Stochastic Mean-Field Limit: Non-Lipschitz Forces \& Swarming

We consider general stochastic systems of interacting particles with noise which are relevant as models for the collective behavior of animals, and rigorously prove that in the mean-field limit the system is close to the solution of a kinetic PDE. Our aim is to include models widely studied in the literature such as the Cucker-Smale model, adding noise to the behavior of individuals. The difficulty, as compared to the classical case of globally Lipschitz potentials, is that in several models the interaction potential between particles is only locally Lipschitz, the local Lipschitz constant growing to infinity with the size of the region considered. With this in mind, we present an extension of the classical theory for globally Lipschitz interactions, which works for only locally Lipschitz ones

Rate of convergence to self-similarity for Smoluchowski's coagulation equation with constant coefficients

We show that solutions to Smoluchowski's equation with a constant coagulation kernel and an initial datum with some regularity and exponentially decaying tail converge exponentially fast to a self-similar profile. This convergence holds in a weighted Sobolev norm which implies the L\^2 convergence of derivatives up to a certain order k depending on the regularity of the initial condition. We prove these results through the study of the linearized coagulation equation in self-similar variables, for which we show a spectral gap in a scale of weighted Sobolev spaces. We also take advantage of the fact that the Laplace or Fourier transforms of this equation can be explicitly solved in this case.Comment: 41 page

Improved energy methods for nonlocal diffusion problems

We prove an energy inequality for nonlocal diffusion operators of the following type, and some of its generalisations: L u ( x ) := ∫ R N K ( x , y ) ( u ( y ) − u ( x ) ) d y , where L acts on a real function u defined on R N , and we assume that K ( x , y ) is uniformly strictly positive in a neighbourhood of x = y . The inequality is a nonlocal analogue of the Nash inequality, and plays a similar role in the study of the asymptotic decay of solutions to the nonlocal diffusion equation ∂ t u = L u as the Nash inequality does for the heat equation. The inequality allows us to give a precise decay rate of the L p norms of u and its derivatives. As compared to existing decay results in the literature, our proof is perhaps simpler and gives new results in some cases.J. A. Cañizo was supported by the Spanish Ministerio de Economía y Competitividad and the European Regional Development Fund (ERDF/FEDER), project MTM2014-52056-P. A. Molino was partially supported by MINECO - FEDER GrantMTM2015-68210-P(Spain), Junta de Andalucía FQM-116 (Spain) andMINECO Grant BES-2013-066595 (Spain)

Three eras of micellization

Micellization is the precipitation of lipids from aqueous solution into aggregates with a broad distribution of aggregation number. Three eras of micellization are characterized in a simple kinetic model of Becker-Döring type. The model asigns the same constant energy to the (k-1) monomer-monomer bonds in a linear chain of k particles. The number of monomers decreases sharply and many clusters of small size are produced during the first era. During the second era, nuclei are increasing steadily in size until their distribution becomes a self-similar solution of the diffusion equation. Lastly, when the average size of the nuclei becomes comparable to its equilibrium value, a simple mean-field Fokker-Planck equation describes the final era until the equilibrium distribution is reached

Regularity and mass conservation for discrete coagulation-fragmentation equations with diffusion

We present a new a-priori estimate for discrete coagulation fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a-priori estimate provides a global L2 bound on the mass density and was previously used, for instance, in the context of reaction-diffusion equations. In this paper we demonstrate two lines of applications for such an estimate: On the one hand, it enables to simplify parts of the known existence theory and allows to show existence of solutions for generalised models involving collision-induced, quadratic fragmentation terms for which the previous existence theory seems difficult to apply. On the other hand and most prominently, it proves mass conservation (and thus the absence of gelation) for almost all the coagulation coefficients for which mass conservation is known to hold true in the space homogeneous case

A well-posedness theory in measures for some kinetic models of collective motion

We present existence, uniqueness and continuous dependence results for some kinetic equations motivated by models for the collective behavior of large groups of individuals. Models of this kind have been recently proposed to study the behavior of large groups of animals, such as flocks of birds, swarms, or schools of fish. Our aim is to give a well-posedness theory for general models which possibly include a variety of effects: an interaction through a potential, such as a short-range repulsion and long-range attraction; a velocity-averaging effect where individuals try to adapt their own velocity to that of other individuals in their surroundings; and self-propulsion effects, which take into account effects on one individual that are independent of the others. We develop our theory in a space of measures, using mass transportation distances. As consequences of our theory we show also the convergence of particle systems to their corresponding kinetic equations, and the local-in-time convergence to the hydrodynamic limit for one of the models

On a new proof of the Harris ergodic theorem and related subexponential convergence results

We revisit a result in probability known as the Harris theorem and give a simple proof which is well-suited for some applications in PDE. The proof is not far from the ideas of Hairer \& Mattingly (2011) but avoids the use of mass transport metrics and can be readily extended to cases where there is no spectral gap and exponential relaxation to equilibrium does not hold. We will also discuss some contexts where this result can be useful, particularly in a model for neuron populations structured by the elapsed time since the last discharge. This talk is based on joint works with Stéphane Mischler and Havva Yolda.Non UBCUnreviewedAuthor affiliation: Universidad de GranadaFacult