Non-uniqueness of integral curves for autonomous Hamiltonian vector fields

Abstract

In this work we prove the existence of an autonomous Hamiltonian vector field in $W^{1,r}(\mathbb T^d; \mathbb R^d)$ with $r < d-1$ and $d \geq 4$ for which the associated transport equation has non-unique positive solutions. As a consequence of Ambrosio's superposition principle [2], we show that this vector field has non-unique integral curves with a positive Lebesgue measure set of initial data and moreover, we show that the Hamiltonian is not constant along these integral curves.