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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!!

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Just to clarify: do you want both existence and uniqueness of solutions for every initial value? –  user31373 Sep 12 '12 at 18:17
    
Yes that'd be good, but it's been a day I'm just not making much progress at all.. –  dynamic89 Sep 12 '12 at 22:32
1  
As DiPerna and Lions remark in their groundbreaking paper, "It has been a permanent question to extend any part of this elementary theory to less regular vector fields... Various (somewhat limited) extensions have been proposed but seemed to be of restricted applicability in view of standard examples." This book has a catalog of such extensions, only slighly beyond Lipschitz in practical terms. By the way, why not impose the Lipschitz condition? It allows solutions that are not $C^2$ in general, but are necessarily in $C^{1,1}=W^{2,\infty}$. –  user31373 Sep 12 '12 at 23:11
    
Thanks so much for your suggestion. This ODE is actually related to a SDE problem I'm working on, and in that problem $a\in W^{1,2}$ is ideal. I'll definitely look into it and get back to you perhaps tomorrow. Many thanks! –  dynamic89 Sep 12 '12 at 23:32

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