sigma_0)} \ell_{\Sigma_0}^{-\alpha'} \right) \right).
\end{aligned}
\end{equation}
$$

Let us fix an arbitrary $\varepsilon \in (0, 1)$ and assume that

$$
(A.13) \qquad \frac{C}{\mu_*} \|[[\mu]]\|_{L^\infty(\Sigma_0)} \le \varepsilon.
$$

The first term on the RHS of (A.12) can then be absorbed into the LHS, yielding:

$$
\|\nabla v\|_{L^p(\mathbb{R}^3)} \le \frac{C_*}{1-\varepsilon} \left( \|F\|_{L^p(\mathbb{R}^3)} + \|\nabla v\|_{L^r(\mathbb{R}^3)} \left( |\mu|_{\dot{\mathcal{E}}_{pw,\Sigma_0}^{\alpha'}(\mathbb{R}^3)} + \|[[\mu]]\|_{L^\infty(\Sigma_0)} \ell_{\Sigma_0}^{-\alpha'} \right) \right).
$$

An energy estimate on (A.3) yields

$$
\|\nabla v\|_{L^2(\mathbb{R}^3)} \le C_* \|F\|_{L^2(\mathbb{R}^3)},
$$

and interpolation inequality gives

$$
\|\nabla v\|_{L^r(\mathbb{R}^3)} \le C \|\nabla u\|_{L^p(\mathbb{R}^3)}^{1-\theta} \|\nabla v\|_{L^2(\mathbb{R}^3)}^{\theta} \le C_* \|\nabla u\|_{L^p(\mathbb{R}^3)}^{1-\theta} \|F\|_{L^2(\mathbb{R}^3)}^{\theta}, \text{ with } \frac{1}{2} - \frac{1}{p} = \frac{\alpha'}{3\theta}.
$$

Finally, applying Young's inequality, we obtain:

$$
(A.14) \qquad \|\nabla v\|_{L^p(\mathbb{R}^3)} \le \frac{C_*}{1-\varepsilon} \|F\|_{L^p(\mathbb{R}^3)} + C_* (1-\varepsilon)^{-\frac{1}{\theta}} \|F\|_{L^2(\mathbb{R}^3)} \left( |\mu|_{\dot{\mathcal{E}}_{pw,\Sigma_0}^{\alpha'}(\mathbb{R}^3)} + \|[[\mu]]\|_{L^\infty(\Sigma_0)} \ell_{\Sigma_0}^{-\alpha'} \right)^{\frac{1}{\theta}}.
$$

We emphasize that only the smallness of the shear viscosity jump is required (see (A.13)), rather than that of the overall shear viscosity fluctuation (see (A.6)). We now turn to deriving an estimate for the piecewise Hölder norm of $\nabla v$.

**Estimate of the** $\mathcal{C}_{pw,\Sigma_0}^{\alpha}(\mathbb{R}^3)$ **norm.** We fix some $\alpha' \in (0, \alpha)$ and $r \in (3/\alpha', \infty)$. First, we apply (A.9) and (A.10) to obtain

$$
(A.15) \qquad \begin{aligned}
\| [\nabla(-\Delta)^{-1} \operatorname{div} \mathbb{P}, \mu - \tilde{\mu}] (2\mathbb{D}v) \|_{L^{\infty}(\mathbb{R}^3)} &\le C \left( |\mu|_{\dot{\mathcal{E}}_{pw,\Sigma_0}^{\alpha'}(\mathbb{R}^3)} + \ell_{\Sigma_0}^{-\alpha'} \|\mu - \tilde{\mu}\|_{L^{\infty}(\mathbb{R}^3)} \right) \|\nabla v\|_{L^r(\mathbb{R}^3)} \\
&\quad + C \|[[\mu]]\|_{L^{\infty}(\Sigma_0)} \|\nabla v\|_{\mathcal{C}_{pw,\Sigma_0}^{\alpha'}(\mathbb{R}^3)},
\end{aligned}
$$
and

$$
\begin{equation} \tag{A.16}
\begin{aligned}
& \left| [\nabla(-\Delta)^{-1} \operatorname{div} \mathbb{P}, \mu - \tilde{\mu}] (2\mathbb{D}\nu) \right|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}(\mathbb{R}^3)} \\
& \le C \left( \left| \mu \right|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}(\mathbb{R}^3)} + (\ell_{\Sigma_0}^{-\alpha} + \mathfrak{P}_{\Sigma_0} |\Sigma_0| \dot{\mathcal{P}}^{1+\alpha}) \left\| [\mu] \right\|_{L^{\infty}(\Sigma_0)} \right) \left\| \nabla(-\Delta)^{-1} \operatorname{div} \mathbb{P} \mathbb{D}\nu, \mathbb{D}\nu \right\|_{L^{\infty}(\mathbb{R}^3)} \\
& \quad + C \left( \left| \mu \right|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha-\alpha'}(\mathbb{R}^3)} + \ell_{\Sigma_0}^{-\alpha+\alpha'} \left\| [\mu] \right\|_{L^{\infty}(\Sigma_0)} \right) \left| \nabla \nu \right|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha'}(\mathbb{R}^3)} + C \left\| [\mu] \right\|_{L^{\infty}(\Sigma_0)} \left| \nabla \nu \right|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)}.
\end{aligned}
\end{equation}
$$

Next, we estimate the second-to-last term in (A.7) as follows:

$$
\begin{align}
& \left| \frac{1}{\mu} [\nabla(-\Delta)^{-1} \operatorname{div} \mathbb{P}, \mu - \tilde{\mu}] (2\mathbb{D}\nu) \right|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)} \nonumber \\
& \le \frac{1}{\mu_*^2} |\mu|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)} \left\| [\nabla(-\Delta)^{-1} \operatorname{div} \mathbb{P}, \mu - \tilde{\mu}] (2\mathbb{D}\nu) \right\|_{L^{\infty}(\mathbb{R}^3)} + \frac{1}{\mu_*} \left| [\nabla(-\Delta)^{-1} \operatorname{div} \mathbb{P}, \mu - \tilde{\mu}] (2\mathbb{D}\nu) \right|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)} \nonumber \\
& \le \frac{C}{\mu_*^2} |\mu|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)} \left\| \nabla \nu \right\|_{L^r(\mathbb{R}^3)} \left( |\mu|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha'}(\mathbb{R}^3)} + \ell_{\Sigma_0}^{-\alpha'} \left\| \mu - \tilde{\mu} \right\|_{L^{\infty}(\mathbb{R}^3)} \right) \nonumber \\
& \quad + \frac{C}{\mu_*} \left\| \nabla u, \nabla \mathbb{P} \nu \right\|_{L^{\infty}(\mathbb{R}^3)} \left[ |\mu|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)} \left( 1 + \frac{1}{\mu_*} \left\| [\mu] \right\|_{L^{\infty}(\Sigma_0)} \right) + (\ell_{\Sigma_0}^{-\alpha} + \mathfrak{P}_{\Sigma_0} |\Sigma_0| \dot{\mathcal{P}}^{1+\alpha}) \left\| [\mu] \right\|_{L^{\infty}(\Sigma_0)} \right] \nonumber \\
& \quad + \frac{C}{\mu_*} \left\| \nabla \nu \right\|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha'}(\mathbb{R}^3)} \left[ |\mu|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha-\alpha'}(\mathbb{R}^3)} + \left\| [\mu] \right\|_{L^{\infty}(\Sigma_0)} \left( \ell_{\Sigma_0}^{-\alpha+\alpha'} + \frac{1}{\mu_*} |\mu|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)} \right) \right] \nonumber \\
& \quad + \frac{C}{\mu_*} \left\| [\mu] \right\|_{L^{\infty}(\Sigma_0)} \left| \nabla \nu \right|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)}. \tag{A.17}
\end{align}
$$

