Pattern forming instabilities driven by non-diffusive interactions

Ivano Primi; Angela Stevens; Juan J. L. Velázquez; Ivano Primi; Angela Stevens; Juan J. L. Velázquez

doi:10.3934/nhm.2013.8.397

Networks and Heterogeneous Media

2013, Volume 8, Issue 1: 397-432. doi: 10.3934/nhm.2013.8.397

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Pattern forming instabilities driven by non-diffusive interactions

1.
Advanced Semiconductor Materials Lithography, ASML B.V., Office 06.C.006, 5500AH Veldhoven
2.
Westfälische-Wilhelms Universität Münster, Applied Mathematics Münster, Einsteinstr. 62, D-48149 Münster
3.
Universität Bonn, Institut für Angewandte Mathematik, Endenicher Allee 60, D-53155 Bonn

Received: 01 January 2012 Revised: 01 March 2013
Primary: 35L45, 35P20, 35B20, 35B36; Secondary: 92C15, 92C17.

In analogy to the analysis of minimal conditions for the formation of diffusion driven instabilities in the sense of Turing, in this paper minimal conditions for a class of kinetic equations with mass conservation are discussed, whose solutions show patterns with a characteristic wavelength. The related linearized systems are analyzed, and the minimal number of equations is derived, which is needed for specific patterns to occur.
- Hyperbolic system,
- pattern formation,
- perturbation of spectra of matrices,
- rippling dynamics of myxobacteria
Citation: Ivano Primi, Angela Stevens, Juan J. L. Velázquez. Pattern forming instabilities driven by non-diffusive interactions[J]. Networks and Heterogeneous Media, 2013, 8(1): 397-432. doi: 10.3934/nhm.2013.8.397

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Abstract

In analogy to the analysis of minimal conditions for the formation of diffusion driven instabilities in the sense of Turing, in this paper minimal conditions for a class of kinetic equations with mass conservation are discussed, whose solutions show patterns with a characteristic wavelength. The related linearized systems are analyzed, and the minimal number of equations is derived, which is needed for specific patterns to occur.

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