Vol. 6, No. 2, 2013

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

Yan Guo and Ian Tice

Vol. 6 (2013), No. 2, 287–369

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.

Navier–Stokes, free boundary problems, surface waves
Mathematical Subject Classification 2010
Primary: 35Q30, 35R35, 76D03, 76E17
Received: 9 June 2011
Revised: 7 March 2012
Accepted: 23 May 2012
Published: 24 June 2013
Yan Guo
Division of Applied Mathematics
Brown University
182 George Street
Providence, RI 02912
United States
Ian Tice
Carnegie Mellon University
Department of Mathematical Sciences
Pittsburgh, PA 15213
United States