The Behavior of Solutions of Multidimensional Aggregation Equations with Mildly Singular Interaction Kernels

The authors consider the multidimensional aggregation equation ∂ _t ρ-div(ρ▿K* 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.