a divergence free vector-field $u$ is completely determined by its vorticity. From we immediately get $$\label{eq:BS_grad}
\nabla u^i\,=\,-\nabla\left(-\Delta\right)^{-1}\sum_{j=1}^N\partial_j\Omega_{ij}\,.$$ Now, as the symbol of the operator $-\d_i\left(-\Delta\right)^{-1}\d_j$ is $\sigma(\xi)=\xi_i\xi_j/|\xi|^2$, the classical Calderon-Zygmund theorem ensures that[^1] for all $q\in\,]1,+\infty[\,$, if $\,\Omega\in L^q$ then $\,\nabla u\in L^q$ and $$\label{est:CZ}
\|\nabla u\|_{L^q}\leq C\,\frac{q^2}{q-1}\,\|\Omega\|_{L^q}\,.$$

In dimension $N=2$ the vorticity equation is simpler than in the general case due to the absence of the stretching term. Nevertheless, as remarked above, the exterior product involving density and pressure terms makes it impossible to get conservation of Lebesgue norms, which was the fundamental issue to get global existence for the classical system $(E)$. So, we immediately focus on the case $N\geq2$ whatever, in which the vorticity equation reads $$\label{eq:vort}
 \d_t\Omega\,+\,u\cdot\nabla\Omega\,+\,\Omega\cdot\nabla u\,+\,^t\nabla u\cdot\Omega\,+\,
\nabla\left(\frac{1}{\rho}\right)\wedge\nabla\Pi\,=\,0\,,$$ where, for two vector-fields $v$ and $w$, we have set $v\wedge w$ to be the skew-symmetric matrix with components $$\left(v\wedge w\right)_{ij}\,=\,v^jw^i\,-\,v^iw^j\,.$$

Finally, recall that we can associate a flow $\psi$ to the velocity field $u$ of the fluid: it is defined by the relation $$\psi(t,x)\,\equiv\,\psi_t(x)\,:=\,x\,+\,\int^t_0u(\tau,\psi_\tau(x))\,d\tau$$ for all $(t,x)\in[0,T]\times{\mathbb{R}}^N$ and it is, for all fixed $t\in[0,T]$, a diffeomorphism over ${\mathbb{R}}^N$, if $\nabla u\in L^\infty$. Let us remark that the flow is still well-defined (in a generalized 