It only remains to estimate the norm of the effective flux $F$ in terms of the functionals $A_1$ and $A_2$ and to do so, we distinguish two cases according to the dimension.

**Case d = 2.** We recall that the effective flux is given by

$$ (2.52) \qquad F = -(\Delta)^{-1} \operatorname{div}(\rho \dot{u}) = \nu \operatorname{div} u - G, \quad G = P(\rho) - \tilde{P}. $$

Hence, interpolation inequality yields the following estimate on the small interval $[0, \sigma(t)]$

$$ \begin{aligned} \int_0^{\sigma(t)} \|F\|_{L^\infty(\mathbb{R}^2)} \le C \int_0^{\sigma(t)} \|F\|_{L^2(\mathbb{R}^2)}^{\frac{1}{2}} \|\nabla F\|_{L^4(\mathbb{R}^2)}^{\frac{2}{3}} &\le \int_0^{\sigma(t)} \|(G, \nu \operatorname{div} u)\|_{L^2(\mathbb{R}^2)}^{\frac{1}{3}} \|\rho \dot{u}\|_{L^4(\mathbb{R}^2)}^{\frac{2}{3}} \\ &\le (C^* E_0^{\frac{1}{6}} + \nu^{\frac{1}{6}} A_1(t)^{\frac{1}{6}}) \int_0^{\sigma(t)} \|\rho \dot{u}\|_{L^4(\mathbb{R}^2)}^{\frac{2}{3}}. \end{aligned} $$

Since the density contains vacuum states, we are not allowed to bound the last factor in the above inequality solely by the Gagliardo-Nirenberg inequality. On $\mathbb{T}^2$, we have at our disposal a logarithmic interpolation inequality (see [16, 17, 21]) which is not valid in the whole space. To address this issue, we prove in **Lemma A.1** an interpolation inequality that will allow us to take into account the vacuum state. Thus, from **Lemma A.1**, we have:

$$ \begin{aligned} \int_0^{\sigma(t)} \|\rho \dot{u}\|_{L^4(\mathbb{R}^2)}^{\frac{2}{3}} &\le C^* \int_0^{\sigma(t)} \|\sqrt{\rho} \dot{u}\|_{L^2(\mathbb{R}^2)}^{\frac{1}{3}} \|\nabla \dot{u}\|_{L^2(\mathbb{R}^2)}^{\frac{1}{3}} + C^* E_0^{\frac{1}{18}} \int_0^{\sigma(t)} \|\sqrt{\rho} \dot{u}\|_{L^2(\mathbb{R}^2)}^{\frac{2}{9}} \|\nabla \dot{u}\|_{L^2(\mathbb{R}^2)}^{\frac{4}{9}} \\ &\le C^* A_1(t)^{\frac{1}{6}} A_2(t)^{\frac{1}{6}} + C^* E_0^{\frac{1}{18}} A_1(t)^{\frac{1}{9}} A_2(t)^{\frac{2}{9}} \end{aligned} $$

and therefore:

$$ (2.53) \qquad \int_0^{\sigma(t)} \|F\|_{L^\infty(\mathbb{R}^2)} \le C^*(1 + E_0^{\frac{1}{18}})(E_0^{\frac{1}{6}} + \nu^{\frac{1}{6}} A_1(t)^{\frac{1}{6}})(A_1(t)^{\frac{1}{3}} + A_2(t)^{\frac{1}{3}}). $$

Similarly, interpolation inequality and **Lemma A.1** yield on the larger time interval $[\sigma(t), t]$

$$ \begin{aligned} \int_{\sigma(t)}^{t} \|F\|_{L^{\infty}(\mathbb{R}^2)}^3 &\le \int_{\sigma(t)}^{t} \|(G, \nu \operatorname{div} u)\|_{L^2(\mathbb{R}^2)} \|\rho \dot{u}\|_{L^4(\mathbb{R}^2)}^2 \\ &\le (C^* E_0^{\frac{1}{2}} + \nu^{\frac{1}{2}} A_1(t)^{\frac{1}{2}}) (C^* \int_0^t \|\sqrt{\rho} \dot{u}\|_{L^2(\mathbb{R}^2)} \|\nabla \dot{u}\|_{L^2(\mathbb{R}^2)} + C^* E_0^{\frac{1}{6}} \int_0^t \|\sqrt{\rho} \dot{u}\|_{L^2(\mathbb{R}^2)}^{\frac{2}{3}} \|\nabla \dot{u}\|_{L^2(\mathbb{R}^2)}^{\frac{4}{3}}) \\ &\le (C^* E_0^{\frac{1}{2}} + \nu^{\frac{1}{2}} A_1(t)^{\frac{1}{2}}) (C^* A_1(t)^{\frac{1}{2}} A_2(t)^{\frac{1}{2}} + C^* E_0^{\frac{1}{6}} A_1(t)^{\frac{1}{3}} A_2(t)^{\frac{2}{3}}). \end{aligned} $$

Hence,

$$ (2.54) \qquad \int_{\sigma(t)}^{t} \|F\|_{L^{\infty}(\mathbb{R}^2)}^3 \le C^*(1 + E_0^{\frac{1}{6}})(C^* E_0^{\frac{1}{2}} + \nu^{\frac{1}{2}} A_1(t)^{\frac{1}{2}})(A_1(t) + A_2(t)). $$

Finally (2.51), (2.53) and (2.54), lead to:

$$ (2.55) \qquad \sup_{[0,t]} \|\rho - \tilde{\rho}\|_{L^\infty(\mathbb{R}^2)} \le \|\rho_0 - \tilde{\rho}\|_{L^\infty(\mathbb{R}^2)} + \frac{C^*}{\nu^{\frac{1}{3}}} (1 + E_0^{\frac{1}{18}})(E_0^{\frac{1}{6}} + \nu^{\frac{1}{6}} A_1(t)^{\frac{1}{6}})(A_1(t)^{\frac{1}{3}} + A_2(t)^{\frac{1}{3}}). $$

**Case d = 3.** From the expression of the effective flux (2.52), we have by Gagliardo-Nirenberg inequality:

$$ \begin{aligned} \int_0^{\sigma(t)} \|F\|_{L^\infty(\mathbb{R}^3)} &\le C \int_0^{\sigma(t)} \|F\|_{L^6(\mathbb{R}^3)}^{\frac{1}{2}} \|\nabla F\|_{L^6(\mathbb{R}^3)}^{\frac{1}{2}} \le C \int_0^{\sigma(t)} \|\rho \dot{u}\|_{L^2(\mathbb{R}^3)}^{\frac{1}{2}} \|\rho \dot{u}\|_{L^6(\mathbb{R}^3)}^{\frac{1}{2}} \le C^* A_1(t)^{\frac{1}{4}} A_2(t)^{\frac{1}{4}}, \\ \int_{\sigma(t)}^{t} \|F\|_{L^\infty(\mathbb{R}^3)}^3 &\le C \int_{\sigma(t)}^{t} \|\rho \dot{u}\|_{L^2(\mathbb{R}^3)}^{\frac{3}{2}} \|\rho \dot{u}\|_{L^6(\mathbb{R}^3)}^{\frac{3}{2}} \le C^* A_1(t)^{\frac{1}{4}} A_2(t)^{\frac{5}{4}}. \end{aligned} $$

Finally, we get:

$$ \sup_{[0,t]} \|\rho - \tilde{\rho}\|_{L^\infty(\mathbb{R}^3)} \le \|\rho_0 - \tilde{\rho}\|_{L^\infty(\mathbb{R}^3)} + \frac{C^*}{\nu^{1/3}} (A_1(t)^{\frac{1}{2}} + A_2(t)^{\frac{1}{2}}), $$
2.2.5. *Proof of Lemma 2.6.* The present section is devoted to the propagation of tangential regularity of the density together with the derivation of the Lipschitz norm of the velocity, provided with the (time-independent) bounds $C_0$ for $A_1(t)$, $A_2(t)$, $\rho(t)$.

We recall that the family of vector fields $\mathcal{X}(t) = (X_v(t))_{1\le v\le m}$ is defined as solution of (1.8)

$$ (2.57) \qquad \left\{ \begin{array}{l} \partial_t X_v + u \cdot \nabla X_v = \partial_{X_v} u = (X_v \cdot \nabla)u, \\ X_{v|t=0} = X_{0,v}, \end{array} \right. $$

and we can estimate the norms of

$$ \|\mathcal{X}(t)\|_{L^{\infty,p}(\mathbb{R}^d)} = \sup_{1\le v\le m} \|X_v(t)\|_{L^{\infty,p}(\mathbb{R}^d)} = \sup_{1\le v\le m} \left( \|X_v(t)\|_{L^{\infty}(\mathbb{R}^d)} + \|\nabla X_v(t)\|_{L^p(\mathbb{R}^d)} \right) $$

by use of the Lipschitz norm of the velocity. On the other side, these norms will help to bound the Lipschitz norm of the velocity, and in particular the pressure-related part $\nabla u_G$ in the decomposition (2.9) of $\nabla u$:

$$ (2.58) \qquad \begin{aligned} \nabla u &= \nabla \tilde{u} + \nabla u_G \\ &:= \left( -\frac{1}{\nu} \mathcal{R}\mathcal{R}(-\Delta)^{-1} \operatorname{div}(\rho \dot{u}) - \frac{1}{\mu} \mathcal{R}\mathcal{R}(-\Delta)^{-1} \operatorname{curl}(\rho \dot{u}) \right) + \left( \frac{1}{\nu} \mathcal{R}\mathcal{R}G \right). \end{aligned} $$

Finally we will close the estimates for

$$ \mathcal{A}_3(t) = \|\mathcal{X}(t)\|_{L^{\infty,p}(\mathbb{R}^d)} + \sup_{1\le v\le m} \|\operatorname{div}(\rho X_v)(t)\|_{L^p(\mathbb{R}^d)} $$

by Gronwall's inequality.

*Proof. Step 1. Preliminary estimates for* $\|\mathcal{X}\|_{L^{\infty,p}(\mathbb{R}^d)}$. From (2.57) we deduce easily:

$$ (2.59) \qquad \|X_v(t)\|_{L^\infty(\mathbb{R}^d)} \le \|X_{0,v}\|_{L^\infty} + \int_0^t \|X_v(s)\|_{L^\infty(\mathbb{R}^d)} \|\nabla u(s)\|_{L^\infty(\mathbb{R}^d)} ds. $$

We now take derivatives in (2.57) and we obtain:

$$ (2.60) \qquad \partial_t \partial_k X_v^j + (u \cdot \nabla) \partial_k X_v^j = \partial_k X_v \cdot \nabla u^j - \partial_k u \cdot \nabla X_v^j + \partial_{X_v} \partial_k u^j. $$

We take the trace in the above equality and we make use of the expression of the divergence of the velocity $\operatorname{div} u = \frac{1}{\nu}(F+G)$, in order to obtain the following equation for $\operatorname{div} X_v$:

$$ \partial_t (\operatorname{div} X_v) + u \cdot \nabla \operatorname{div} X_v = \frac{1}{\nu} \partial_{X_v} G + \frac{1}{\nu} \partial_{X_v} F. $$

Hence, it is straightforward to show the following:

$$ (2.61) \qquad \begin{aligned} \|\operatorname{div} X_v(t)\|_{L^p(\mathbb{R}^d)} \le \|\operatorname{div} X_{0,v}\|_{L^p(\mathbb{R}^d)} + \frac{1}{\nu} \int_0^t \|\partial_{X_v} G\|_{L^p(\mathbb{R}^d)} + \frac{1}{\nu} \int_0^t \|X_v\|_{L^\infty(\mathbb{R}^d)} \|\nabla F\|_{L^p(\mathbb{R}^d)} \\ + \int_0^t \|\operatorname{div} u(s)\|_{L^\infty(\mathbb{R}^d)} \|\operatorname{div} X_v(s)\|_{L^p(\mathbb{R}^d)} ds. \end{aligned} $$

As well, we take the antisymmetric part in (2.60) to get the following equation for $\operatorname{curl} X_v$:

$$ \partial_t (\partial_k X_v^j - \partial_j X_v^k) + u \cdot \nabla (\partial_k X_v^j - \partial_j X_v^k) = \partial_k X_v \cdot \nabla u^j - \partial_j X_v \cdot \nabla u^k \\ + \partial_j u \cdot \nabla X_v^k - \partial_k u \cdot \nabla X_v^j + \partial_{X_v} (\partial_k u^j - \partial_j u^k), $$

from which we deduce easily:

$$ (2.62) \qquad \begin{aligned} \|\operatorname{curl} X_v(t)\|_{L^p(\mathbb{R}^d)} \le \|\operatorname{curl} X_{0,v}\|_{L^p(\mathbb{R}^d)} + \int_0^t \|\nabla X_v\|_{L^p(\mathbb{R}^d)} \|\nabla u\|_{L^\infty(\mathbb{R}^d)} \\ + \int_0^t \|X_v\|_{L^\infty(\mathbb{R}^d)} \|\nabla \operatorname{curl} u\|_{L^p(\mathbb{R}^d)}. \end{aligned} $$
and this ends the proof of Lemma 2.3.

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

2.2.5. *Proof of Lemma 2.6.* The present section is devoted to the propagation of tangential regularity of the density together with the derivation of the Lipschitz norm of the velocity, provided with the (time-independent) bounds $C_0$ for $A_1(t)$, $A_2(t)$, $\rho(t)$.

We recall that the family of vector fields $\mathcal{X}(t) = (X_v(t))_{1\le v\le m}$ is defined as solution of (1.8)

$$ (2.57) \qquad \begin{cases} \partial_t X_v + u \cdot \nabla X_v = \partial_{X_v} u = (X_v \cdot \nabla)u, \\ X_{v|t=0} = X_{0,v}, \end{cases} $$

and we can estimate the norms of

$$ \|\mathcal{X}(t)\|_{L^{\infty,p}(\mathbb{R}^d)} = \sup_{1\le v\le m} \|X_v(t)\|_{L^{\infty,p}(\mathbb{R}^d)} = \sup_{1\le v\le m} \left( \|X_v(t)\|_{L^{\infty}(\mathbb{R}^d)} + \|\nabla X_v(t)\|_{L^p(\mathbb{R}^d)} \right) $$

by use of the Lipschitz norm of the velocity. On the other side, these norms will help to bound the Lipschitz norm of the velocity, and in particular the pressure-related part $\nabla u_G$ in the decomposition (2.9) of $\nabla u$:

$$ (2.58) \qquad \begin{aligned} \nabla u &= \nabla \tilde{u} + \nabla u_G \\ &:= \left( -\frac{1}{\nu} \mathcal{R} \mathcal{R} (-\Delta)^{-1} \operatorname{div}(\rho \dot{u}) - \frac{1}{\mu} \mathcal{R} \mathcal{R} (-\Delta)^{-1} \operatorname{curl}(\rho \dot{u}) \right) + \left( \frac{1}{\nu} \mathcal{R} \mathcal{R} G \right). \end{aligned} $$

Finally we will close the estimates for

$$ \mathcal{A}_3(t) = \|\mathcal{X}(t)\|_{L^{\infty,p}(\mathbb{R}^d)} + \sup_{1\le v\le m} \|\operatorname{div}(\rho X_v)(t)\|_{L^p(\mathbb{R}^d)} $$

by Grönwall's inequality.

