[CvGmt News] CVGMT seminar/6 April/17:30

Valentino Magnani magnani at dm.unipi.it
Fri Mar 31 12:23:10 CEST 2006

Dear All,

the next CVGMT seminar will be held by

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

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

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.

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