Let $b:\mathbb R_+ \times \mathbb R^d \to \mathbb R^d$ be measurable such that for all $n \in \mathbb N$ we have
$$
\sup_{t \ge 0} |b(t, 0)| + \sup_{t \ge 0} \sup_{x \in \mathbb R^d} |\nabla^n_x b (t, x)| < \infty.
$$

Fix $T>0$. For a fixed $(s, x) \in \mathbb R_+ \times \mathbb R^d$, we consider the ODE
$$
\frac{\mathrm d \theta_{s, t} (x)}{\mathrm d t} =: \dot \theta_{s, t} (x) = b(t, \theta_{s, t} (x)),
\quad t \in [0, T],
$$
under the condition $\theta_{s, s} (x) = x$. For $0 \le s <t \le T$, we define
$$
\psi_{s, t}:\mathbb R^d \to \mathbb R^d, x \mapsto \theta_{s, t} (x).
$$

1. This [paper](https://www.sciencedirect.com/science/article/pii/S0022039620304988) said that $\psi_{s, t}$ is a $C^\infty$-diffeomorphism.
2. This [paper](https://www.sciencedirect.com/science/article/pii/S0022039623002255) said that $\sup_{0 \le s <t \le T} \|\nabla \psi_{s, t} \|_\infty < \infty$.

I'm aware of Carathéodory's existence [theorem](https://www.wikiwand.com/en/Carath%C3%A9odory%27s_existence_theorem#introduction) but it seems not address above statements.

>Could you elaborate on references for above claims?

Thank you so much for your help!