The regularity of the boundary of a multidimensional aggregation patch

Abstract

Let $d \geq 2$ and let $N(y)$ be the fundamental solution of the Laplace equation in $R^d$ We consider the aggregation equation $$ \frac{\partial ρ}{\partial t} + \operatorname{div}(ρv) =0, v = -\nabla N * ρ$$ with initial data $ρ(x,0) = χ_{D_0}$, where $χ_{D_0}$ is the indicator function of a bounded domain $D_0 \subset R^d.$ We now fix $0 < γ< 1$ and take $D_0$ to be a bounded $C^{1+γ}$ domain (a domain with smooth boundary of class $C^{1+γ}$). Then we have Theorem: If $D_0$ is a $C^{1 + γ}$ domain, then the initial value problem above has a solution given by $$ρ(x,t) = \frac{1}{1 -t} χ_{D_t}(x), \quad x \in R^d, \quad 0 \le t < 1$$ where $D_t$ is a $C^{1 + γ}$ domain for all $0 \leq t < 1$.