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

We present a divergence free vector field in the Sobolev space H¹ 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.