vorticity $\Omega_0\in L^\infty\cap L^q$, with $q\in\,[2,+\infty[$ such that $1/p\,+\,1/q\,\geq\,1/2$.\
Assume that the initial density $\rho_0\in W^{1,\infty}$ be such that $0<\rho_*\leq\rho_0\leq\rho^*$.\
Finally, let $\Omega_0$ and $\nabla\rho_0$ belong to ${\mathcal}{C}^{{\varepsilon}}_{\Sigma_0}$.

Then there exist a time $T>0$ and a unique solution $(\rho,u,\nabla\Pi)$ of system , which verifies the same properties of theorem \[t:stri-N\].\
Moreover, if we define $$\Sigma(t)\,:=\,\psi_t\left(\Sigma_0\right)\,,$$ $\Sigma(t)$ is, at every time $t\in[0,T]$, a hypersurface of class ${\mathcal}{C}^{1,{\varepsilon}}$ of ${\mathbb{R}}^N$, and conormal regularity is preserved: at every time $t\in[0,T]$, one has $$\nabla\rho(t)\,,\,\,\Omega(t)\,\,\in\;{\mathcal}{C}^{{\varepsilon}}_{\Sigma(t)}\qquad\mbox{ and }\qquad
u(t)\,,\,\,\nabla\Pi(t)\,\,\in\;{\mathcal}{C}^{1+{\varepsilon}}_{\Sigma(t)}$$ uniformly on $[0,T]$.

Tools {#s:tools}
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In this section we will introduce the main tools we used to prove our results; they are mostly based on Fourier analysis techniques. Unless otherwise specified, one can find the proof of all the results quoted here in [@B-C-D], chapter 2.

Littlewood-Paley decomposition and Besov spaces
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Let us first define the so called “Littlewood-Paley decomposition”, based on a non-homogeneous dyadic partition of unity with respect to the Fourier variable. So, fix a smooth radial function $\chi$ supported in (say) the ball $B(0,\frac43),$ equal to $1$ in a neighborhood of $B(0,\frac34)$ and such that $r\mapsto\chi(r\,e)$ is nonincreasing over ${\mathbb{R}}_+$ for all unitary vector $e\in{\mathbb{R}}^N$. Moreover, set $\varphi\left(\xi\right)=\chi\left(\xi/2\right)-\chi\left(\xi\right).$ The dyadic blocks $(\Delta_j)_{j\in{\mathbb{Z}}}$ 