Calculus of Variations and Geometric Measure Theory

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

Gianluca Crippa (Departement Mathematik und Informatik, Universit├Ąt Basel)

created by magnani on 31 Mar 2006

6 apr 2006


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.