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

Abstract

The two-phase free boundary problem for the Navier-Stokes system is considered in a situation where the initial interface is close to a halfplane. By means of Lp-maximal regularity of the underlying linear problem we show local well-posedness of the problem, and prove that the solution, in particular the interface, becomes instantaneously real analytic. 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 1(0) ⊂ R n+1 (n ≥ 1) be a region occupied by a viscous incompressible fluid, fluid1, and let 2(0) be the complement of the closure of 1(0) in R n+1 , corre- sponding to the region occupied by a second incompressible viscous fluid, fluid2. We assume that the two fluids are immiscible. Let 0 be the hypersurface that bounds 1(0) (and hence also 2(0)) and let ( t) denote the position of 0 at time t. Thus, ( t) is a sharp interface which separates the fluids occupying the regions 1(t) and 2(t), respectively, where 2(t) := R n+1 1(t). We denote the normal field on ( t), pointing from 1(t) into 2(t), by ν(t, � ). Moreover, we de- note by V (t, � ) and κ(t, � ) the normal velocity and the mean curvature of ( t) with respect to ν(t, � ), respectively. Here the curvature κ(x, t) is assumed to be negative when 1(t) is convex in a neighborhood of x ∈ ( t). The motion of the fluids is governed by the following system of equations for i = 1,2 :    