TY - JOUR

T1 - Commutators of flow maps of nonsmooth vector fields

AU - Rampazzo, Franco

AU - Sussmann, Héctor J.

N1 - Funding Information:
* Corresponding author. E-mail addresses: [email protected] (F. Rampazzo), [email protected] (H.J. Sussmann). 1 Research supported in part by the Research Project Alcuni aspetti analitici e geometrico-differenziali nei problemi di minimo a crescita lenta, University of Padova, 2003. 2 Research supported in part by NSF Grants DMS01-03901 and DMS-05-09930.

PY - 2007/1/1

Y1 - 2007/1/1

N2 - Relying on the notion of set-valued Lie bracket introduced in an earlier paper, we extend some classical results valid for smooth vector fields to the case when the vector fields are just Lipschitz. In particular, we prove that the flows of two Lipschitz vector fields commute for small times if and only if their Lie bracket vanishes everywhere (i.e., equivalently, if their classical Lie bracket vanishes almost everywhere). We also extend the asymptotic formula that gives an estimate of the lack of commutativity of two vector fields in terms of their Lie bracket, and prove a simultaneous flow box theorem for commuting families of Lipschitz vector fields.

AB - Relying on the notion of set-valued Lie bracket introduced in an earlier paper, we extend some classical results valid for smooth vector fields to the case when the vector fields are just Lipschitz. In particular, we prove that the flows of two Lipschitz vector fields commute for small times if and only if their Lie bracket vanishes everywhere (i.e., equivalently, if their classical Lie bracket vanishes almost everywhere). We also extend the asymptotic formula that gives an estimate of the lack of commutativity of two vector fields in terms of their Lie bracket, and prove a simultaneous flow box theorem for commuting families of Lipschitz vector fields.

KW - Asymptotic formula

KW - Commutativity

KW - Higher order bracket

KW - Lie bracket

KW - Lipschitz vector field

KW - Simultaneous flow-box

UR - http://www.scopus.com/inward/record.url?scp=33750180905&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750180905&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2006.04.016

DO - 10.1016/j.jde.2006.04.016

M3 - Article

AN - SCOPUS:33750180905

SN - 0022-0396

VL - 232

SP - 134

EP - 175

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 1

ER -