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ODEs with Sobolev coefficients: the Eulerian and the Lagrangian approach


De Lellis, Camillo (2008). ODEs with Sobolev coefficients: the Eulerian and the Lagrangian approach. Discrete and Continuous Dynamical Systems. Series S, 1(3):405-426.

Abstract

In this paper we describe two approaches to the well-posedness of Lagrangian flows of Sobolev vector fields. One is the theory of renormalized solutions which was introduced by DiPerna and Lions in the eighties. In this framework the well-posedness of the flow is a corollary of an analogous result for the corresponding transport equation. The second approach has been recently introduced by Gianluca Crippa and the author and it is instead based on suitable estimates performed directly on the lagrangian formulation.

Abstract

In this paper we describe two approaches to the well-posedness of Lagrangian flows of Sobolev vector fields. One is the theory of renormalized solutions which was introduced by DiPerna and Lions in the eighties. In this framework the well-posedness of the flow is a corollary of an analogous result for the corresponding transport equation. The second approach has been recently introduced by Gianluca Crippa and the author and it is instead based on suitable estimates performed directly on the lagrangian formulation.

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Additional indexing

Item Type:Journal Article, not_refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > Analysis
Physical Sciences > Discrete Mathematics and Combinatorics
Physical Sciences > Applied Mathematics
Language:English
Date:2008
Deposited On:14 Jan 2009 09:08
Last Modified:01 Dec 2023 02:47
Publisher:American Institute of Mathematical Sciences
ISSN:1937-1179
Additional Information:First published in Discrete and Continuous Dynamical Systems. Series S in Vol 1 No 3(2008), published by the American Institute of Mathematical Sciences.
OA Status:Closed
Publisher DOI:https://doi.org/10.3934/dcdss.2008.1.405
Related URLs:http://www.ams.org/mathscinet-getitem?mr=2425023