, Volume 30, Issue 5, pp 463-482
Date: 10 Aug 2009

The behavior of solutions of multidimensional aggregation equations with mildly singular interaction kernels

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Abstract

The authors consider the multidimensional aggregation equation t ρ-div(ρK* ρ) = 0 in which the radially symmetric attractive interaction kernel has a mild singularity at the origin (Lipschitz or better), and review recent results on this problem concerning well-posedness of nonnegative solutions and finite time blowup in multiple space dimensions depending on the behavior of the kernel at the origin. The problem with bounded initial data, data in L p L 1, and measure solutions are also considered.

Dedicated to Professor Andrew Majda on the Occasion of his 60th Birthday
Project supported by the Office of Naval Research, the Army Research Office and the National Science Foundation (No. DMS-0907931).