will introduce the notion of paravector-field, as defined in [@D1999]: it will play a fundamental role in our analysis, because it is, in a certain sense, the principal part of the derivation operator along a fixed vector-field. Moreover, we will also quote some results about transport equations in Hölder spaces and about elliptic equations in divergence form with low regularity coefficients.

In section \[s:striated\], we will finally be able to tackle the proof of our result about striated regularity. First of all, we will state a priori estimates for suitable smooth solutions of the Euler system . Then from them we will get, in a quite classical way, the existence of a solution with the required properties: we will construct a sequence of regular solutions of system with approximated data, and, using a compactness argument, we will show the convergence of this sequence to a “real” solution. Proving preservation of the geometric structure requires instead strong convergence in rough spaces of type ${\mathcal}{C}^{-\alpha}$ (for some $\alpha>0$). The uniqueness of the solution will follow from a stability result for our equations. In the following section, we will also give an estimate from below for the lifespan of the solution.

Finally, we 