Almost Everywhere Nonuniqueness of Integral Curves for Divergence-Free Sobolev Vector Fields
Jules Pitcho, Massimo Sorella
Source abstract
Abstract. We construct divergence-free Sobolev vector fields in [Formula: see text] with [Formula: see text] and [Formula: see text] 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 [Formula: see text]-a.e. point of [Formula: see text] as the number of distinct positive solutions to the continuity equation 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.
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