Critical non-Sobolev regularity for continuity equations with rough velocity fields

We present a divergence free vector field in the Sobolev space H1H1 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.