Almost everywhere non-uniqueness of integral curves for divergence-free Sobolev vector fields

TL;DR

This paper demonstrates that divergence-free Sobolev vector fields can have multiple integral curves starting from almost every point, showing non-uniqueness of solutions to the associated continuity equation using convex integration techniques.

Contribution

The authors construct specific divergence-free Sobolev vector fields with non-unique integral curves, extending the understanding of non-uniqueness phenomena in the context of the continuity equation.

Findings

Constructed vector fields admit multiple solutions to the continuity equation.

Established non-uniqueness of integral curves for a.e. point in the domain.

Applied convex integration techniques to demonstrate non-uniqueness.

Abstract

We construct divergence-free Sobolev vector fields in C([0,1];W^{1,r}(T^d;R^d)) with r < d and d >=2 which simultaneously admit any finite number of distinct positive solutions to the continuity equation. We then show that the vector fields we produce have at least as many integral curves starting from a.e. point of T^d as the number of distinct positive solutions to the continuity equation these vector fields admit. Our work uses convex integration techniques for the continuity equation introduced in [L. Szekelyhidi and S. Modena, Annals of PDE, 2018] and [E. Brue, M. Colombo, C. De Lellis, Archive for Rational Mechanics and Analysis, 2021 ] to study non-uniqueness for positive solutions. We then infer non-uniqueness for integral curves from Ambrosio superposition principle.