On similarity solutions to the multidimensional aggregation equation

TL;DR

This paper investigates similarity solutions to the multidimensional aggregation equation with power-law kernels, characterizing radially symmetric solutions and demonstrating the existence of multi delta-ring solutions depending on the kernel parameter.

Contribution

It provides a complete characterization of first kind radially symmetric similarity solutions and proves the existence of multi delta-ring solutions for certain kernel parameters.

Findings

All first kind radially symmetric solutions are a combination of delta ring and delta mass.

Multi delta-ring solutions exist in R^d for specific alpha ranges.

Existence of multi delta-ring solutions in 3D when alpha is just below 1.

Abstract

We study similarity solutions to the multidimensional aggregation equation $u_t+\Div(uv)=0$, $v=-\nabla K*u$ with general power-law kernels $K(x)=|x|^\alpha,\alpha\in (2-d,2)$. We analyze the equation in different regimes of the parameter $\alpha$. In the case when $\alpha\in [4-d,2)$, we give a characterization 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 $R^d$. In particular, our results imply that multi delta-ring similarity solutions exist in 3D if $\alpha$ is just a little bit below 1.