Applying (A.45), the last term in (A.7) can be estimated as

$$
\begin{align*}
& \left| \nabla(-\Delta)^{-1} \nabla \left( \frac{1}{2\mu + \lambda} [(-\Delta)^{-1} \operatorname{div} \operatorname{div}, \mu - \tilde{\mu}] (2\mathbb{D}\nu) \right) \right|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)} \\
& \le \frac{C}{\mu_*} \left( \frac{1}{\mu_*} \left| 2\mu + \lambda \right|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)} + \mathfrak{P}_{\Sigma_0} |\Sigma_0| \dot{\mathcal{P}}^{1+\alpha} \right) \left\| [(-\Delta)^{-1} \operatorname{div} \operatorname{div}, \mu - \tilde{\mu}] (2\mathbb{D}\nu) \right\|_{L^{\infty}(\mathbb{R}^3)} \\
& \quad + \frac{C}{\mu_*} \left| [(-\Delta)^{-1} \operatorname{div} \operatorname{div}, \mu - \tilde{\mu}] (2\mathbb{D}\nu) \right|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)},
\end{align*}
$$

and hence (similar estimates to (A.15)-(A.16) also hold for $[(-\Delta)^{-1} \operatorname{div} \operatorname{div}, \mu - \tilde{\mu}] (2\mathbb{D}\nu)$)

$$
\begin{align}
& \left| \nabla(-\Delta)^{-1} \nabla \left( \frac{1}{2\mu + \lambda} [(-\Delta)^{-1} \operatorname{div} \operatorname{div}, \mu - \tilde{\mu}] (2\mathbb{D}\nu) \right) \right|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)} \nonumber \\
& \le \frac{C}{\mu_*} \|\nabla \nu\|_{L^r(\mathbb{R}^3)} \left( \frac{1}{\mu_*} |\nu|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)} + \mathfrak{P}_{\Sigma_0} |\Sigma_0| \dot{\mathcal{P}}^{1+\alpha} \right) \left( |\mu|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha'}(\mathbb{R}^3)} + \ell_{\Sigma_0}^{-\alpha'} \|\mu - \tilde{\mu}\|_{L^{\infty}(\mathbb{R}^3)} \right) \nonumber \\
& \quad + \frac{C}{\mu_*} \|\nabla \nu\|_{L^{\infty}(\mathbb{R}^3)} \left[ |\mu|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)} + \|\llbracket \mu \rrbracket\|_{L^{\infty}(\Sigma_0)} \left( \frac{1}{\mu_*} |\nu|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)} + \mathfrak{P}_{\Sigma_0} |\Sigma_0| \dot{\mathcal{P}}^{1+\alpha} + \ell_{\Sigma_0}^{-\alpha} \right) \right] \nonumber \\
& \quad + \frac{C}{\mu_*} \|\nabla \nu\|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha'}(\mathbb{R}^3)} \left[ |\mu|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha-\alpha'}(\mathbb{R}^3)} + \|\llbracket \mu \rrbracket\|_{L^{\infty}(\Sigma_0)} \left( \frac{1}{\mu_*} |\nu|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)} + \mathfrak{P}_{\Sigma_0} |\Sigma_0| \dot{\mathcal{P}}^{1+\alpha} + \ell_{\Sigma_0}^{-\alpha+\alpha'} \right) \right] \nonumber \\
& \quad + \frac{C}{\mu_*} \|\llbracket \mu \rrbracket\|_{L^{\infty}(\Sigma_0)} \left| \nabla \nu \right|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)}. \tag{A.18}
\end{align}
$$

Combining (A.7), (A.17), and (A.18), we deduce that

$$
\begin{align*}
|\nabla \nu|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)} & \le |\nabla u_G|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)} + \frac{C}{\mu_*} \|\nabla \nu\|_{L^r(\mathbb{R}^3)} \left( \frac{1}{\mu_*} |\mu, \nu|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)} + \mathfrak{P}_{\Sigma_0} |\Sigma_0| \dot{\mathcal{P}}^{1+\alpha} \right) \left( |\mu|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha'}(\mathbb{R}^3)} + \ell_{\Sigma_0}^{-\alpha'} \|\mu - \tilde{\mu}\|_{L^{\infty}(\mathbb{R}^3)} \right) \\
& \quad + \frac{C}{\mu_*} \|\nabla \nu, \nabla \mathbb{P} \nu\|_{L^{\infty}(\mathbb{R}^3)} \left[ |\mu|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)} + \|\llbracket \mu \rrbracket\|_{L^{\infty}(\Sigma_0)} \left( \frac{1}{\mu_*} |\mu, \nu|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)} + \mathfrak{P}_{\Sigma_0} |\Sigma_0| \dot{\mathcal{P}}^{1+\alpha} + \ell_{\Sigma_0}^{-\alpha} \right) \right] \\
& \quad + \frac{C}{\mu_*} \|\nabla \nu\|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha'}(\mathbb{R}^3)} \left[ |\mu|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha-\alpha'}(\mathbb{R}^3)} + \|\llbracket \mu \rrbracket\|_{L^{\infty}(\Sigma_0)} \left( \frac{1}{\mu_*} |\mu, \nu|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)} + \mathfrak{P}_{\Sigma_0} |\Sigma_0| \dot{\mathcal{P}}^{1+\alpha} + \ell_{\Sigma_0}^{-\alpha+\alpha'} \right) \right] \\
& \quad + \frac{C}{\mu_*} \|\llbracket \mu \rrbracket\|_{L^{\infty}(\Sigma_0)} \left| \nabla \nu \right|_{\dot{\mathcal{P}}_{\text{pw}, \Sigma_0}^{\alpha}(\mathbb{R}^3)}.
\end{align*}
$$
Hereafter, $\nabla u_G$ denotes

$$ \nabla u_G = \frac{1}{\mu} \nabla (-\Delta)^{-1} \mathrm{div} \mathbb{P}G - \nabla (-\Delta)^{-1} \nabla \left( \frac{1}{2\mu + \lambda} (-\Delta)^{-1} \mathrm{div} \mathrm{div} G \right). $$

