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Just the simple ODE with Lipschitz coefficient $a$ \begin{align} &\frac{d}{dx}h(x)=a(h(x))\\ &h(0)=x_0 \end{align} We know that the existence and uniqueness holds and the solution is in $C^1$. Does the second derivative of the solution exist (in distributional sense perhaps)? How do I show it? So the solution is in $W^{2,p}$, what's $p$?

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Since $h$ is $C^1$, the composition $a(h(x))$ is Lipschitz. Hence $h' $ is Lipschitz. We can write this as $h\in C^{1,1}$. Or, if you prefer, $h\in W^{2, \infty }$ which says the same thing. All of this applies to finite time intervals, of course.

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