The purpose of this project is to improve our understanding of the effect of complicated background geometry on solutions to certain wave or dispersive equations. The geometry may result, for example, from the introduction of a boundary, from the influence of a nonflat metric, or from consideration of nonhomogeneous materials. One major aspect of this project is to study global-in-time measures of dispersion, such as localized energy estimates and Strichartz estimates, for variable coefficient wave equations. As compared to the flat cases, one encounters additional difficulties that arise, for instance, because of the focusing of Hamiltonian rays, trapped rays, and the possibility of eigenfunctions or resonances. A particular example of interest is to study the wave equation on nonflat solutions to Einstein's equations, such as the Schwarzschild space-time. A second major component of this project is to study the long-time existence of solutions to certain nonlinear wave or elastic wave equations. Of special interest here are problems in anisotropic elasticity or existence questions for nonlinear wave equations in exterior domains where certain invariances, which are typically used to prove long-time existence, are not available.
A class of fundamental open questions in theoretical physics revolves around the stability of solutions to Einstein's equations. Namely, if the universe starts close to a certain known solution, one might seek to prove that it remains close to that solution for all later times. The only known rigorous proofs of stability are for the so-called flat solution. It is believed that a good understanding of the dispersive nature of wave equations will play an essential role in any proof of stability, and the first set of questions described above seeks to provide some insights into this. Understanding the lifespan of solutions to partial differential equations is of basic interest. The objective is to estimate the amount of time that a solution may exist prior to, say, becoming infinitely large (blow-up phenomena). Of particular interest in this project is the lifespan of solutions to certain problems in nonlinear elasticity. These problems resemble certain questions in nonlinear wave equations where the proofs of long-time existence rely heavily on the many invariances of the linear wave equation. In nonhomogeneous materials, however, these invariances may be lost, so different techniques are required. As a model problem for problems where isotropy is maintained in planes, such as hexagonal crystals, certain two-dimensional isotropic problems will first be studied. Here one must make more delicate use of the nonlinear structure of the equations in order to show such long-time existence.
