Non-uniqueness of integral curves for autonomous Hamiltonian vector fields

TL;DR

This paper demonstrates that certain autonomous Hamiltonian vector fields in specific Sobolev spaces can have non-unique integral curves and solutions to the associated transport equation, challenging assumptions of uniqueness.

Contribution

It constructs explicit examples of Hamiltonian vector fields in W^{1,r} spaces with non-unique integral curves and solutions, revealing non-uniqueness phenomena.

Findings

Existence of non-unique solutions to the transport equation.

Non-uniqueness of integral curves with positive measure initial data.

Hamiltonian is not constant along these non-unique integral curves.

Abstract

In this work we prove the existence of an autonomous Hamiltonian vector field in W^{1,r}(T^d;R^d) with r< d-1and d>=4 for which the associated transport equation has non-unique positive solutions. As a consequence of Ambrosio superposition principle, 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.