ODEs with Sobolev coefficients: the Eulerian and the Lagrangian approach

De Lellis, C (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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