The smallness of the shear viscosity jump (see (A.13)) helps absorb the last term above into the LHS. Next, we use the interpolation inequality

$$ (A.19) \qquad |\nabla v|_{\dot{\varphi}_{\text{pw},\Sigma_0}^{\alpha'}(\mathbb{R}^3)} \le 2 \|\nabla v\|_{L^\infty(\mathbb{R}^3)} \frac{1-\frac{\alpha'}{\alpha}}{|\nabla v|_{\dot{\varphi}_{\text{pw},\Sigma_0}^{\alpha}(\mathbb{R}^3)}} $$

and Young's inequality to obtain

$$ |\nabla v|_{\dot{\varphi}_{\text{pw},\Sigma_0}^{\alpha}(\mathbb{R}^3)} \le \frac{C}{1-\varepsilon} |\nabla u_G|_{\dot{\varphi}_{\text{pw},\Sigma_0}^{\alpha}(\mathbb{R}^3)} + C \Psi_{\varepsilon} \|\nabla v, \nabla \mathbb{P} v\|_{L^{\infty}(\mathbb{R}^3)} \\ + \frac{C}{\mu_*(1-\varepsilon)} \|\nabla v\|_{L^r(\mathbb{R}^3)} \left( \frac{1}{\mu_*} |\mu, v|_{\dot{\varphi}_{\text{pw},\Sigma_0}^{\alpha}(\mathbb{R}^3)} + \mathfrak{P}_{\Sigma_0} |\Sigma_0|_{\dot{\varphi}^{1+\alpha}} \right) \left( |\mu|_{\dot{\varphi}_{\text{pw},\Sigma_0}^{\alpha'}(\mathbb{R}^3)} + \ell_{\Sigma_0}^{-\alpha'} \|\mu - \tilde{\mu}\|_{L^{\infty}(\mathbb{R}^3)} \right), $$

where,

$$ \Psi_{\varepsilon} = \frac{1}{\mu_*(1-\varepsilon)} \left[ |\mu|_{\dot{\varphi}_{\text{pw},\Sigma_0}^{\alpha}(\mathbb{R}^3)} + \|[[\mu]]\|_{L^{\infty}(\Sigma_0)} \left( \frac{1}{\mu_*} |\mu, v|_{\dot{\varphi}_{\text{pw},\Sigma_0}^{\alpha}(\mathbb{R}^3)} + \mathfrak{P}_{\Sigma_0} |\Sigma_0|_{\dot{\varphi}^{1+\alpha}} + \ell_{\Sigma_0}^{-\alpha} \right) \right] \\ + \frac{1}{(\mu_*(1-\varepsilon))^{\alpha/(\alpha-\alpha')}} \left[ |\mu|_{\dot{\varphi}_{\text{pw},\Sigma_0}^{\alpha-\alpha'}(\mathbb{R}^3)} + \|[[\mu]]\|_{L^{\infty}(\Sigma_0)} \left( \frac{1}{\mu_*} |\mu, v|_{\dot{\varphi}_{\text{pw},\Sigma_0}^{\alpha}(\mathbb{R}^3)} + \mathfrak{P}_{\Sigma_0} |\Sigma_0|_{\dot{\varphi}^{1+\alpha}} + \ell_{\Sigma_0}^{-\alpha+\alpha'} \right) \right]^{\frac{\alpha}{\alpha-\alpha'}}. $$

For the reader convenience, we shall denote by $C_{**}$ a generic constant that depends on the piecewise regularity of $\mu, \lambda$, as well $\ell_{\Sigma_0}, \mathfrak{P}_{\Sigma_0}, |\Sigma_0|_{\dot{\varphi}^{1+\alpha}}$. For instance the above estimate can be written as

$$ (A.20) \qquad |\nabla v|_{\dot{\varphi}_{\text{pw},\Sigma_0}^{\alpha}(\mathbb{R}^3)} \le \frac{C}{1-\varepsilon} |\nabla u_G|_{\dot{\varphi}_{\text{pw},\Sigma_0}^{\alpha}(\mathbb{R}^3)} + \frac{C_{**}}{1-\varepsilon} \|\nabla v\|_{L^r(\mathbb{R}^3)} + C \Psi_{\varepsilon} \|\nabla v, \nabla \mathbb{P} v\|_{L^{\infty}(\mathbb{R}^3)}. $$

We will now derive an estimate for the $L^\infty$-norm of $\nabla v, \nabla \mathbb{P} v$.

Observe that (A.15) already provides a bound for the second term in the expression (A.7). We express the last term as

$$ (A.21) \qquad -\nabla(-\Delta)^{-1}\nabla\left(\frac{1}{2\mu + \lambda} [(-\Delta)^{-1} \mathrm{div} \mathrm{div}, \mu - \tilde{\mu}] (2\mathbb{D}v)\right) = -\left[\nabla(-\Delta)^{-1}\nabla, \frac{1}{2\mu + \lambda}\right] [(-\Delta)^{-1} \mathrm{div} \mathrm{div}, \mu - \tilde{\mu}] (2\mathbb{D}v) \\ -\frac{1}{2\mu + \lambda} \nabla(-\Delta)^{-1}\nabla\left([(-\Delta)^{-1} \mathrm{div} \mathrm{div}, \mu - \tilde{\mu}] (2\mathbb{D}v)\right), $$

and applying (A.9)-(A.46), we obtain

$$ \begin{aligned} & \left\| \left[ \nabla(-\Delta)^{-1} \nabla, \frac{1}{2\mu + \lambda} \right] [(-\Delta)^{-1} \mathrm{div} \mathrm{div}, \mu - \tilde{\mu}] (2\mathbb{D}v) \right\|_{L^{\infty}(\mathbb{R}^3)} \\ & \le \frac{C}{\mu_*} \left( \frac{1}{\mu_*} |v|_{\dot{\varphi}_{\text{pw},\Sigma_0}^{\alpha'}(\mathbb{R}^3)} + \ell_{\Sigma_0}^{-\alpha'} \right) \left\| [(-\Delta)^{-1} \mathrm{div} \mathrm{div}, \mu - \tilde{\mu}] (2\mathbb{D}v) \right\|_{L^r(\mathbb{R}^3)} \\ & \quad + \frac{C}{\mu_*^2} \left\| [[v]] \right\|_{L^{\infty}(\Sigma_0)} \left\| [(-\Delta)^{-1} \mathrm{div} \mathrm{div}, \mu - \tilde{\mu}] (2\mathbb{D}v) \right\|_{\dot{\varphi}_{\text{pw},\Sigma_0}^{\alpha'}(\mathbb{R}^3)} \\ & \le C_{**} \left( \|\nabla v\|_{L^r(\mathbb{R}^3)} + \|\nabla v\|_{L^{3/(\alpha-\alpha')}}(\mathbb{R}^3) \right) + \frac{C}{\mu_*^2} \left\| [[v]] \right\|_{L^{\infty}(\Sigma_0)} \left\| [[\mu]] \right\|_{L^{\infty}(\Sigma_0)} \|\nabla v\|_{\dot{\varphi}_{\text{pw},\Sigma_0}^{\alpha'}(\mathbb{R}^3)} \\ & \quad + \frac{C}{\mu_*^2} \left\| [[v]] \right\|_{L^{\infty}(\Sigma_0)} \left( \|[[\mu]]|_{\dot{\varphi}^{\alpha'}(\Sigma_0)} + (\ell_{\Sigma_0}^{-\alpha'} + \mathfrak{P}_{\Sigma_0} |\Sigma_0|_{\dot{\varphi}^{1+\alpha'}}) \|[[\mu]] \right) \|\nabla v\|_{L^{\infty}(\mathbb{R}^3)}. \end{aligned} $$
Using (A.44)-(A.9)-(A.46) we can estimate the last term of (A.21) as follows

