Calculus of Variations and Geometric Measure Theory

J. A. Carrillo - M. Di Francesco - A. Figalli - T. Laurent - D. SlepĨev

Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations

created by figalli on 14 Apr 2010
modified on 25 Nov 2010


Accepted Paper

Inserted: 14 apr 2010
Last Updated: 25 nov 2010

Journal: Duke Math. J.
Year: 2010


In this paper we 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. We develop an existence theory that enables one 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, we show the finite time total collapse of the solution onto a single point for compactly supported initial measures. Our approach is based on the theory of gradient flows in the space of probability measures endowed with the Wasserstein metric. In addition to classical tools, we exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations.