Transport equation and Cauchy problem for $$BV$$ vector fields and applications.(English)Zbl 1069.35022

Proceedings of the conference on partial differential equations, Forges-les-Eaux, France, June 7–11, 2004. Exp. I–XIII. Paris: Centre National de la Recherche Scientifique, Groupement de Recherche 2434 (ISBN 2-7302-1221-3/pbk). Exp. No. I, 11 p. (2004).
Summary: In this talk I am going to describe the main results of [Invent. Math. 158, No. 2, 227–260 (2004; Zbl 1075.35087)], where the DiPerna-Lions theory is extended to the case of a BV dependence of the vector field with respect to the spatial variables. I will also illustrate some differences between my approach and the DiPerna-Lions one in the treatment of the uniqueness of the flow, and some applications obtained in [L. Ambrosio, F. Bouchut, and C. De Lellis, Comm. Partial Differ. Equ. 29, No. 9–10, 1635–1651 (2004; Zbl 1072.35116)] and [L. Amborio and C. De Lellis, Int. Math. Res. Not. 2003, No. 41, 2205–2220 (2003; Zbl 1061.35048)] to PDE’s. Finally, I will also mention some open problems and some work in progress.
For the entire collection see [Zbl 1055.00015].

MSC:

 35F20 Nonlinear first-order PDEs

Keywords:

DiPerna-Lions theory
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