initial data $x(T_n), v(T_n)$.\
We say that $x$ solves (\[cs\]) on the time interval $[0,T]$, with weight given by (\[psi\]) and arbitrary initial data $x(0), v(0)$ if and only if for all $T_n$ and all $t\in(T_n,T_{n+1})$ the function $x\in (C^1([0,T]))^{Nd}$ is a weak in $(W^{2,1}([T_{n},t]))^{Nd}$ solution of (\[cs\]), the initial data are correct (i.e. $x(0)=x(T_{-1})$ and $v(0)=v(T_{-1})$) and for some $n$, we have $T\leq T_n$.

This definition may not be clear at the first glance. It is somewhat weaker than a weak solution but stronger than an a.e. solution. Such definition is based on the idea described in section \[outline\]: the solution exists in a weak sense between two collision times $T_{n-1}$ and $T_n$. However as it approaches $T_n$, the second derivative of $x$ may blow up. Despite this $v$ is still continuous in a left sided neighborhood of $T_n$ and has a limit at $T_n^-$. Therefore we may continuously define it at $T_n$ ensuring existence of unique initial data for local weak existence in $[T_n,T_{n+1})$.

Existence up to the time of collision {#exi}
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Before we begin let us state the following simple remark.

\[sususus\] Property 4 from Proposition \[properties\] implies equicontinuity of $x^n$, thus by Arzela-Ascoli theorem there 