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arXiv · 2108.05050

Non-uniqueness of integral curves for autonomous Hamiltonian vector fields

Vikram Giri·Massimo Sorella
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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.

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Vikram Giri, Massimo Sorella. 2021-08-11. Non-uniqueness of integral curves for autonomous Hamiltonian vector fields. https://arxiv.org/abs/2108.05050

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