Abstract
We consider the incompressible viscous surface wave problem in the setting that the fluid domain is a horizontal infinite layer in 3D. The fluid dynamics is governed by the gravity-driven incompressible Navier–Stokes equations, and the effect of surface tension is ignored on the upper free surface. We prove the optimal time-decay rate of the low-order energy of the solution with minimal derivative count 3, which implies that the Lipschitz norm of the velocity decays at the rate (1+t)−1. This together with a time-weighted estimate for the highest order spatial derivatives of the free surface function leads to the boundedness of the high-order energy, which improves the result of Wang [9].
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