Vol. 7, No. 5, 2014

Download this article
Download this article For screen
For printing
Recent Issues

Volume 10
Issue 7, 1539–1791
Issue 6, 1285–1538
Issue 5, 1017–1284
Issue 4, 757–1015
Issue 3, 513–756
Issue 2, 253–512
Issue 1, 1–252

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
Cover
About the Cover
Editorial Board
Editors’ Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Subscriptions
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Well-posedness of Lagrangian flows and continuity equations in metric measure spaces

Luigi Ambrosio and Dario Trevisan

Vol. 7 (2014), No. 5, 1179–1234
Abstract

We establish, in a rather general setting, an analogue of DiPerna–Lions theory on well-posedness of flows of ODEs associated to Sobolev vector fields. Key results are a well-posedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into .

When specialized to the setting of Euclidean or infinite-dimensional (e.g., Gaussian) spaces, large parts of previously known results are recovered at once. Moreover, the class of RCD(K,) metric measure spaces, introduced by Ambrosio, Gigli and Savaré [Duke Math. J. 163:7 (2014) 1405–1490] and the object of extensive recent research, fits into our framework. Therefore we provide, for the first time, well-posedness results for ODEs under low regularity assumptions on the velocity and in a nonsmooth context.

Keywords
continuity equation, flows, DiPerna–Lions theory, $\Gamma$-calculus
Mathematical Subject Classification 2010
Primary: 49J52
Secondary: 35K90
Milestones
Received: 19 February 2014
Accepted: 12 July 2014
Published: 27 September 2014
Authors
Luigi Ambrosio
Classe di Scienze
Scuola Normale Superiore
I-56126 Pisa
Italy
Dario Trevisan
Classe di Scienze
Scuola Normale Superiore
I-56126 Pisa
Italy