Asymptotics of Hele-Shaw flows with multiple point sources

We study the asymptotic behaviour of a He le-Shaw flow produced by the injection of fluid from a finite number of points at different speeds. We prove that, as time tends to infinity, the boundary of the fluid domain approaches the circle centred at the barycentre of the injection points with weights proportional to the injection rates. The distances from the barycentre to the boundary points are estimated from both above and below.