$$
\begin{align*}
& \left\| \frac{1}{2\mu + \lambda} \nabla(-\Delta)^{-1} \nabla \left( [(-\Delta)^{-1} \operatorname{div} \operatorname{div}, \mu - \tilde{\mu}] (2\mathbb{D}\nu) \right) \right\|_{L^{\infty}(\mathbb{R}^3)} \\
& \le \frac{C}{\mu_*} \ell_{\Sigma_0}^{-\frac{3}{r}} \left\| [(-\Delta)^{-1} \operatorname{div} \operatorname{div}, \mu - \tilde{\mu}] (2\mathbb{D}\nu) \right\|_{L^r(\mathbb{R}^3)} + \frac{C}{\mu_*} \left\| [(-\Delta)^{-1} \operatorname{div} \operatorname{div}, \mu - \tilde{\mu}] (2\mathbb{D}\nu) \right\|_{\mathcal{C}_{\text{pw},\Sigma_0}^{\alpha'}(\mathbb{R}^3)} \\
& \le C_{**} \left( \|\nabla \nu\|_{L^r(\mathbb{R}^3)} + \|\nabla \nu\|_{L^{3/(a-\alpha')}(\mathbb{R}^3)} \right) + \frac{C}{\mu_*} \left\| \llbracket \mu \rrbracket \right\|_{L^{\infty}(\Sigma_0)} \|\nabla \nu\|_{\dot{\mathcal{C}}_{\text{pw},\Sigma_0}^{\alpha'}(\mathbb{R}^3)} \\
& \qquad + \frac{C}{\mu_*} \left( \left\| \llbracket \mu \rrbracket \right\|_{\dot{\mathcal{C}}^{\alpha'}(\Sigma_0)} + (\ell_{\Sigma_0}^{-\alpha'} + \mathfrak{P}_{\Sigma_0} |\Sigma_0| \dot{\epsilon}_{1+\alpha'}) \left\| \llbracket \mu \rrbracket \right\|_{L^{\infty}(\Sigma_0)} \right) \|\nabla \nu\|_{L^{\infty}(\mathbb{R}^3)}.
\end{align*}
$$

By combining all these computations, we obtain the following estimate (we freely use $\frac{1}{\mu_*} \|\llbracket v \rrbracket\|_{L^\infty(\Sigma_0)} \le C$):

$$
\begin{align*}
\|\nabla v\|_{L^{\infty}(\mathbb{R}^3)} &\le \|\nabla u_G\|_{L^{\infty}(\mathbb{R}^3)} + C_{**} \left( \|\nabla v\|_{L^r(\mathbb{R}^3)} + \|\nabla v\|_{L^{3/(a-\alpha')}(\mathbb{R}^3)} \right) + \frac{C}{\mu_*} \|\llbracket \mu \rrbracket\|_{L^{\infty}(\Sigma_0)} \|\nabla v\|_{\dot{\mathcal{C}}_{\text{pw},\Sigma_0}^{\alpha'}(\mathbb{R}^3)} \\
&\quad + \frac{C}{\mu_*} \left( \left\| \llbracket \mu \rrbracket \right\|_{\dot{\mathcal{C}}^{\alpha'}(\Sigma_0)} + (\ell_{\Sigma_0}^{-\alpha'} + \mathfrak{P}_{\Sigma_0} |\Sigma_0| \dot{\epsilon}_{1+\alpha'}) \left\| \llbracket \mu \rrbracket \right\|_{L^{\infty}(\Sigma_0)} \right) \|\nabla v\|_{L^{\infty}(\mathbb{R}^3)}.
\end{align*}
$$

Applying the Leray projector $\mathbb{P}$ to (A.27) and repeating the above computations, we observe that a similar estimate also holds for $\nabla \mathbb{P}$. Next, (A.19) and Young's inequality yield

$$
\begin{align*}
\|\nabla v, \nabla \mathbb{P} v\|_{L^{\infty}(\mathbb{R}^3)} &\le C \|\nabla u_G, \mathbb{P} \nabla u_G\|_{L^{\infty}(\mathbb{R}^3)} + C_{**} \left( \|\nabla v\|_{L^r(\mathbb{R}^3)} + \|\nabla v\|_{L^{3/(a-\alpha')}(\mathbb{R}^3)} \right) \\
&\quad + \left( \frac{C}{\mu_*} \|\llbracket \mu \rrbracket\|_{L^{\infty}(\Sigma_0)} \right)^{\frac{\alpha}{\alpha'}} \|\nabla v\|_{\dot{\mathcal{C}}_{\text{pw},\Sigma_0}^{\alpha}(\mathbb{R}^3)} \\
&\quad + \frac{C}{\mu_*} \left( \left\| \llbracket \mu \rrbracket \right\|_{\dot{\mathcal{C}}^{\alpha'}(\Sigma_0)} + (\ell_{\Sigma_0}^{-\alpha'} + \mathfrak{P}_{\Sigma_0} |\Sigma_0| \dot{\epsilon}_{1+\alpha'}) \left\| \llbracket \mu \rrbracket \right\|_{L^{\infty}(\Sigma_0)} \right) \|\nabla v\|_{L^{\infty}(\mathbb{R}^3)},
\end{align*}
$$

