m}\,{\mathcal}{C}^\eta_{X_\lambda}\qquad\mbox{ and }
\qquad
\left\|f\right\|_{{\mathcal}{C}^\eta_X}\,:=\,\frac{1}{I(X)}\,\left(\|f\|_{L^\infty}\,{\widetilde}{|||}X|||_{{\mathcal}{C}^{\varepsilon}}\,+\,
|||\,{{\rm div}\,}\left(f\,X\right)|||_{{\mathcal}{C}^{\eta-1}}\right)\,.$$

\[r:div\] Our aim is to investigate Hölder regularity of the derivation of $f$ along a fixed vector-field (say) $Y$, i.e. the quantity $$\d_Yf\,:=\,\,\sum_{i=1}^N\,Y^i\,\d_if\,.$$ If $f$ is only bounded, however, this expression has no meaning: this is why we decided to focus on $\,{{\rm div}\,}(f\,Y)$, as done in the literature about this topic (see also [@D1999], section 1). Lemma \[l:div\] below will clarify the relation between these two quantities.

Now, let us take a vector-field $X_0$ and define its time evolution $X(t)$: $$\label{def:X}
X(t,x)\,\equiv\,X_t(x)\,:=\,\d_{X_0(x)}\psi_t\left(\psi^{-1}_t(x)\right)\,,$$ that is $X(t)$ is the vector-field $X_0$ transported by the flow associated to $u$. From this definition, it immediately follows that $\left[X(t)\,,\,\d_t+u\cdot\nabla\right]=0$, i.e. $X(t)$ satisfies the following system: $$\label{eq:X}
 \left\{\begin{array}{l}
         \left(\d_t\,+\,u\cdot\nabla\right)X\,=\,\d_Xu \\[1ex]
         X_{|t=0}\,=\,X_0\,.
        \end{array}\right.$$

We are now ready for stating our first result, on striated regularity.

\[t:stri-N\] Fix ${\varepsilon}\in\,]0,1[$ and take a non-degenerate family of vector-fields $X_0=\left(X_{0,\lambda}\right)_{1\leq\lambda\leq m}$ over ${\mathbb{R}}^N$, whose components and divergence are in ${\mathcal}{C}^{\varepsilon}$.\
Let the initial velocity field $u_0\in L^p$, with $p\in\,]2,+\infty]$, and its vorticity $\Omega_0\in L^\infty\cap L^q$, with $q\in\,[2,+\infty[$ such that $1/p\,+\,1/q\,\geq\,1/2$.\
Suppose the initial density $\rho_0\in W^{1,\infty}$ to be such that $0<\rho_*\leq\rho_0\leq\rho^*$.\
Finally, let us assume that $\Omega_0$ and $\nabla\rho_0$ both belong to ${\mathcal}{C}^{\varepsilon}_{X_0}$.

