Seminario di Equazioni Differenziali e Applicazioni: “Loss of regularity for linear transport equations”

Mercoledì 8 Giugno 2016, ore 12:00 - Aula 2BC30 - Gianluca Crippa


Mercoledì 8 Giugno 2016 alle ore 12:00 in Aula 2BC30, Gianluca Crippa (Università di Basilea) terrà un seminario dal titolo “Loss of regularity for linear transport equations”.

For a linear transport equation $$ \partial_t u + b \cdot \nabla u = 0 $$ with a Lipschitz velocity field $b$, the classical Cauchy-Lipschitz theory ensures propagation in time of the (Lipschitz) regularity of the initial datum. Although for less regular (Sobolev or $BV$, for instance) velocity fields a well-posedness theory for this equation is by now available (based on seminal results by DiPerna-Lions and Ambrosio), it turns out that the issue of the propagation in time of the regularity is much more delicate. In this talk I will report on a joint work with Alberti and Mazzucato, in which Sobolev velocity fields and smooth initial data are constructed, in such a way that any fractional regularity of the solution is instantaneously destroyed. Connections to mixing phenomena in fluids will also be mentioned.

Rif. Int. M. Bardi, C. Marchi, L. Caravenna

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