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Pattern formation of a nonlocal, anisotropic interaction model

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journal contribution
posted on 2023-06-09, 08:20 authored by Martin Burger, Bertram Duering, Lisa Maria Kreusser, Peter A Markowich, Carola-Bibiane Schönlieb
We consider a class of interacting particle models with anisotropic, repulsive-attractive interaction forces whose orientations depend on an underlying tensor field. An example of this class of models is the so-called Kücken-Champod model describing the formation of fingerprint patterns. This class of models can be regarded as a generalization of a gradient flow of a nonlocal interaction potential which has a local repulsion and a longrange attraction structure. In contrast to isotropic interaction models the anisotropic forces in our class of models cannot be derived from a potential. The underlying tensor field introduces an anisotropy leading to complex patterns which do not occur in isotropic models. This anisotropy is characterized by one parameter in the model. We study the variation of this parameter, describing the transition between the isotropic and the anisotropic model, analytically and numerically. We analyze the equilibria of the corresponding mean-field partial differential equation and investigate pattern formation numerically in two dimensions by studying the dependence of the parameters in the model on the resulting patterns.

Funding

Novel discretisations of higher-order nonlinear PDE; G1603; LEVERHULME TRUST; RPG-2015-069

History

Publication status

  • Published

File Version

  • Accepted version

Journal

Mathematical Models and Methods in Applied Sciences

ISSN

0218-2025

Publisher

World Scientific Publishing

Issue

3

Volume

28

Page range

409-451

Department affiliated with

  • Mathematics Publications

Research groups affiliated with

  • Numerical Analysis and Scientific Computing Research Group Publications

Notes

Electronic version of an article published as Mathematical Models and Methods in Applied Sciences, 28 (3), 2018, pp. 409-451 https://doi.org/10.1142/S0218202518500112 © [copyright World Scientific Publishing Company] https://www.worldscientific.com/worldscinet/m3as

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2017-10-18

First Open Access (FOA) Date

2018-12-29

First Compliant Deposit (FCD) Date

2017-10-18

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