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Stability analysis of line patterns of an anisotropic interaction model

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Version 2 2023-06-07, 08:27
Version 1 2023-06-07, 06:42
journal contribution
posted on 2023-06-07, 08:27 authored by José A Carrilo, Bertram Duering, Lisa Maria Kreusser, Carola-Bibiane Schöenlieb
Motivated by the formation of fingerprint patterns we consider a class of interacting particle models with anisotropic, repulsive-attractive interaction forces whose orientations depend on an underlying tensor field. 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 long-range attraction structure. In addition, the underlying tensor field introduces an anisotropy leading to complex patterns which do not occur in isotropic models. Central to this pattern formation are straight line patterns. For a given spatially homogeneous tensor field, we show that there exists a preferred direction of straight lines, i.e. straight vertical lines can be stable for sufficiently many particles, while many other rotations of the straight lines are unstable steady states, both for a sufficiently large number of particles and in the continuum limit. For straight vertical lines we consider specific force coefficients for the stability analysis of steady states, show that stability can be achieved for exponentially decaying force coefficients for a sufficiently large number of particles and relate these results to the Kuecken-Champod model for simulating fingerprint patterns. The mathematical analysis of the steady states is completed with numerical results.

Funding

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

History

Publication status

  • Published

File Version

  • Published version

Journal

SIAM Journal on Applied Dynamical Systems

ISSN

1536-0040

Publisher

Society for Industrial and Applied Mathematics

Issue

4

Volume

18

Page range

1798-1845

Department affiliated with

  • Mathematics Publications

Research groups affiliated with

  • Numerical Analysis and Scientific Computing Research Group Publications

Full text available

  • No

Peer reviewed?

  • Yes

Legacy Posted Date

2019-08-21

First Open Access (FOA) Date

2019-10-16

First Compliant Deposit (FCD) Date

2019-08-20

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