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CATEGORIES:Conference on Mathematical Topics in Kinetic Theor
y
SUMMARY:Nonlocal transport equations and systems: from par
ticle description to large time asymptotics - Marc
o Di Francesco (University of Bath)
DTSTART;TZID=Europe/London:20130620T105000
DTEND;TZID=Europe/London:20130620T114000
UID:TALK45895AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/45895
DESCRIPTION:Aggregation phenomena in microbiology and animal b
iology can be often described by PDEs of "transpor
t" type\, with a "nonlocal" velocity field. I shal
l quickly provide a formal derivation of those PDE
s from particle-based ODEs. I shall then highlight
their variational structure\, which often leads t
o well-posedness in a probability-measure sense. A
major issue is providing a mathematical descripti
on of the emergence (or not) of collective behavio
ur\, or "multiple" behaviour in the large-time asy
mptotics\, depending on the choice of the initial
conditions or other parameters. This issue has bee
n partly investigated in the recent literature (cf
. chemotaxis with two species). I will briefly des
cribe recent results on the existence and uniquene
ss of non-trivial steady states for a model with q
uadratic diffusion (in collaboration with M. Burge
r)\, and a recent work in preparation on the finit
e time blow up and "multiple collapse" for a "pure
ly nonlocal" model with two species of agents (wit
h S. Fagioli\, PhD student from L'Aquila). Finally
\, I shall focus on the derivation of a "mildly" s
ingular repulsive model as "large particle limit"
of discrete ODE systems in one space dimension (in
collaboration with G. A. Bonaschi\, J. A. Carrill
o\, and M. Peletier)\, and its interplay with the
theory of entropy solutions for scalar conservatio
n laws.
LOCATION:MR12
CONTACT:HoD Secretary\, DPMMS
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