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We consider the viscous incompressible fluids in a three-dimensional horizontally periodic domain bounded below by a fixed smooth boundary and above by a free moving surface. The fluid dynamics are governed by the Navier-Stokes equations with the effect of gravity and surface tension on the free surface. We develop a global well-posedness theory by a nonlinear energy method in low regular Sobolev spaces with several techniques, including: the horizontal energy-dissipation estimates, a new tripled bootstrap argument inspired by Guo and Tice Arch. Ration. Mech. Anal.(2018). Moreover, the solution decays asymptotically to the equilibrium in an exponential rate.
Cheng et al. (Thu,) studied this question.
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