The Aggregation Equation with Power-Law Kernels: Ill-Posedness, Mass Concentration and Similarity Solutions

Abstract

We study the multidimensional aggregation equation ${u_t+{\rm div}(uv)=0, v=-\nabla K*u}$ with initial data in ${\fancyscript{P}_2\left(\mathbb R^d\right)\cap L_{p} \left(\mathbb R^d\right)}$. We prove that with biological relevant potential K(x) = |x|, the equation is ill-posed in the critical Lebesgue space ${L_{d/(d-1)}\left(\mathbb R^d\right)}$ in the sense that there exists initial data in ${\fancyscript{P}_2\left(\mathbb R^d\right)\cap L_{d/(d-1)}\left(\mathbb R^d\right)}$ such that the unique measure-valued solution leaves ${L_{d/(d-1)}\left(\mathbb R^d\right)}$ immediately. We also extend this result to more general power-law kernels K(x) = |x|^α, 0 < α < 2 for p = p _s := d/(d + α − 2), and prove a conjecture in Bertozzi et al. (Comm Pure Appl Math 64(1):45–83, 2010) about instantaneous mass concentration for initial data in ${\fancyscript{P}_2\left(\mathbb R^d\right)\cap L_{p}\left(\mathbb R^d\right)}$ with p < p _s . Finally, we characterize all the “first kind” radially symmetric similarity solutions in dimension greater than two.