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SUMMARY:"The structure of transport equations and the Vlasov-Poisson syste
 m"
DTSTART:20170228T141500
DTEND:20170228T151500
DTSTAMP:20261001T160908Z
UID:d426f1a0c9803d1b187d1ce8b0057da357e04db7851b4e136a81e6fe
CATEGORIES:Conferences - Seminars
DESCRIPTION:Dr. Maria Colombo (ETHZ - ITS) \nThe transport equation descri
 bes the evolution of a distribution of particles moving along the flow of 
 a prescribed smooth vector field. An accurate description of its solutions
 \, even when the smoothness assumption is dropped\, is motivated by severa
 l applications\, among which the study of kinetic equations such as the Vl
 asov-Poisson system.\n\nGiven a vector field in R^d\, the classical Cauchy
 -Lipschitz theorem shows existence and uniqueness of its flow provided the
  vector field is sufficiently smooth\; this\, in turn\, translates in exis
 tence and uniqueness results for the transport equation. In 1989\, Di Pern
 a and Lions proved that Sobolev regularity for vector fields\, with bounde
 d divergence and a growth assumption\, is sufficient to establish existenc
 e\, uniqueness and stability of a generalized notion of flow\, consisting 
 of a suitable selection among the trajectories of the associated ODE. Thei
 r theory relies on a growth assumption which prevents the trajectories fro
 m blowing up in finite time. In this seminar we give an overview of the to
 pic and we introduce a new notion of maximal flow for non-smooth vector fi
 elds which allows for finite-time blow up of the trajectories. We show str
 ucture results for the transport equation under only local assumptions on 
 the vector field and we apply them to the Vlasov-Poisson system\, where we
  describe the solutions as transported by a suitable flow.
LOCATION:CIB - BI A0 448 http://plan.epfl.ch/?room=BIA0448
STATUS:CONFIRMED
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