the space. Approximate solutions $x^n$ and $v^n$ are defined in section \[ap\]. Moreover by $B_i(t)$ we will denote the set of all indexes $j$, such that the trajectory of $x_j$ does not coincide with the trajectory of $x_i$ as of the time $t$. Assuming that the trajectories, once coinciding cannot separate, we may define it as $$\begin{aligned}
\label{bi}
B_i(t):=\{k=1,...,N: x^n_k(t)\neq x^n_i(t)\ {\rm or}\ v^n_k(t)\neq v^n_i(t)\},\end{aligned}$$ since any two particles with sufficiently smooth trajectories have the same position and velocity at the time $t$, if and only if they move on the same trajectory. Further, by $W^{k,p}(\Omega)$ we denote the Sobolev space of the functions with up to $k$-th weak derivative belonging to the space $L^p(\Omega)$ and by $C^\infty_c(\Omega)$ we denote the space of smooth and compactly supported functions.\
In the sequel we use the following lemmas, which are well known results from measure theory.

\[vit\] Let $\Omega\subset\r^d$ be bounded and $f_n$ be a sequence of functions converging a.e. to an a.e. finite function $f$. Then if $f_n$ are uniformly integrable, then $f\in L^1(\Omega)$ and $$\begin{aligned}
\lim_{n\to\infty}\int_\Omega|f_n-f|d\mu = 0.\end{aligned}$$

\[weakstrong\] Let $\Omega\subset\r^d$ and $f_n,f,g_n,g:\Omega\to\r$ be measurable functions. If $f_n\to f$ a.e. in $\Omega$, $f_n$ is uniformly bounded in $L^\infty(\Omega)$ and $g_n\rightharpoonup 