the divergence-free constraint over the velocity.

We will always suppose that the variable $x$ belongs to the whole space $\,{\mathbb{R}}^N$.

The problem of preserving geometric structures came out already in the homogeneous case, for which $\rho\equiv1$ and system becomes $$\left\{\begin{array}{l}
\d_tu\,+\,u\cdot\nabla u\,+\,\nabla\Pi\,=\,0\\[1ex]
{{\rm div}\,}u\,=\,0\,,
\end{array}\right.
\leqno(E)$$ in studying $2$-dimensional vortex patches, that is to say the initial vorticity $\Omega_0$ is the characteristic function of a bounded domain $D_0$. As we will explain below, in the case of higher dimension $N\geq3$ this notion is generalized by the properties of striated and conormal regularity.

The vorticity of the fluid is defined as the skew-symmetric matrix $$\label{def:vort}
\Omega\,:=\,\nabla u\,-\,^t\nabla u$$ and in the homogenous case it satisfies the equation $$\d_t\Omega\,+\,u\cdot\nabla\Omega\,+\,\Omega\cdot\nabla u\,+\,^t\nabla u\cdot\Omega\,=\,0\,.$$ In dimension $N=2$ it can be identified with the scalar function $\omega\,=\,\partial_1u^2\,-\,\partial_2u^1$, while for $N=3$ with the vector-field $\omega\,=\,\nabla\times u$. Let us recall also that in the bidimensional case this quantity is transported by the velocity field: $$\d_t\omega\,+\,u\cdot\nabla\omega\,=\,0\,.$$

The notion of vortex patches was introduced in [@Y] and gained new interest after the survey paper [@M] of Majda. In the case $\,N=2\,$ Yudovich’s theorem ensures the existence of a unique global solution of the homogeneous Euler system, which preserves the geometric 