Phase boundaries motion preserving the volume of each connected component

Abstract. We consider a finite number of particles initially close to spherical but of varying size exhibiting the property of preserving the measure of each connected component. We show that under the assumptions i) particle size/interparticle distance 1 and ii) initial deviation from sphericity/particle size 1 the particles retain their almost spherical shape and the dynamics of the system is determined by the motion of the centers. This is in sharp contrast with the Mullins-Sekerka free boundary problem where the particle centers remain almost fixed and the dynamics of the system is determined by the evolution of the radii.