Proposal: DMS-9801094 Principal Investigator: Sijue Wu Abstract: Wu will study the long time behavior of water waves, as well as the motion of a general two-layered fluid flow. To understand the long time behavior of a water wave, Wu would like to determine whether the free surface of such a wave remains non-self-intersecting and whether solutions of water wave problems exist for all time, given that the initial free surface of the water wave is a small perturbation of still water. To be more precise, she would like to characterize those initial free surfaces that guarantee the non-self-intersection of the free surface of a water wave for all time. She also wants to study the blow-up mechanism for those solutions of the water wave problem that fail to exist for all time. The methods used to tackle these problems arise from harmonic analysis and the theory of partial differential equations. They include: decay estimates, maximal principle arguments, and the construction of a self-similar solution to the problem. To understand the motion of the interface between any two superposed fluids, Wu intends to study such fundamental questions as the existence of solutions to the equation governing the motion and the singularity profiles of solutions. To achieve this goal, she hopes to adapt the methods in her earlier work on water waves to derive a quasilinear equation equivalent to the equation that describes the motion of the interface. The reason for this approach is that the equation which models the motion of a two-fluid interface is highly nonlinear. In general, one has a better understanding of quasilinear equations than of fully nonlinear equations. The proposed research is a continuation of the work started by Nalimov, Yosihara and Craig and later advanced by the principal investigator on the existence and uniqueness of solutions of water wave problems and the work of Sulem, Dochun and Robert, Caflisch and Orellana, and Ebin on the well-posedness of vortex sheet motion. Fluid waves, in numerous guises, are present in some of the most familiar experiences of daily life, from the jarring impact of loud noises on our eardrums to the soothing ebb and flow of surf at a beach. The rich variety of phenomena associated with wave motion have provided generations of physicists and mathematicians with an important and challenging research subject. The general problem of motion of the free interface of two superposed fluids has applications to a wide variety of concrete physical problems. It has been used to understand the mixing of fluids, the separation of boundary layers, the generation of sounds, and coherent structures in models of turbulence. The long term objective of the proposed work is to understand wave-breaking in surface waves and the singularity mechanism in vortex sheets, both topics that have significant ramifications for physics and engineering.
