# Does ODE uniqueness unconditionally implies the flow continuity?

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.

• 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. Nov 2 '20 at 9:28

## 1 Answer

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

• Do you mean using Arzela-Ascoli to find a convergent subsequence and show that the limit solution is as desired? Nov 2 '20 at 4:13
• @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. Nov 2 '20 at 15:31