*Proof. Step 1. Preliminary estimates for* $\|\mathcal{X}\|_{L^{\infty,p}(\mathbb{R}^d)}$. From (2.57) we deduce easily:

$$ (2.59) \qquad \|X_v(t)\|_{L^{\infty}(\mathbb{R}^d)} \le \|X_{0,v}\|_{L^{\infty}} + \int_0^t \|X_v(s)\|_{L^{\infty}(\mathbb{R}^d)} \|\nabla u(s)\|_{L^{\infty}(\mathbb{R}^d)} ds. $$

We now take derivatives in (2.57) and we obtain:

$$ (2.60) \qquad \partial_t \partial_k X_v^j + (u \cdot \nabla) \partial_k X_v^j = \partial_k X_v \cdot \nabla u^j - \partial_k u \cdot \nabla X_v^j + \partial_{X_v} \partial_k u^j. $$

We take the trace in the above equality and we make use of the expression of the divergence of the velocity $\operatorname{div} u = \frac{1}{\nu} (F + G)$, in order to obtain the following equation for $\operatorname{div} X_v$:

$$ \partial_t (\operatorname{div} X_v) + u \cdot \nabla \operatorname{div} X_v = \frac{1}{\nu} \partial_{X_v} G + \frac{1}{\nu} \partial_{X_v} F. $$

Hence, it is straightforward to show the following:

$$ (2.61) \qquad \begin{aligned} \|\operatorname{div} X_v(t)\|_{L^p(\mathbb{R}^d)} \le \|\operatorname{div} X_{0,v}\|_{L^p(\mathbb{R}^d)} &+ \frac{1}{\nu} \int_0^t \|\partial_{X_v} G\|_{L^p(\mathbb{R}^d)} + \frac{1}{\nu} \int_0^t \|X_v\|_{L^{\infty}(\mathbb{R}^d)} \|\nabla F\|_{L^p(\mathbb{R}^d)} \\ &+ \int_0^t \|\operatorname{div} u(s)\|_{L^{\infty}(\mathbb{R}^d)} \|\operatorname{div} X_v(s)\|_{L^p(\mathbb{R}^d)} ds. \end{aligned} $$

As well, we take the antisymmetric part in (2.60) to get the following equation for $\operatorname{curl} X_v$:

$$ \begin{aligned} \partial_t (\partial_k X_v^j - \partial_j X_v^k) + u \cdot \nabla (\partial_k X_v^j - \partial_j X_v^k) = \partial_k X_v \cdot \nabla u^j - \partial_j X_v \cdot \nabla u^k \\ + \partial_j u \cdot \nabla X_v^k - \partial_k u \cdot \nabla X_v^j + \partial_{X_v} (\partial_k u^j - \partial_j u^k), \end{aligned} $$
By combining (2.61) and (2.62), we obtain the following estimate for the vector field gradient:

$$ (2.63) \qquad \begin{aligned} \|\nabla X_v(t)\|_{L^p(\mathbb{R}^d)} &\le \|\nabla X_{0,v}\|_{L^p(\mathbb{R}^d)} + \frac{1}{\nu} \int_0^t \|\partial_{X_v} G\|_{L^p(\mathbb{R}^d)} + \int_0^t \|\nabla X_v(s)\|_{L^p(\mathbb{R}^d)} \|\nabla u(s)\|_{L^\infty(\mathbb{R}^d)} ds \\ &\quad + \int_0^t \|X_v(s)\|_{L^\infty(\mathbb{R}^d)} \left\|\left(\frac{1}{\nu}\nabla F(s), \nabla \text{curl } u(s)\right)\right\|_{L^p(\mathbb{R}^d)} ds. \end{aligned} $$

**Step 2.** Estimates for $A_3(t) = \|\chi(t)\|_{L^\infty, p(\mathbb{R}^d)} + \sup_{1 \le v \le m} \|\text{div}(\rho X_v)(t)\|_{L^p(\mathbb{R}^d)}$. In order to estimate the $L^p(\mathbb{R}^d)$ norm of $\text{div}(\rho X_v)$, we combine the equation on the density (1.1)$_1$ and the equation on the vector field $X_v$ (2.57) in order to obtain

$$ \partial_t(\text{div}(\rho X_v)) + \text{div}(u \text{div}(\rho X_v)) = 0, $$

from which we deduce the following estimates:

$$ (2.64) \qquad \begin{aligned} \|\text{div}(\rho X_v)(t)\|_{L^p(\mathbb{R}^d)} &\le \|\text{div}(\rho_0 X_{0,v})\|_{L^p(\mathbb{R}^d)} + \int_0^t \|\text{div } u(s)\|_{L^\infty(\mathbb{R}^d)} \|\text{div}(\rho X_v)(s)\|_{L^p(\mathbb{R}^d)} ds. \end{aligned} $$

Consequently we can estimate

$$ \partial_{X_v} G = P'(\rho) \text{div}(\rho X_v) - \rho P'(\rho) \text{div } X_v $$

by

$$ \|\partial_{X_v} G\|_{L^p(\mathbb{R}^d)} \le C_0 \left( \|\text{div}(\rho X_v)\|_{L^p(\mathbb{R}^d)} + \|\text{div } X_v\|_{L^p(\mathbb{R}^d)} \right). $$

We combine (2.59), (2.63) and (2.64) together with (2.18) to get

$$ A_3(t) \le A_3(0) + C_0 \int_0^t A_3(s) \left[ \frac{1}{\nu} + \|\nabla u(s)\|_{L^\infty(\mathbb{R}^d)} + \|\rho \dot{u}(s)\|_{L^p(\mathbb{R}^d)} \right] ds. $$

Gronwall's Lemma yields:

$$ (2.65) \qquad A_3(t) \le A_3(0) \exp \left( C_0 \int_0^t \left[ 1 + \|\nabla u(s)\|_{L^\infty(\mathbb{R}^d)} + \|\rho \dot{u}(s)\|_{L^p(\mathbb{R}^d)} \right] ds \right). $$

**Step 3.** Estimates for $\|\nabla u_G\|_{L^1_t L^\infty_x}$.

We consider $\nabla u_G$ first. We use **Proposition 2.5** to bound the second part $\nabla u_G = \frac{1}{\nu} \mathcal{R}\mathcal{R}G$:

$$ \|\nabla u_G(t)\|_{L^\infty(\mathbb{R}^d)} \le \frac{C}{\nu} \|G(t)\|_{L^2(\mathbb{R}^d)} + \frac{C}{\nu} \|G(t)\|_{L^\infty(\mathbb{R}^d)} \left[ 1 + \log \left( e + \frac{\|G(t)\|_{L^p(\mathcal{X}(t)(\mathbb{R}^d))}}{\|G(t)\|_{L^\infty(\mathbb{R}^d)}} \right) \right]. $$

Now we focus on the estimate for

$$ \|G(t)\|_{L^p_{\mathcal{X}(t)}(\mathbb{R}^d)} \sim \frac{1}{I(\mathcal{X}(t))} \left[ \|G(t)\|_{L^\infty(\mathbb{R}^d)} \|\chi(t)\|_{L^\infty, p(\mathbb{R}^d)} + \sup_{1 \le v \le m} \|\partial_{X_v} G(t)\|_{L^p(\mathbb{R}^d)} \right] \le \frac{C_0}{I(\mathcal{X}(t))} A_3(t). $$

From (2.57), it is obvious that this quantity comes with the following lower bound:

$$ I(\mathcal{X}(t)) \ge I(\mathcal{X}_0) \exp \left( - \int_0^t \|\nabla u(s)\|_{L^\infty(\mathbb{R}^d)} ds \right). $$

We hence have

$$ (2.66) \qquad \|\nabla u_G(t)\|_{L^\infty(\mathbb{R}^d)} \le C_0 \left( 1 + \frac{A_3(0)}{I(\mathcal{X}_0)} + \int_0^t \left[ 1 + \|\rho \dot{u}\|_{L^p(\mathbb{R}^d)} + \|\nabla u\|_{L^\infty(\mathbb{R}^d)} \right] (s) ds \right). $$

**Step 4.** Final estimates. In the following we continue with the estimates for $\|\rho \dot{u}\|_{L^1_t L^p(\mathbb{R}^d)}$ and $\|\nabla \tilde{u}\|_{L^1_t L^\infty(\mathbb{R}^d)}$, taking the dimension into account.
from which we deduce easily:

$$
(2.62) \quad \|\mathrm{curl} X_v(t)\|_{L^p(\mathbb{R}^d)} \le \|\mathrm{curl} X_{0,v}\|_{L^p(\mathbb{R}^d)} + \int_0^t \|\nabla X_v\|_{L^p(\mathbb{R}^d)} \|\nabla u\|_{L^\infty(\mathbb{R}^d)} \\
\qquad \qquad + \int_0^t \|X_v\|_{L^\infty(\mathbb{R}^d)} \|\nabla \mathrm{curl} u\|_{L^p(\mathbb{R}^d)} \, ds.
$$

By combining (2.61) and (2.62), we obtain the following estimate for the vector field gradient:

$$
(2.63) \quad \|\nabla X_v(t)\|_{L^p(\mathbb{R}^d)} \le \|\nabla X_{0,v}\|_{L^p(\mathbb{R}^d)} + \frac{1}{\nu} \int_0^t \|\partial_{X_v} G\|_{L^p(\mathbb{R}^d)} + \int_0^t \|\nabla X_v(s)\|_{L^p(\mathbb{R}^d)} \|\nabla u(s)\|_{L^\infty(\mathbb{R}^d)} ds \\
\qquad + \int_0^t \|X_v(s)\|_{L^\infty(\mathbb{R}^d)} \left\| \left( \frac{1}{\nu} \nabla F(s), \nabla \mathrm{curl} u(s) \right) \right\|_{L^p(\mathbb{R}^d)} ds.
$$

Step 2. Estimates for $\mathcal{A}_3(t) = \|\mathcal{X}(t)\|_{L^\infty, p(\mathbb{R}^d)} + \sup_{1 \le v \le m} \|\mathrm{div}(\rho X_v)(t)\|_{L^p(\mathbb{R}^d)}$. In order to estimate the $L^p(\mathbb{R}^d)$ norm of $\mathrm{div}(\rho X_v)$, we combine the equation on the density $(1.1)_1$ and the equation on the vector field $X_v$ (2.57) in order to obtain

$$
\partial_t(\mathrm{div}(\rho X_v)) + \mathrm{div}(u \mathrm{div}(\rho X_v)) = 0,
$$

from which we deduce the following estimates:

$$
(2.64) \quad \|\mathrm{div}(\rho X_v)(t)\|_{L^p(\mathbb{R}^d)} \le \|\mathrm{div}(\rho_0 X_{0,v})\|_{L^p(\mathbb{R}^d)} + \int_0^t \|\mathrm{div} \, u(s)\|_{L^\infty(\mathbb{R}^d)} \|\mathrm{div}(\rho X_v)(s)\|_{L^p(\mathbb{R}^d)} \, ds.
$$

Consequently we can estimate

$$
\partial_{X_v} G = P'(\rho) \operatorname{div}(\rho X_v) - \rho P'(\rho) \operatorname{div} X_v
$$

by

$$
\| \partial_{X_v} G \|_{L^p(\mathbb{R}^d)} \le C_0 \left( \| \mathrm{div}(\rho X_v) \|_{L^p(\mathbb{R}^d)} + \| \mathrm{div} X_v \|_{L^p(\mathbb{R}^d)} \right).
$$

We combine (2.59), (2.63) and (2.64) together with (2.18) to get

$$
\mathcal{A}_3(t) \le \mathcal{A}_3(0) + C_0 \int_0^t \mathcal{A}_3(s) \left[ \frac{1}{\nu} + \|\nabla u(s)\|_{L^\infty(\mathbb{R}^d)} + \|\rho \dot{u}(s)\|_{L^p(\mathbb{R}^d)} \right] ds.
$$

Grönwall’s Lemma yields:

$$
(2.65) \quad \mathcal{A}_3(t) \le \mathcal{A}_3(0) \exp \left( C_0 \int_0^t \left[ 1 + \|\nabla u(s)\|_{L^\infty(\mathbb{R}^d)} + \|\rho \dot{u}(s)\|_{L^p(\mathbb{R}^d)} \right] ds \right).
$$

Step 3. Estimates for $\|\nabla u_G\|_{L^1_t L^\infty_x}$.

