Decay of viscous surface waves without surface tension in horizontally infinite domains

Abstract

We consider a viscous fluid of finite depth below the air, occupying a three-dimensional domain bounded below by a fixed solid boundary and above by a free moving boundary. The fluid dynamics are governed by the gravity-driven incompressible Navier–Stokes equations, and the effect of surface tension is neglected on the free surface. The long-time behavior of solutions near equilibrium has been an intriguing question since the work of Beale (1981).

This is the second in a series of three papers by the authors that answers the question. Here we consider the case in which the free interface is horizontally infinite; we prove that the problem is globally well-posed and that solutions decay to equilibrium at an algebraic rate. In particular, the free interface decays to a flat surface.

Our framework utilizes several techniques, which include

1. a priori estimates that utilize a “geometric” reformulation of the equations;

2. a two-tier energy method that couples the boundedness of high-order energy to the decay of low-order energy, the latter of which is necessary to balance out the growth of the highest derivatives of the free interface;

3. control of both negative and positive Sobolev norms, which enhances interpolation estimates and allows for the decay of infinite surface waves.

Our decay estimates lead to the construction of global-in-time solutions to the surface wave problem.