|Title||ODEs with Sobolev coefficients: the Eulerian and the Lagrangian approach.|
|Publication Type||Journal Article|
|Year of Publication||2008|
|Authors||De Lellis C.|
|Journal||Discrete and Continuous Dynamical Systems. Series S|
|Publisher||American Institute of Mathematical Sciences|
|Type of Article||Expository PDE|
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.
Discrete Contin. Dyn. Syst. Ser. S 1 (2008), no. 3, 405–426.