} \| u - v\|_\infty.$$  0.  Given $u \in C_D([0,0])$ and $t \in (0,\tau^u)$ there exist $C_{u,t}, \delta_{u,t} > 0$ such that      -   $\tau^v > t$ for any $v \in B_{\delta_{u,t}}(u)$      -   For any $v \in B_{\delta_{u,t}}(u)$ $$\| {\partial}_x U^{u}(t,\cdot) - {\partial}_x U^v (t,\cdot) \|_\infty \leq C_{