Stability and clustering of self-similar solutions of aggregation equations

作者：Sun Hui^*; Uminsky David; Bertozzi Andrea L

来源：Journal of Mathematical Physics, 2012, 53(11): 115610.

DOI：10.1063/1.4745180

• aggregation
• fractals
• nonlinear differential equations
• partial differential equations

In this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation rho(t) = del . (rho del K * rho) in R-d, d >= 2, where K(r) = r(gamma)/gamma with gamma > 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)] that radially symmetric solutions are attracted to a self-similar collapsing shell profile in infinite time for gamma > 2. In this paper we compute the stability of the similarity solution and show that the collapsing shell solution is stable for 2 < gamma < 4. For gamma > 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.

• 出版日期2012-11