also in the vorticity equation and has a smoothing effect; however, he still got global in time results. Moreover, in both his works he had to handle with spaces of type $B^{1+{\varepsilon}}_{p,\infty}$ (with $p\in\,]2,+\infty[$ and ${\varepsilon}\in\,]2/p,1[\,$) due to technical reasons which come out with a viscous fluid. Let us immediately clarify that these problems have been recently solved by Hmidi in [@Hm] (see also [@B-C-D]). In the above mentioned works Danchin proved also a priori estimates for solutions of $(N\!S_\nu)$ independent of the viscosity $\nu$, therefore preservation of the geometric structures in passing from solutions of $(N\!S_\nu)$ to solutions of $(E)$ in the limit $\nu{\rightarrow}0$.

In this paper we come back to the inviscid case and we study the non-homogeneous incompressible Euler system . We will prove that preservation of geometric properties of initial data, such as striated and conormal regularity, still holds in this setting, as in the classical (homogeneous) one.

Let us point out that considering geometric structures is not only a generalization of the classical problem of vortex patches. It can be also seen as an improvement to the well-posedness result for in critical Besov spaces $B^s_{\infty,r}$ (see paper [@D-F]). As a matter of fact, here we 