Stability and clustering of self-similar solutions of aggregation equations

• Published in: Journal of mathematical physics 2012-11, Vol.53 (11), p.1
• Main Authors: Sun, Hui, Uminsky, David, Bertozzi, Andrea L.
• Format: Article
• Language: English
• Subjects: Agglomeration; Collapsing; Mathematical analysis; Mathematical models; Nonlinear equations; Partial differential equations; Self-similarity; Shells; Similarity solutions; Simplex method; Stability
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.4745180

In this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation ρ t = ∇ · (ρ∇K * ρ) in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^d$\end{document} R d , d ⩾ 2, where K(r) = r γ/γ with γ > 2. It was previously observed [Y. Huang and A. L. Bertozzi, “Self-similar blowup solutions to an aggregation equation in Rn,” J. SIAM Appl. Math. 70, 2582–2603 (2010)]10.1137/090774495 that radially symmetric solutions are attracted to a self-similar collapsing shell profile in infinite time for γ > 2. In this paper we compute the stability of the similarity solution and show that the collapsing shell solution is stable for 2 < γ < 4. For γ > 4, we show that the shell solution is always unstable and destabilizes into clusters that form a simplex which we observe to be the long time attractor. We then classify the stability of these simplex solutions and prove that two-dimensional (in-)stability implies n-dimensional (in-)stability.