Well-posedness of Lagrangian flows and continuity equations in metric measure spaces

Ambrosio, Luigi; Trevisan, Dario

doi:10.2140/apde.2014.7.1179

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Well-posedness of Lagrangian flows and continuity equations in metric measure spaces

Luigi Ambrosio and Dario Trevisan

Vol. 7 (2014), No. 5, 1179–1234

DOI: 10.2140/apde.2014.7.1179

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 $ℝ^{\infty}$ .

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 $R C D (K, \infty)$ 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

Vol. 7, No. 5, 2014