Kreusser, Lisa Maria
Markowich, Peter A.
KAUST DepartmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Applied Mathematics and Computational Science Program
Computer Science Program
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AbstractWe 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 long-range 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.
CitationBurger M, Düring B, Kreusser LM, Markowich PA, Schönlieb C-B (2018) Pattern formation of a nonlocal, anisotropic interaction model. Mathematical Models and Methods in Applied Sciences 28: 409–451. Available: http://dx.doi.org/10.1142/S0218202518500112.
SponsorsM.B. acknowledges support by ERC via Grant EU FP 7—ERC Consolidator Grant 615216 LifeInverse and by the German Science Foundation DFG via EXC 1003 Cells in Motion Cluster of Excellence, Munster, Germany. B.D. has been supported by the Leverhulme Trust Research Project Grant “Novel discretizations for higherorder nonlinear PDE” (RPG-2015-69). L.M.K. was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) Grant EP/L016516/1. C.-B.S. acknowledges support from Leverhulme Trust Project on breaking the non-convexity barrier, EPSRC Grant No. EP/M00483X/1, the EPSRC Center No. EP/N014588/1 and the Cantab Capital Institute for the Mathematics of Information. The authors would like to thank Carsten Gottschlich and Stephan Huckemann for introducing them to the Kucken–Champod model and for very useful discussions on the dynamics required for simulating fingerprints. The authors are grateful to the referees for their thorough review and valuable remarks that helped to improve the paper.
PublisherWorld Scientific Pub Co Pte Lt