Inserted: 1 nov 2013
Last Updated: 9 jul 2015
Journal: Proceedings of the Royal Society of Edinburgh: Section A
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.