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Large-time behavior of viscous surface waves. (English) Zbl 0642.76048
Recent topics in nonlinear PDE II, Lect. 2nd Meet., Sendai/Jap. 1984, Lect. Notes Numer. Appl. Anal. 8, 1-14 (1985).
[For the entire collection see Zbl 0604.00009.]
The authors are concerned with global-in-time solutions to a free surface problem for viscous incompressible fluids. The motion in a time-dependent domain \(\Omega (t)=\{x\in {\mathbb{R}}^ 2:\) \(-b<y<\eta (t,x)\}\) is governed by the Navier-Stokes equations; on the free surface \(y=\eta (t,x)\) the usual kinematic and dynamic conditions are assumed, while the velocity of the fluid is assumed to be zero on the fixed lower boundary \(y=-b\). Surface tension on \(y=\eta (t,x)\) is taken into account. Recently [Arch. Ration. Mech. Anal. 84, 307-352 (1984; Zbl 0545.76029)] the first author proved the existence of a global-in-time solution to this problem, assuming that the initial data are near the equilibrium. The aim of this paper is to give the asymptotic decay rate for this problem as \(t\to +\infty\).

MSC:
76D33 Waves for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations