Vol. 6, No. 6, 2013

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Decay of viscous surface waves without surface tension in horizontally infinite domains

Yan Guo and Ian Tice

Vol. 6 (2013), No. 6, 1429–1533
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.

Keywords
Navier–Stokes equations, free boundary problems, global existence
Mathematical Subject Classification 2010
Primary: 35Q30, 35R35, 76D03
Secondary: 35B40, 76E17
Milestones
Received: 15 October 2012
Accepted: 15 November 2012
Published: 18 November 2013
Authors
Yan Guo
Division of Applied Mathematics
Brown University
182 George Street
Providence, Rhode Island 02912
USA
Ian Tice
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh PA 15213
USA