-\,{{\rm div}\,}\left(\frac{\nabla\Pi}{\rho}\right)\,=\,{{\rm div}\,}\left(u\cdot\nabla u\right)\,.$$ From this, remembering our hypothesis and remark \[r:q\], estimate and lemma \[l:laxmilgram\], the control of $L^2$ norm immeditely follows: $$\label{est:Pi_L^2}
 \frac{1}{\rho^*}\,\|\nabla\Pi\|_{L^2}\,\leq\,C\,\|u\|_{L^p}\,\|\Omega\|_{L^q\cap L^\infty}\,.$$

Moreover, we have that $\nabla\Pi$ belongs also to $L^\infty$, and so $\nabla\Pi\in L^a$ for all $a\in[2,+\infty]$. In fact, we are going to show a stronger claim, that is to say $\nabla\Pi\in{\mathcal}{C}^1_*$. Cutting in low and high frequencies, we have that $$\|\nabla\Pi\|_{{\mathcal}{C}^1_*}\,\leq\,\|\Delta_{-1}\nabla\Pi\|_{{\mathcal}{C}^1_*}\,+\,\|({{\rm Id}\,}-\Delta_{-1})\nabla\Pi\|_{{\mathcal}{C}^1_*}\,\leq\,
C\left(\|\nabla\Pi\|_{L^2}\,+\,\|\Delta\Pi\|_{{\mathcal}{C}^0_*}\right)\,.$$ Now, from we obtain $$\label{eq:Lapl-Pi}
 -\,\Delta\Pi\,=\,\nabla\left(\log\rho\right)\cdot\nabla\Pi\,+\,\rho\,\,{{\rm div}\,}\left(u\cdot\nabla u\right)\,.$$ From this relation, the fact that ${{\rm div}\,}(u\cdot\nabla u)=\nabla u:\nabla u$ and the immersion $L^\infty\hookrightarrow{\mathcal}{C}^0_*$, we get $$\begin{aligned}
\|\Delta\Pi\|_{{\mathcal}{C}^0_*}\;\;\leq\;\;\|\Delta\Pi\|_{L^\infty} & \leq &
\left\|\nabla\left(\log\rho\right)\cdot\nabla\Pi\right\|_{L^\infty}\,+\,
\left\|\rho\,\,{{\rm div}\,}\left(u\cdot\nabla u\right)\right\|_{L^\infty} \\
 & \leq & C\left(\left\|\nabla\rho\right\|_{L^\infty}\,\|\nabla\Pi\|_{L^\infty}\,+\,\rho^*\,
\left\|\nabla u\right\|^2_{L^\infty}\right)\,.\end{aligned}$$ Now, by interpolation we have, for some $\beta\in\,]0,1[\,$ depending only on the dimension $N$, $$\|\nabla\Pi\|_{L^\infty}\,\leq\,C\,\|\nabla\Pi\|^\beta_{{\mathcal}{C}^{-N/2}}\,\|\nabla\Pi\|^{1-\beta}_{{\mathcal}{C}^1_*}\,\leq\,
C\,\|\nabla\Pi\|^\beta_{L^2}\,\|\nabla\Pi\|^{1-\beta}_{{\mathcal}{C}^1_*}\,.$$ Thanks to Young’s inequality, from this relation and one finally gets $$\label{est:Pi_C^1_*}
 \left\|\nabla\Pi\right\|_{{\mathcal}{C}^1_*}\,\leq\,C\,\left(\left(1+\|\nabla\rho\|^\delta_{L^\infty}\right)
\|u\|_{L^p}\,\|\Omega\|_{L^q\cap L^\infty}\,+\,
\rho^*\,\left\|\nabla u\right\|^2_{L^\infty}\right)\,,$$ for some $\delta$ depending only on $\beta$, and so finally on the space dimension $N$. So our claim is now proved.

Finally, we want to find a bound on the second derivatives of the pressure term. For doing this, we will need the striated norm of $\nabla\Pi$. In fact, passing in Fourier variables and using lemma \[l:Du\_L\^inf\], 