Abstract

In this lecture, we will discuss the irrotational water wave problem. We will place the problem in the larger context of vortex sheets; the vortex sheet is the interface between two irrotational fluids, allowing for a jump in the tangential components of velocity across the interface. We will present the equations of motion in both 2D and 3D, for the water wave problem both with and without surface tension. We will present the numerical method of Hou, Lowengrub, and Shelley (HLS) for computing solutions of the initial value problem in 2D, and how the HLS ideas can be used to prove short-time well-posedness of the initial value problem. The corresponding numerical method and well-posedness proofs for the three-dimensional problem will also be discussed. If time allows, we will go beyond the initial value problem, and discuss how the vortex sheet formulation with the HLS ideas can be extended to treat other problems, such as the traveling wave problem. This talk includes joint work with Nader Masmoudi, Michael Siegel, Svetlana Tlupova, and possibly others.