Lie brackets of nonsmooth vector fields and commutation of their flows

Standard

Lie brackets of nonsmooth vector fields and commutation of their flows. / Rigoni, Chiara; Stepanov, Eugene; Trevisan, Dario.

In: Journal of the London Mathematical Society, Vol. 106, No. 2, 09.2022, p. 1232-1256.

Research output: Contribution to journal › Article › peer-review

Author

Rigoni, Chiara ; Stepanov, Eugene ; Trevisan, Dario. / Lie brackets of nonsmooth vector fields and commutation of their flows. In: Journal of the London Mathematical Society. 2022 ; Vol. 106, No. 2. pp. 1232-1256.

BibTeX

@article{1fb84350a45a4b1cad02734e2613e1ef,

title = "Lie brackets of nonsmooth vector fields and commutation of their flows",

abstract = "It is well-known that the flows generated by two smooth vector fields commute, if the Lie bracket of these vector fields vanishes. This assertion is known to extend to Lipschitz continuous vector fields, up to interpreting the vanishing of their Lie bracket in the sense of almost everywhere equality. We show that this cannot be extended to general a.e. differentiable vector fields admitting a.e. unique flows. We show, however, that the extension holds when one field is Lipschitz continuous and the other one is merely Sobolev regular (but admitting a regular Lagrangian flow).",

author = "Chiara Rigoni and Eugene Stepanov and Dario Trevisan",

note = "Publisher Copyright: {\textcopyright} 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.",

year = "2022",

month = sep,

doi = "10.1112/jlms.12597",

language = "English",

volume = "106",

pages = "1232--1256",

journal = "Journal of the London Mathematical Society",

issn = "0024-6107",

publisher = "Oxford University Press",

number = "2",

}

RIS

TY - JOUR

T1 - Lie brackets of nonsmooth vector fields and commutation of their flows

AU - Rigoni, Chiara

AU - Stepanov, Eugene

AU - Trevisan, Dario

PY - 2022/9

Y1 - 2022/9

N2 - It is well-known that the flows generated by two smooth vector fields commute, if the Lie bracket of these vector fields vanishes. This assertion is known to extend to Lipschitz continuous vector fields, up to interpreting the vanishing of their Lie bracket in the sense of almost everywhere equality. We show that this cannot be extended to general a.e. differentiable vector fields admitting a.e. unique flows. We show, however, that the extension holds when one field is Lipschitz continuous and the other one is merely Sobolev regular (but admitting a regular Lagrangian flow).

AB - It is well-known that the flows generated by two smooth vector fields commute, if the Lie bracket of these vector fields vanishes. This assertion is known to extend to Lipschitz continuous vector fields, up to interpreting the vanishing of their Lie bracket in the sense of almost everywhere equality. We show that this cannot be extended to general a.e. differentiable vector fields admitting a.e. unique flows. We show, however, that the extension holds when one field is Lipschitz continuous and the other one is merely Sobolev regular (but admitting a regular Lagrangian flow).

UR - http://www.scopus.com/inward/record.url?scp=85129577504&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/a33a2010-084d-3480-b8d7-fdbe55567f0a/

U2 - 10.1112/jlms.12597

DO - 10.1112/jlms.12597

M3 - Article

AN - SCOPUS:85129577504

VL - 106

SP - 1232

EP - 1256

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 2

ER -

ID: 100611519