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Flows of measures generated by vector fields

Published online by Cambridge University Press:  14 February 2018

Emanuele Paolini
Affiliation:
Dipartimento di Matematica ‘U. Dini‘, Università di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy
Eugene Stepanov
Affiliation:
St Petersburg Branch of the Steklov Mathematical Institute of the Russian Academy of Sciences, Fontanka 27, 191023 St Petersburg, Russia Department of Mathematical Physics, Faculty of Mathematics and Mechanics, St Petersburg State University, Universitetskij pr. 28, Old Peterhof, 198504 St Petersburg, Russia ITMO University, Russia ([email protected])

Abstract

The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ordinary differential equation. We further show that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ‘flows along’ the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2018 

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