  We present a divergence free vector field in the Sobolev space $H^1$ such
that the flow associated to the field does not belong to any Sobolev space. The
vector field is deterministic but constructed as the realization of a random
field combining simple elements. This construction illustrates the optimality
of recent quantitative regularity estimates as it gives a straightforward
example of a well-posed flow which has nevertheless only very weak regularity.
