This project will explore several related areas of research into the qualitative properties of solutions to partial differential equations (PDE). The water-wave problem with surface tension concerns the time evolution of the interface between two different fluids, which is naturally posed as an initial value PDE. By formulating the problem in the right fashion, the evolution of the interface is governed by a quasi-linear dispersive equation weakly coupled to a nonlinear transport equation. The water-wave projects outlined in this proposal focus on properties of solutions to this system that follow from the dispersion relation. Specifically, solutions to the nonlinear water-wave problem with surface tension are, on average in time, smoother than the initial data. In addition, it will be interesting to study mixed space-time dispersive properties (Strichartz estimates), higher-order smoothing properties, and the long-time well-posedness of the initial value problem. In addition to the water-wave problem, this project is concerned with spectral properties of the Laplacian on smooth manifolds: eigenfunction scarring in the compact case, and local smoothing for solutions to the Schroedinger equation in the noncompact case. The common thread running through all of these projects is the extensive use of microlocal analysis, the rough idea of which is to localize solutions of equations in space and in frequency (to the extent allowed by the uncertainty principle), after which these solutions are simpler to understand.
Dispersive equations and equations on curved spaces (manifolds) provide a rich area of interaction between various branches of mathematics as well as between different sciences. The work on the water-wave problem, a problem coming from mathematical hydrodynamics, represents a cross-disciplinary collaboration between pure math, applied math, and physics. The major goal of the project is to provide mathematical statements of physically intuitive ideas, such as "surface tension is a regularizing effect." The study of equations on curved spaces is of interest to geometers, analysts, and number theorists in mathematics, as well as to theoretical physicists working in quantum chaos and general relativity.
