EVENT DETAILS AND ABSTRACT

Title: Some recent results on the aggregation equations
Speaker: Hongjie Dong
Speaker Info: Brown University
Brief Description:
Special Note:
Abstract:

We study the multidimensional aggregation equations with power-law kernels $K$. We prove that with biological relevant potential $K(x)=|x|$, the equation is ill-posed in the critical Lebesgue space $L_{d/(d-1)}(R^d)$. We then 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 of Bertozzi, Laurent, and Rosado about an instantaneous-mass-concentration phenomenon. Finally, we classify all the ``first kind'' radially symmetric similarity solutions when $d+\alpha\ge 4$, and construct non-trivial similarity solutions when $d+\alpha< 4$.