We place the irrotational water wave problem in the larger context of vortex sheets. We describe the evolution equations for vortex sheets in 2D or in 3D. The numerical method of Hou, Lowengrub, and Shelley (HLS) for the solution of the initial value problem for the vortex sheet with surface tension in 2D is discussed; furthermore, we indicate how the HLS formulation of the problem is useful for a proof of well-posedness. We then show how one may take the zero surface tension limit in the water wave case. We close with a brief discussion of the extension of the HLS ideas to 3D, for both analysis and computing.
The irrotational water wave is a special case of the irrotational vortex sheet. For the vortex sheet problem, two fluids whose motions are described by the incompressible, irrotational Euler equations meet at an interface. This interface, the vortex sheet, is free to move, and moves according to the velocities of the two fluids restricted to the interface. Each fluid has its own non-negative, constant density. Different geometries are possible, but to be definite, at present we consider the case in which the fluids are two-dimensional and such that each fluid region has one component, which is of infinite vertical extent and horizontally periodic. Thus, we may say that we have an upper fluid and a lower fluid. In the water wave case, the density of the upper fluid is equal to zero.
Without surface tension, if each of the two fluids has positive density, then the vortex sheet is known to have an ill-posed initial value problem; this has been demonstrated by several authors. We note that when discussing ill-posedness of a problem, to be precise, one should mention the function spaces under consideration; for example, Caflisch and Orellana have shown that the vortex sheet initial value problem is ill-posed in Sobolev spaces . In analytic function spaces, however, solutions of the vortex sheet problem have been shown to exist by a Cauchy-Kowalewski argument .
The ill-posedness of the vortex sheet initial value problem (when the two fluids have positive densities) is caused by the presence of the Kelvin-Helmholtz instability.