Inserted: 12 sep 2005
Lecture Notes of the CIME Summer school in Cetrary, June 27-July 2, 2005
In these notes we describe some recent progress on the well-posedness of the continuity equation and the Cauchy problem for non-smooth vector fields. In the first part of the paper we describe, at an abstract level, the link between the well-posedness of the two problems. Then, we analyze this issue in the specific context of possibly time-dependent vector fields having a Sobolev or a BV regularity with respect to the spatial variables. Finally, we illustrate some applications to conservation laws and to PDE's arising in fluid mechanics.
Keywords: conservation laws, Renormalized solutions, continuity equation, Transport equation