In this paper we investigate the existence of global weak solutions for the multilayer model introduced by Audusse et al. which is related to incompressible free surface flows. We prove the global stability of weak solutions over the torus. We observe that this model admits the so called BD-entropy and a gain of integrability on the velocity in the spirit of the work of Mellet and Vasseur. It allows us to obtain enough compactness estimates in order to show the stability of global weak solutions.