Solvability of the free boundary value problem of the Navier-Stokes equations
Hantaek Bae
Discrete & Continuous Dynamical Systems - A 2011, 29(3): 769-801 doi: 10.3934/dcds.2011.29.769
In this paper, we study the incompressible Navier-Stokes equations on a moving domain in $\mathbb{R}^{3}$ of finite depth, bounded above by the free surface and bounded below by a solid flat bottom. We prove that there exists a unique, global-in-time solution to the problem provided that the initial velocity field and the initial profile of the boundary are sufficiently small in Sobolev spaces.
keywords: Navier-Stokes equations Free Boundary Surface Tension.

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