Critical non Sobolev regularity for continuity equations with rough force fields

TL;DR

This paper constructs a divergence-free vector field in Sobolev space that generates a flow with even weaker regularity, demonstrating the limits of current regularity estimates for continuity equations with rough force fields.

Contribution

It provides a deterministic example of a Sobolev regular vector field whose flow lacks Sobolev regularity, highlighting the optimality of recent regularity estimates.

Findings

Flow does not belong to any Sobolev space

Constructed vector field is divergence-free and in H^1

Shows limits of regularity estimates for rough force fields

Abstract

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.