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Flows of measures generated by vector fields

Part of: General first-order equations and systems Manifolds Miscellaneous topics - Partial differential equations Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  14 February 2018

Emanuele Paolini  and
Eugene Stepanov
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 (stepanov.eugene@gmail.com)
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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.

Keywords

flow of measurescontinuity equationmetric currentmeasurable vector fieldintegral curve

MSC classification

Secondary: 35R05: Partial differential equations with discontinuous coefficients or data 35R06: Partial differential equations with measure 49Q15: Geometric measure and integration theory, integral and normal currents 35F05: Linear first-order equations 35Q35: PDEs in connection with fluid mechanics 35R15: Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 4 , August 2018 , pp. 773 - 818
Copyright
Copyright © Royal Society of Edinburgh 2018 

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