# Regularity and compactness for the flow associated to weakly differentiable vector fields

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Gianluca Crippa
(Departement Mathematik und Informatik, Universität Basel)

created by magnani on 31 Mar 2006

6 apr 2006

**Abstract.**

Gianluca Crippa, from S.N.S.,
on Thursday, 6 April,
in ``Sala dei Seminari'' of the Mathematics Department,
at 17:30 will present

``Regularity and compactness for the flow associated to weakly
differentiable vector fields''

ABSTRACT
Given a vector field with Sobolev or BV regularity and with bounded
divergence, thanks to the results of DiPerna-Lions and of Ambrosio it is
possible to give a good notion of solution to the ordinary differential
equation, encoded in the concept of regular Lagrangian flow. Roughly
speaking, the regular Lagrangian flow is the unique solution of the ODE
which is stable with respect to smooth approximations of the vector field.

Several questions about the nature of the regular Lagrangian flow are
possible: in connection with the Cauchy-Lipschitz theory it is reasonable
to investigate the approximate differentiability with respect to the
initial datum, and in view of some applications to conservation laws it is
interesting to discuss the compactness of the flow under natural bounds on
the BV norm of the vector field and on the compressibility coefficient of
the flow.

During the talk, we will give a general overwiev of the problem, first
recollecting some results present in the recent literature (due to
Le Bris and Lions and to Ambrosio, Lecumberry and Maniglia), and then
presenting the new approach contained in a work in collaboration with
Camillo De Lellis. This method leads to a Lusin-type approximation
of the flow relative to W^{{1,p}} vector fields (p>1) with Lipschitz
maps, with quantitative estimates on the Lipschitz constant. We will
indicate how this implies some new compactness and stability results.