The regularity of the boundary of a multidimensional aggregation patch

Abstract

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