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

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

2010-05-13

arxiv.org/abs/1005.2232v2

SIAM J. Math. Anal. 43, pp. 1995-2008

Mathematics

Analysis of PDEs

typos corrected, 18 pages, to appear in Comm. Math. Phys

Scientific paper

10.1137/110820427

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

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