Non-uniqueness of integral curves for autonomous Hamiltonian vector fields

2021/08/11 by Giri, Vikram, Sorella, Massimo · 1 citation
#70H33 - 35A02 - 35D30 - 35Q49 - 34A12 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2108.05050

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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