We consider $\nabla u_G$ first. We use **Proposition 2.5** to bound the second part $\nabla u_G = \frac{1}{\nu} \mathcal{R}\mathcal{R}G$:

$$
\|\nabla u_G(t)\|_{L^\infty(\mathbb{R}^d)} \le \frac{C}{\nu} \|G(t)\|_{L^2(\mathbb{R}^d)} + \frac{C}{\nu} \|G(t)\|_{L^\infty(\mathbb{R}^d)} \left[ 1 + \log \left( e + \frac{\|G(t)\|_{L^p_{\mathcal{X}(t)}(\mathbb{R}^d)}}{\|G(t)\|_{L^\infty(\mathbb{R}^d)}} \right) \right].
$$

Now we focus on the estimate for

$$
\|G(t)\|_{L^p_{\mathcal{X}(t)}(\mathbb{R}^d)} = \frac{1}{I(\mathcal{X}(t))} \left[ \|G(t)\|_{L^\infty(\mathbb{R}^d)} \|\mathcal{X}(t)\|_{L^\infty, p(\mathbb{R}^d)} + \sup_{1 \le v \le m} \|\partial_{X_v} G(t)\|_{L^p(\mathbb{R}^d)} \right] \le \frac{C_0}{I(\mathcal{X}(t))} \mathcal{A}_3(t).
$$

It is well-known that the denominator $I(\mathcal{X}(t))$ has a positive lower bound as follows, see e.g. [9, (4.3)]
(one has to make use of the flow associated to $u$):

$$
I(\mathcal{X}(t)) \ge I(\mathcal{X}_0) \exp \left( -C \int_0^t \|\nabla u(s)\|_{L^\infty(\mathbb{R}^d)} ds \right).
$$

We hence have

$$
(2.66) \quad \|\nabla u_G(t)\|_{L^\infty(\mathbb{R}^d)} \le C_0 \left( 1 + \frac{\mathcal{A}_3(0)}{I(\mathcal{X}_0)} + \int_0^t \left[ 1 + \|\rho \dot{u}\|_{L^p(\mathbb{R}^d)} + \|\nabla u\|_{L^\infty(\mathbb{R}^d)} \right] (s) ds \right).
$$
Case $d=2$. Similar as in the proof of Lemma 2.3, by using our interpolation inequality in Lemma A.1, we obtain:

$$ \int_0^t \|\rho \dot{u}\|_{L^p(\mathbb{R}^2)} \le C_0 \int_0^t \left[ \|\sqrt{\rho} \dot{u}\|_{L^2(\mathbb{R}^2)}^{\frac{2}{p}} \|\nabla \dot{u}\|_{L^2(\mathbb{R}^2)}^{\frac{1}{p'}-\frac{1}{p}} + E_0^{\frac{1}{p}(1-\frac{p'}{2})} \|\sqrt{\rho} \dot{u}\|_{L^2(\mathbb{R}^2)}^{\frac{p'}{p}} \|\nabla \dot{u}\|_{L^2(\mathbb{R}^2)}^{1-\frac{p'}{p}} \right], $$

where the first integral of the right hand side above is bounded as

$$ \begin{aligned} \int_0^t \|\sqrt{\rho} \dot{u}\|_{L^2(\mathbb{R}^2)}^{\frac{2}{p}} \|\nabla \dot{u}\|_{L^2(\mathbb{R}^2)}^{\frac{1}{p'}-\frac{1}{p}} &= \int_0^t \|\sqrt{\rho} \dot{u}\|_{L^2(\mathbb{R}^2)}^{\frac{2}{p}} \left[ \sigma \|\nabla \dot{u}\|_{L^2(\mathbb{R}^2)}^2 \right]^{\frac{1}{2}-\frac{1}{p}} \sigma^{\frac{1}{p}-\frac{1}{2}} \\ &\le \left[ \int_0^t \|\sqrt{\rho} \dot{u}\|_{L^2(\mathbb{R}^2)}^2 \right]^{\frac{1}{p}} \left[ \int_0^t \sigma \|\nabla \dot{u}\|_{L^2(\mathbb{R}^2)}^2 \right]^{\frac{1}{2}-\frac{1}{p}} \left[ \int_0^t \sigma^{\frac{2}{p}-1} \right]^{\frac{1}{2}} \le C(p)(1+\sqrt{t}) A_1(t)^{\frac{1}{p}} A_2(t)^{\frac{1}{2}-\frac{1}{p}}, \end{aligned} $$

and the second integral of the right hand side above is bounded as

$$ \begin{aligned} \int_0^t \|\sqrt{\rho} \dot{u}\|_{L^2(\mathbb{R}^2)}^{\frac{p'}{p}} \|\nabla \dot{u}\|_{L^2(\mathbb{R}^2)}^{1-\frac{p'}{p}} &= \int_0^t \left[ \|\sqrt{\rho} \dot{u}\|_{L^2(\mathbb{R}^2)}^2 \right]^{\frac{p'}{2p}} \left[ \sigma \|\nabla \dot{u}\|_{L^2(\mathbb{R}^2)}^2 \right]^{\frac{1}{2}-\frac{p'}{2p}} \sigma^{\frac{p'}{2p}-\frac{1}{2}} \\ &\le \left[ \int_0^t \|\sqrt{\rho} \dot{u}\|_{L^2(\mathbb{R}^2)}^2 \right]^{\frac{p'}{2p}} \left[ \int_0^t \sigma \|\nabla \dot{u}\|_{L^2(\mathbb{R}^2)}^2 \right]^{\frac{1}{2}-\frac{p'}{2p}} \left[ \int_0^t \sigma^{\frac{p'}{p}-1} \right]^{\frac{1}{2}} \le C(p)(1+\sqrt{t}) A_1(t)^{\frac{p'}{2p}} A_2(t)^{\frac{1}{2}-\frac{p'}{2p}}. \end{aligned} $$

In sum, for all $2 < p < \infty$:

$$ (2.67) \qquad \int_0^t \|\rho \dot{u}\|_{L^p(\mathbb{R}^2)} \le C_0(1+\sqrt{t}). $$

Now following the computations leading to (2.53), it is straightforward to obtain the following

$$ \begin{aligned} (2.68) \quad \int_0^t \|\nabla \tilde{u}\|_{L^\infty(\mathbb{R}^2)} ds &\le \frac{1}{\nu} \int_0^t \|\mathcal{R}\mathcal{R}(-\Delta)^{-1} \operatorname{div}(\rho \dot{u})\|_{L^\infty(\mathbb{R}^2)} + \frac{1}{\mu} \int_0^t \|\mathcal{R}\mathcal{R}(-\Delta)^{-1} \operatorname{curl}_{jk}(\rho \dot{u})\|_{L^\infty(\mathbb{R}^2)} \\ &\le C^* \left(\frac{1}{\nu^{5/6}} + 1\right) (1 + t^{\frac{2}{3}}) \left(1 + E_0^{\frac{1}{18}}\right) \left(A_1(t)^{\frac{1}{2}} + A_2(t)^{\frac{1}{2}}\right) \le C_0(1 + t^{\frac{2}{3}}). \end{aligned} $$

We plug (2.67) into (2.66), sum (2.66) and (2.68) up, and finally use Gronwall's inequality to get the estimate $(2.13)_2$ for $d=2$. The estimate $(2.13)_1$ for $A_3(t)$ follows correspondingly from (2.65).

Case $d=3$. Similarly as the proof of (2.67), for $3 < p < 6$ we interpolate the $L^p(\mathbb{R}^3)$ norm of $\sqrt{\rho} \dot{u}$ between $L^2(\mathbb{R}^3)$ and $L^6(\mathbb{R}^3)$, next we make use of the embedding $\dot{H}^1(\mathbb{R}^3) \hookrightarrow L^6(\mathbb{R}^3)$ to derive

$$ (2.69) \qquad \int_0^t \|\rho \dot{u}\|_{L^p(\mathbb{R}^3)} \le C_0 \int_0^t \|\sqrt{\rho} \dot{u}\|_{L^2(\mathbb{R}^3)}^{\frac{3}{p}-\frac{1}{2}} \left[ \sigma \|\nabla \dot{u}\|_{L^2(\mathbb{R}^3)}^2 \right]^{\frac{3}{4}-\frac{3}{2p}} \sigma^{\frac{3}{2p}-\frac{3}{4}} \le C_0(1+\sqrt{t}). $$

Owing to interpolation inequality and Sobolev embedding, we have

$$ (2.70) \qquad \int_0^t \|\nabla \tilde{u}\|_{L^\infty(\mathbb{R}^3)} \le \int_0^t \|\rho \dot{u}\|_{L^2(\mathbb{R}^3)}^{\frac{1}{2}} \|\rho \dot{u}\|_{L^6(\mathbb{R}^3)}^{\frac{1}{2}} \le C_0(1+\sqrt{t}). $$

Similar as above, we plug (2.69) into (2.66), sum (2.66) and (2.70) up, and finally use Gronwall's inequality to get the estimate $(2.13)_2$ for $d=3$. The estimate $(2.13)_1$ for $A_3(t)$ follows from (2.65) in dimension three. This ends the proof of Lemma 2.6. $\square$

**2.3. Proof of Theorem 1.6 and Corollary 1.10.** This section is devoted to the final step in the proof of the main result, Theorem 1.6. We recall that we are considering the Cauchy problem associated with equations (1.1) and with initial data (1.2) verifying (1.3) and (1.4).

Usually the sequence of initial data $(\rho_0^\delta, u_0^\delta)$ is obtained by mollifying $(\rho_0, u_0)$ with a smooth kernel. This regularization procedure has the unfortunate effect of destroying the density's structure. As observed in [29, 49], the most effective approach is to construct the approximate solutions in a class that is very close to the limit. From this point of view, the local result obtained by Danchin, Fanelli, Paicu in [10] should be appropriate. However, the argument of the maximum regularity of the heat equations requires
**Step 4. Final estimates.** In the following we continue with the estimates for $||\rho\dot{u}||_{L^1_t L^p(\mathbb{R}^d)}$ and $||\nabla \tilde{u}||_{L^1_t L^\infty(\mathbb{R}^d)}$, taking the dimension into account.

Case $d=2$. Similar as in the proof of Lemma 2.3, by using our interpolation inequality in Lemma A.1, we obtain:

$$ \int_0^t ||\rho\dot{u}||_{L^p(\mathbb{R}^2)} \le C_0 \int_0^t \left[ ||\sqrt{\rho}\dot{u}||_{L^2(\mathbb{R}^2)}^{\frac{2}{p}} ||\nabla\dot{u}||_{L^2(\mathbb{R}^2)}^{\frac{1}{p'}-\frac{1}{p}} + E_0^{\frac{1}{p}(1-\frac{p'}{2})} ||\sqrt{\rho}\dot{u}||_{L^2(\mathbb{R}^2)}^{\frac{p'}{p}} ||\nabla\dot{u}||_{L^2(\mathbb{R}^2)}^{1-\frac{p'}{p}} \right], $$

where the first integral of the right hand side above is bounded as

$$ \begin{aligned} \int_0^t ||\sqrt{\rho}\dot{u}||_{L^2(\mathbb{R}^2)}^{\frac{2}{p}} ||\nabla\dot{u}||_{L^2(\mathbb{R}^2)}^{\frac{1}{p'}-\frac{1}{p}} &= \int_0^t ||\sqrt{\rho}\dot{u}||_{L^2(\mathbb{R}^2)}^{\frac{2}{p}} \left[ \sigma ||\nabla\dot{u}||_{L^2(\mathbb{R}^2)}^2 \right]^{\frac{1}{2}-\frac{1}{p}} \sigma^{\frac{1}{p}-\frac{1}{2}} \\ &\le \left[ \int_0^t ||\sqrt{\rho}\dot{u}||_{L^2(\mathbb{R}^2)}^2 \right]^{\frac{1}{p}} \left[ \int_0^t \sigma ||\nabla\dot{u}||_{L^2(\mathbb{R}^2)}^2 \right]^{\frac{1}{2}-\frac{1}{p}} \left[ \int_0^t \sigma^{\frac{2}{p}-1} \right]^{\frac{1}{2}} \le C(p)(1+\sqrt{t}) A_1(t)^{\frac{1}{p}} A_2(t)^{\frac{1}{2}-\frac{1}{p}}, \end{aligned} $$

and the second integral of the right hand side above is bounded as

$$ \begin{aligned} \int_0^t ||\sqrt{\rho}\dot{u}||_{L^2(\mathbb{R}^2)}^{\frac{p'}{p}} ||\nabla\dot{u}||_{L^2(\mathbb{R}^2)}^{1-\frac{p'}{p}} &= \int_0^t \left[ ||\sqrt{\rho}\dot{u}||_{L^2(\mathbb{R}^2)}^2 \right]^{\frac{p'}{2p}} \left[ \sigma ||\nabla\dot{u}||_{L^2(\mathbb{R}^2)}^2 \right]^{\frac{1}{2}-\frac{p'}{2p}} \sigma^{\frac{p'}{2p}-\frac{1}{2}} \\ &\le \left[ \int_0^t ||\sqrt{\rho}\dot{u}||_{L^2(\mathbb{R}^2)}^2 \right]^{\frac{p'}{2p}} \left[ \int_0^t \sigma ||\nabla\dot{u}||_{L^2(\mathbb{R}^2)}^2 \right]^{\frac{1}{2}-\frac{p'}{2p}} \left[ \int_0^t \sigma^{\frac{p'}{p}-1} \right]^{\frac{1}{2}} \le C(p)(1+\sqrt{t}) A_1(t)^{\frac{p'}{2p}} A_2(t)^{\frac{1}{2}-\frac{p'}{2p}}. \end{aligned} $$

In sum, for all $2 < p < \infty$:

$$ (2.67) \qquad \int_0^t ||\rho\dot{u}||_{L^p(\mathbb{R}^2)} \le C_0(1+\sqrt{t}). $$

Now following the computations leading to (2.53), it is straightforward to obtain the following

$$ \begin{aligned} (2.68) \quad &\int_0^t ||\nabla \tilde{u}||_{L^\infty(\mathbb{R}^2)} ds \le \frac{1}{\nu} \int_0^t ||\mathcal{R}\mathcal{R}(-\Delta)^{-1} \operatorname{div}(\rho\dot{u})||_{L^\infty(\mathbb{R}^2)} + \frac{1}{\mu} \int_0^t ||\mathcal{R}\mathcal{R}(-\Delta)^{-1} \operatorname{curl}_{jk}(\rho\dot{u})||_{L^\infty(\mathbb{R}^2)} \\ &\le C^* \left( \frac{1}{\nu^{5/6}} + 1 \right) (1 + t^{\frac{2}{3}}) \left( 1 + E_0^{\frac{1}{18}} \right) \left( A_1(t)^{\frac{1}{2}} + A_2(t)^{\frac{1}{2}} \right) \le C_0(1 + t^{\frac{2}{3}}). \end{aligned} $$

We plug (2.67) into (2.66), sum (2.66) and (2.68) up, and finally use Grönwall's inequality to get the estimate $(2.13)_2$ for $d=2$. The estimate $(2.13)_1$ for $A_3(t)$ follows correspondingly from (2.65).

