that the vorticity is more regular along some fixed directions, given by a nondegenerate family of vector-fields (see definition \[d:stri\] below). This notion was introduced first by Bony in [@Bo1979] in studying hyperbolic equations, and then adapted by Alinhac (see [@A]) and Chemin (see [@Ch1988]) for nonlinear partial differential equations.

In [@G-SR], Gamblin and Saint-Raymond proved that striated regularity is preserved during the evolution in any dimension $N\geq3$, but, as already remarked, only locally in time (see also [@S]). They also obtained global results if the initial data have other nice properties (e.g., if the initial velocity is axisymmetric).

As Euler system is, in a certain sense, a limit case of the Navier-Stokes system as the viscosity of the fluid goes to $0$, it’s interesting to study if there is also “convergence” of the geometric properties of the solutions. Recently Danchin proved results on striated regularity for the solutions of the Navier-Stokes system $$\left\{\begin{array}{l}
\d_tu\,+\,u\cdot\nabla u\,-\,\nu\,\Delta u\,+\,\nabla\Pi\,=\,0 \\[1ex]
{{\rm div}\,}u\,=\,0\,,
\end{array}\right.
\leqno(N\!S_\nu)$$ in [@D1997] for the $2$-dimensional case, in [@D1999] for the general one. Already in the former paper, he had to dismiss the vortex patch structure “stricto sensu” due to the presence of the viscous term, which comes out 