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

We construct divergence-free Sobolev vector fields in C([0,1];W-1,W-r(T-d;Rd)) with r < d and d\geq 2 which simultaneously admit any finite number of distinct positive solutions to the continuity equation. These vector fields are then shown to have at least as many integral curves starting from 2d-a.e. point of Td as the number of distinct positive solutions to the continuity equa-tion these vector fields admit. This work uses convex integration techniques to study nonuniqueness for positive solutions of the continuity equation. Nonuniqueness for integral curves is then inferred from Ambrosio's superposition principle.