Case $d=3$. Similarly as the proof of (2.67), for $3 < p < 6$ we interpolate the $L^p(\mathbb{R}^3)$ norm of $\sqrt{\rho}\dot{u}$ between $L^2(\mathbb{R}^3)$ and $L^6(\mathbb{R}^3)$, next we make use of the embedding $\dot{H}^1(\mathbb{R}^3) \hookrightarrow L^6(\mathbb{R}^3)$ to derive

$$ (2.69) \qquad \int_0^t ||\rho\dot{u}||_{L^p(\mathbb{R}^3)} \le C_0 \int_0^t ||\sqrt{\rho}\dot{u}||_{L^2(\mathbb{R}^3)}^{\frac{3}{p}-\frac{1}{2}} \left[ \sigma ||\nabla\dot{u}||_{L^2(\mathbb{R}^3)}^2 \right]^{\frac{3}{4}-\frac{3}{2p}} \sigma^{\frac{3}{2p}-\frac{3}{4}} \le C_0(1+\sqrt{t}). $$

Owing to interpolation inequality and Sobolev embedding, we have

$$ (2.70) \qquad \int_0^t ||\nabla \tilde{u}||_{L^\infty(\mathbb{R}^3)} \le \int_0^t ||\rho\dot{u}||_{L^2(\mathbb{R}^3)}^{\frac{1}{2}} ||\rho\dot{u}||_{L^6(\mathbb{R}^3)}^{\frac{1}{2}} \le C_0(1+\sqrt{t}). $$

Similar as above, we plug (2.69) into (2.66), sum (2.66) and (2.70) up, and finally use Grönwall's inequality to get the estimate $(2.13)_2$ for $d=3$. The estimate $(2.13)_1$ for $A_3(t)$ follows from (2.65) in dimension three. This ends the proof of Lemma 2.6. $\square$

**2.3. Proof of Theorem 1.6 and Corollary 1.10.** This section is devoted to the final step in the proof of the main result, Theorem 1.6. We recall that we are considering the Cauchy problem associated with equations (1.1) and with initial data (1.2) verifying (1.3) and (1.4).

Usually the sequence of initial data $(\rho_0^\delta, u_0^\delta)$ is obtained by mollifying $(\rho_0, u_0)$ with a smooth kernel. This regularization procedure has the unfortunate effect of destroying the density's structure. As observed in [29, 50], the most effective approach is to construct the approximate solutions in a class that is very
the density to be a small perturbation of a constant state, even for the local solution. We are therefore
led to prove the local well-posedness of equations (1.1) stated in **Theorem 1.4** in **Appendix B**.

*Proof of Theorem 1.6.* In order to apply the local-in-time well-posedness results in **Theorem 1.4**, we consider a sequence of initial densities $(\rho_0^\delta)_\delta$ verifying: for all $0 < \delta < 1$:

$$
(2.71) \qquad \rho_0^\delta \ge \delta, \quad \rho_0^\delta - \tilde{\rho} \in L^2(\mathbb{R}^d) \cap L^\infty(\mathbb{R}^d) \cap \mathbb{L}_{\chi_0}^p(\mathbb{R}^d)
$$

such that

$$
\rho_0^\delta - \tilde{\rho} \xrightarrow{\delta \to 0} \rho_0 - \tilde{\rho} \quad \text{in } L^2(\mathbb{R}^d).
$$

The construction of the sequence of initial velocities $(u_0^\delta)_\delta \subset H^1(\mathbb{R}^d)$, converging strongly to $u_0$ in $H^1(\mathbb{R}^d)$ and satisfying the compatibility condition

$$
(2.72) \qquad \operatorname{div}\{2\mu\mathbb{D}u_0^\delta + (\lambda \operatorname{div} u_0^\delta - P(\rho_0^\delta) + \tilde{P})\} \in L^2(\mathbb{R}^d)
$$

can be found in [49, Section 3.5]. Now we can apply **Theorem 1.4** to get the existence of a unique solution $(\rho^\delta, u^\delta)$ that satisfies:

$$
(2.73) \qquad \begin{cases} \partial_t \rho^\delta + \operatorname{div}(\rho^\delta u^\delta) = 0, \\ \partial_t (\rho^\delta u^\delta) + \operatorname{div}(\rho^\delta u^\delta \otimes u^\delta) + \nabla P(\rho^\delta) = \mu \Delta u^\delta + (\mu + \lambda) \nabla \operatorname{div} u^\delta \end{cases}
$$

with initial data

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

The solution is defined up to a maximal time $T^\delta$ and enjoys the regularities outlined in **Theorem 1.4** which are sufficient for the computations performed in the preceding sections to be meaningful, leading to **Lemma 2.4** and **Lemma 2.6**. In particular, all the conditions outlined in the blow-up criterion (1.9) are satisfied, implying that $T^\delta = +\infty$. Finally, employing classical arguments involving compact embedding, Aubin-Lions Lemma and leveraging the regularity of the effective flux, one can establish the strong convergence of a subsequence of $(\rho^\delta, u^\delta)$ to $(\rho, u)$ satisfying the regularity in **Theorem 1.6**. Furthermore, given the improved-in-time Lipschitz bound of the velocity field in **Remark 2.7**, a change of variables into Lagrangian coordinates ensures the uniqueness of such a solution. We refer for example to [10] for the computations.

□

*Proof of Corollary 1.10.* At this level, we obtain a sequence $(\rho^{(\nu)}, u^{(\nu)})_{\nu \ge \underline{\nu}}$ satisfying:

$$
(2.74) \qquad \left\{
\begin{array}{l}
\displaystyle \partial_t \rho^{(\nu)} + \operatorname{div}(\rho^{(\nu)} u^{(\nu)}) = 0, \\
\displaystyle \partial_t (\rho^{(\nu)} u^{(\nu)}) + \operatorname{div}(\rho^{(\nu)} u^{(\nu)} \otimes u^{(\nu)}) - \nabla F^{(\nu)} - \mu \Delta u^{(\nu)} = -\frac{\mu}{\nu} \nabla F^{(\nu)} - \frac{\mu}{\nu} \nabla P(\rho^{(\nu)}),
\end{array}
\right.
$$

with initial data (1.2) verifying (1.3) and (1.4) and $\operatorname{div} u_0 = 0$. Above the effective flux $F^{(\nu)}$ solves the
following elliptic equation:

$$
\Delta F^{(\nu)} = \operatorname{div}(\rho^{(\nu)} \dot{u}^{(\nu)}).
$$

Given that $(\rho^{(\nu)})_\nu$ is bounded in $L^\infty((0, \infty) \times \mathbb{R}^d)$ and $(\sqrt{\rho^{(\nu)}} \dot{u}^{(\nu)})$ is bounded in $L^2((0, \infty) \times \mathbb{R}^d)$ it follows that $(-\nabla F^{(\nu)})_\nu$ is bounded in $L^2((0, \infty) \times \mathbb{R}^d)$, resulting in weak convergence to some $\nabla \Pi \in L^2((0, \infty) \times \mathbb{R}^d)$. Obviously, the right hand side of (2.74)$_2$ converges strongly to zero in $L^\infty((0, \infty), \dot{H}^{-1}(\mathbb{R}^d))$, given that $(\nu^{-1/2}F^{(\nu)})$ is bounded in $L^\infty((0, \infty), L^2(\mathbb{R}^d))$ and $(P(\rho^{(\nu)}) - \tilde{P})_\nu$ is bounded in $L^\infty((0, \infty), L^2(\mathbb{R}^d))$.

The regularity of the sequence $(u^{(\nu)})$ ensures that, up to a subsequence, $(u^{(\nu)})$ converges strongly in $L^2_{\text{loc}}((0, \infty) \times \mathbb{R}^d)$ to some $v \in L^\infty((0, T), H^1(\mathbb{R}^d))$. Furthermore, since the sequence $(\rho^{(\nu)})_\nu$ is bounded in $L^\infty((0, \infty) \times \mathbb{R}^d)$, it converges weakly-$^*$ to some $\varrho \in L^\infty((0, \infty) \times \mathbb{R}^d)$. Additionally, $(\operatorname{div} u^{(\nu)})_\nu$ converges strongly to zero in $L^\infty((0, \infty), L^2(\mathbb{R}^d))$, since from the bound of functional $\mathcal{A}_1$, the sequence $(\nu \|\operatorname{div} u^{(\nu)}\|_{L^2(\mathbb{R}^d)}^2)_\nu$ is bounded. These convergences, along with the Aubin-Lions Lemma, are sufficient to pass to the limit in (2.74) and establish that $(\varrho, v)$ solves the incompressible model (1.24). The uniqueness result for (1.24) in [44] and the uniform bounds in **Theorem 1.6** implies the convergence of
close to the limit. From this point of view, the local result obtained by Danchin, Fanelli, Paicu in [10] should be appropriate. However, the argument of the maximum regularity of the heat equation requires the density to be a small perturbation of a constant state, even for the local solution. We are therefore led to prove the local well-posedness of equations (1.1) stated in Theorem 1.4 in Appendix B.

**Proof of Theorem 1.6.** In order to apply the local-in-time well-posedness results in Theorem 1.4, we consider a sequence of initial densities $(\rho_0^\delta)_\delta$ verifying: for all $0 < \delta < 1$:

$$ (2.71) \qquad \rho_0^\delta \ge \delta, \quad \rho_0^\delta - \tilde{\rho} \in L^2(\mathbb{R}^d) \cap L^\infty(\mathbb{R}^d) \cap L_{\chi_0}^p(\mathbb{R}^d) $$

such that

$$ \rho_0^\delta - \tilde{\rho} \xrightarrow{\delta \to 0} \rho_0 - \tilde{\rho} \quad \text{in } L^2(\mathbb{R}^d). $$

The construction of the sequence of initial velocities $(u_0^\delta)_\delta \subset H^1(\mathbb{R}^d)$, converging strongly to $u_0$ in $H^1(\mathbb{R}^d)$ and satisfying the compatibility condition

$$ (2.72) \qquad \mathrm{div}\{2\mu\mathbb{D}u_0^\delta + (\lambda \mathrm{div} \, u_0^\delta - P(\rho_0^\delta) + \tilde{P})\} \in L^2(\mathbb{R}^d) $$

can be found in [50, Section 3.5]. Now we can apply Theorem 1.4 to get the existence of a unique solution $(\rho^\delta, u^\delta)$ that satisfies:

$$ (2.73) \qquad \left\{ \begin{array}{l} \partial_t \rho^\delta + \mathrm{div}(\rho^\delta u^\delta) = 0, \\ \partial_t (\rho^\delta u^\delta) + \mathrm{div}(\rho^\delta u^\delta \otimes u^\delta) + \nabla P(\rho^\delta) = \mu \Delta u^\delta + (\mu + \lambda) \nabla \mathrm{div} \, u^\delta \end{array} \right. $$

with initial data

$$ (\rho^\delta)|_{t=0} = \rho_0^\delta \quad \text{and} \quad (u^\delta)|_{t=0} = u_0^\delta. $$

The solution is defined up to a maximal time $T^\delta$ and enjoys the regularities outlined in Theorem 1.4 which are sufficient for the computations performed in the preceding sections to be meaningful, leading to Lemma 2.4 and Lemma 2.6. In particular, all the conditions outlined in the blow-up criterion (1.9) are satisfied, implying that $T^\delta = +\infty$. Finally, employing classical arguments involving compact embedding, Aubin-Lions Lemma and leveraging the regularity of the effective flux, one can establish the strong convergence of a subsequence of $(\rho^\delta, u^\delta)$ to $(\rho, u)$ satisfying the regularity in Theorem 1.6. Furthermore, given the improved-in-time Lipschitz bound of the velocity field in Remark 2.7, a change of variables into Lagrangian coordinates ensures the uniqueness of such a solution. We refer for example to [10] for the computations.

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

**Proof of Corollary 1.10.** At this level, we obtain a sequence $(\rho^{(\nu)}, u^{(\nu)})_{\nu \ge \nu}$ satisfying:

$$ (2.74) \qquad \left\{ \begin{array}{l} \partial_t \rho^{(\nu)} + \mathrm{div}(\rho^{(\nu)} u^{(\nu)}) = 0, \\ \partial_t (\rho^{(\nu)} u^{(\nu)}) + \mathrm{div}(\rho^{(\nu)} u^{(\nu)} \otimes u^{(\nu)}) - \nabla F^{(\nu)} - \mu \Delta u^{(\nu)} = -\frac{\mu}{\nu} \nabla F^{(\nu)} - \frac{\mu}{\nu} \nabla P(\rho^{(\nu)}), \end{array} \right. $$

with initial data (1.2) verifying (1.3) and (1.4) and $\mathrm{div} \, u_0 = 0$. Above the effective flux $F^{(\nu)}$ solves the following elliptic equation:

$$ \Delta F^{(\nu)} = \mathrm{div}(\rho^{(\nu)} \dot{u}^{(\nu)}). $$

Given that $(\rho^{(\nu)})_\nu$ is bounded in $L^\infty((0, \infty) \times \mathbb{R}^d)$ and $(\sqrt{\rho^{(\nu)}} \dot{u}^{(\nu)})$ is bounded in $L^2((0, \infty) \times \mathbb{R}^d)$ it follows that $(-\nabla F^{(\nu)})_\nu$ is bounded in $L^2((0, \infty) \times \mathbb{R}^d)$, resulting in weak convergence to some $\nabla \Pi \in L^2((0, \infty) \times \mathbb{R}^d)$. Obviously, the right hand side of (2.74)$_2$ converges strongly to zero in $L^\infty((0, \infty), \dot{H}^{-1}(\mathbb{R}^d))$, given that $(\nu^{-1/2} F^{(\nu)})$ is bounded in $L^\infty((0, \infty), L^2(\mathbb{R}^d))$ and $(P(\rho^{(\nu)}) - \tilde{P})_\nu$ is bounded in $L^\infty((0, \infty), L^2(\mathbb{R}^d))$.

