Inserted: 17 jan 2006
Last Updated: 5 mar 2009
Journal: In: Transport Equations and Multi-D Hyperbolic Conservation Laws, Lecture Notes of the Unione Matematica Italiana
In these notes we illustrate some recent developments of the DiPerna-Lions theory in the case of vector fields having a BV regularity with respect to the spatial variables. In the Sobolev case we discuss also the (weak) differentiability properties of the flow X(t,x) with respect to the x variable, and some recent quantitative regularity estimates leading to a new proof of the existence and the uniqueness of the flow, as well as to quantitative stability results.
Keywords: Flow theory, Approximate differentiability, Maximal functions