Decay and Global Well-Posedness of the Free-Boundary Incompressible Euler Equations with Damping

We consider the free boundary problem for a layer of incompressible fluid lying below the atmosphere and above a rigid bottom in the horizontally infinite setting. The fluid dynamics is governed by the incompressible Euler equations with damping and gravity, and the effect of surface tension is neglected on the upper free boundary. We prove the global well-posedness of the problem with the small initial data in both 2D and 3D. One of key ideas here is to make use of the time-weighted dissipation estimates to close the nonlinear energy estimates; in particular, this implies that the Lipschitz norm of the velocity is integrable-in-time, which is significantly different from that of viscous surface waves (Guo and Tice in Anal PDE 6(6):1429–1533, 2013; Wang in Adv Math 374:107330, 2020).