An Osgood modulus of continuity is an increasing function $\omega:(0,1]\to(0,1]$ such that $\int_0^1\frac{dt}{\omega(t)}=\infty$.

We say a vector field $X$ satisfies Osgood condition with modulus $\omega$ if locally there is a and $|X(x)-X(y)|\le\omega(|x-y|)$ (here I don't want a constant).

When $X$ satisfies Osgood condition, we know the ODE has uniqueness hence the flow $\mathfrak F_X(t,x)$ of $X$, given by $\begin{cases}\frac d{dt}\mathfrak F_X(t;x)=X(\mathfrak F_X(t;x))\\\mathfrak F_X(t;x)=x\end{cases}$ is well-defined locally in a small time interval.

**My question** is, given a Osgood modulus $\omega$, is there any estimate under modulus of continuity, that for any Osgood vector fields $X$ of modulus $\omega$, what is the modulus of continuity for $\mathfrak F_X$?

In DiPerna-Lions people mostly treat the cases where the vector field has bounded divergence, and the flow is not a regular Lagrange flow. But this is not true for general Osgood vector fields. I found a paper here of Clop the talk about the general Osgood vector fields but the regularity result seems still rough if we focus on some "smoother" modulus.

Consider a log-Lipschitz ODE $x'=x\log|x|$, $\begin{cases}\frac d{dt}\phi(t;s)=\phi\log|\phi|\\\phi(0;s)=s\end{cases}$, then $\phi(t;s)=|s|^{e^t}\operatorname{sgn}s$. This flow is not regular Lagrange.

Here globally $\phi$ is $\bigcup_{\alpha>0}C^\alpha$. And locally near 0 we have for any $\epsilon>0$ there is a neighborhood $0\in U_\epsilon\subset\mathbb R\times\mathbb R$, such that $\phi$ is $C^{1-\epsilon}(U_\epsilon)$.

For general log-Lipschitz vector field, is the flow (global or local) regularity in this case sharp?

I think in Saric's paper Here, for Zygmund vector field in 1-dim, the global result is $\bigcup_{\alpha>0}C^\alpha$ holds (so-called a **quasisymmetric** flow). What about the higher dimension?

If we consider log-log-Lipschitz case: Consider $x'=x\log|x|\log|\log|x||$, $\begin{cases}\frac d{dt}\phi(t;s)=\phi\log|\phi|\log|\log|\phi||\\\phi(0;s)=s\end{cases}$, then near 0 we have $\phi(t;s)=|s|^{|\log s|^{e^t-1}}\operatorname{sgn}s$. It's not even locally Holder near 0.

It's likely that the modulus of the flow depends on the asymtoptic behavior of $\int_\delta^1\frac{dt}{\omega(t)}$ as $\delta\to0$.