On the two-phase Navier–Stokes equations with surface tension

In this paper we consider a free boundary problem that describes the motion of two viscous incompressible capillary Newtonian fluids. The fluids are separated by an interface that is unknown and has to be determined as part of the problem. Let Ω1(0) ⊂ Rn+1 (n > 1) be a region occupied by a viscous incompressible fluid, fluid1, and letΩ2(0) be the complement of the closure ofΩ1(0) in Rn+1, corresponding to the region occupied by a second incompressible viscous fluid, fluid2. We assume that the two fluids are immiscible. Let Γ0 be the hypersurface that bounds Ω1(0) (and hence also Ω2(0)) and let Γ (t) denote the position of Γ0 at time t . Thus, Γ (t) is a sharp interface which separates the fluids occupying the regions Ω1(t) and Ω2(t), respectively, where Ω2(t) := Rn+1 \Ω1(t). We denote the normal field on Γ (t), pointing from Ω1(t) into Ω2(t), by ν(t, ·). Moreover, we denote by V (t, ·) and κ(t, ·) the normal velocity and the mean curvature of Γ (t) with respect to ν(t, ·), respectively. Here the curvature κ(x, t) is assumed to be negative when Ω1(t) is convex in a neighborhood of x ∈ Γ (t). The motion of the fluids is governed by the following system of equations for i = 1, 2: