A Grandient Flow Approach to Non-Local Interaction Equations
speaker: Alessio Figalli (ETH Zurich)abstract: In this talk, we will see how the ``Wasserstein gradient flow approach" allows to provide a well-posedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interacting particle systems with attractiverepulsive pairwise interaction potentials. The main phenomenon of interest is that, even with smooth initial data, the solutions can concentrate mass in finite time. Thanks to the Wasserstein approach, one can develop an existence theory that enables to go beyond the blow-up time in classical norms and allows for solutions to form atomic parts of the measure in finite time. The weak measure solutions are shown to be unique and exist globally in time. Moreover, in the case of sufficiently attractive potentials, one can prove the total collapse of the solution onto a single point in finite time, for compactly supported initial data.
timetable:
Sat 16 Oct, 10:00 - 10:50, Aula Dini
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