On Similarity Solutions to the Multidimensional Aggregation Equation

We study similarity solutions to the multidimensional aggregation equation $u_t+{div}(uv)=0$, $v=-\nabla K*u$ with general power-law kernels K such that $\nabla K(x)=x|x|^{\alpha-2}$, $\alpha\in(2-d,2)$. We analyze the equation in different regimes of the parameter $\alpha$. In the case when $\alpha\in[4-d,2)$, we give a characterization of 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 $\alpha\in(2-d,4-d)$, we show that there exist multi delta-ring similarity solutions in $\mathbb{R}^d$. In particular, our results imply that multi delta-ring similarity solutions exist in three dimensions if $\alpha$ is just a little bit below 1.