Here's a simple ODE \begin{align} &\frac{d}{dx}h(x)=a(h(x))\\ &h(0)=x_0 \end{align} I want the solution $h(x)$ to be (at least) continuous with its first and second order derivative exist only in the sense of distribution. What condition should $a$ satisfy?
The only result in this direction I'm aware is the DiPerna-Lions theory considering $a\in W^{1,1}$, it gives $C$ solutions in the renormalised sense..
I would like some ideas on how to deal with this problem..
Many thanks!!