and the velocity field

From first equation of , it follows that $$\rho(t,x)\,=\,\rho_0\left(\psi^{-1}_t(x)\right)\,,$$ so, as the flow $\psi_t$ is a diffeomorphism over ${\mathbb{R}}^N$ at all fixed time, we have that $$0\,<\,\rho_*\,\leq\,\rho(t)\,\leq\,\rho^* \label{est:rho_L^inf}\,.$$ Applying the operator $\,\d_i$ to the same equation, using classical $L^p$ estimates for the transport equation and Gronwall’s lemma, we get $$\label{est:Drho_L^inf}
 \|\nabla\rho(t)\|_{L^\infty}\,\leq\,\|\nabla\rho_0\|_{L^\infty}\,\exp\left(C\int^t_0\|\nabla u\|_{L^\infty}\,d\tau\right)\,.$$

From the equation for the velocity, instead, we get in the same way $$\|u(t)\|_{L^p}\,\leq\,\|u_0\|_{L^p}\,+\,\int^t_0\left\|\frac{\nabla\Pi}{\rho}\right\|_{L^p}\,d\tau\,;$$ so, using and Hölder inequalities, the following estimate holds for some $\theta\in\,]0,1[\,$: $$\label{est:u_L^p}
 \|u(t)\|_{L^p}\,\leq\,\|u_0\|_{L^p}\,+\,\frac{C}{\rho_*}\,\int^t_0\|\nabla\Pi\|^\theta_{L^2}\,
\|\nabla\Pi\|^{1-\theta}_{L^\infty}\,d\tau\,.$$

Let us observe that, as regularity of the pressure goes like that of the velocity field, one can try to estimate directly the $L^p$ norm of the pressure term. Unfortunately, we can’t solve its (elliptic) equation in this space without assuming a smallness condition on the gradient of the density. So, we will prove that $\nabla\Pi$ is in $L^2\cap L^\infty$, which is actually stronger than the previous property and requires no other hypothesis on the density term.

From it’s clear that we need an estimate for the $L^\infty$ norm for the gradient of the velocity. As remarked before, we can’t expect to get it from the hypothesis $\Omega\in L^\infty$; the key will be the further 