and (A.20) leads to

$$
\begin{align}
\|\nabla v, \nabla \mathbb{P} v\|_{L^{\infty}(\mathbb{R}^3)} &\le C_{**}(\varepsilon) \left( \|\nabla v\|_{L^r(\mathbb{R}^3)} + \|\nabla v\|_{L^{3/(a-\alpha')}(\mathbb{R}^3)} \right) \nonumber \\
&\quad + C \|\nabla u_G, \mathbb{P} \nabla u_G\|_{L^{\infty}(\mathbb{R}^3)} + \frac{1}{1-\varepsilon} \left( \frac{C}{\mu_*} \|\llbracket \mu \rrbracket\|_{L^{\infty}(\Sigma_0)} \right)^{\frac{\alpha}{\alpha'}} |\nabla u_G|_{\dot{\mathcal{C}}_{\text{pw},\Sigma_0}^{\alpha}(\mathbb{R}^3)} \nonumber \\
&\quad + \left[ \Psi_{\varepsilon} \left( \frac{C}{\mu_*} \|\llbracket \mu \rrbracket\|_{L^{\infty}(\Sigma_0)} \right)^{\frac{\alpha}{\alpha'}} + \frac{C}{\mu_*} \left( \left\| \llbracket \mu \rrbracket \right\|_{\dot{\mathcal{C}}^{\alpha'}(\Sigma_0)} + (\ell_{\Sigma_0}^{-\alpha'} + \mathfrak{P}_{\Sigma_0} |\Sigma_0| \dot{\epsilon}_{1+\alpha'}) \left\| \llbracket \mu \rrbracket \right\|_{L^{\infty}(\Sigma_0)} \right) \right] \|\nabla v, \nabla \mathbb{P} v\|_{L^{\infty}(\mathbb{R}^3)}. \tag{A.22}
\end{align}
$$

Assuming that

$$
(A.23) \qquad \Psi_{\varepsilon} \left( \frac{C}{\mu_*} \|\llbracket \mu \rrbracket\|_{L^{\infty}(\Sigma_0)} \right)^{\frac{\alpha}{\alpha'}} + \frac{C}{\mu_*} \left( \left\| \llbracket \mu \rrbracket \right\|_{\dot{\mathcal{C}}^{\alpha'}(\Sigma_0)} + (\ell_{\Sigma_0}^{-\alpha'} + \mathfrak{P}_{\Sigma_0} |\Sigma_0| \dot{\epsilon}_{1+\alpha'}) \left\| \llbracket \mu \rrbracket \right\|_{L^{\infty}(\Sigma_0)} \right) < \varepsilon' < 1,
$$

the last term can be absorbed in the LHS and interpolation and Young's inequalities imply:

$$
(A.24) \quad \|\nabla v, \nabla \mathbb{P} v\|_{L^{\infty}(\mathbb{R}^3)} \le C_{**}(\varepsilon, \varepsilon') \|\nabla v\|_{L^2(\mathbb{R}^3)} + C(\varepsilon, \varepsilon') \left( \|\nabla u_G, \mathbb{P} \nabla u_G\|_{L^{\infty}(\mathbb{R}^3)} + |\nabla u_G|_{\dot{\mathcal{C}}_{\text{pw},\Sigma_0}^{\alpha}(\mathbb{R}^3)} \right).
$$

Upon inserting this estimate into (A.20), we obtain

$$
(A.25) \quad \|\nabla v, \nabla \mathbb{P} v\|_{L^{\infty}(\mathbb{R}^3)} \le C_{**}(\varepsilon, \varepsilon') \|\nabla v\|_{L^2(\mathbb{R}^3)} + C(\varepsilon, \varepsilon') \left( \|\nabla u_G, \mathbb{P} \nabla u_G\|_{L^{\infty}(\mathbb{R}^3)} + |\nabla u_G|_{\dot{\mathcal{C}}_{\text{pw},\Sigma_0}^{\alpha}(\mathbb{R}^3)} \right).
$$

The above two estimate with div G = div F - ρ̃u, will replace estimate (3.11) in [44]. We shall use (A.14) to derive an estimate for the L^p(\mathbb{R}^3)-norm of ∇u and ∇u̇, and hence close the energy estimates for u̇ and u̇. Here, we only rely on (A.13)-(A.23), which is significantly more favorable than the smallness condition (3.5) in [44], in the sense that only the smallness of the viscosities jump is required, not that of the full fluctuation of the viscosities. The remainder of the proof proceeds identically.
**Proof of Lemma A.2.** To prove Lemma A.2, we first observe that since $w \in C_{pw,\Sigma}^{\alpha}(\mathbb{R}^3)$, there exists a $C^{\alpha}$-regular extension $w^e$ of $w$ that satisfies the following properties:

$$ (A.26) \qquad \|w^e\|_{L^\infty(\mathbb{R}^3)} \le C \|w\|_{L^\infty(\mathbb{R}^3)}, \quad w^e = w \text{ on } D^c, $$

$$ (A.27) \qquad \|w - w^e\|_{L^q(\mathbb{R}^3)} \le C \ell_{\Sigma}^{\frac{1}{q}} \|\llbracket w \rrbracket\|_{L^q(\Sigma)} \quad \text{for all } q \in [1, \infty], $$

$$ (A.28) \qquad |w - w^e|_{\dot{C}_{pw,\Sigma}^{\beta}(\mathbb{R}^3)} \le C |\llbracket w \rrbracket|_{\dot{C}^{\beta}(\Sigma)} + C \ell_{\Sigma}^{-\beta} \|\llbracket w \rrbracket\|_{L^{\infty}(\Sigma)}, \quad \beta \in (0, \alpha], $$

$$ (A.29) \qquad |w^e|_{\dot{C}^{\beta}(\mathbb{R}^3)} \le C|w|_{\dot{C}_{pw,\Sigma}^{\beta}(\mathbb{R}^3)} + C \ell_{\Sigma}^{-\beta} \|\llbracket w \rrbracket\|_{L^{\infty}(\Sigma)}, \quad \beta \in (0, \alpha]. $$

The above extension result is an updated version of Lemma A.2 in [44], originally established in [24, Lemma 5.2] for the two-dimensional case.

**Proof of (A.8).** Our approach consists in substituting $w$ with $w^e + (w - w^e)$, which yields

$$ (A.30) \qquad [\mathcal{T}, w]g = [\mathcal{T}, w^e]g + [\mathcal{T}, (w - w^e)]g. $$

Given that $w^e$ is $C^\alpha(\mathbb{R}^3)$-regular (in the whole space), the first term is now more regular than the LHS. In particular, we have

$$ \begin{aligned} |[\mathcal{T}, w^e]g(x)| &= \left| \int_{\mathbb{R}^3} K(x-y)(w^e(y) - w^e(x))g(y)dy \right| \\ &\le C|w^e|_{\dot{C}^{\alpha'}(\mathbb{R}^3)} (|\cdot|^{-3+\alpha'} * |g|)(x), \quad \text{for all } \alpha' \in (0, \alpha]. \end{aligned} $$

