Calculus of Variations and Geometric Measure Theory

L. Ambrosio - G. Crippa

Continuity equations and ODE flows with non-smooth velocity

created by crippa on 01 Nov 2013
modified on 19 Jun 2018


Published Paper

Inserted: 1 nov 2013
Last Updated: 19 jun 2018

Journal: Proceedings of the Royal Society of Edinburgh: Section A
Volume: 144
Number: 6
Pages: 1191-1244
Year: 2014


In this paper we review many aspects of the well-posedness theory for the Cauchy problem for the continuity and the transport equations, and for the ordinary differential equation, in the context of velocity fields which are not smooth, but enjoy suitable ``weak differentiability'' assumptions. We first explore the connection between the PDE and the ODE in the non-smooth setting. After, we address the renormalization property for the PDE and prove that such property holds for Sobolev velocity fields and for bounded variation velocity fields. Finally, we present an approach to the ODE theory based on quantitative estimates.