Analysis of Linear and Nonlinear Wave Equations: regularity, asymptotics and blowup.
Abstract of Proposed Research Jacob K Sterbenz
This project is to investigate a variety of problems involving both the regularity and the long time behavior of solutions of linear and non-linear wave equations. The fundamental goal is to understand the asymptotic behavior of such systems in a wide range of contexts, from local phenomena such as the spontaneous formation of singularities, to delicate dispersive properties which take place over infinite space-time scales. The main themes of recent research in this area has been critical regularity results for geometric wave equations, blowup phenomena for such equations in critical dimensions, and the asymptotic stability of key linear and non-linear models. This is a very active field of current research. Effective strategies and techniques which yield insight into these problems will have non-trivial overlap with many other areas of classical and modern mathematics such as spectral and scattering theory, differential geometry, real variable methods of harmonic analysis, and the construction and implementation of numerical simulations.
These systems arise as models of many different problems in physics, including the underlying classical field theories of electro-magnetism, nonlinear elasticity, the equations of linearized gravitational radiation, quantum mechanics and quantum field theory. The mathematical issues to be studied here all are related to the basic question of describing the dynamical properties of linear and nonlinear physical systems. Progress on the mathematical analysis of these problems should lead to a much better understanding, as well as prediction and perhaps control, of these physical phenomena.
