Analysis & PDE

  • Anal. PDE
  • Volume 6, Number 6 (2013), 1429-1533.

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

Yan Guo and Ian Tice

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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.

Article information

Anal. PDE, Volume 6, Number 6 (2013), 1429-1533.

Received: 15 October 2012
Accepted: 15 November 2012
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 35R35: Free boundary problems 76D03: Existence, uniqueness, and regularity theory [See also 35Q30]
Secondary: 35B40: Asymptotic behavior of solutions 76E17: Interfacial stability and instability

Navier–Stokes equations free boundary problems global existence


Guo, Yan; Tice, Ian. Decay of viscous surface waves without surface tension in horizontally infinite domains. Anal. PDE 6 (2013), no. 6, 1429--1533. doi:10.2140/apde.2013.6.1429.

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