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

By José Antonio Carrillo, M. Difrancesco, Alessio Figalli, Thomas Laurent, Dejan Slepcev. 2011

309 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/4e37c5722c3c35aba311e2cfa3a189cc92648189

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Citations in the corpus

A review of the mean field limits for Vlasov equations (2014, unknown): “However gradient flow techniques can be used effectively even if F is singular but typically assuming some type of convexity on V ; see [38] and [39].”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, background): “If W is not smooth enough, the aggregation into a unique point can also occur in finite time; see [118].”
Quantitative estimates of propagation of chaos for stochastic systems with W-1,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{d (2017, background): “(2), even without diffusion, as in [18,19] and in [10] for the mean field limit.”
{Euclidean, metric, and Wasserstein} gradient flows: an overview (2016, background): “If W is not smooth enough, the aggregation into a unique point can also occur in finite time, see [30].”
Stationary states of quadratic diffusion equations with long-range attraction (2011, background): “In the purely nonlocal case, namely when ε = 0, this issue has been studied extensively in many papers [37, 13, 35, 5, 4, 16, 6, 7, 29, 25, 24], combined with the study of the regularity of solutions compared to the attractive singularity of the interaction kernel.”
On minimizers of interaction functionals with competing attractive and repulsive potentials (2015, background): “This partial differential equation is the gradient flow of the energy (1) with respect to the 2-Wasserstein metric [14,15].”
The derivation of swarming models: Mean-field limit and Wasserstein distances (2013, methodology): “In these works, the potential is assumed to be smooth except at the origin, where the allowed singularity cannot be worse that Lipschitz and the potential has to be λ -convex, see [23] for details.”
Fractional Orlicz-Sobolev embeddings (2020, background): “A touch of recent contributions in this connection is furnished by [7, 8, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 41, 44, 45, 51, 52, 53, 58, 59, 61, 62, 63, 64, 65, 68, 70, 71, 77, 76, 78, 83, 86].”
Mean Field Limit for Stochastic Particle Systems (2017, background): “See for instance [9, 10, 13, 57, 58] for the mathematical analysis of the particle system (6), [19, 20, 25] for the analysis of the limiting PDE.”