TY - JOUR

T1 - The saga of a fish: from a survival guide to closing lemmas

AU - Крыжевич, Сергей Геннадьевич

AU - Степанов, Евгений Олегович

PY - 2019/9/5

Y1 - 2019/9/5

N2 - In the paper by D. Burago, S. Ivanov and A. Novikov, “A survival guide for feeble fish”, it has been shown that a fish with limited velocity can reach any point in the (possibly unbounded)ocean provided that the fluid velocity field is incompressible, bounded and has vanishing mean drift. This result extends some known global controllability theorems though being substantially nonconstructive. We give a fish a different recipe of how to survive in a turbulent ocean, and show its relationship to structural stability of dynamical systems by providing a constructive way to change slightly the velocity field to produce conservative (in the sense of not having wandering sets of positive measure)dynamics. In particular, this leads to the extension of C. Pugh's closing lemma to incompressible vector fields over unbounded domains. The results are based on an extension of the Poincaré recurrence theorem to some σ-finite measures and on specially constructed Newtonian potentials.

AB - In the paper by D. Burago, S. Ivanov and A. Novikov, “A survival guide for feeble fish”, it has been shown that a fish with limited velocity can reach any point in the (possibly unbounded)ocean provided that the fluid velocity field is incompressible, bounded and has vanishing mean drift. This result extends some known global controllability theorems though being substantially nonconstructive. We give a fish a different recipe of how to survive in a turbulent ocean, and show its relationship to structural stability of dynamical systems by providing a constructive way to change slightly the velocity field to produce conservative (in the sense of not having wandering sets of positive measure)dynamics. In particular, this leads to the extension of C. Pugh's closing lemma to incompressible vector fields over unbounded domains. The results are based on an extension of the Poincaré recurrence theorem to some σ-finite measures and on specially constructed Newtonian potentials.

KW - Global controllability

KW - Pugh closing lemma

KW - Structural stability

KW - CONTINUITY EQUATIONS

UR - http://www.scopus.com/inward/record.url?scp=85064621938&partnerID=8YFLogxK

UR - http://www.mendeley.com/research/saga-fish-survival-guide-closing-lemmas

U2 - 10.1016/j.jde.2019.04.010

DO - 10.1016/j.jde.2019.04.010

M3 - Article

VL - 267

SP - 3442

EP - 3474

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 6

ER -