Abstract
We consider the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a general rigid bottom in a three-dimensional horizontally periodic setting. We establish the global well-posedness of the problem both with and without surface tension. We prove that without surface tension the solution decays to the equilibrium state at an almost exponential rate; with surface tension, we show that the solution decays at an exponential rate. Our results include the case in which a heavier fluid lies above a lighter one, provided that the surface tension at the free internal interface is above a critical value, which we identify. This means that sufficiently large surface tension stabilizes the Rayleigh–Taylor instability in the nonlinear setting. As a part of our analysis, we establish elliptic estimates for the two-phase stationary Stokes problem.
Highlights
In this paper we study the viscous surface-internal wave problem, which concerns the dynamics of two layers of distinct, immiscible, viscous, incompressible fluid lying above a general rigid bottom and below an atmosphere of constant pressure
We assume that a uniform gravitational field points in the direction of the rigid bottom
This is a free boundary problem since both the upper surface, where the upper fluid meets the atmosphere, and the internal interface, where the upper and lower fluids meet, are free to evolve in time with the motion of the fluids
Summary
We will transform the free boundary problem in Ω(t) to one in the fixed domain Ω by using the unknown free surface functions η± Hataya [16] proved an existence result for a periodic free interface problem with surface tension as a perturbation around the plane Couette flow of two fluids; he showed the local existence of small smooth solution for any physical constants, and the existence of exponentially decaying small solution if the viscosities of the two fluids are sufficiently large and their difference is small. The purpose of this paper is to investigate in the energy spaces the well-posedness and decay of solutions to the viscous surface-internal wave problem under the horizontally periodic assumption, both with and without surface tension and without any constraints on the viscosities
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