Let $X$ be a manifold with corners. Let $\vec v$ be an inner vector field on $X$. The existence and uniqueness theorem for ODE says there's a domain of flow $\mathfrak D(\vec v)\subset \mathbb R\times X$ containing $\left\{ 0 \right\} \times X$, $[0,\varepsilon_x)\times \left\{ x \right\} $ for each $x\in \partial X$ and $(-a_x,b_x)\times \left\{ x \right\}$ for $x\in X\setminus \partial X$.
Question 1. Is the domain of flow $\mathfrak D(\vec v)\subset \mathbb R\times X$ an embedded submanifold with corners?
Question 2. For each $x\in X$, does $(0,x)\in \mathfrak D(\vec v)$ have a product neighborhood $[0,\varepsilon _x)\times U_x\subset \mathfrak D(\vec v)$ with $U_x\subset X$ open?