Hölder continuous dependence on parameters for solutions of ODE

Question

This is a cross-post from Stackexchange Mathematics (https://math.stackexchange.com/questions/3893961/h%c3%b6lder-continuous-dependence-on-parameters-for-solutions-of-ode).

We have the following standard result for continuous dependence of the initial value for ODEs with a continuous right-hand side (Satz 8.18 in https://www.mathematik.hu-berlin.de/~baum/Skript/DGL-2012.pdf):

Let $F:U \subset \mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}^n$ on $U$ be continuous and Lipschitz continuous with respect to the $\mathbb{R}^n$-variable with Lipschitz constant $L$. Let $(x_0,t_0),(x_0^*,t_0) \in U$ and $\varphi_{x_0},\varphi_{x_0^*} : [t_0 - \epsilon, t_0 + \epsilon] \rightarrow \mathbb{R}^n$ be solutions of the ODE $x'=F(x,t)$ with initial values $\varphi_{x_0}(t_0)=x_0$ and $\varphi_{x_0^*}(t_0)=x_0^*$. Then: $$| \varphi_{x_0}(t)-\varphi_{x_0^*}(t) | \leq |x_0-x_0^*| \cdot e^{L|t-t_0|} \forall t \in [t_0-\epsilon,t_0+\epsilon].$$

Question: is there a similar result for Holder continuity? My dream result would be

Let $F:U \subset \mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}^n$ on $U$ be smooth and $\alpha$-Holder continuous with respect to the $\mathbb{R}^n$-variable, with $\alpha$-Holder norm bounded by $L$. Let $(x_0,t_0),(x_0^*,t_0) \in U$ and $\varphi_{x_0},\varphi_{x_0^*} : [t_0 - \epsilon, t_0 + \epsilon] \rightarrow \mathbb{R}^n$ be solutions of the ODE $x'=F(x,t)$ with initial values $\varphi_{x_0}(t_0)=x_0$ and $\varphi_{x_0^*}(t_0)=x_0^*$. Then there exist a universal constant $c$ independent of $F$, and $\beta \in (0,1)$ such that $$| \varphi_{x_0}(t)-\varphi_{x_0^*}(t) | \leq c|x_0-x_0^*|^{\beta} \cdot e^{L|t-t_0|} \forall t \in [t_0-\epsilon,t_0+\epsilon].$$

Note that I am happy to assume my function is smooth in order to have a unique solution. I also note that I can use a $C^k$-bound for $F$ to get a $C^k$ bound for the solution depending on the initial value. But what I need is a $C^{0,\beta}$-bound for the solution that only depends on the $C^{0,\alpha}$-bound for $F$. I can imagine something like this exists for $\beta=\alpha/2$, but maybe even $\beta=\alpha$ is possible.

I know that the proof of cited theorem cannot be adapted to prove my dream result. I also found "Agarwal, Lakshmikantham: Uniqueness and nonuniqueness criteria for ordinary differential equations" to make some statements about uniqueness of the solution for a Holder-continuous right-hand side $F$. But I did not find anything resembling the estimate I need.

Context: I have two metrics on a compact manifold, $g_1, g_2$, satisfying the estimate $||g_1-g_2||_{C^{1,\alpha},g_1}<c_1$. I also have a vector $\eta$ satisfying $|| \eta ||_{C^{1,\alpha},g_1} < c_2$. I would like to have an estimate $|| \eta ||_{C^{1,\beta},g_2} < F(c_1,c_2)$, where $\beta$ can depend on $\alpha$, but should not depend on $\eta$, and $F(c_1,c_2)$ is some universal expression in $c_1$ and $c_2$. I believe that my dream ODE result from above would give me such an estimate.

Answer 1

No, what you want is not possible. (Basically, you can approximate the nonunique example with smooth examples which have slightly moved initial data. This gives you two nearby initial data points that move apart arbitrarily fast.)

Consider the case $n = 1$, let $F_0(x) = \begin{cases} 0 & x \leq 0 \newline \sqrt{x} & x> 0\end{cases}$. Let $F_\delta$ be smooth functions that equal $F_0$ outside $(-\delta,\delta)$; you can choose $F_\delta$ to have uniform $C^{\alpha}$ norm for $\alpha < \frac12$. (Possibly also $\alpha = \frac12$, have to check.)

If you take $\delta \ll \epsilon^2$, and let $x_0 = -\delta$ and $x_0^* = +\delta$, you have that $\varphi_{x_0} \equiv -\delta$ but $\varphi_{x_0^*}(\epsilon) > \frac14 \epsilon^2$. Taking $\delta\to 0$ you see an arbitrarily large factor of inflation even compared against $(2\delta)^\beta$ for any $\beta > 0$.