On smooth approximations of rough vector fields and the selection of flows

In this work we deal with the selection problem of flows of an irregular vector field. We first summarize an example from \cite{CCS} of a vector field $b$ and a smooth approximation $b_\epsilon$ for which the sequence $X^\epsilon$ of flows of $b_\epsilon$ has subsequences converging to different flows of the limit vector field $b$. Furthermore, we give some heuristic ideas on the selection of a subclass of flows in our specific case.