For $0 < \alpha' < \frac{3}{p'}$, we can apply Hardy-Littlewood-Sobolev's inequality (see [1, Theorem 1.7]) to obtain:

$$ \begin{aligned} \| [\mathcal{T}, w^e] g \|_{L^p(\mathbb{R}^3)} &\le C |w^e|_{\dot{C}^{\alpha'}(\mathbb{R}^3)} \|g\|_{L^r(\mathbb{R}^3)}, \quad \text{recall } \frac{1}{r} = \frac{\alpha'}{3} + \frac{1}{p} < 1 \\ &\le C (|w|_{\dot{C}^{\alpha'}_{pw,\Sigma}(\mathbb{R}^3)} + \ell_{\Sigma}^{-\alpha'} \|\llbracket w \rrbracket\|_{L^{\infty}(\Sigma)}) \|g\|_{L^r(\mathbb{R}^3)}, \quad \text{recall (A.29)}. \end{aligned} \quad (A.31) $$

To estimate the $L^p(\mathbb{R}^3)$-norm of the last term of (A.30), we rely on the boundedness of commutators between Calderón-Zygmund operators and BMO($\mathbb{R}^3$) functions on $L^p(\mathbb{R}^3)$:

$$ \begin{aligned} \| [\mathcal{T}, (w - w^e)] g \|_{L^p(\mathbb{R}^3)} &\le C \|w - w^e\|_{L^{\infty}(\mathbb{R}^3)} \|g\|_{L^p(\mathbb{R}^3)} \\ &\le C \|\llbracket w \rrbracket\|_{L^{\infty}(\Sigma)} \|g\|_{L^p(\mathbb{R}^3)}, \quad \text{recall (A.27)}. \end{aligned} \quad (A.32) $$

Estimate (A.8) follows directly from (A.31)-(A.32). We now proceed to the proof of (A.9).

**Proof of (A.9).** We recall that the first term of the expression (A.30) reads:

$$ \begin{aligned} &([\mathcal{T}, w^e]g)(x) = \int_{\mathbb{R}^3} K(x-z)(w^e(z) - w^e(x))g(z)dz \\ &= \int_{B(x,1)} K(x-z)(w^e(z) - w^e(x))g(z)dz + \int_{\mathbb{R}^3 \setminus B(x,1)} K(x-z)(w^e(z) - w^e(x))g(z)dz. \end{aligned} $$

To estimate the first term, we use the $C^{\alpha'}(\mathbb{R}^3)$ regularity of $w^e$ to mitigate the singularity of the kernel $K$:

$$ \begin{aligned} \left| \int_{B(x,1)} K(x-z)(w^e(z) - w^e(x))g(z)dz \right| &\le C |w^e|_{\dot{C}^{\alpha'}(\mathbb{R}^3)} \int_{B(x,1)} |x-z|^{-3+\alpha'} |g(z)|dz \\ &\le C |w^e|_{\dot{C}^{\alpha'}(\mathbb{R}^3)} \|g\|_{L^r(\mathbb{R}^3)} \left( \int_0^1 \tau^{(-3+\alpha')r'+2} d\tau \right)^{\frac{1}{r}} \\ &\le C |w^e|_{\dot{C}^{\alpha'}(\mathbb{R}^3)} \|g\|_{L^r(\mathbb{R}^3)}, \quad \text{recall that } \frac{1}{r} < \frac{\alpha'}{3}, \\ &\le C(|w|_{\dot{C}^{\alpha'}_{pw,\Sigma}(\mathbb{R}^3)} + \ell_{\Sigma}^{-\alpha'} \|\llbracket w \rrbracket\|_{L^{\infty}(\Sigma)}) \|g\|_{L^r(\mathbb{R}^3)} \quad \text{recall (A.29)}. \end{aligned} $$
Using the decay of the kernel $K$, we derive:

$$
\begin{aligned}
\left| \int_{\mathbb{R}^3 \setminus B(x,1)} K(x-z)(w^e(z) - w^e(x))g(z)dz \right| &\le C \|w^e\|_{L^\infty(\mathbb{R}^3)} \int_{\mathbb{R}^3 \setminus B(x,1)} |x-z|^{-3} |g(z)|dz \\
&\le C \|w^e\|_{L^\infty(\mathbb{R}^3)} \|g\|_{L^p(\mathbb{R}^3)} \left( \int_1^\infty \tau^{-3p'+2} d\tau \right)^{\frac{1}{p'}} \\
&\le C \|w^e\|_{L^\infty(\mathbb{R}^3)} \|g\|_{L^p(\mathbb{R}^3)}, \quad \text{recall that } p' > 1, \\
&\le C \|w\|_{L^\infty(\mathbb{R}^3)} \|g\|_{L^p(\mathbb{R}^3)}, \quad \text{recall (A.26).}
\end{aligned}
$$

It follows that

$$
(A.33) \quad \| [\mathcal{T}, w^e] g \|_{L^\infty(\mathbb{R}^3)} \le C \left( |w|_{\dot{\epsilon}_{\text{pw},\Sigma}^{\alpha'}(\mathbb{R}^3)} + \ell_{\Sigma}^{-\alpha'} \|[[w]]\|_{L^\infty(\Sigma)} \right) \|g\|_{L^r(\mathbb{R}^3)} + C \|w\|_{L^\infty(\mathbb{R}^3)} \|g\|_{L^p(\mathbb{R}^3)}.
$$

To estimate the $L^\infty(\mathbb{R}^3)$-norm of the last term of (A.30), we use the fact that $(w - w^e)\mathbb{1}_{D^c} = 0$ to express:

♦ for $x \in D$,

$$
(A.34) \quad [\mathcal{T}, (w - w^e)] g(x) = [\mathcal{T}, (w - w^e)] (g\mathbb{1}_D)(x) - (w - w^e)(x) \mathcal{T}(g\mathbb{1}_{D^c})(x).
$$

♦ for $x \in D^c$,

$$
(A.35) \quad [\mathcal{T}, (w - w^e)] g(x) = \mathcal{T}((w - w^e)g\mathbb{1}_D)(x).
$$

Since $(w - w^e) \in C^{\alpha'}(\bar{D})$, we follow the computations leading to (A.33) to estimate the first term in the expression (A.34):

$$
\begin{aligned}
& \|[\mathcal{T}, (w - w^e)](g\mathbb{1}_D)\|_{L^\infty(D)} \\
&\le C(|w - w^e|_{\dot{\epsilon}^{\alpha'}(\bar{D})} + \ell_{\Sigma}^{-\alpha'} \|[w - w^e]\|_{L^\infty(\Sigma)}) \|g\|_{L^r(D)} + C \|w - w^e\|_{L^\infty(D)} \|g\|_{L^p(D)} \\
&\le C(|[[w]]|_{\dot{\epsilon}^{\alpha'}(\Sigma)} + \ell_{\Sigma}^{-\alpha'} \|[w]\|_{L^\infty(\Sigma)}) \|g\|_{L^r(\mathbb{R}^3)} + C \|[w]\|_{L^\infty(\Sigma)} \|g\|_{L^p(\mathbb{R}^3)}, \text{ recall (A.27) -- (A.28).}
\end{aligned}
$$

