assumption of more regularity of the vorticity along the directions given by the family $X_0$.\
Here we quote also a fundamental lemma, whose proof can be found in [@B-C-D] (chapter 7) for the $2$-dimensional case, in [@D1999] (section 3) and [@G-SR] (again section 3) for the general one. It is the key point to get the velocity field to be Lipschitz and it turns out to be immediately useful in the sequel.

\[l:Du\_L\^inf\] Fix ${\varepsilon}\in\,]0,1[$ and an integer $m\geq N-1$ and take a non-degenerate family $Y=\left(Y_\lambda\right)_{1\leq\lambda\leq m}$ of ${\mathcal}{C}^{\varepsilon}$ vector-fields over ${\mathbb{R}}^N$ such that also their divergences are in $\,{\mathcal}{C}^{\varepsilon}$.

Then, for all indices $1\leq i,j\leq N$, there exist ${\mathcal}{C}^{\varepsilon}$ functions $a_{ij}$, $b^{k\lambda}_{ij}$ (with $1\leq k\leq N$, $1\leq\lambda\leq m$) such that, for all $(x,\xi)\in{\mathbb{R}}^N\times{\mathbb{R}}^N$, the following equality holds: $$\xi_i\,\xi_j\,=\,a_{ij}(x)|\xi|^2\,+\,\sum_{k,\lambda}b^{k\lambda}_{ij}(x)\left(Y_\lambda(x)\cdot\xi\right)\xi_k\,.$$ Moreover, the functions in the previous relation could be chosen such that $$\left\|a_{ij}\right\|_{L^\infty}\;\leq\;1\qquad\mbox{ and }\qquad
\left\|b^{k\lambda}_{ij}\right\|_{{\mathcal}{C}^{\varepsilon}}\;\leq\;C\,\frac{m^{2N-2}}{I(Y)}\,\,|||Y|||\,^{9N-10}_{{\mathcal}{C}^{\varepsilon}}\,.$$

Now, we can state the stationary estimate which says that the velocity field $u$ is Lipschitz. This can be done as in the classical case, because it’s based only on the Biot-Savart law.

\[p:Du\_L\^inf\] Fix ${\varepsilon}\in\,]0,1[$ and $q\in\,]1,+\infty[$ and take a non-degenerate family $Y=\left(Y_\lambda\right)_{1\leq\lambda\leq m}$ of ${\mathcal}{C}^{\varepsilon}$ vector-fields over ${\mathbb{R}}^N$ such that also their divergences 