suppose $\rho_0$ to be bounded with its gradient and that it satisfies geometric assumptions analogous to those for $\Omega_0$. Let us point out that proving the velocity field to be Lipschitz, which was the key part in the homogeneous case, works as in this setting: it relies on Biot-Savart law and it requires no further hypothesis on the density term. Let us also remark that no smallness conditions over the density are needed. Of course, we will get only local in time results. Moreover, we will see that geometric structures propagate also to the velocity field and to the pressure term.

Our paper is organized in the following way.

In the first part, we will recall basic facts about Euler system: some properties of the vorticity and how to associate a flow to the velocity field. In this section we will also give the definition of the geometric properties we are studying and we will state the main results we got about striated and conormal regularity.

In section \[s:tools\], we will explain the mathematical tools, from Fourier Analysis, we need to prove our claims: we will present the Littlewood-Paley decomposition and some techniques coming from paradifferential calculus. In particular, we 