To estimate the last term of (A.34) we express:

$$
\begin{aligned}
\mathcal{T}(g\mathbb{1}_{D^c})(x) &= \int_{D^c \cap B(\check{x}, \ell_\Sigma)} K(x-y)(g(y)-g(\check{x}))dy + g(\check{x}) \int_{D^c \cap B(\check{x}, \ell_\Sigma)} K(x-y)dy \\
&\quad + \int_{D^c \cap B^c(\check{x}, \ell_\Sigma)} K(x-y)g(y)dy,
\end{aligned}
$$

where $\check{x} \in \partial D$ such that $d(x, D^c) = |x - \check{x}|$. It follows that

$$
\left| \int_{D^c \cap B(\check{x}, \ell_\Sigma)} K(x-y)(g(y)-g(\check{x}))dy \right| \le C|g|_{\dot{\epsilon}_{\text{pw},\Sigma}^{\alpha'}(\mathbb{R}^3)},
$$

and

$$
\left| \int_{D^c \cap B^c(\check{x}, \ell_\Sigma)} K(x-y)g(y)dy \right| \le C \ell_\Sigma^{-\frac{3}{p}} \|g\|_{L^p(\mathbb{R}^3)}.
$$

Using the extra-cancellation of even-order Riesz kernels ([2, 36]), we bound the remaining term as

$$
\left| g(\check{x}) \int_{D^c \cap B(\check{x}, \ell_\Sigma)} K(x-y) dy \right| \le C \|g\|_{L^\infty(\mathbb{R}^3)}.
$$

We thus obtain the following bound for the last term of (A.34):

$$
(A.36) \quad \left| (w - w^e)(x) \mathcal{T}(g\mathbb{1}_{D^c})(x) \right| \le C \left\| [[w]] \right\|_{L^\infty(\Sigma)} \left( \left\| g \right\|_{C^{\alpha'}_{\text{pw},\Sigma}(\mathbb{R}^3)} + \ell_{\Sigma}^{-\frac{3}{p}} \left\| g \right\|_{L^p(\mathbb{R}^3)} \right).
$$
where $\tilde{x} \in \partial D$ is such that $d(x, \partial D) = |x - \tilde{x}|$. We observe that that the estimate (A.40) also holds for the last term of (A.39) and we obtain:

$$ \left| (w^e(y) - w^e(x)) \int_{\mathbb{R}^3 \setminus B(y, 2\delta)} K(x-z)g(z)dz \right| \\ \le C \left( |w^e|_{\dot{\mathcal{C}}^\beta(\mathbb{R}^3)} \|g, \mathcal{F}g\|_{L^\infty(\mathbb{R}^3)} + |w^e|_{\dot{\mathcal{C}}^{\beta-a'}(\mathbb{R}^3)} |g|_{\dot{\mathcal{C}}^{a'}_{\text{pw},\Sigma}(\mathbb{R}^3)} \right) |x-y|^{\beta}. $$

To estimate the last term of (A.38), we observe that

$$ |K(x-z) - K(y-z)| \le |x-y| \int_0^1 |\nabla K(\tau x + (1-\tau)y-z)| d\tau \\ \le C \int_0^1 \frac{|x-y|}{|\tau x + (1-\tau)y-z|^4} d\tau. $$

Hence, for all $z \notin B(y, 2\delta)$, and $\tau \in (0, 1)$, we have:

$$ \begin{aligned} |\tau x + (1-\tau)y-z| &\ge |y-z| - \tau|x-y| \\ &\ge \frac{1}{2}|y-z| + (\frac{1}{2}|y-z| - \tau\delta) \\ &\ge \frac{1}{2}|y-z| + (\frac{1}{2}(2\delta) - \tau\delta) \\ &\ge \frac{1}{2}|y-z|. \end{aligned} $$

In consequence, we have:

$$ \begin{aligned} \left| \int_{\mathbb{R}^3 \setminus B(y, 2\delta)} (K(x-z) - K(y-z))(w^e(z) - w^e(y))g(z)dz \right| &\le C|w^e|_{\dot{\mathcal{C}}^\beta(\mathbb{R}^3)} \|g\|_{L^\infty(\mathbb{R}^3)} |x-y| \int_{\mathbb{R}^3 \setminus B(y, 2\delta)} |y-z|^{\beta-4} dz \\ &\le C|w^e|_{\dot{\mathcal{C}}^\beta(\mathbb{R}^3)} \|g\|_{L^\infty(\mathbb{R}^3)} |x-y|^{\beta}. \end{aligned} $$

We finally obtain:

$$ \begin{aligned} |[\mathcal{F}, w^e]g|_{\dot{\mathcal{C}}^{\beta}_{\text{pw},\Sigma}(\mathbb{R}^3)} &\le C(|w^e|_{\dot{\mathcal{C}}^{\beta}(\mathbb{R}^3)} \|g, \mathcal{F}g\|_{L^\infty(\mathbb{R}^3)} + |w^e|_{\dot{\mathcal{C}}^{\beta-a'}(\mathbb{R}^3)} |g|_{\dot{\mathcal{C}}^{a'}_{\text{pw},\Sigma}(\mathbb{R}^3)}) \\ &\le C(|w|_{\dot{\mathcal{C}}^{\beta}_{\text{pw},\Sigma}(\mathbb{R}^3)} + \ell_{\Sigma}^{-\beta} \|[[w]]\|_{L^\infty(\Sigma)}) \|g, \mathcal{F}g\|_{L^\infty(\mathbb{R}^3)} \\ &\quad + C(|w|_{\dot{\mathcal{C}}^{\beta-a'}_{\text{pw},\Sigma}(\mathbb{R}^3)} + \ell_{\Sigma}^{-\beta+\alpha'} \|[[w]]\|_{L^\infty(\Sigma)}) |g|_{\dot{\mathcal{C}}^{a'}_{\text{pw},\Sigma}(\mathbb{R}^3)}, \end{aligned} \quad (A.41) $$

where we make use of (A.29).

