This project is concerned with basic mathematical questions about systems of nonlinear hyperbolic differential equations in mathematical physics. These include many important equations in classical field theory and continuum mechanics (e.g., Einstein's equations of general relativity, Euler's equations for fluids). The basic questions are: (i) Do we have existence and uniqueness of solutions, and continuous dependence on data, in a certain class? (ii) Can solutions blow up (e.g., black holes in general relativity)? (iii) What is the long-time behavior of solutions? More specifically, the principal investigator is working in two main areas. One part of the project is to study the problem of existence of global solutions of Einstein's equations and of other related equations. The first goal is to simplify, generalize, and refine the existence results for Einstein's equations. A long-term objective is to study the stability of large solutions like black holes in general relativity. This is related to one of the central problems in mathematical relativity, namely, the cosmic censorship conjecture of Penrose. The question of stability or blow-up of large solutions is also the main question now in the area of nonlinear wave equations. A second component of the project involves studying a class of problems that occur in fluid dynamics and general relativity, in particular, proving the well-posedness for the free boundary problem of the motion of the surface of a fluid in a vacuum. The first goal in this area is to prove local existence. A longer-range goal is to study the long-time behavior of astrophysical bodies such as gaseous stars, along with other problems related to the interfaces between fluids and solids. To solve these problems the principal investigator and his collaborators are developing new techniques that could be useful for studying many other problems as well. In particular, they are using geometric methods to study hyperbolic differential equations.
The principal investigator and his collaborators have recently simplified greatly the existence proof for solutions to Einstein's equations and their generalizations. The continuing refinement of those ideas that constitute part of the current project could have a significant impact. To name just one, the new approach should make it much easier for graduate students to study mathematical relativity. Moreover, the detailed asymptotic behavior that these methods reveal through the introduction of so-called harmonic coordinates will be useful to the physics and astronomical communities. Physicists are in the process of constructing large gravitational wave detectors to observe the universe. In order for the scientists to know what to look for with these instruments, there is a need for a large-scale effort in doing numerical calculations and simulations based on Einstein's equations. The only successful attempts hitherto to do so have been with the aid of harmonic coordinates. It is also conceivable that understanding the properties of and controlling the interface between two fluids could have industrial applications. In particular, there is a version of the problem for plasma physics in magneto-hydrodynamics. As is well known, the ability to control a plasma is essential to the construction of fusion reactors.
