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Regularity of Lagrangian flows over RCD*(K, N) spaces

Elia Brué and Daniele Semola

Abstract

The aim of this note is to provide regularity results for Regular Lagrangian flows of Sobolev vector fields over compact metric measure spaces verifying the Riemannian curvature dimension condition. We first prove, borrowing some ideas already present in the literature, that flows generated by vector fields with bounded symmetric derivative are Lipschitz, providing the natural extension of the standard Cauchy–Lipschitz theorem to this setting. Then we prove a Lusin-type regularity result in the Sobolev case (under the additional assumption that the m.m.s. is Ahlfors regular) therefore extending the already known Euclidean result.

Funding statement: The authors acknowledge the support of the PRIN2015 MIUR Project “Calcolo delle Variazioni”.

Acknowledgements

The authors would like to thank Luigi Ambrosio for suggesting them the study of this problem and for kind and numerous comments and suggestions. They are grateful to the anonymous reviewer for the detailed report which greatly improved this note.

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Received: 2018-04-13
Revised: 2019-05-22
Published Online: 2019-08-14
Published in Print: 2020-08-01

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