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SUMMARY:Nonlocal transport equations and systems: from particle descriptio
 n to large time asymptotics - Marco Di Francesco  (University of Bath) 
DTSTART:20130620T095000Z
DTEND:20130620T104000Z
UID:TALK45895@talks.cam.ac.uk
CONTACT:HoD Secretary\, DPMMS
DESCRIPTION:Aggregation phenomena in microbiology and animal biology can b
 e often described by PDEs of "transport" type\, with a "nonlocal" velocity
  field. I shall quickly provide a formal derivation of those PDEs from par
 ticle-based ODEs. I shall then highlight their variational structure\, whi
 ch often leads to well-posedness in a probability-measure sense. A major i
 ssue is providing a mathematical description of the emergence (or not) of 
 collective behaviour\, or "multiple" behaviour in the large-time asymptoti
 cs\, depending on the choice of the initial conditions or other parameters
 . This issue has been partly investigated in the recent literature (cf. ch
 emotaxis with two species). I will briefly describe recent results on the 
 existence and uniqueness of non-trivial steady states for a model with qua
 dratic diffusion (in collaboration with M. Burger)\, and a recent work in 
 preparation on the finite time blow up and "multiple collapse" for a "pure
 ly nonlocal" model with two species of agents (with S. Fagioli\, PhD stude
 nt from L'Aquila). Finally\, I shall focus on the derivation of a "mildly"
  singular repulsive model as "large particle limit" of discrete ODE system
 s in one space dimension (in collaboration with G. A. Bonaschi\, J. A. Car
 rillo\, and M. Peletier)\, and its interplay with the theory of entropy so
 lutions for scalar conservation laws. 
LOCATION:MR12
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