Abstract : From biological swarming and n-body dynamics to self-assembly of nanoparticles, crystallization and
granular media, many physical and biological systems are described by mathematical models involving nonlocal interactions. Mostly due to their purely nonlocal character, these models present mathematical challenges that require a combination of different techniques of applied mathematics. With this scientific session we aim to bring together young researchers and leading scholars who study nonlocal interaction models and their applications. In particular, we hope that by inviting applied and pure analysts we will create a platform that will lead to a more complete and reliable understanding of these models.
Abstract : We study a nonlocal isoperimetric problem for several interacting phase domains which consists of a local interface energy and of a longer-range Coulomb interaction energy. We consider global minimizers on the two-dimensional torus, in a limit in which some of the species have vanishingly small mass. Depending on the relative strengths of the coefficients we may see different structures for the global minimizers. This represents work with S. Alama, X. Lu, and C. Wang.
[00152] Patterns in tri-block copolymers: droplets, double-bubbles and core-shells.
Format : Talk at Waseda University
Author(s) :
Stan Alama (McMaster Univ)
Lia Bronsard (McMaster University)
Xin Yang Lu (Lakehead Univ)
Chong Wang (Washington and Lee Univ)
Abstract : We study the Nakazawa-Ohta ternary inhibitory system, which describes domain morphologies in a triblock copolymer as a nonlocal isoperimetric problem for three interacting phase domains. The free energy consists of two parts: the local interface energy measures the total perimeter of the phase boundaries, while a longer-range Coulomb interaction energy reflects the connectivity of the polymer chains and promotes splitting into micro-domains. We consider global minimizers on the two-dimensional torus, in a limit in which two of the species have vanishingly small mass but the interaction strength is correspondingly large. In this limit there is splitting of the masses, and each vanishing component rescales to a minimizer of an isoperimetric problem for clusters in 2D. Depending on the relative strengths of the coefficients of the interaction terms we may see different structures for the global minimizers, ranging from a lattice of isolated simple droplets of each minority species to double-bubbles or core-shells. This represents work with S. Alama, X. Lu, and C. Wang.
[00113] Ground states for aggregation-diffusion models on Cartan-Hadamard manifolds
Format : Talk at Waseda University
Author(s) :
Hansol Park (Simon Fraser University)
Abstract : We investigate the existence of ground states of a free energy functional defined on Cartan-Hadamard manifolds. There are two competing effects in the free energy: repulsion modelled by linear diffusion and attraction modelled by a nonlocal interaction term. Nonexistence of energy minimizers can occur if either the diffusion is too strong (spreading) or attraction is dominant (blow-up). Variational approaches have been used to provide sufficient conditions of the attractive interaction to prevent the two scenarios from happening, and thus establishing the existence of global minimizers of the free energy.
[00101] Well-posedness and asymptotic behaviour of an interaction model on Riemannian manifolds
Format : Talk at Waseda University
Author(s) :
Razvan C Fetecau (Simon Fraser University)
Abstract : We consider a model for collective behaviour with intrinsic interactions on Riemannian manifolds. We
establish the well-posedness of measure solutions and study the long-time behaviour of solutions. For
the latter, the primary goal is to establish sufficient conditions for a consensus state to form
asymptotically. The analytical results are illustrated with numerical experiments that exhibit various asymptotic patterns.
[00131] Mean field games with aggregating interaction potentials of nonlocal type
Format : Talk at Waseda University
Author(s) :
Annalisa Cesaroni (University of Padova)
Abstract : We discuss existence/non existence of solutions to ergodic mean field game systems in the whole space
with interactions of aggregative Riesz type, in dependance on the strength of the interaction term.
Moreover, we present qualitative properties of the solutions and concentration phenomena, as the diffusion term vanishes. Finally we discuss some open problems related to stability of equilibria.
[00105] Deterministic particle approximation for a nonlocal interaction equation with repulsive singular potential
Format : Talk at Waseda University
Author(s) :
Marco Di Francesco (University of L'Aquila)
Markus Schmidtchen (TU Dresden)
Abstract : We consider a variant of a deterministic particle approximation for a nonlocal interaction equation with repulsive Morse potential in 1d, which is not covered by previous similar results in the literature. We prove convergence in a weak sense towards solutions to the corresponding continuum PDE. We prove our scheme is able to capture L^p contractivity and a smoothing effect which allows to extend the result to initial data in the set of probability measures.
