## 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

### Citations:

Zbl 0604.00009; Zbl 0545.76029