Analysis & PDE
- Anal. PDE
- Volume 6, Number 2 (2013), 287-369.
Local well-posedness of the viscous surface wave problem without surface tension
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.
Anal. PDE, Volume 6, Number 2 (2013), 287-369.
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
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] 76E17: Interfacial stability and instability
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