Well-posedness of linearized motion for 3-D water waves far from equilibrium

In this paper, we study the motion of a free surface separating two different layers of fluid in three dimensions. We assume the flow to be inviscid, irrotational, and incompressible. In this case, one can reduce the entire motion by variables on the surface alone. In general, without additional regularizing effects such as surface alone. In general, without additional regularizing effects such as surface tension or viscosity, the flow can be subject to Rayleigh-Taylor or Kelvin-Helmholtz instabilities which will lead to unbounded growth in high frequency wave numbers. In this case, the problem is not well-posed in the Hadamard sense. The problem of water wave with no fluid above is a special case. It is well-known that such motion is well-posed when the free surface is sufficiently close to equilibrium. Beale, Hous and Lowengrub derived a general condition which ensures well-posedness of the linearization about a presumed time-dependent motion in two dimensional case. The linearized equations, when formulated in a proper coordinate system are found to have a qualitative structure surprisingly like that for the simple case of linear waves near equilbrium. Such an analysis is essential in analyzing stability of boundary integral methods for computing free interface problems. 19more » refs.« less