University of Cambridge > Talks.cam > Applied and Computational Analysis Graduate Seminar > Measure solutions, finite time blow up, global confinement and multiple blow up for nonlocal transport PDE's on Rn

Measure solutions, finite time blow up, global confinement and multiple blow up for nonlocal transport PDE's on Rn

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Nonlocal transport PDE ’s arise very often nowadays, from population dynamics, swarming models, aggregative phenomena in social sciences, cell biology. Typically, these equations feature finite time “blow-up” 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 Wasserstein 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 finite number of interacting particles (corresponding to the combination of deltas moving along the orbits of a system of ODE ’s) is then included in our set of solutions. Our theory, which is set in any spatial dimension, covers interaction potentials featuring a “pointy” attractive singularity at the origin and possibly repulsive-attractive ranges of interaction. As a byproduct of our existence theory, we recover a stability result which allows to prove finite time blow up and confinement results by simply detecting these phenomena at the level of particles (somehow an abstract particle method). In particular, we show that any initially compactly supported measure collapses to a delta in a finite time (possible occurrence of multiple 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 interaction potential at infinity. Finally, we investigate the behaviour of the minimizing movement scheme of the interaction energy in the case of an absolutely continuous perturbation of a finite number of atoms. The presented work is done in collaboration with Jose Antonio Carrillo, Alessio Figalli, Thomas Laurent and Dejan Slepcev.

This talk is part of the Applied and Computational Analysis Graduate Seminar series.

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