Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Two-species particle aggregation and stability of co-dimension one solutions

Pages: 1411 - 1436, Volume 19, Issue 5, July 2014      doi:10.3934/dcdsb.2014.19.1411

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Alan Mackey - University of California, Los Angeles, Department of Mathematics, Box 951555, Los Angeles, CA 90095-1555, United States (email)
Theodore Kolokolnikov - Dalhousie University, Department of Mathematics and Statistics, Halifax, Nova Scotia, B3H 3J5, Canada (email)
Andrea L. Bertozzi - University of California Los Angeles, Department of Mathematics, 520 Portola Plaza Box 951555, Los Angeles, CA 90095-1555, United States (email)

Abstract: Systems of pairwise-interacting particles model a cornucopia of physical systems, from insect swarms and bacterial colonies to nanoparticle self-assembly. We study a continuum model with densities supported on co-dimension one curves for two-species particle interaction in $\mathbb{R}^2$, and apply linear stability analysis of concentric ring steady states to characterize the steady state patterns and instabilities which form. Conditions for linear well-posedness are determined and these results are compared to simulations of the discrete particle dynamics, showing predictive power of the linear theory. Some intriguing steady state patterns are shown through numerical examples.

Keywords:  Aggregation equation, pattern formation, measure solutions, continuum limit, linear stability, linear well-posedness, asymptotics.
Mathematics Subject Classification:  Primary: 35B35, 25B20, 35B36; Secondary: 35D30, 41A60.

Received: May 2013;      Revised: January 2014;      Available Online: April 2014.