Long-time asymptotics of Hele–Shaw flow for perturbed balls with injection and suction

Abstract

We discuss long-time behaviour of Hele-Shaw flow with injection and suction, for domains that are small perturbations of balls. An evolution equation for the motion of these domains is derived and linearised. We use spectral properties of the linearisation to show that in the case of injection, perturbations of balls decay algebraically. For classical Hele-Shaw flow, convergence turns out to be faster if low Richardson moments vanish. If for the three-dimensional case surface tension is included, all liquid can be removed by suction if the suction point and the geometric centre coincide and the ratio of suction speed and surface tension is small enough. An arbitrarily large portion of the liquid can be removed if the initial domain is sufficiently close to a ball. The main tools are the principle of linearised stability and the theory of H. Amann for abstract quasilinear parabolic evolution equations.