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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 $$ \frac{\mathrm d}{\mathrm d t} \theta_{s, t} (x) =: \dot \theta_{s, t} (x) = b(t, \theta_{s, t} (x)), \quad t \in [0, T], $$ 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 said that $\psi_{s, t}$ is a $C^\infty$-diffeomorphism.
2. This paper said that $\sup_{0 \le s <t \le T} \|\nabla \psi_{s, t} \|_\infty < \infty$.

I'm aware of Carathéodory's existence theorem but it seems not address above statements.

Could you elaborate on references for above claims?

Thank you so much for your help!

Update Chapter V Dependence on Initial Conditions and Parameters in Hartman's Ordinary Differential Equations only concerns the case $b$ is jointly continuous in $(t, x)$.

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 $$ \frac{\mathrm d}{\mathrm d t} \theta_{s, t} (x) =: \dot \theta_{s, t} (x) = b(t, \theta_{s, t} (x)), \quad t \in [0, T], $$ 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 said that $\psi_{s, t}$ is a $C^\infty$-diffeomorphism.
2. This paper said that $\sup_{0 \le s <t \le T} \|\nabla \psi_{s, t} \|_\infty < \infty$.

I'm aware of Carathéodory's existence theorem but it seems not address above statements.

Could you elaborate on references for above claims?

Thank you so much for your help!

Update I have found the following textbooks that concern only the case where $b$ is jointly continuous in $(t, x)$.

1. Chapter V Dependence on Initial Conditions and Parameters in Hartman's Ordinary Differential Equations

2. Section 3.5. Theorem of differentiable dependence in Viana/Espinar's Differential Equations A Dynamical Systems Approach to Theory and Practice.

3. Section 8.9 Differentiating Solutions with Respect to ICs in Kelley/Peterson's The Theory of Differential Equations Classical and Qualitative.

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 $$ \frac{\mathrm d}{\mathrm d t} \theta_{s, t} (x) =: \dot \theta_{s, t} (x) = b(t, \theta_{s, t} (x)), \quad t \in [0, T], $$ 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 said that $\psi_{s, t}$ is a $C^\infty$-diffeomorphism.
2. This paper said that $\sup_{0 \le s <t \le T} \|\nabla \psi_{s, t} \|_\infty < \infty$.

I'm aware of Carathéodory's existence theorem but it seems not address above statements.

Could you elaborate on references for above claims?

Thank you so much for your help!

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 $$ \frac{\mathrm d}{\mathrm d t} \theta_{s, t} (x) =: \dot \theta_{s, t} (x) = b(t, \theta_{s, t} (x)), \quad t \in [0, T], $$ 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 said that $\psi_{s, t}$ is a $C^\infty$-diffeomorphism.
2. This paper said that $\sup_{0 \le s <t \le T} \|\nabla \psi_{s, t} \|_\infty < \infty$.

I'm aware of Carathéodory's existence theorem but it seems not address above statements.

Could you elaborate on references for above claims?

Thank you so much for your help!

Update Chapter V Dependence on Initial Conditions and Parameters in Hartman's Ordinary Differential Equations only concerns the case $b$ is jointly continuous in $(t, x)$.

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 said that $\psi_{s, t}$ is a $C^\infty$-diffeomorphism.
2. This paper said that $\sup_{0 \le s <t \le T} \|\nabla \psi_{s, t} \|_\infty < \infty$.

I'm aware of Carathéodory's existence theorem but it seems not address above statements.

Could you elaborate on references for above claims?

Thank you so much for your help!

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 $$ \frac{\mathrm d}{\mathrm d t} \theta_{s, t} (x) =: \dot \theta_{s, t} (x) = b(t, \theta_{s, t} (x)), \quad t \in [0, T], $$ 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 said that $\psi_{s, t}$ is a $C^\infty$-diffeomorphism.
2. This paper said that $\sup_{0 \le s <t \le T} \|\nabla \psi_{s, t} \|_\infty < \infty$.

I'm aware of Carathéodory's existence theorem but it seems not address above statements.

Could you elaborate on references for above claims?

Thank you so much for your help!