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Stability analysis of line patterns of an anisotropic interaction model
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 SchöenliebMotivated 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 SystemsISSN
1536-0040Publisher
Society for Industrial and Applied MathematicsExternal DOI
Issue
4Volume
18Page range
1798-1845Department 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
2019-08-21First Open Access (FOA) Date
2019-10-16First Compliant Deposit (FCD) Date
2019-08-20Usage metrics
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