Suppose we have a (say compactly supported) $C^0$-vector field $X:\mathbb R^n\to\mathbb R^n$ such that for every $x\in\mathbb R^n$ there is a unique $C^1$-curve $\gamma:\mathbb R\to\mathbb R^n$ solving $\dot\gamma\_x(t)=X(\gamma\_x(t))$ with $\gamma\_x(0)=x$.


Then the ode flow $\mathcal F\_X$ is pointwisely defined in the way that $\mathcal F\_X(t,x)=\gamma\_x(t)$.


My **question** is: Must $\mathcal F\_X$ be a continuous map with respect to $x$?


Certainly there is no problem if $X$ is Lipschitz or just satisfies the Osgood condition. Since we have the regularity estimate of ODE flows with respect to their modulus of continuity.


When $X$ is H"older and somehow its ODE is uniquely solvable at every point, could their be continuous dependence? And if not how does the blow up occur.



Does ODE uniqueness unconditionally implies the flow continuity?
Yes, the uniqueness of solutions implies continuous dependence on initial conditions and parameters. See Theorem 3.2 in Hartman's "Ordinary differential equations".


