structure: the vorticity remains the characteristic function of the evolution (by the flow associated to this solution) of the domain $D_0$. Vortex patches in bounded domains of ${\mathbb{R}}^2$ were also studied by Depauw (see [@Dep]), while Dutrifoy in [@Dut] focused on the case of domains in ${\mathbb{R}}^3$. Moreover, in [@Ch1991] Chemin proved that, if the initial domain has boundary $\d D_0$ of class ${\mathcal}{C}^{1,{\varepsilon}}$ for some ${\varepsilon}>0$, then this regularity is preserved during the evolution for small times; in [@Ch1993] he also showed a global in time persistence issue. In [@D1997-sing] Danchin considered instead the case in which initial data of the Euler system are vortex patches with singular boundary: he proved that if $\d D_0$ is regular apart from a closed subset, then it remains regular for all times, apart from the closed subset transported by the flow associated to the solution.

In the case $N\geq3$ one can’t expect to have global results anymore, nor to preserve the initial vortex patch structure, because of the presence of the stretching term in the vorticity equation. Nevertheless, it’s possible to introduce the definition of *striated regularity*, which generalizes in a quite natural way the previous one of vortex patch: it means 