then, for all $\sigma>0$ ($\sigma>-1$ if $\,{{\rm div}\,}v=0$), estimate holds with $V'(t):=\|\nabla f(t)\|_{L^\infty}$.

Finally, we shall make an extensive use of energy estimates for the following elliptic equation: $$\label{eq:elliptic}
-\,{{\rm div}\,}(a\,\nabla\Pi)={{\rm div}\,}F\quad\hbox{in }\ {\mathbb{R}}^N\,,$$ where $a=a(x)$ is a given suitably smooth bounded function satisfying $$\label{eq:ellipticity}
a_*\,:=\,\inf_{x\in{\mathbb{R}}^N}a(x)\,>\,0\,.$$

We shall use the following result based on Lax-Milgram’s theorem (see the proof in e.g. [@D2010]).

\[l:laxmilgram\] For all vector field $F$ with coefficients in $L^2$, there exists a tempered distribution $\Pi$, unique up to constant functions, such that $\nabla\Pi\in L^2$ and equation $\eqref{eq:elliptic}$ is satisfied. In addition, we have $$a_*\,\|\nabla\Pi\|_{L^2}\,\leq\,\|F\|_{L^2}\,.$$

Propagation of striated regularity {#s:striated}
==================================

Now we are ready to tackle the proof of theorem \[t:stri-N\]. We will carry out it in a standard way: first of all we will prove a priori estimates for smooth solutions to . Then, we will construct a sequence of regular approximated solutions. Finally, thanks to the the just proved upper bounds, we will get convergence of this sequence to a solution of our initial system, with the required properties.

A priori estimates
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First of all, we will prove a priori estimates for a smooth solution $(\rho,u,\nabla\Pi)$ to system .

### Estimates for the density 