On similarity solutions to the multidimensional aggregation equation

Abstract

We study similarity solutions to the multidimensional aggregation equation $u_t+\Div(uv)=0$, $v=-\nabla K*u$ with general power-law kernels $K(x)=|x|^α,α\in (2-d,2)$. We analyze the equation in different regimes of the parameter $α$. In the case when $α\in [4-d,2)$, we give a characterization all the "first kind" radially symmetric similarity solutions. We prove that any such solution is a linear combination of a delta ring and a delta mass at the origin. On the other hand, when $α\in (2-d,4-d)$, we show that there exist multi delta-ring similarity solutions in $R^d$. In particular, our results imply that multi delta-ring similarity solutions exist in 3D if $α$ is just a little bit below 1.

Minor revision, 17 pages, to appear in CMP