TY - JOUR

T1 - Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations

AU - Carrillo, J.A.

AU - Di Francesco, Marco

AU - Figalli, A.

AU - Laurent, T.

AU - Slepčev, D.

PY - 2011/2/1

Y1 - 2011/2/1

N2 - 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 attractive/repulsive 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.

AB - 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 attractive/repulsive 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.

UR - http://www.scopus.com/inward/record.url?scp=79551709781&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1215/00127094-2010-211

U2 - 10.1215/00127094-2010-211

DO - 10.1215/00127094-2010-211

M3 - Article

AN - SCOPUS:79551709781

VL - 156

SP - 229

EP - 271

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 2

ER -