The regularity of the sequence $(u^{(\nu)})$ ensures that, up to a subsequence, $(u^{(\nu)})$ converges strongly in $L^2_{\mathrm{loc}}((0, \infty) \times \mathbb{R}^d)$ to some $v \in L^\infty((0, T), H^1(\mathbb{R}^d))$. Furthermore, since the sequence $(\rho^{(\nu)})_\nu$ is bounded in $L^\infty((0, \infty) \times \mathbb{R}^d)$, it converges weakly-* to some $\varrho \in L^\infty((0, \infty) \times \mathbb{R}^d)$. Additionally, $(\mathrm{div} \, u^{(\nu)})_\nu$ converges strongly to zero in $L^\infty((0, \infty), L^2(\mathbb{R}^d))$, since from the bound of functional $A_1$, the sequence $(\nu \|\mathrm{div} \, u^{(\nu)}\|_{L^2(\mathbb{R}^d)}^2)_\nu$ is bounded. These convergences, along with the Aubin-Lions Lemma, are sufficient to pass to the limit in (2.74) and establish that $(\varrho, v)$ solves the incompressible model (1.25). The
the whole sequence $(\rho^{(\nu)}, u^{(\nu)})_\nu$ (instead of some subsequence). This completes the proof of Corollary 1.10. □

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

## ACKNOWLEDGMENT

X. Liao is partially supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)- Project-ID 258734477-SFB 1173. S. M. Zodji is funded by the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 945332. This work is partially supported by the project CRISIS (ANR-20-CE40-0020-01), operated by the French National Research Agency (ANR).

This work is partially finished during S.M. Zodji's research visit at the Karlsruhe Institute of Technology (KIT), and he acknowledges the hospitality of KIT. The authors would like to thank Raphaël Danchin, Francesco Fanelli and Cosmin Burtea and David Gérard-Varet (the two PhD supervisors of S.M. Zodji) for their fruitful discussions and valuable comments.

## APPENDIX A. INTERPOLATION INEQUALITY

**Lemma A.1 (Density-weighted interpolation inequality).** Let $v \in \dot{H}^1(\mathbb{R}^2)$, $\rho \ge 0$ such that $\sqrt{\rho v} \in L^2(\mathbb{R}^2)$ and $\rho - \tilde{\rho} \in L^p(\mathbb{R}^2)$ for some $1 < p < \infty$, with $\tilde{\rho} > 0$. Then $v \in L^2(\mathbb{R}^2)$ and there exists a constant $C > 0$ depending only on $\tilde{\rho}$ and $p$ such that the following estimate holds true:

$$ (A.1) \qquad \|v\|_{L^2(\mathbb{R}^2)} \le C \left( \|\rho - \tilde{\rho}\|_{L^p(\mathbb{R}^2)}^{\frac{p}{2}} \|\nabla v\|_{L^2(\mathbb{R}^2)} + \|\sqrt{\rho v}\|_{L^2(\mathbb{R}^2)} \right). $$

Moreover, for all $2 < q < \infty$, we have:

$$ (A.2) \qquad \|\rho^{\frac{q'}{2q}} v\|_{L^q(\mathbb{R}^2)} \le C \left( \|\sqrt{\rho v}\|_{L^2(\mathbb{R}^2)}^{\frac{2}{q}} \|\nabla v\|_{L^2(\mathbb{R}^2)}^{\frac{1}{q'} - \frac{1}{q}} + \|\rho - \tilde{\rho}\|_{L^p(\mathbb{R}^2)}^{\frac{p}{q}(1-\frac{q'}{2})} \|\sqrt{\rho v}\|_{L^2(\mathbb{R}^2)}^{\frac{q'}{q}} \|\nabla v\|_{L^2(\mathbb{R}^2)}^{1-\frac{q'}{q}} \right). $$

*Proof.* In the first step of the proof, we emulate the approach in [44, Proposition A.6] by expressing:

$$ (A.3) \qquad \tilde{\rho}|v|^2 = (\tilde{\rho} - \rho)|v|^2 + \rho|v|^2. $$

Due to the assumption that $\sqrt{\rho v} \in L^2(\mathbb{R}^2)$, we only need to compute the integral of the first term of the right hand above. With the help of interpolation, Hölder and Young inequalities, we have:

$$ \begin{aligned} \int_{\mathbb{R}^2} (\tilde{\rho} - \rho) |v|^2 &\le \|\rho - \tilde{\rho}\|_{L^p(\mathbb{R}^2)} \|v\|_{L^{2p'}(\mathbb{R}^2)}^2 \\ &\le C_p \|\rho - \tilde{\rho}\|_{L^p(\mathbb{R}^2)} \|v\|_{L^2(\mathbb{R}^2)}^{\frac{2}{p'}} \|\nabla v\|_{L^2(\mathbb{R}^2)}^{\frac{2}{p}} \\ &\le \frac{1}{2} \tilde{\rho} \|v\|_{L^2(\mathbb{R}^2)}^2 + C_{p, \tilde{\rho}} \|\rho - \tilde{\rho}\|_{L^p(\mathbb{R}^2)}^p \|\nabla v\|_{L^2(\mathbb{R}^2)}^2, \end{aligned} $$

and (A.1) just follows. Next, for all $2 < q < \infty$, Hölder, Gagliardo Nirenberg inequalities yield:

$$ \begin{aligned} \|\rho^{\frac{q'}{2q}} v\|_{L^q(\mathbb{R}^2)} &\le \|\sqrt{\rho v}\|_{L^2(\mathbb{R}^2)}^{\frac{q'}{q}} \|v\|_{L^{2q}(\mathbb{R}^2)}^{1-\frac{q'}{q}} \\ &\le C \|\sqrt{\rho v}\|_{L^2(\mathbb{R}^2)}^{\frac{q'}{q}} \|\nabla v\|_{L^2(\mathbb{R}^2)}^{\frac{1}{q'} - \frac{1}{q}} \|v\|_{L^2(\mathbb{R}^2)}^{\frac{1}{q} - \frac{q'}{q^2}} \\ &\le C \|\sqrt{\rho v}\|_{L^2(\mathbb{R}^2)}^{\frac{q'}{q}} \|\nabla v\|_{L^2(\mathbb{R}^2)}^{\frac{1}{q'} - \frac{1}{q}} \|v\|_{L^2(\mathbb{R}^2)}^{\frac{2}{q} - \frac{q'}{q}}. \end{aligned} $$

Hence (A.2) holds true while replacing the $L^2(\mathbb{R}^2)$ norm of the velocity by (A.1). □

<figure><img src="image_5.png" /></figure>
uniqueness result for (1.25) in [44] and the uniform bounds in Theorem 1.6 implies the convergence of the whole sequence $(\rho^{(\nu)}, u^{(\nu)})_\nu$ (instead of some subsequence). This completes the proof of Corollary 1.10. □

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

ACKNOWLEDGMENT

X. Liao is partially supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)- Project-ID 258734477-SFB 1173. S. M. Zodji is funded by the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 945332. This work is partially supported by the project CRISIS (ANR-20-CE40-0020-01), operated by the French National Research Agency (ANR).

This work is partially finished during S.M. Zodji's research visit at the Karlsruhe Institute of Technology (KIT), and he acknowledges the hospitality of KIT. The authors would like to thank Raphaël Danchin, Francesco Fanelli and Cosmin Burtea and David Gérard-Varet (the two PhD supervisors of S.M. Zodji) for their fruitful discussions and valuable comments.

APPENDIX A. INTERPOLATION INEQUALITY

**Lemma A.1** (Density-weighted interpolation inequality). Let $v \in \dot{H}^1(\mathbb{R}^2)$, $\rho \ge 0$ such that $\sqrt{\rho v} \in L^2(\mathbb{R}^2)$ and $\rho - \tilde{\rho} \in L^p(\mathbb{R}^2)$ for some $1 < p < \infty$, with $\tilde{\rho} > 0$. Then $v \in L^2(\mathbb{R}^2)$ and there exists a constant $C > 0$ depending only on $\tilde{\rho}$ and $p$ such that the following estimate holds true:

$$ (A.1) \qquad \|v\|_{L^2(\mathbb{R}^2)} \le C \left( \|\rho - \tilde{\rho}\|_{L^p(\mathbb{R}^2)}^{\frac{p}{2}} \|\nabla v\|_{L^2(\mathbb{R}^2)} + \|\sqrt{\rho v}\|_{L^2(\mathbb{R}^2)} \right). $$

Moreover, for all $2 < q < \infty$, we have:

$$ (A.2) \qquad \|\rho^{\frac{q'}{2q}} v\|_{L^q(\mathbb{R}^2)} \le C \left( \|\sqrt{\rho v}\|_{L^2(\mathbb{R}^2)}^{\frac{2}{q}} \|\nabla v\|_{L^2(\mathbb{R}^2)}^{\frac{1}{q'} - \frac{1}{q}} + \|\rho - \tilde{\rho}\|_{L^p(\mathbb{R}^2)}^{\frac{p}{q}(1-\frac{q'}{2})} \|\sqrt{\rho v}\|_{L^2(\mathbb{R}^2)}^{\frac{q'}{q}} \|\nabla v\|_{L^2(\mathbb{R}^2)}^{1-\frac{q'}{q}} \right). $$

*Proof.* In the first step of the proof, we emulate the approach in [44, Proposition A.6] by expressing:

$$ (A.3) \qquad \tilde{\rho}|v|^2 = (\tilde{\rho} - \rho)|v|^2 + \rho|v|^2. $$

Due to the assumption that $\sqrt{\rho v} \in L^2(\mathbb{R}^2)$, we only need to compute the integral of the first term of the right hand above. With the help of interpolation, Hölder and Young inequalities, we have:

$$ \begin{aligned} \int_{\mathbb{R}^2} (\tilde{\rho} - \rho) |v|^2 &\le \|\rho - \tilde{\rho}\|_{L^p(\mathbb{R}^2)} \|v\|_{L^{2p'}(\mathbb{R}^2)}^2 \\ &\le C_p \|\rho - \tilde{\rho}\|_{L^p(\mathbb{R}^2)} \|v\|_{L^{2q}(\mathbb{R}^2)}^{\frac{2}{p'}} \|\nabla v\|_{L^2(\mathbb{R}^2)}^{\frac{2}{p}} \\ &\le \frac{1}{2} \tilde{\rho} \|v\|_{L^{2q}(\mathbb{R}^2)}^2 + C_{p, \tilde{\rho}} \|\rho - \tilde{\rho}\|_{L^p(\mathbb{R}^2)}^p \|\nabla v\|_{L^2(\mathbb{R}^2)}^2, \end{aligned} $$

and (A.1) just follows. Next, for all $2 < q < \infty$, Hölder, Gagliardo Nirenberg inequalities yield:

$$ \begin{aligned} \|\rho^{\frac{q'}{2q}} v\|_{L^q(\mathbb{R}^2)} &\le \|\sqrt{\rho v}\|_{L^2(\mathbb{R}^2)}^{\frac{q'}{q}} \|v\|_{L^{2q}(\mathbb{R}^2)}^{1-\frac{q'}{q}} \\ &\le C \|\sqrt{\rho v}\|_{L^2(\mathbb{R}^2)}^{\frac{q'}{q}} \|\nabla v\|_{L^2(\mathbb{R}^2)}^{\frac{1}{q'} - \frac{1}{q}} \|v\|_{L^2(\mathbb{R}^2)}^{\frac{1}{q} - \frac{q'}{q^2}} \\ &\le C \|\sqrt{\rho v}\|_{L^2(\mathbb{R}^2)}^{\frac{q'}{q}} \|\nabla v\|_{L^2(\mathbb{R}^2)}^{\frac{1}{q'} - \frac{1}{q}} \|v\|_{L^2(\mathbb{R}^2)}^{\frac{2}{q} - \frac{q'}{q}}. \end{aligned} $$

Hence (A.2) holds true while replacing the $L^2(\mathbb{R}^2)$ norm of the velocity by (A.1). □

<figure><img src="image_6.png" /></figure>
## APPENDIX B. LOCAL WELL-POSEDNESS

In this section, we prove the local well-posedness result in Theorem 1.4 of the Navier-Stokes equations for a compressible fluid with an initial density having tangential regularity. Our method relies on a change of variables into Lagrangian coordinates, followed by the study of the linearized system and the full nonlinear system, in a similar way as in [50]. In particular, we do not require the density to be a small perturbation around a constant state in $L^\infty(\mathbb{R}^d)$.

More precisely, we consider the Cauchy problem of the system (1.1):

$$ (B.1) \qquad \begin{cases} \partial_t \rho + \operatorname{div}(\rho u) = 0, \\ \partial_t (\rho u) + \operatorname{div}(\rho u \otimes u) + \nabla P(\rho) = \mu \Delta u + (\mu + \lambda) \nabla \operatorname{div} u, \end{cases} $$

equipped with initial data (1.2):

$$ (B.2) \qquad \rho|_{t=0} = \rho_0 \quad \text{and} \quad (\rho u)|_{t=0} = \rho_0 u_0, $$

verifying (1.3), (1.4) and (1.6):

$$ (B.3) \qquad \begin{cases} 0 < \underline{\rho} \le \rho_0(x), & \rho_0 - \tilde{\rho} \in L^2(\mathbb{R}^d) \cap L^\infty(\mathbb{R}^d) \cap \mathbb{L}_{\mathcal{X}_0}^p(\mathbb{R}^d), \\ u_0 \in H^1(\mathbb{R}^d), & \mu \Delta u_0 + (\mu + \lambda) \nabla \operatorname{div} u_0 - \nabla P(\rho_0) \in L^2(\mathbb{R}^d). \end{cases} $$

Above $\tilde{\rho} > 0$ is a constant and $\mathcal{X}_0 = (X_{0,v})_{1\le v \le m} \subset \mathbb{L}^{\infty,p}(\mathbb{R}^d)$, $d < p < \infty$, is a non-degenerate family of $m \in \mathbb{N}^*$ vectors fields, with $m \ge d - 1$.

Here, we present the main lines of the proof of Theorem 1.4; detailed computations can be found in [50], in the more involved case of density-dependent viscosity.

