are still in ${\mathcal}{C}^{\varepsilon}$.

Then there exists a constant $C$, depending only on the space dimension $N$ and on the number of vector-fields $m$, such that, for all skew-symmetric matrices $\Omega$ with coefficients in $L^q\cap{\mathcal}{C}^{\varepsilon}_Y$, the corresponding (by ) divergence-free vector-field $u$ satisfies $$\left\|\nabla u\right\|_{L^\infty}\,\leq\,C\,\left(\frac{q^2}{q-1}\,\|\Omega\|_{L^q}\,+\,\frac{1}{{\varepsilon}\,(1-{\varepsilon})}\,\|\Omega\|_{L^\infty}
\log\left(e\,+\,\frac{\left\|\Omega\right\|_{{\mathcal}{C}^{\varepsilon}_Y}}{\|\Omega\|_{L^\infty}}\right)\right)\,. \label{est:Du_L^inf}$$

### Estimates for the vorticity

As in [@D-F], using the well-known $L^q$ estimates for transport equation and taking advantage of Gronwall’s lemma and Hölder inequality in Lebesgue spaces, from we obtain, for some $\gamma\in\,]0,1[\,$, $$\begin{aligned}
 \|\Omega(t)\|_{L^q} & \leq & C\,\exp\left(\int^t_0\|\nabla u\|_{L^\infty}d\tau\right)\times \label{est:Om_L^q} \\
& & \times\left(\|\Omega_0\|_{L^q}\,+\,\frac{1}{\left(\rho_*\right)^2}\int^t_0e^{-\int^\tau_0\|\nabla u\|_{L^\infty}d\tau'}
\left\|\nabla\rho\right\|_{L^\infty}\,\left\|\nabla\Pi\right\|^\gamma_{L^2}\,\left\|\nabla\Pi\right\|^{1-\gamma}_{L^\infty}
d\tau\right). \nonumber\end{aligned}$$

Moreover, of course an analogue estimate holds also for the $L^\infty$ norm: $$\begin{aligned}
 \|\Omega(t)\|_{L^\infty} & \leq & C\,\exp\left(\int^t_0\|\nabla u\|_{L^\infty}d\tau\right)\times \label{est:Om_L^inf} \\
& & \times\left(\|\Omega_0\|_{L^\infty}\,+\,\frac{1}{\left(\rho_*\right)^2}\int^t_0e^{-\int^\tau_0\|\nabla u\|_{L^\infty}d\tau'}
\left\|\nabla\rho\right\|_{L^\infty}\,\left\|\nabla\Pi\right\|_{L^\infty}\,d\tau\right)\,. \nonumber\end{aligned}$$

\[r:q\] Let us fix the index $p$ pertaining to $u$ and let us call ${\overline}{q}$ the real number in $[2,+\infty[$ such that $1/p\,+\,1/{\overline}{q}\,=\,1/2$. From our hypothesis, it’s clear that $q\leq{\overline}{q}$; therefore, thanks to Hölder and Young inequalities, we have $$\left\|\Omega\right\|_{L^{{\overline}{q}}}\,\leq\,\left\|\Omega\right\|^\eta_{L^q}\,\left\|\Omega\right\|^{1-\eta}_{L^\infty}\,\leq\,
\left\|\Omega\right\|_{L^q\cap L^\infty}\,.$$

### Estimates for the pressure term {#sss:est-p}

Now, let us focus on the pressure term: taking the divergence of the second equation of system , we discover that it solves the elliptic equation $$\label{eq:Pi}
 