This project is concerned with a variety of problems involving the long-time behavior of solutions to linear and nonlinear wave equations. Such systems arise in a variety of problems in mathematical physics, including the underlying geometric field theories of special and general relativity. The basic concern of this project 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. Such questions are among the most central and difficult in the theory of partial differential equations, and the work proposed here is a continuation of recent investigations and advances of the principal investigator on several of the most important problems in geometric wave equations, including the blowup vs. global regularity question for energy critical wave-maps.
Linear and nonlinear wave equations arise as models in many different areas of physics, from the underlying classical field theories of electro-magnetism, elasticity, the equations of linearized gravitational radiation, and the full Einstein field equations themselves, to many equations of quantum mechanics and quantum field theory. The mathematical issues to be studied by the principal investigator are all related to the basic question of describing the dynamical properties such physical systems. Progress on the mathematical understanding of these systems should lead to better modeling, prediction, and control of the physical phenomena they govern. The PI plans not only to continue his recent work, but also to train several new graduate students in this exciting and fast paced field of study.
This project was concerned with investigating the long time behavior of solutions to linear and non-linear wave equations. Such equations arise in a variety of contexts in mathematical physics, including the underlying geometric field theories of special and general relativity. A deeper knowledge of the mathematical structure of these equations is relevant to our understanding of physical phenomena on a wide range of length scales; from microscopic world of sub-atomic particles all the way to the astrophysical regime of gravitational waves. The findings of the PI break down into roughly two categories. The first provides the basis for general description of "all" solutions to certain non-linear gauge field equations in a gravitational vacuum where the dominant force is electro-magnetic. The relevance of such findings belongs to the realm of atomic physics. The second category of results provides a basis for describing solutions to certain non-linear wave equations in a situation where gravity exerts a significant force. The relevance here is to large scale phenomena involving gravity as the dominant force. Underlying both sets of findings is an analysis of the wave-like nature of the forces and particles involved. A significant portion of the research carried out in this project was done by graduate students at the University of California San Diego. The findings of the PI and collaborators forms a basis for ongoing investigations.
