Decay of the 2D Periodic Stationary Viscous Surface Waves

We consider the evolution of viscous fluids in a 2D horizontally periodic slab bounded above by a free top surface and below by a fixed flat bottom. This is a free boundary problem. The dynamics of the fluid are governed by the incompressible stationary Navier.Stokes equations under the influence of gravity and the effect of surface tension. We develop the global theory of solutions in low regularity Sobolev spaces for small data by nonlinear energy estimates.