[00155] Nonlocal deterministic and stochastic models for collective movement in biology
Format : Talk at Waseda University
Author(s) :
Raluca EFTIMIE (University of Franche-Comté)
Abstract : The collective movement of animals occurs as a result of communication between the members of the community. However, inter-individual communication can be affected by the stochasticity of the environment, leading to changes in the perception of neighbours and subsequent changes in individual behaviour, which then influence the overall behaviour of the animal aggregations. To investigate the effect of noise on the overall behaviour of animal aggregations, we consider a class of nonlocal hyperbolic models for the collective movement of animals. We show numerically that for some sets of model parameters associated with individual communication, strong noise does not influence the spatio-temporal pattern (i.e., travelling aggregations) observed when all neighbours are perceived with the same intensity (i.e., the environment is homogeneous). However, when neighbours ahead/behind are perceived differently by a reference individual, noise can lead to the destruction of the spatio-temporal pattern.
[00117] Zero-Inertia Limit: from Particle Swarm Optimization to Consensus-Based Optimization
Format : Talk at Waseda University
Author(s) :
Hui Huang (University of Graz)
Abstract : Large systems of interacting particles are widely used to investigate self-organization and collective behavior. They have also been used in metaheuristic methods, which can provide empirically robust solutions to tackle hard optimization problems with fast algorithms. In this talk, we will focus on two examples of metaheuristics, i.e. Particle Swarm Optimization $(PSO)$ and Consensus-Based Optimization $(CBO)$. In particular, we shall provide a rigorous derivation of CBO from PSO through the limit of zero inertia. This is also related to the problems of overdamped limit and large-friction limit.
[00093] Many-spike limits of reaction-diffusion systems of PDEs
Format : Talk at Waseda University
Author(s) :
Theodore Kolokolnikov (Dalhousie University)
Abstract : Many reaction-diffusion have solutions consisting of spots or spikes. We consider the problem of describing the density distribution of these spikes when the number of spikes is large. This naturally leads to integral equation for spike density.
[00157] Pattern formation in particle systems: spherical shells to regular simplices
Format : Talk at Waseda University
Author(s) :
Robert John McCann (University of Toronto, Department of Mathematics)
Abstract : Flocking and swarming models address mathematical biological pattern formation. When organisms interact through a difference of power laws attractive over large distances yet repulsive at short distances, we detail a phase transition which separates a region where the minimum energy configuration is uniquely attained by a uniform distribution of organisms over a spherical shell, from a region in which it is uniquely attained by equidistributing the organisms over the vertices of a regular top-dimensional simplex.
[05071] Some remarks on minimization of nonlocal attracting repulsing energies
Abstract : We will discuss some results on the minimization of nonlocal energies of attraction-repulsion type. In particular, we will be interested in existence and regularity of minimizing sets, and of generalised solutions, i.e., functions or measures. Some of the presented results are well-known, some others are recent or very recent developments. Some of the results have been obtained in various collaborations with D. Carazzato, N. Fusco, I. Topaloglu.
[00145] On a Becker-Döring model for prions and an associated nonlocal problem.
Format : Online Talk on Zoom
Author(s) :
Klemens Fellner (University of Graz)
Abstract : Prions are able to self-propagate biological information through the transfer of structural information from a misfolded/infectious protein in a prion state to a protein in a non-prion state. Prions cause diseases like Creuzfeldt-Jakob. Prion-like mechanisms are associated to Alzheimer, Parkinson ans Huntington diseases. We present a fundamental bi-monomeric, nonlinear Becker-Döring type model, which aims to explain experiments in the lab of Human Rezaei showing sustained oscillatory behaviour over multiple hours. We exemplify a mechanism of oscillatory behaviour and show numerical simulations. An interesting non-local problem describes an associated self-similar structure.
[04453] Shape Optimization for nonlocal anisotropic energies
Author(s) :
Lucia Scardia (Heriot-Watt University)
Abstract : We address the problem of shape optimisation for sets with fixed mass, in the case of attractive-repulsive nonlocal energies. More precisely, we focus on energies whose repulsive part is an anisotropic Newtonian potential, and whose attractive part is radially symmetric, and quadratic.
For the fully radially symmetric case, it is known that the existence of minimisers depends on the value of the mass: there is a critical value such that minimisers
are balls above it, and do not exist below it. We show that a similar dichotomy occurs also in the anisotropic case.
The anisotropy, however, introduces an additional critical value that makes the analysis subtle.
This is work in collaboration with Riccardo Cristoferi and Maria Giovanna Mora.
