Asymptotics of blowup solutions for the aggregation equation
Yanghong Huang Andrea Bertozzi
Discrete & Continuous Dynamical Systems - B 2012, 17(4): 1309-1331 doi: 10.3934/dcdsb.2012.17.1309
We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation $ u_t = \nabla\cdot(u\nabla K*u) $ in $\mathbb{R}^n$, for homogeneous potentials $K(x) = |x|^\gamma$, $\gamma>0$. For $\gamma>2$, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing $\delta$-ring. We develop an asymptotic theory for the approach to this singular solution. For $\gamma < 2$, the solution blows up in finite time and we present careful numerics of second type similarity solutions for all $\gamma$ in this range, including additional asymptotic behaviors in the limits $\gamma \to 0^+$ and $\gamma\to 2^-$.
keywords: asymptotic behavior self-similar solutions. blowup Aggregation equation
Thomas P. Witelski David M. Ambrose Andrea Bertozzi Anita T. Layton Zhilin Li Michael L. Minion
Discrete & Continuous Dynamical Systems - B 2012, 17(4): i-ii doi: 10.3934/dcdsb.2012.17.4i
Studies of problems in fluid dynamics have spurred research in many areas of mathematics, from rigorous analysis of nonlinear partial differential equations, to numerical analysis, to modeling and applied analysis of related physical systems. This special issue of Discrete and Continuous Dynamical Systems Series B is dedicated to our friend and colleague Tom Beale in recognition of his important contributions to these areas.

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