Then there exist a time $T>0$ and a unique solution $\left(\rho,u,\nabla\Pi\right)$ of system , such that:

-   $\rho\,\in\,L^\infty([0,T];W^{1,\infty})\cap{\mathcal}{C}_b([0,T]\times{\mathbb{R}}^N)$, such that $0<\rho_*\leq\rho\leq\rho^*$ at every time;

-   $u\,\in\,{\mathcal}{C}([0,T];L^p)\cap L^\infty([0,T];{\mathcal}{C}^{0,1})$, with $\d_tu\,\in\,{\mathcal}{C}([0,T];L^2)$ and vorticity $\,\Omega\,\in\,{\mathcal}{C}([0,T];L^q)$;

-   $\nabla\Pi\,\in\,{\mathcal}{C}([0,T];L^2)$, with $\nabla^2\Pi\,\in\,L^\infty([0,T];L^\infty)$.

Moreover, the family of vector-fields transported by the flow still remains, at every time, non-degenerate and with components and divergence in ${\mathcal}{C}^{\varepsilon}$, and striated regularity is preserved: for all $t\in[0,T]$, one has $$\nabla\rho(t)\,,\,\,\Omega(t)\,\,\in\;{\mathcal}{C}^{{\varepsilon}}_{X(t)}\qquad\mbox{ and }\qquad
u(t)\,,\,\,\nabla\Pi(t)\,\,\in\;{\mathcal}{C}^{1+{\varepsilon}}_{X(t)}$$ uniformly on $[0,T]$.

Another interesting notion, strictly related to the previous one, is that of *conormal regularity*. First of all, we have to recall a definition (see again section 1 of [@D1999]).

Let $\Sigma\subset{\mathbb}{R}^N$ be a compact hypersurface of class ${\mathcal}{C}^{1,{\varepsilon}}$. Let us denote by ${\mathcal}{T}^{\varepsilon}_\Sigma$ the set of all vector-fields $X$ with components and divergence in ${\mathcal}{C}^{\varepsilon}$, which are tangent to $\Sigma$, i.e. $\d_XH\,_{|\Sigma}\equiv0$ for all local equations $H$ of $\Sigma$.

Given a $\eta\in[{\varepsilon},1+{\varepsilon}]$, we say that a function $f\in L^\infty$ belongs to the space ${\mathcal}{C}^{\eta}_\Sigma$ if $$\forall\,X\,\in\,{\mathcal}{T}^{\varepsilon}_\Sigma\;\;,\qquad{{\rm div}\,}\left(f\,X\right)\,\in\,{\mathcal}{C}^{\eta-1}\,.$$

Similarly to what happens for striated regularity, also conormal structure propagates during the time evolution.

\[t:conorm-N\] Fix ${\varepsilon}\in\,]0,1[$ and take a ${\mathcal}{C}^{1,{\varepsilon}}$ compact hypersurface $\Sigma_0\subset{\mathbb{R}}^N$.\
Let us suppose the initial velocity field $u_0\in L^p$, with $p\in\,]2,+\infty]$, and its 