  Let $d \geq 2$ and let $N(y)$ be the fundamental solution of the Laplace
equation in $R^d$ We consider the aggregation equation $$ \frac{\partial
\rho}{\partial t} + \operatorname{div}(\rho v) =0, v = -\nabla N * \rho$$ 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
$$\rho(x,t) = \frac{1}{1 -t} \chi_{D_t}(x), \quad x \in R^d, \quad 0 \le t <
1$$ where $D_t$ is a $C^{1 + \gamma}$ domain for all $0 \leq t < 1$.