**Step 1.** **Lagrangian coordinates.** Let $0 < T \le \infty$, $u$ be a Lipschitz vector field such that $\nabla u \in L^1((0,T), L^\infty(\mathbb{R}^d))$ and let us consider its flow map $\mathcal{X}$ given by

$$ \mathcal{X}_{\tilde{u}}(t, y) = y + \int_{0}^{t} u(\tau, \mathcal{X}_{\tilde{u}}(\tau, y)) d\tau := y + \int_{0}^{t} \bar{u}(\tau, y) d\tau $$

where, hereafter, for all $g = g(t, x)$, we define $\bar{g} = g(t, y)$ by:

$$ \bar{g}(t, y) = g(t, \mathcal{X}(t, y)). $$

By performing this change of variables, the equations (B.1) reads:

$$ (B.4) \qquad \begin{cases} \partial_t (\bar{\rho} J_{\bar{u}}) = 0, \\ \rho_0 \partial_t \bar{u} = \operatorname{div} \left( \operatorname{Adj}(D\mathcal{X}_{\bar{u}}) \left\{ 2\mu \mathbb{D}_{A_{\bar{u}}} \bar{u} + (\lambda \operatorname{div}_{A_{\bar{u}}} \bar{u} - P(\bar{\rho}) + \tilde{P}) \right\} \right), \end{cases} $$

where

$$ J_{\bar{u}} = \det(D\mathcal{X}_{\bar{u}}), \quad A_{\bar{u}} = (D\mathcal{X}_{\bar{u}})^{-1}, \quad \operatorname{div}_{A_{\bar{u}}} w = A_{\bar{u}}^T: \nabla w = Dw: A_{\bar{u}}, \quad 2\mathbb{D}_{A_{\bar{u}}} w = Dw \cdot A_{\bar{u}} + A_{\bar{u}}^T \nabla w. $$

**Step 2.** **Well-posedness of the linearized system.** Motivated by (B.4), we are led to consider the linear system

$$ (B.5) \qquad \begin{cases} \rho_0 \partial_t v - \mu \Delta v - (\mu + \lambda) \nabla \operatorname{div} v = \operatorname{div} f, \\ v|_{t=0} = v_0, \end{cases} $$

where the source term $f$ and the initial datum $v_0$ belong to the following space:

$$ Y_T := \left\{ \begin{array}{l} (f, v_0) \in L^\infty((0,T), L^2(\mathbb{R}^d)) \times H^1(\mathbb{R}^d) : f, \partial_t f, \sigma \partial_{tt} f \in L^2((0,T) \times \mathbb{R}^d); \\ \sqrt{\sigma} \partial_t f \in L^\infty((0,T), L^2(\mathbb{R}^d)), \mu \Delta v_0 + (\mu + \lambda) \nabla \operatorname{div} v_0 + \operatorname{div} f|_{t=0} \in L^2(\mathbb{R}^d) \end{array} \right\}. $$

The solution of the linearized system (B.5) is constructed in the following space:

$$ Z_T := \left\{ v \in L^\infty((0,T), H^1(\mathbb{R}^d)) : \begin{array}{l} \nabla v, \partial_t v, \nabla \partial_t v, \sqrt{\sigma} \partial_{tt} v, \sigma \nabla \partial_{tt} v \in L^2((0,T) \times \mathbb{R}^d); \\ \partial_t v, \sqrt{\sigma} \nabla \partial_t v, \sigma \partial_{tt} v \in L^\infty((0,T), L^2(\mathbb{R}^d)) \end{array} \right\}. $$
## APPENDIX B. LOCAL WELL-POSEDNESS

In this section, we prove the local well-posedness result in Theorem 1.4 of the Navier-Stokes equations for a compressible fluid with an initial density having tangential regularity. Our method relies on a change of variables into Lagrangian coordinates, followed by the study of the linearized system and the full nonlinear system, in a similar way as in [49]. In particular, we do not require the density to be a small perturbation around a constant state in $L^\infty(\mathbb{R}^d)$.

More precisely, we consider the Cauchy problem of the system (1.1):

$$ (B.1) \qquad \left\{ \begin{array}{l} \partial_t \rho + \operatorname{div}(\rho u) = 0, \\ \partial_t (\rho u) + \operatorname{div}(\rho u \otimes u) + \nabla P(\rho) = \mu \Delta u + (\mu + \lambda) \nabla \operatorname{div} u, \end{array} \right. $$

equipped with initial data (1.2):

$$ (B.2) \qquad \rho|_{t=0} = \rho_0 \quad \text{and} \quad (\rho u)|_{t=0} = \rho_0 u_0, $$

verifying (1.3), (1.4) and (1.6):

$$ (B.3) \qquad \left\{ \begin{array}{l} 0 < \underline{\rho} \le \rho_0(x), \quad \rho_0 - \tilde{\rho} \in L^2(\mathbb{R}^d) \cap L^\infty(\mathbb{R}^d) \cap \mathbb{L}_{\mathcal{X}_0}^p(\mathbb{R}^d), \\ u_0 \in H^1(\mathbb{R}^d), \quad \mu \Delta u_0 + (\mu + \lambda) \nabla \operatorname{div} u_0 - \nabla P(\rho_0) \in L^2(\mathbb{R}^d). \end{array} \right. $$

Above $\tilde{\rho} > 0$ is a constant and $\mathcal{X}_0 = (X_{0,v})_{1\le v \le m} \subset \mathbb{L}^{\infty,p}(\mathbb{R}^d)$, $d < p < \infty$, is a non-degenerate family of $m \in \mathbb{N}^*$ vectors fields, with $m \ge d - 1$.

Here, we present the main lines of the proof of Theorem 1.4; detailed computations can be found in [49], in the more involved case of density-dependent viscosity.

**Step 1.** **Lagrangian coordinates.** Let $0 < T \le \infty$, $u$ be a Lipschitz vector field such that $\nabla u \in L^1((0,T), L^\infty(\mathbb{R}^d))$ and let us consider its flow map $\mathcal{X}$ given by

$$ \mathcal{X}_{\tilde{u}}(t, y) = y + \int_{0}^{t} u(\tau, \mathcal{X}_{\tilde{u}}(\tau, y)) d\tau := y + \int_{0}^{t} \bar{u}(\tau, y) d\tau $$

where, hereafter, for all $g = g(t, x)$, we define $\bar{g} = g(t, y)$ by:

$$ \bar{g}(t, y) = g(t, \mathcal{X}(t, y)). $$

By performing this change of variables, the equations (B.1) reads:

$$ (B.4) \qquad \left\{ \begin{array}{l} \partial_t (\bar{\rho} J_{\bar{u}}) = 0, \\ \rho_0 \partial_t \bar{u} = \operatorname{div} \left( \operatorname{Adj}(D\mathcal{X}_{\bar{u}}) \left\{ 2\mu \mathbb{D}_{A_{\bar{u}}} \bar{u} + (\lambda \operatorname{div}_{A_{\bar{u}}} \bar{u} - P(\bar{\rho}) + \tilde{P}) \right\} \right), \end{array} \right. $$

where

$$ J_{\bar{u}} = \det(D\mathcal{X}_{\bar{u}}), \quad A_{\bar{u}} = (D\mathcal{X}_{\bar{u}})^{-1}, \quad \operatorname{div}_{A_{\bar{u}}} w = A_{\bar{u}}^T: \nabla w = Dw: A_{\bar{u}}, \quad 2\mathbb{D}_{A_{\bar{u}}} w = Dw \cdot A_{\bar{u}} + A_{\bar{u}}^T \cdot \nabla w. $$

**Step 2.** **Well-posedness of the linearized system.** Motivated by (B.4), we are led to consider the linear system

$$ (B.5) \qquad \left\{ \begin{array}{l} \rho_0 \partial_t v - \mu \Delta v - (\mu + \lambda) \nabla \operatorname{div} v = \operatorname{div} f, \\ v|_{t=0} = v_0, \end{array} \right. $$

where the source term $f$ and the initial datum $v_0$ belong to the following space:

$$ Y_T := \left\{ \begin{array}{l} (f, v_0) \in L^\infty((0, T), L^2(\mathbb{R}^d)) \times H^1(\mathbb{R}^d): f, \partial_t f, \sigma \partial_{tt} f \in L^2((0, T) \times \mathbb{R}^d); \\ \sqrt{\sigma} \partial_t f \in L^\infty((0, T), L^2(\mathbb{R}^d)); \mu \Delta v_0 + (\mu + \lambda) \nabla \operatorname{div} v_0 + \operatorname{div} f|_{t=0} \in L^2(\mathbb{R}^d) \end{array} \right\}. $$

The solution of the linearized system (B.5) is constructed in the following space:

$$ Z_T := \left\{ v \in L^\infty((0, T), H^1(\mathbb{R}^d)) : \begin{array}{l} \nabla v, \partial_t v, \nabla \partial_t v, \sqrt{\sigma} \partial_{tt} v, \sigma \nabla \partial_{tt} v \in L^2((0, T) \times \mathbb{R}^d); \\ \partial_t v, \sqrt{\sigma} \nabla \partial_t v, \sigma \partial_{tt} v \in L^\infty((0, T), L^2(\mathbb{R}^d)) \end{array} \right\}. $$
It is straightforward to check that for $T < \infty$ every $v \in Z_T$ satisfies:

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

The well-posedness result for the linearized system (B.5) reads as follows.

**Proposition B.1.** Let $0 < T \le \infty$. For all $(f, v_0) \in Y_T$, there exists a unique solution $v \in Z_T$ of the Cauchy problem (B.5). Moreover, the following estimates hold true for $v$.

(1) Basic energy estimates:

$$(B.6) \quad \begin{aligned} & \sup_{[0,T]} \|(\sqrt{\rho_0}v, \sqrt{\rho_0}\partial_t v, \nabla v)\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|(\nabla v, \sqrt{\rho_0}\partial_t v, \nabla \partial_t v)\|_{L^2(\mathbb{R}^d)}^2 \\ & \lesssim \|(\sqrt{\rho_0}v_0, \sqrt{\rho_0}\partial_t v|_{t=0}, \nabla v_0)\|_{L^2(\mathbb{R}^d)}^2 + \sup_{[0,T]} \|f\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|(f, \partial_t f)\|_{L^2(\mathbb{R}^d)}^2. \end{aligned}$$

(2) Higher energy estimates:

$$(B.7) \quad \begin{aligned} & \sup_{[0,T]} \|(\sqrt{\sigma}\nabla\partial_t v, \sigma\sqrt{\rho_0}\partial_{tt}v)\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \|(\sqrt{\sigma\rho_0}\partial_{tt}v, \sigma\nabla(\partial_{tt}v))\|_{L^2(\mathbb{R}^d)}^2 \\ & \lesssim \int_0^T \|\nabla\partial_t v\|_{L^2(\mathbb{R}^d)}^2 + \sup_{[0,T]} \sigma \|\partial_t f\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \sigma^2 \|\partial_{tt} f\|_{L^2(\mathbb{R}^d)}^2. \end{aligned}$$

The constant appearing in the above estimates does not depend on the upper or lower bound of the density $\rho_0$.

The proof of Proposition B.1 is not part of the classical theory of parabolic systems due to the roughness of the density. However, it can be achieved by a regularization process followed by a compactness argument. We refer, for example, to [50, Theorem 3.1] for the derivation of estimates (B.6) and (B.7).

**Step 3. Further estimates of the linearized system.** By interpolating the estimates (B.6) and (B.7), we observe that the following estimates hold true for the velocity gradient and its time derivative.

**Corollary B.2.** The following estimates hold true.

(1) Assuming that $f \in L^r((0,T), L^p(\mathbb{R}^d))$ for $2 \le r \le \infty$ and $2 < p < \infty$ if $d = 2$ or $2 < p \le 6$ if $d = 3$, we have:

$$(B.8) \quad \begin{aligned} & \|\nabla v\|_{L^r((0,T), L^p(\mathbb{R}^d))}^2 \lesssim \|(f, v_0)\|_{Y_T}^2 + \|f\|_{L^r((0,T), L^p(\mathbb{R}^d))}^2. \end{aligned}$$

(2) For all $2 \le r \le \infty$ and $2 < p < \infty$ if $d = 2$ and $2 < p \le 6$ if $d = 3$ we have:

$$(B.9) \quad \begin{aligned} & \|\sigma^s \nabla \partial_t v\|_{L^r((0,T), L^p(\mathbb{R}^d))}^2 \lesssim \|(f, v_0)\|_{Y_T}^2 + \|\sigma^s \partial_t f\|_{L^r((0,T), L^p(\mathbb{R}^d))}^2 \end{aligned}$$

where

$$s = \begin{cases} 1 - \frac{1}{p} - \frac{1}{r} & \text{if } d = 2, \\ \frac{5}{4} - \frac{3}{2p} - \frac{1}{r} & \text{if } d = 3. \end{cases}$$

For $d = 3$, the same estimate also holds true for all $6 < p < \infty$ and $2 \le r \le 4p/(p - 6)$.

(3) Let $\chi_0 = (X_{0,v})_{1 \le v \le m} \subset L^{\infty,p}(\mathbb{R}^d)$, $d < p < \infty$, be a non-degenerate family of $m \in \mathbb{N}^*$ vectors fields, with $m \ge d - 1$.

(a) Assuming that $f \in L^r((0,T), L^\infty(\mathbb{R}^d) \cap L^p_\chi(\mathbb{R}^d))$, with $2 \le r \le 8$ if $d = 2$ and $2 \le r \le 32/9$ if $d = 3$, then $\nabla v \in L^r((0,T), L^\infty(\mathbb{R}^d))$ and the following estimate holds true.

$$(B.10) \quad \begin{aligned} & \|\nabla v\|_{L^r((0,T), L^\infty(\mathbb{R}^d))}^2 \lesssim \|(f, v_0)\|_{Y_T}^2 + \|f\|_{L^r((0,T), L^2(\mathbb{R}^d) \cap L^\infty(\mathbb{R}^d) \cap L^p_\chi(\mathbb{R}^d))}^2. \end{aligned}$$
It is straightforward to check that for $T < \infty$ every $v \in Z_T$ satisfies:

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

The well-posedness result for the linearized system (B.5) reads as follows.

**Proposition B.1.** Let $0 < T \le \infty$. For all $(f, v_0) \in Y_T$, there exists a unique solution $v \in Z_T$ of the Cauchy problem (B.5). Moreover, the following estimates hold true for $v$.

(1) Basic energy estimates:

$$ \begin{aligned} \text{(B.6)} \quad & \sup_{[0,T]} \| (\sqrt{\rho_0}v, \sqrt{\rho_0}\partial_t v, \nabla v) \|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \| (\nabla v, \sqrt{\rho_0}\partial_t v, \nabla \partial_t v) \|_{L^2(\mathbb{R}^d)}^2 \\ & \lesssim \| (\sqrt{\rho_0}v_0, \sqrt{\rho_0}\partial_t v|_{t=0}, \nabla v_0) \|_{L^2(\mathbb{R}^d)}^2 + \sup_{[0,T]} \|f\|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \| (f, \partial_t f) \|_{L^2(\mathbb{R}^d)}^2. \end{aligned} $$

(2) Higher order energy estimates:

$$ \begin{aligned} \text{(B.7)} \quad & \sup_{[0,T]} \| (\sqrt{\sigma} \nabla \partial_t v, \sigma \sqrt{\rho_0} \partial_t v) \|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \| (\sqrt{\sigma \rho_0} \partial_t v, \sigma \nabla (\partial_t v)) \|_{L^2(\mathbb{R}^d)}^2 \\ & \lesssim \int_0^T \| \nabla \partial_t v \|_{L^2(\mathbb{R}^d)}^2 + \sup_{[0,T]} \| \sigma \partial_t f \|_{L^2(\mathbb{R}^d)}^2 + \int_0^T \| \sigma^2 \partial_t f \|_{L^2(\mathbb{R}^d)}^2. \end{aligned} $$

The constant appearing in the above estimates does not depend on the upper or lower bound of the density $\rho_0$.

