In this paper, we consider the motion of two immiscible, incompressible, viscous fluids, \(\text {fluid}_+\) and \(\text {fluid}_-\) , with different constant densities in the presence of a uniform gravitational field acting vertically downward in the N-dimensional Euclidean space \({\textbf {R}} ^N\) for \(N\ge 3\) . The motion is governed by the two-phase Navier–Stokes equations with a sharp interface \(x_N=\eta (x',t)\) , where \(x'=(x_1,\dots ,x_{N-1})\in {\textbf {R}} ^{N-1}\) and \(t>0\) . Surface tension is taken into account on the interface, while \(\text {fluid}_+\) and \(\text {fluid}_-\) occupy the upper region \(x_N>\eta (x',t)\) of the interface and the lower region \(x_N<\eta (x',t)\) of the interface, respectively. One thus calls \(\text {fluid}_+\) the upper fluid and \(\text {fluid}_-\) the lower fluid. It is well-known that the trivial steady state, i.e., the motionless state with the flat interface \(x_N=0\) and a constant pressure, is unstable if the upper fluid is heavier than the lower fluid due to the effect of gravity. This instability is called the Rayleigh–Taylor instability and was studied from a physical point of view by Harrison (Proc. Lond. Math. Soc. (2) 6:396–405, 1908), Bellman and Pennington (Quart. Appl. Math. 12:151–162, 1954), and Chandrasekhar (Internat. Ser. Monogr. Phys., Clarendon Press, Oxford, 1961) for a linearized problem in a horizontally periodic setting. That result was extended by Pr \(\ddot{\textrm{u}}\) ss and Simonett (Indiana Univ. Math. J. 59:1853–1871, 2010) to the nonlinear Rayleigh–Taylor instability without assuming periodicity in the horizontal direction. On the other hand, we are interested in the stability of the trivial steady state in the case where the upper fluid is lighter than the lower fluid. Under this stability condition, the present paper treats the same nonlinear problem as in Pr \(\ddot{\textrm{u}}\) ss and Simonett (Indiana Univ. Math. J. 59:1853–1871, 2010) and shows the asymptotic stability of the trivial steady state.
