In this paper we propose a strategy to approximate incompressible hydrostatic free surface Euler and Navier-Stokes models. The main advantage of the proposed models is that the water depth is a dynamical variable of the system and hence the model is formulated over a fixed domain.The proposed strategy extends previous works approximating the Euler and Navier-Stokes systems using a multilayer description. Here, the needed closure relations are obtained using an energy-based optimality criterion instead of an asymptotic expansion. Moreover, the layer-averaged description is successfully applied to the Navier-Stokes system with a general form of the Cauchy stress tensor.

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This paper concerns the global stability of weak solutions for the multilayer system introduced by Audusse et al. which models incompressible free surface flows. To do this, it is proven that this model admits the so called BD-entropy and a gain of integrability on the velocity. It allows to obtain enough compactness estimates in order to show the stability of global weak solutions.

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