will make lower regularity hypothesis on the initial data, which will be compensated by the additional geometric assumptions.

Note that in the $2$-dimensional case the equation for the vorticity reads $$\d_t\omega\,+\,u\cdot\nabla\omega\,+\,\nabla\left(\frac{1}{\rho}\right)\wedge\nabla\Pi\,=\,0\,,$$ so it’s not better than in higher dimension due to the presence of the density term, which doesn’t allow us to get conservation of Lebesgue norms. This is also the reason why it’s not clear if Yudovich’s theorem still holds true for non-homogeneous fluids: having $\omega_0\in L^q\cap L^\infty$, combined with suitable hypothesis on $\rho_0$, doesn’t give rise to a local solution.\
So, we will immediately focus on the general case $N\geq2$. We will assume the initial velocity $u_0$ and the initial vorticity $\Omega_0$ to be in some Lebesgue spaces, in order to assure the pressure term to belong to $L^2$, a requirement we could not bypass. As a matter of fact, $\nabla\Pi$ satisfies an elliptic equation with low regularity coefficient, $$-\,{{\rm div}\,}\left(a\,\nabla\Pi\right)\,=\,{{\rm div}\,}F\,,$$ and it can be solved independently of $a$ only in the energy space $L^2$. Moreover, we will suppose $\Omega_0$ to have regularity properties of geometric type. Obviously, we will require some natural but quite general hypothesis also on the initial density $\rho_0$ of the fluid: we 