Using (A.34) we express

$$ \begin{aligned} & [\mathcal{F}, (w - w^e)]g(x) - [\mathcal{F}, (w - w^e)]g(y) \\ &= \int_{D \cap B(x, \delta)} K(x-z)((w - w^e)(z) - (w - w^e)(x))g(z)dz \\ &\quad - \int_{D \cap B(x, \delta)} K(y-z)((w - w^e)(z) - (w - w^e)(y))g(z)dz \\ &\quad + \int_{D \cap B^c(x, \delta)} (K(x-z) - K(y-z))((w - w^e)(z) - (w - w^e)(x))g(z)dz \\ &\quad + ((w - w^e)(y) - (w - w^e)(x))\left(\int_{D \cap B^c(x, \delta)} K(y-z)g(z)dz + \int_{D^c} K(y-z)g(z)dz\right) \\ &\quad - (w - w^e)(x)\int_{D^c} (K(x-z) - K(y-z))g(z)dz. \end{aligned} \quad (A.42) $$
As in the previous step, we estimate the first three terms on the RHS as follows:

$$ \left| \int_{D \cap B(x, \delta)} K(x-z)((w-w^e)(z) - (w-w^e)(x))g(z)dz \right| + \left| \int_{D \cap B(x, \delta)} K(y-z)((w-w^e)(z) - (w-w^e)(y))g(z)dz \right| \\ \left| \int_{D \cap B^c(x, \delta)} (K(x-z) - K(y-z))((w-w^e)(z) - (w-w^e)(x))g(z)dz \right| \le C |w-w^e|_{\dot{\phi}_{\text{pw},\Sigma}^{\beta}(\mathbb{R}^3)} \|g\|_{L^{\infty}(\mathbb{R}^3)} |x-y|^{\beta}. $$

To bound the last term, we express:

$$ \begin{aligned} & \int_{D^c} (K(x-z) - K(y-z))g(z)dz \\ &= \int_{D^c \cap B(\check{x}, 2\delta)} K(x-z)(g(z) - g(\check{x}))dz - \int_{D^c \cap B(\check{x}, 2\delta)} K(y-z)(g(z) - g(\check{y}))dz \\ &\quad + (g(\check{y}) - g(\check{x})) \int_{D \cap B(\check{x}, 2\delta)} K(y-z)dz + \int_{D^c \cap B^c(\check{x}, 2\delta)} (K(x-z) - K(y-z))(g(z) - g(\check{x}))dz \\ &\quad - g(\check{x}) \int_D (K(x-z) - K(y-z))dz. \end{aligned} $$

Following the preceding step, all terms on the RHS, except for the last, are bounded by

$$ C|g|_{\dot{\phi}^{\beta}(D^c)} |x-y|^{\beta}. $$

Next, we apply Theorem 2.3 of [16] to estimate the last term as

$$ |g(\check{x}) \int_D (K(x-z) - K(y-z)) dz| \le C \|g\|_{L^{\infty}(\mathbb{R}^3)} \mathfrak{P}_{\Sigma} |\Sigma|^{1+\beta} |x-y|^{\beta}, $$

where $\mathfrak{P}_{\Sigma}$ is a polynomial in $|\Sigma|$, $\|\Sigma\|_{\text{Lip}}$ and $|\Sigma|_{\text{inf}}^{-1}$. For the remaining term of (A.42), we express

$$ \int_{D \cap B^c(x, \delta)} K(y-z)g(z)dz + \int_{D^c} K(y-z)g(z)dz = \mathcal{T}g - \int_{D \cap B(x, \delta)} K(y-z)g(z)dz, $$

and we estimate

$$ \left| \int_{D \cap B^c(x, \delta)} K(y-z)g(z)dz + \int_{D^c} K(y-z)g(z)dz \right| \le C \left( \|g, \mathcal{T}g\|_{L^\infty(\mathbb{R}^3)} + |g|_{\dot{\phi}_{\text{pw},\Sigma}^{\beta}(\mathbb{R}^3)} |x-y|^{\beta} \right). $$

In sum, we obtain:

$$ \begin{aligned} & |[\mathcal{T}, (w-w^e)]g|_{\dot{\phi}_{\text{pw},\Sigma}^{\beta}(\mathbb{R}^3)} \le C \|[[w]]\|_{L^{\infty}(\Sigma)} |g|_{\dot{\phi}_{\text{pw},\Sigma}^{\beta}(\mathbb{R}^3)} \\ & \qquad + C(|[[w]]|_{\dot{\phi}^{\beta}(\Sigma)} + (\ell_{\Sigma}^{-\beta} + \mathfrak{P}_{\Sigma}|\Sigma|^{1+\beta}) \|[[w]]\|_{L^{\infty}(\Sigma)}) \|\mathcal{T}g, g\|_{L^{\infty}(\mathbb{R}^3)}. \end{aligned} \quad (A.43) $$

Estimate (A.10) simply follows from (A.41)-(A.43). This completes the proof of Lemma A.2.

Let us observe that following the same computations as above, one obtains $(r \in [1, \infty)$ and $\alpha' \in (0, \alpha]$):

$$ (A.44) \qquad \|\mathcal{T}g\|_{L^{\infty}(\mathbb{R}^3)} \le C \left( \ell_{\Sigma}^{-\frac{3}{r}} \|g\|_{L^r(\mathbb{R}^3)} + \|g\|_{C_{\text{pw},\Sigma}^{\alpha'}(\mathbb{R}^3)} \right), $$

and

$$ (A.45) \qquad \|\mathcal{T}g\|_{\dot{\phi}_{\text{pw},\Sigma}^{\alpha}(\mathbb{R}^3)} \le C \left( |g|_{\dot{\phi}_{\text{pw},\Sigma}^{\alpha}(\mathbb{R}^3)} + \|g\|_{L^{\infty}(\mathbb{R}^3)} \mathfrak{P}_{\Sigma} |\Sigma|^{1+\alpha} \right). $$

Also, we may notice that in the case $\beta \in (0, \alpha)$, the third term on the RHS of (A.38) can be estimated as

$$ \begin{aligned} \left| (w^e(y) - w^e(x)) \int_{\mathbb{R}^3 \setminus B(y, 2\delta)} K(x-z)g(z)dz \right| &\le C|w^e|_{\dot{\phi}^{\alpha}(\mathbb{R}^3)} |x-y|^{\alpha} \int_{|x-z| \ge 2\delta} |x-z|^{-3}|g(z)|dz \\ &\le C|w^e|_{\dot{\phi}^{\alpha}(\mathbb{R}^3)} \|g\|_{L^{3/(\alpha-\beta)}(\mathbb{R}^3)} |x-y|^{\beta} \end{aligned} $$

leading to

$$ \begin{aligned} & |[\mathcal{T}, w]g|_{\dot{\phi}_{\text{pw},\Sigma}^{\beta}(\mathbb{R}^3)} \le C \|[[w]]\|_{L^{\infty}(\Sigma)} |g|_{\dot{\phi}_{\text{pw},\Sigma}^{\beta}(\mathbb{R}^3)} + C(|w|_{\dot{\phi}^{\alpha}(\mathbb{R}^3)} + \ell_{\Sigma}^{-\alpha} \|[[w]]\|_{L^{\infty}(\Sigma)}) \|g\|_{L^{3/(\alpha-\beta)}(\mathbb{R}^3)} \\ & \qquad + C(|[[w]]|_{\dot{\phi}^{\beta}(\Sigma)} + (\ell_{\Sigma}^{-\beta} + \mathfrak{P}_{\Sigma}|\Sigma|^{1+\beta}) \|[[w]]\|_{L^{\infty}(\Sigma)}) \|\mathcal{T}g, g\|_{L^{\infty}(\mathbb{R}^3)}. \end{aligned} \quad (A.46) $$

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