In the first part of the talk I will give an overview on some recent

advances on the study of the continuity equation when the velocity

field is non-smooth. This kind of equation appears very often in

problems originating from the dynamics of fluids, and the lack of

regularity of the velocity field is due "irregular" physical

behaviours, like shocks or turbulence. I will motivate the need for

the use of geometric measure theory for this kind of analysis, and I

will illustrate the approach based on the notion of renormalized

solutions used by DiPerna-Lions and by Ambrosio to study the Sobolev

and the bounded variation cases, respectively.

In the second part, I will present some results from a project in collaboration

with Giovanni Alberti (University of Pisa) and Stefano Bianchini

(SISSA, Trieste). We focus on the two-dimensional case. In the

simplest form, our result gives a characterization of (bounded,

autonomous and divergence-free) vector fields on the plane such that

uniqueness for the continuity equation holds. The proof relies on a

dimension-reduction argument which reduces the problem to a family of

one-dimensional problems. I will try to convey to the audience some

flavour of the techniques in our proof.

When? | 22.11.2011 17:15 |
---|---|

Where? | PER 08 Phys 2.52 Chemin du Musée 3 1700 Fribourg |

Contact | Department of Mathematics isabella.schmutz@unifr.ch |