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We consider a coupled two-phase Navier-Stokes/Mullins-Sekerka systemdescribing the motion of two immiscible, incompressible fluids inside a boundedcontainer. The moving interface separating the liquids meets the boundary ofthe container at a constant ninety degree angle. This common interface isunknown and has to be determined as a part of the problem.  We show well-posedness and investigate the long-time behaviour of solutionsstarting close to certain equilibria. We prove that for equal densities thesesolutions exist globally in time, are stable, and converge to an equilibriumsolution at an exponential rate.

Well-Posedness and Qualitative Behaviour of a Two-phase Navier-Stokes/Mullins-Sekerka system with boundary contact