sense) even if $u$ is only log-Lipschitz continuous, but it is no more a diffeomorphism (see e.g. chapter 3 of [@B-C-D], or [@C1995], for more details).

Let us now introduce the geometric properties we are handling throughout this paper. The first notion we are interested in is the *striated regularity*, that is to say initial data are more regular along some given directions.

So, let us take a family $X=\left(X_\lambda\right)_{1\leq\lambda\leq m}$ of $m$ vector-fields with components and divergence of class ${\mathcal}{C}^{\varepsilon}$ for some fixed ${\varepsilon}\in\,]0,1[\,$. We also suppose this family to be non-degenerate, i.e. $$I(X)\,:=\,\inf_{x\in{\mathbb}{R}^N}\,\sup_{\Lambda\in\Lambda^m_{N-1}}\,\left|\stackrel{N-1}{\wedge}X_\Lambda(x)\right|^{\frac{1}{N-1}}
\,\,\,>\,0\,.$$ Here $\Lambda\in\Lambda^m_{N-1}$ means that $\Lambda=\left(\lambda_1,\ldots,\lambda_{N-1}\right)$, with each $\lambda_i\in\left\{1,\ldots,m\right\}$ and $\lambda_i<\lambda_j$ for $i<j$, while the symbol $\stackrel{N-1}{\wedge}X_\Lambda$ stands for the element of ${\mathbb{R}}^N$ such that $$\forall\,\,Y\in{\mathbb{R}}^N\,,\qquad\left(\stackrel{N-1}{\wedge}X_\Lambda\right)\cdot Y\,=\,
{{\rm det}}\left(X_{\lambda_1}\ldots X_{\lambda_{N-1}},Y\right)\,.$$ For each vector-field of this family we put $${\widetilde}{\|}X_\lambda\|_{{\mathcal}{C}^{\varepsilon}}\,:=\,\|X_\lambda\|_{{\mathcal}{C}^{\varepsilon}}\,+\,\|{{\rm div}\,}X_\lambda\|_{{\mathcal}{C}^{\varepsilon}}\,,$$ while we will use the symbol $|||\,\cdot\,|||$ in considering the supremum over all indices $\lambda\in\Lambda^m_1=\{1\ldots m\}$.

\[d:stri\] Take a vector-field $Y$ with components and divergence in ${\mathcal}{C}^{\varepsilon}$ and fix a $\eta\in[{\varepsilon},1+{\varepsilon}]$. A function $f\in L^\infty$ is said to be of class ${\mathcal}{C}^\eta$ along $Y$, and we write $\,f\in{\mathcal}{C}^\eta_Y$, if ${{\rm div}\,}\left(f\,Y\right)\in\,{\mathcal}{C}^{\eta-1}\left({\mathbb{R}}^N\right)$.

If $X=\left(X_\lambda\right)_{1\leq\lambda\leq m}$ is a non-degenerate family of vector-fields as above, we define $${\mathcal}{C}^\eta_X\,:=\,\bigcap_{1\leq\lambda\leq 