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Local well-posedness of
the viscous surface wave problem without surface tension
Yan Guo and Ian Tice |
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Vol. 6 (2013), No. 2, 287–369
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Abstract |
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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 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. |
Keywords
Navier–Stokes, free boundary problems, surface waves
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Mathematical Subject Classification 2010
Primary: 35Q30, 35R35, 76D03, 76E17
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Milestones
Received: 9 June 2011
Revised: 7 March 2012
Accepted: 23 May 2012
Published: 24 June 2013
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Authors |
| Yan Guo | |
| Division of Applied
Mathematics Brown University 182 George Street Providence, RI 02912 United States |
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| Ian Tice | |
| Carnegie Mellon University Department of Mathematical Sciences Pittsburgh, PA 15213 United States |
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