Analysis & PDE

  • Anal. PDE
  • Volume 6, Number 2 (2013), 287-369.

Local well-posedness of the viscous surface wave problem without surface tension

Yan Guo and Ian Tice

Full-text: Open access

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 domain is allowed to have a horizontal cross-section that is either periodic or infinite in extent. 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. This paper is the first in a series of three on the global well-posedness and decay of the viscous surface wave problem without surface tension. Here we develop a local well-posedness theory for the equations in the framework of the nonlinear energy method, which is based on the natural energy structure of the problem. Our proof involves several novel techniques, including: energy estimates in a “geometric” reformulation of the equations, a well-posedness theory of the linearized Navier–Stokes equations in moving domains, and a time-dependent functional framework, which couples to a Galerkin method with a time-dependent basis.

Article information

Source
Anal. PDE, Volume 6, Number 2 (2013), 287-369.

Dates
Received: 9 June 2011
Revised: 7 March 2012
Accepted: 23 May 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731312

Digital Object Identifier
doi:10.2140/apde.2013.6.287

Mathematical Reviews number (MathSciNet)
MR3071393

Zentralblatt MATH identifier
1273.35209

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

Keywords
Navier–Stokes free boundary problems surface waves

Citation

Guo, Yan; Tice, Ian. Local well-posedness of the viscous surface wave problem without surface tension. Anal. PDE 6 (2013), no. 2, 287--369. doi:10.2140/apde.2013.6.287. https://projecteuclid.org/euclid.apde/1513731312


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