The proof of Proposition B.1 is not part of the classical theory of parabolic systems due to the roughness of the density. However, it can be achieved by a regularization process followed by a compactness argument. We refer, for example, to [49, Theorem 3.1] for the derivation of estimates (B.6) and (B.7).

**Step 3. Further estimates of the linearized system.** By interpolating the estimates (B.6) and (B.7), we observe that the following estimates hold true for the velocity gradient and its time derivative.

**Corollary B.2.** The following estimates hold true.

(1) Assuming that $f \in L^r((0,T), L^p(\mathbb{R}^d))$ for $2 \le r \le \infty$ and $2 < p < \infty$ if $d = 2$ or $2 < p \le 6$ if $d = 3$, we have:

$$ \begin{aligned} \text{(B.8)} \quad & \| \nabla v \|_{L^r((0,T), L^p(\mathbb{R}^d))}^2 \lesssim \| (f, v_0) \|_{Y_T}^2 + \| f \|_{L^r((0,T), L^p(\mathbb{R}^d))}^2. \end{aligned} $$

(2) For all $2 \le r \le \infty$ and $2 < p < \infty$ if $d = 2$ and $2 < p \le 6$ if $d = 3$ we have:

$$ \text{(B.9)} \quad \| \sigma^s \nabla \partial_t v \|_{L^r((0,T), L^p(\mathbb{R}^d))}^2 \lesssim \| (f, v_0) \|_{Y_T}^2 + \| \sigma^s \partial_t f \|_{L^r((0,T), L^p(\mathbb{R}^d))}^2 $$

where

$$ s = \begin{cases} 1 - \frac{1}{p} - \frac{1}{r} & \text{if } d = 2, \\ \frac{5}{4} - \frac{3}{2p} - \frac{1}{r} & \text{if } d = 3. \end{cases} $$

For $d = 3$, the same estimate also holds true for all $6 < p < \infty$ and $2 \le r \le 4p/(p-6)$.

(3) Let $\chi_0 = (X_{0,v})_{1 \le v \le m} \subset L^{\infty,p}(\mathbb{R}^d)$, $d < p < \infty$, be a non-degenerate family of $m \in \mathbb{N}^*$ vectors fields, with $m \ge d - 1$.

(a) Assuming that $f \in L^r((0,T), L^\infty(\mathbb{R}^d) \cap L^p_\chi(\mathbb{R}^d))$, with $2 \le r \le 8$ if $d = 2$ and $2 \le r \le 32/9$ if $d = 3$, then $\nabla v \in L^r((0,T), L^\infty(\mathbb{R}^d))$ and the following estimate holds true.

$$ \text{(B.10)} \quad \| \nabla v \|_{L^r((0,T), L^\infty(\mathbb{R}^d))}^2 \lesssim \| (f, v_0) \|_{Y_T}^2 + \| f \|_{L^r((0,T), L^2(\mathbb{R}^d) \cap L^\infty(\mathbb{R}^d) \cap L^p_\chi(\mathbb{R}^d))}^2. $$
(b) Assuming that $f \in L^r((0,T), L^\infty(\mathbb{R}^d) \cap \mathbb{L}_{\mathcal{X}_0}^p(\mathbb{R}^d))$, with $2 < p < \infty$, $2 \le r \le 2p/(p-2)$ if $d=2$ and $3 < p \le 6$, $2 \le r \le 4p/(3p-6)$ if $d=3$. Then we have:

$$ (B.11) \quad \|\partial_{\mathcal{X}_0} \nabla v\|_{L^r((0,T), L^p(\mathbb{R}^d))}^2 \lesssim \|\mathcal{X}_0\|_{L^\infty(\mathbb{R}^d)}^2 \|(f, v_0)\|_{Y_T}^2 + \|\partial_{\mathcal{X}_0} f\|_{L^r((0,T), L^p(\mathbb{R}^d))}^2 \\ \qquad + \|\nabla \mathcal{X}_0\|_{L^p(\mathbb{R}^d)}^2 \|f\|_{L^r((0,T), L^2(\mathbb{R}^d) \cap L^\infty(\mathbb{R}^d) \cap \mathbb{L}_{\mathcal{X}_0}^p(\mathbb{R}^d))}^2. $$

Let $d=3$ and $6 < p < \infty$, $2 \le r \le \infty$. If $\sigma^s f \in L^r((0,T), L^\infty(\mathbb{R}^3) \cap \mathbb{L}_{\mathcal{X}_0}^p(\mathbb{R}^3))$ and $\sigma^{3/4} - \frac{1}{r} \partial_t f \in L^3(\mathbb{R}^3)$ with

$$ s = \frac{3}{4} - \frac{1}{r} - \frac{3}{2p} $$

then, from (2), we have $\sigma^{3/4} - \frac{1}{r} \nabla \partial_t v \in L^3(\mathbb{R}^3)$ and

$$ (B.12) \quad \|\sigma^s \partial_{\mathcal{X}_0} \nabla v\|_{L^r((0,T), L^p(\mathbb{R}^3))}^2 \lesssim \|\mathcal{X}_0\|_{L^\infty(\mathbb{R}^3)}^2 \left( \|(f, v_0)\|_{Y_T}^2 + \|\sigma^{3/4} - \frac{1}{r} \partial_t f\|_{L^r((0,T), L^3(\mathbb{R}^3))}^2 \right) \\ \qquad + \|\sigma^s \partial_{\mathcal{X}_0} f\|_{L^r((0,T), L^p(\mathbb{R}^3))}^2 + \|\nabla \mathcal{X}_0\|_{L^p(\mathbb{R}^3)}^2 \|\sigma^s f\|_{L^r((0,T), L^2(\mathbb{R}^3) \cap L^\infty(\mathbb{R}^3) \cap \mathbb{L}_{\mathcal{X}_0}^p(\mathbb{R}^3))}^2. $$

The constant appearing in the above estimates depends on both the lower and upper bounds of the density.

Indeed, all the estimates in Corollary B.2 are based in the following expression of the velocity gradient:

$$ \begin{align} \nabla v &= \nabla \mathcal{P} v + \nabla \mathcal{Q} v \nonumber \\ &= -\frac{1}{\mu} (-\Delta)^{-1} \nabla \mathcal{P}(\rho_0 \partial_t v) - \frac{1}{\nu} (-\Delta)^{-1} \nabla \mathcal{Q}(\rho_0 \partial_t v) \tag{B.13} \\ &\quad + \frac{1}{\mu} (-\Delta)^{-1} \nabla \mathcal{P} \operatorname{div} f + \frac{1}{\nu} (-\Delta)^{-1} \nabla \mathcal{Q} \operatorname{div} f. \nonumber \end{align} $$

The first two terms associated with $\partial_t v$ exhibit regularity due to the regularity of $\partial_t v$. In particular, their $L^r((0,T), L^p(\mathbb{R}^d))$ norm estimates can be obtained by interpolating the estimate (B.6). The $L^r((0,T), L^p(\mathbb{R}^d))$ norm estimate for the last two terms in the expression of the velocity gradient (B.13) follows from the continuity of Riesz operators on $L^p(\mathbb{R}^d)$ for all $1 < p < \infty$. These computations lead to (B.8).

The estimate (B.10) is obtained similarly: the $L^r((0,T), L^\infty(\mathbb{R}^d))$ norm of the terms associated with $\partial_t v$ can be estimated by interpolating the estimate (B.6), while the norm of the remaining terms is obtained using Proposition 2.5 since Riesz operators fail to be continuous on $L^\infty(\mathbb{R}^d)$.

To derive the estimate (B.9), we take time derivative of (B.13) and apply the continuity of Riesz transforms on $L^p(\mathbb{R}^d)$, $1 < p < \infty$, to obtain norms for the terms associated with $\partial_t f$. The norm of the first two terms, associated with $\partial_t v$, can be obtained by interpolating estimate (B.7).

For the last estimates (B.11) and (B.12), we take the derivative along $\mathcal{X}_0$ in (B.13) and we obtain:

$$ \begin{align*} \partial_{\mathcal{X}_0} \nabla v = & -\frac{1}{\mu} \mathcal{X}_0 \cdot \nabla (-\Delta)^{-1} \nabla \mathcal{P}(\rho_0 \partial_t v) - \frac{1}{\nu} \mathcal{X}_0 \cdot \nabla (-\Delta)^{-1} \nabla \mathcal{Q}(\rho_0 \partial_t v) \\ & + \frac{1}{\mu} \partial_{\mathcal{X}_0} (-\Delta)^{-1} \nabla \mathcal{P} \operatorname{div} f + \frac{1}{\nu} \partial_{\mathcal{X}_0} (-\Delta)^{-1} \nabla \mathcal{Q} \operatorname{div} f. \end{align*} $$

Once again, the norms of the first two terms are obtained using Hölder's inequality and by interpolating estimates (B.6) and (B.7). For the remaining terms, we use Lemma A.1 of [10]. This completes this step of the study of the linear system (B.5).

**Step 4. Final conclusion.** Once we conclude the study of the linear system associated with (B.4), the next step is to define a map that is contracting for some small time $T > 0$, such that it admits a unique fixed point, which serves as a solution to (B.1) after reverting to Eulerian coordinates. With
(b) Assuming that $f \in L^r((0,T), L^\infty(\mathbb{R}^d) \cap \mathbb{L}_{\chi_0}^p(\mathbb{R}^d))$, with $2 < p < \infty$, $2 \le r \le 2p/(p-2)$ if $d=2$ and $3 < p \le 6$, $2 \le r \le 4p/(3p-6)$ if $d=3$. Then we have:

$$ (B.11) \quad \|\partial_{\chi_0} \nabla v\|_{L^r((0,T), L^p(\mathbb{R}^d))}^2 \lesssim \|\chi_0\|_{L^\infty(\mathbb{R}^d)}^2 \|(f, v_0)\|_{Y_T}^2 + \|\partial_{\chi_0} f\|_{L^r((0,T), L^p(\mathbb{R}^d))}^2 \\ \qquad + \|\nabla \chi_0\|_{L^p(\mathbb{R}^d)}^2 \|f\|_{L^r((0,T), L^2(\mathbb{R}^d) \cap L^\infty(\mathbb{R}^d) \cap \mathbb{L}_{\chi_0}^p(\mathbb{R}^d))}^2. $$

Let $d=3$ and $6 < p < \infty$, $2 \le r \le \infty$. If $\sigma^s f \in L^r((0,T), L^\infty(\mathbb{R}^3) \cap \mathbb{L}_{\chi_0}^p(\mathbb{R}^3))$ and $\sigma^{3/4} - \frac{1}{r} \partial_t f \in L^3(\mathbb{R}^3)$ with

$$ s = \frac{3}{4} - \frac{1}{r} - \frac{3}{2p} $$

then, from (2), we have $\sigma^{3/4} - \frac{1}{r} \nabla \partial_t v \in L^3(\mathbb{R}^3)$ and

$$ (B.12) \quad \|\sigma^s \partial_{\chi_0} \nabla v\|_{L^r((0,T), L^p(\mathbb{R}^3))}^2 \lesssim \|\chi_0\|_{L^\infty(\mathbb{R}^3)}^2 \left( \|(f, v_0)\|_{Y_T}^2 + \|\sigma^{3/4} - \frac{1}{r} \partial_t f\|_{L^r((0,T), L^3(\mathbb{R}^3))}^2 \right) \\ \qquad + \|\sigma^s \partial_{\chi_0} f\|_{L^r((0,T), L^p(\mathbb{R}^3))}^2 + \|\nabla \chi_0\|_{L^p(\mathbb{R}^3)}^2 \|\sigma^s f\|_{L^r((0,T), L^2(\mathbb{R}^3) \cap L^\infty(\mathbb{R}^3) \cap \mathbb{L}_{\chi_0}^p(\mathbb{R}^3))}^2. $$

The constant appearing in the above estimates depends on both the lower and upper bounds of the density.

Indeed, all the estimates in Corollary B.2 are based on the following expression of the velocity gradient:

$$ \begin{align} \nabla v &= \nabla \mathcal{P} v + \nabla \mathcal{Q} v \nonumber \\ &= -\frac{1}{\mu} (-\Delta)^{-1} \nabla \mathcal{P}(\rho_0 \partial_t v) - \frac{1}{\nu} (-\Delta)^{-1} \nabla \mathcal{Q}(\rho_0 \partial_t v) \tag{B.13} \\ &\quad + \frac{1}{\mu} (-\Delta)^{-1} \nabla \mathcal{P} \operatorname{div} f + \frac{1}{\nu} (-\Delta)^{-1} \nabla \mathcal{Q} \operatorname{div} f. \nonumber \end{align} $$

The first two terms associated with $\partial_t v$ exhibit regularity due to the regularity of $\partial_t v$. In particular, their $L^r((0,T), L^p(\mathbb{R}^d))$ norm estimates can be obtained by interpolating the estimate (B.6). The $L^r((0,T), L^p(\mathbb{R}^d))$ norm estimate for the last two terms in the expression of the velocity gradient (B.13) follows from the continuity of Riesz operators on $L^p(\mathbb{R}^d)$ for all $1 < p < \infty$. These computations lead to (B.8).

The estimate (B.10) is obtained similarly: The $L^r((0,T), L^\infty(\mathbb{R}^d))$ norm of the terms associated with $\partial_t v$ can be estimated by interpolating the estimate (B.6), while the norm of the remaining terms is obtained using Proposition 2.5 since Riesz operators fail to be continuous on $L^\infty(\mathbb{R}^d)$.

To derive the estimate (B.9), we take time derivative of (B.13) and apply the continuity of Riesz transforms on $L^p(\mathbb{R}^d)$, $1 < p < \infty$, to obtain norms for the terms associated with $\partial_t f$. The norm of the first two terms, associated with $\partial_t v$, can be obtained by interpolating estimate (B.7).

