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SUMMARY:Measure solutions\, finite time blow up\, global confinement and m
 ultiple blow up for nonlocal transport PDE's on Rn - Marco Di Francesco
DTSTART:20090529T130000Z
DTEND:20090529T140000Z
UID:TALK18484@talks.cam.ac.uk
CONTACT:Carola-Bibiane Schoenlieb
DESCRIPTION:Nonlocal transport PDE's arise very often nowadays\, from popu
 lation dynamics\, swarming models\, aggregative phenomena in social scienc
 es\, cell biology. Typically\, these equations feature finite time "blow-u
 p" of the solution (possibly depending on initial parameters)\, as well as
  the formation of delta type singularities. In view of that\, a global-in-
 time existence theory in a "measure" sense is needed. We perform this task
  by casting our problem (without diffusion) in the context of the Wasserst
 ein gradient flow theory recently developed by Ambrosio\, Gigli and Savare
 \, inspired by a basic idea due to Felix Otto in 2000. The motion of a fin
 ite number of interacting particles (corresponding to the combination of d
 eltas moving along the orbits of a system of ODE's) is then included in ou
 r set of solutions. Our theory\, which is set in any spatial dimension\, c
 overs interaction potentials featuring a "pointy" attractive singularity a
 t the origin and possibly repulsive-attractive ranges of interaction. As a
  byproduct of our existence theory\, we recover a stability result which a
 llows to prove finite time blow up and confinement results by simply detec
 ting these phenomena at the level of particles (somehow an abstract partic
 le method). In particular\, we show that any initially compactly supported
  measure collapses to a delta in a finite time (possible occurrence of mul
 tiple blow up is also shown in a similar way). Moreover\, we prove that a 
 global confinement property of the support holds\, in a possibly repulsive
 -attractive framework\, by requiring a coercivity assumption of the intera
 ction potential at infinity. Finally\, we investigate the behaviour of the
  minimizing movement scheme of the interaction energy in the case of an ab
 solutely continuous perturbation of a finite number of atoms. The presente
 d work is done in collaboration with Jose Antonio Carrillo\, Alessio Figal
 li\, Thomas Laurent and Dejan Slepcev.
LOCATION:MR12\,  Centre for Mathematical Sciences\, Wilberforce Road\, Cam
 bridge
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