Weak Solutions to a Sharp Interface Model for a Two-Phase Flow of Incompressible Viscous Fluids with Different Densities

In this paper we consider the flow of two incompressible, viscous and immiscible fluids in a bounded domain, with different densities and viscosities. This model consists of a coupled system of Navier-Stokes and Mullins-Sekerka type parts, and can be obtained from the sharp interface limit of the diffuse interface model proposed by the first author, Garcke, and Gr\"{u}n (Math. Models Methods Appl. Sci. 22, 2012). We introduce a new notion of weak solutions and prove its global in time existence, together with a consistency result of smooth weak solutions with the classical Navier-Stokes-Mullins-Sekerka system. Our new notion of solution allows to include the case of different densities of the two fluids, a sharp energy dissipation principle \`a la De Giorgi, together with a weak formulation of the constant contact angle condition at the boundary, which were left open in the previous notion of solution proposed by the first author and R\"{o}ger (Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire 26, 2009).