For the last estimates (B.11) and (B.12), we take the derivative along $\chi_0$ in (B.13) and we obtain:

$$ \begin{align*} \partial_{\chi_0} \nabla v = & -\frac{1}{\mu} \chi_0 \cdot \nabla (-\Delta)^{-1} \nabla \mathcal{P}(\rho_0 \partial_t v) - \frac{1}{\nu} \chi_0 \cdot \nabla (-\Delta)^{-1} \nabla \mathcal{Q}(\rho_0 \partial_t v) \\ & + \frac{1}{\mu} \partial_{\chi_0} (-\Delta)^{-1} \nabla \mathcal{P} \operatorname{div} f + \frac{1}{\nu} \partial_{\chi_0} (-\Delta)^{-1} \nabla \mathcal{Q} \operatorname{div} f. \end{align*} $$

Once again, the norms of the first two terms are obtained using Hölder's inequality and by interpolating estimates (B.6) and (B.7). For the remaining terms, we use Lemma A.1 of [10]. This completes this step of the study of the linear system (B.5).

**Step 4. Final conclusion.** Once we conclude the study of the linear system associated with (B.4), the next step is to define a map that is contracting for some small time $T > 0$, such that it admits a unique fixed point, which serves as a solution to (B.1) after reverting to Eulerian coordinates. With
**Proposition B.1** and **Corollary B.2** in mind, we can verify that the unique solution of the full nonlinear system can be constructed in

$$ \tilde{Z}_T := \{ v \in Z_T : \nabla v \in L^2((0,T), L^\infty(\mathbb{R}^d) \cap \mathbb{L}_{\chi_0}^p(\mathbb{R}^d)) \} $$

for $2 < p < \infty$ if $d = 2$ and $3 < p \le 6$ if $d = 3$ by following the steps outlined in [50, Section 4]. The only argument we need to specify is the following lemmas.

**Lemma B.3.** Let $v$ be a vector field verifying $\nabla v \in L^1((0,t), L^\infty(\mathbb{R}^d))$ and $\partial_{\chi_0} \nabla v \in L^1((0,t), L^p(\mathbb{R}^d))$ for some $t > 0$. Assuming that

$$ (B.14) \qquad V := \int_0^t \left[ \|\nabla v\|_{L^\infty(\mathbb{R}^d)} + \|\partial_{\chi_0} \nabla v\|_{L^p(\mathbb{R}^d)} \right] < 1, $$

then, there exists a constant $K = K(V)$ such that the following estimate holds true:

$$ \|\partial_{\chi_0} \operatorname{Adj}(D\mathcal{X}_v(t)), \partial_{\chi_0} A_v(t), \partial_{\chi_0} J_v^{\pm 1}(t)\|_{L^p(\mathbb{R}^d)} \le K \|\partial_{\chi_0} Dv\|_{L^1((0,t), L^p(\mathbb{R}^d))}. $$

Moreover, we have for all $Dw \in \mathbb{L}_{\chi_0}^p(\mathbb{R}^d)$,

$$ \begin{aligned} & \|\partial_{\chi_0}(\operatorname{Adj}(D\mathcal{X}_v(t))D_{A_v(t)}w) - \partial_{\chi_0}Dw\|_{L^p(\mathbb{R}^d)} + \|\partial_{\chi_0}(\operatorname{Adj}(D\mathcal{X}_v(t))\operatorname{div}_{A_v(t)}w) - \partial_{\chi_0}\operatorname{div}w\|_{L^p(\mathbb{R}^d)} \\ & \le K \left( \|Dw\|_{L^\infty(\mathbb{R}^d)} + \|\partial_{\chi_0}Dw\|_{L^p(\mathbb{R}^d)} \right) \int_0^t \left( \|Dv\|_{L^\infty(\mathbb{R}^d)} + \|\partial_{\chi_0}Dv\|_{L^p(\mathbb{R}^d)} \right). \end{aligned} $$

**Lemma B.4.** Let $v_1$ and $v_2$ two vector fields verifying (B.14): $V_1, V_2 < 1$, and let $\delta v := v_2 - v_1$. Then, there exists a constant $K = K(V_1, V_2)$ such that the following estimate holds true:

$$ \begin{aligned} & \| (\partial_{\chi_0} A_{v_2}(t) - \partial_{\chi_0} A_{v_1}(t), \partial_{\chi_0} \operatorname{Adj}(D\mathcal{X}_{v_2}(t)) - \partial_{\chi_0} \operatorname{Adj}(D\mathcal{X}_{v_1}(t)), \partial_{\chi_0} J_{v_2}^{\pm 1}(t) - \partial_{\chi_0} J_{v_1}^{\pm 1}(t)) \|_{L^p(\mathbb{R}^d)} \\ & \le K \int_0^t \left( \|D\delta v\|_{L^\infty(\mathbb{R}^d)} + \|\partial_{\chi_0} D\delta v\|_{L^p(\mathbb{R}^d)} \right). \end{aligned} $$

The particular case of $3 < p \le 6$ for $d = 3$ is sufficient for constructing blocks for the global solution of **Theorem 1.6**. For $6 < p < \infty$ in three dimension, the fixed point theorem may be applied in a closed subset of the following space:

$$ \tilde{Z}_T := \left\{ v \in Z_T : \sigma^{\frac{3}{4}-\frac{1}{r}} \nabla \partial_t v \in L^r((0,T), L^3(\mathbb{R}^d)); \sigma^{\frac{3}{4}-\frac{1}{r}-\frac{3}{2p}} \nabla v \in L^r((0,T), L^\infty(\mathbb{R}^d) \cap \mathbb{L}_{\chi_0}^p(\mathbb{R}^d)) \right\}. $$

This ends the sketchy proof of **Theorem 1.4**.

## REFERENCES

[1] BAHOURI, H., CHEMIN, J., AND DANCHIN, R. *Fourier Analysis and Nonlinear Partial Differential Equations*. Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg, 2011.

[2] BERTOZZI, A. L., AND CONSTANTIN, P. Global regularity for vortex patches. *Communications in Mathematical Physics* 152, 1 (1993), 19–28.

[3] BRESCH, D., AND BURTEA, C. Extension of the Hoff solutions framework to cover compressible Navier-Stokes equations with possible anisotropic viscous tensor. *Indiana University Mathematics Journal* 72, 5 (2023), 2145–2189.

[4] CHARVE, F., AND DANCHIN, R. A global existence result for the compressible Navier-Stokes equations in the critical $L^p$ framework. *Archive for Rational Mechanics and Analysis* 198, 1 (2010), 233–271.

[5] CHEMIN, J. Y. Sur le mouvement des particules d’un fluide parfait incompressible bidimensionnel. *Inventiones mathematicae* 103 (1991), 599–629.

[6] CHEMIN, J.-Y. Persistance de structures géométriques dans les fluides incompressibles bidimensionnels. In *Annales scientifiques de l’Ecole normale supérieure* (1993), vol. 26, pp. 517–542.

[7] CHEN, Q., MIAO, C., AND ZHANG, Z. Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity. *Communications on Pure and Applied Mathematics* 63, 9 (2010), 1173–1224.

[8] COIFMAN, R., LIONS, P.-L., MEYER, Y., AND SEMMES, S. Compensated compactness and Hardy spaces. *Journal de mathématiques pures et appliquées* 72, 3 (1993), 247–286.

[9] DANCHIN, R. Persistance de structures géométriques et limite non visqueuse pour les fluides incompressibles en dimension quelconque. *Bulletin de la Société Mathématique de France* 127, 2 (1999), 179–227.

[10] DANCHIN, R., FANELLI, F., AND PAICU, M. A well-posedness result for viscous compressible fluids with only bounded density. *Analysis & PDE* 13, 1 (2020), 275–316. available online.
**Proposition B.1** and **Corollary B.2** in mind, we can verify that the unique solution of the full nonlinear system can be constructed in

$$ \tilde{Z}_T := \{ v \in Z_T : \nabla v \in L^2((0,T), L^\infty(\mathbb{R}^d) \cap \mathbb{L}_{\mathcal{X}_0}^p(\mathbb{R}^d)) \} $$

for $2 < p < \infty$ if $d = 2$ and $3 < p \le 6$ if $d = 3$ by following the steps outlined in [49, Section 4]. The only argument we need to specify is the following lemmas.

**Lemma B.3.** Let $v$ be a vector field verifying $\nabla v \in L^1((0,t), L^\infty(\mathbb{R}^d))$ and $\partial_{\mathcal{X}_0} \nabla v \in L^1((0,t), L^p(\mathbb{R}^d))$ for some $t > 0$. Assuming that

$$ (B.14) \qquad V := \int_0^t \left[ \|\nabla v\|_{L^\infty(\mathbb{R}^d)} + \|\partial_{\mathcal{X}_0} \nabla v\|_{L^p(\mathbb{R}^d)} \right] < 1, $$

then, there exists a constant $K = K(V)$ such that the following estimate holds true:

$$ \|\partial_{\mathcal{X}_0} \operatorname{Adj}(D\mathcal{X}_v(t)), \partial_{\mathcal{X}_0} A_v(t), \partial_{\mathcal{X}_0} J_v^{\pm 1}(t)\|_{L^p(\mathbb{R}^d)} \le K \|\partial_{\mathcal{X}_0} Dv\|_{L^1((0,t), L^p(\mathbb{R}^d))}. $$

Moreover, we have for all $Dw \in \mathbb{L}_{\mathcal{X}_0}^p(\mathbb{R}^d)$,

$$ \begin{aligned} & \|\partial_{\mathcal{X}_0}(\operatorname{Adj}(D\mathcal{X}_v(t))D_{A_v(t)}w) - \partial_{\mathcal{X}_0}Dw\|_{L^p(\mathbb{R}^d)} + \|\partial_{\mathcal{X}_0}(\operatorname{Adj}(D\mathcal{X}_v(t))\operatorname{div}_{A_v(t)}w) - \partial_{\mathcal{X}_0}\operatorname{div}w\|_{L^p(\mathbb{R}^d)} \\ & \le K \left( \|Dw\|_{L^\infty(\mathbb{R}^d)} + \|\partial_{\mathcal{X}_0}Dw\|_{L^p(\mathbb{R}^d)} \right) \int_0^t \left( \|Dv\|_{L^\infty(\mathbb{R}^d)} + \|\partial_{\mathcal{X}_0}Dv\|_{L^p(\mathbb{R}^d)} \right). \end{aligned} $$

**Lemma B.4.** Let $v_1$ and $v_2$ two vector fields verifying (B.14): $V_1, V_2 < 1$, and let $\delta v := v_2 - v_1$. Then, there exists a constant $K = K(V_1, V_2)$ such that the following estimate holds true:

$$ \begin{aligned} & \| (\partial_{\mathcal{X}_0} A_{v_2}(t) - \partial_{\mathcal{X}_0} A_{v_1}(t), \partial_{\mathcal{X}_0} \operatorname{Adj}(D\mathcal{X}_{v_2}(t)) - \partial_{\mathcal{X}_0} \operatorname{Adj}(D\mathcal{X}_{v_1}(t)), \partial_{\mathcal{X}_0} J_{v_2}^{\pm 1}(t) - \partial_{\mathcal{X}_0} J_{v_1}^{\pm 1}(t)) \|_{L^p(\mathbb{R}^d)} \\ & \le K \int_0^t \left( \|D\delta v\|_{L^\infty(\mathbb{R}^d)} + \|\partial_{\mathcal{X}_0} D\delta v\|_{L^p(\mathbb{R}^d)} \right). \end{aligned} $$

The particular case of $3 < p \le 6$ for $d = 3$ is sufficient for constructing blocks for the global solution in **Theorem 1.6**. For $6 < p < \infty$ in three dimensions, the fixed point theorem may be applied in a closed subset of the following space:

$$ \tilde{Z}_T := \left\{ v \in Z_T : \sigma^{\frac{3}{4}-\frac{1}{r}} \nabla \partial_t v \in L^r((0,T), L^3(\mathbb{R}^3)); \sigma^{\frac{3}{4}-\frac{1}{r}-\frac{3}{2p}} \nabla v \in L^r((0,T), L^\infty(\mathbb{R}^d) \cap \mathbb{L}_{\mathcal{X}_0}^p(\mathbb{R}^3)) \right\}. $$

This ends the sketchy proof of **Theorem 1.4**.

**Data availability.** There are no data sets generated or analyzed in this study, therefore data sharing does not apply.

**Conflict of interest.** The authors declare that they have no conflict of interest.

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(X. LIAO) KARLSRUHE INSTITUT OF TECHNOLOGY, KARLSRUHE, GERMANY

Email address: xian.liao@kit.edu

(S. M. ZODJI) INSTITUT DE MATHÉMATIQUES DE JUSSIEU PARIS RIVE GAUCHE, UNIVERSITÉ PARIS CITÉ, PARIS, FRANCE

Email address: marcel.zodji@imj-prg.fr
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[42] NASH, J. Le problème de Cauchy pour les équations différentielles d'un fluide général. *Bull. Soc. Math. France* **90** (1962), 487-497.

[43] PAICU, M., AND ZHANG, P. Striated regularity of 2-D inhomogeneous incompressible Navier-Stokes system with variable viscosity. *Comm. Math. Phys.* **376**, 1 (2020), 385-439.

[44] PRANGE, C., AND TAN, J. Free boundary regularity of vacuum states for incompressible viscous flows in unbounded domains. *Preprint, arXiv:2310.09288v3 [math.AP]* (2023).

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[49] ZODJI, S. M. A well-posedness result for the compressible two-fluid model with density-dependent viscosity. *Preprint, arXiv:2310.12525v3 [math.AP]* (2023).

[50] ZODJI, S. M. Discontinuous solutions for the Navier-Stokes equations with density-dependent viscosity. *To appear in Ann. Inst. H. Poincaré C Anal. Non Linéaire* (2024).

(X. LIAO) KARLSRUHE INSTITUT OF TECHNOLOGY, KARLSRUHE, GERMANY

Email address: xian.liao@kit.edu

(S. M. ZODJI) INSTITUT DE MATHÉMATIQUES DE JUSSIEU PARIS RIVE GAUCHE, UNIVERSITÉ PARIS CITÉ, PARIS, FRANCE

Email address: marcel.zodji@imj-prg.fr