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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.

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    $\begingroup$ Uniqueness allows to define a solution as a function of the initial datum, $x\mapsto \gamma$. The graph of this map is locally compact by Ascoli-Arzelà. Local compactness of the graph implies continuity. $\endgroup$ Nov 2 '20 at 9:28
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Yes, the uniqueness of solutions implies continuous dependence on initial conditions and parameters. See Theorem 3.2 in Hartman's "Ordinary differential equations".

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  • $\begingroup$ Do you mean using Arzela-Ascoli to find a convergent subsequence and show that the limit solution is as desired? $\endgroup$
    – yaoliding
    Nov 2 '20 at 4:13
  • $\begingroup$ @yaoliding, solutions with sufficiently close (to a given one) initial conditions exist on a common sufficiently small time interval and lie in a common bounded set. Thus, Arzelà–Ascoli theorem gives the precompactness of any sequence of solutions, for which initial conditions converge to a given one. To show the convergence of solutions one uses a simple criterion: a sequence converges (to some limit) if and only if any of its subsequences has a convergent (to the same limit) subsequence. $\endgroup$
    – demolishka
    Nov 2 '20 at 15:31

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