ODEs with Sobolev coefficients: The eulerian and the lagrangian approach

Camillo De Lellis

doi:10.3934/dcdss.2008.1.405

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2008, Volume 1, Issue 3: 405-426. Doi: 10.3934/dcdss.2008.1.405

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ODEs with Sobolev coefficients: The eulerian and the lagrangian approach

Camillo De Lellis^1,

1.
Universität Zürich, Institut für Mathematik, Winterthurerstrasse 190, CH–8057 Zürich

Received: November 2007

Revised: March 2008

Published: September 2008

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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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Mathematics Subject Classification: Primary: 34A12; Secondary: 35L65, 58D25, 35F05.

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