ALMOST EVERYWHERE NON-UNIQUENESS OF INTEGRAL CURVES FOR DIVERGENCE-FREE SOBOLEV VECTOR FIELDS

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 L d-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 introduced in [4, 20] to study nonuniqueness for positive solutions of the continuity equation. We then infer non-uniqueness for integral curves from Ambrosio's superposition principle