The aggregation equation with power-law kernels: ill-posedness, mass concentration and similarity solutions

TL;DR

This paper investigates the ill-posedness and mass concentration phenomena in the multidimensional aggregation equation with power-law kernels, establishing new results on solution behavior and classifying similarity solutions.

Contribution

It proves ill-posedness in critical Lebesgue spaces for the aggregation equation with biological kernels, extends results to general power-law kernels, and classifies radially symmetric similarity solutions.

Findings

Ill-posedness in critical Lebesgue space for the aggregation equation with kernel K(x)=|x|

Proof of instantaneous mass concentration for certain initial data

Classification of radially symmetric similarity solutions in higher dimensions

Abstract

We study the multidimensional aggregation equation $u_t+\Div(uv)=0$, $v=-\nabla K*u$ with initial data in $\cP_2(\bR^d)\cap L_{p}(\bR^d)$. We prove that with biological relevant potential $K(x)=|x|$, the equation is ill-posed in the critical Lebesgue space $L_{d/(d-1)}(\bR^d)$ in the sense that there exists initial data in $\cP_2(\bR^d)\cap L_{d/(d-1)}(\bR^d)$ such that the unique measure-valued solution leaves $L_{d/(d-1)}(\bR^d)$ immediately. We also extend this result to more general power-law kernels $K(x)=|x|^\alpha$, $0<\alpha<2$ for $p=p_s:=d/(d+\alpha-2)$, and prove a conjecture in Bertozzi, Laurent and Rosado [5] about instantaneous mass concentration for initial data in $\cP_2(\bR^d)\cap L_{p}(\bR^d)$ with $p<p_s$. Finally, we classify all the "first kind" radially symmetric similarity solutions in dimension greater than two.