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

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.Comment: typos corrected, 18 pages, to appear in Comm. Math. Phy