Existence of Solutions to Hyperbolic Differential Equations in Mathematical Physics

Johns Hopkins University, Baltimore MD

Investigators

Abstract

The PI's research concerns 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 and Euler's equations of fluids. The basic questions concern existence, uniqueness and stability of solutions, as well as the question if solutions blow up (e.g. black holes in general relativity) and if not what the long time behavior of solutions are? More specifically, the PI is mainly working in two areas. One project is to study if Einstein's and related equations have solutions for all times or if the solutions blow up. A long term goal is to study the stability of large solutions like black holes and the big bang in general relativity. The motivation is to understand the large scale structure of our universe. The Physicists are building large gravitational wave detectors to observe the universe and the theory has to be developed together with observations. Another project is to study a class of problems that occur in fluid dynamics and general relativity, in particular, proving existence and stability for the free boundary problem of the motion of the surface of a fluid in vacuum (such as the surface of the ocean). A long term goal is to study the long time behavior of astrophysical bodies such as gaseous stars as well as other interface problems of fluids and solids. It is conceivable that understanding the properties of and controlling the interface between two fluids could have important applications. In particular there is a version of the problem in magneto-hydrodynamics and controlling the plasma is needed for constructing fusion reactors. To solve these problems the PI and collaborators are developing new techniques that could be useful for studying many other problems as well. In particular, they are using geometric methods combined with frequency decomposition methods to study hyperbolic differential equations. The PI's and collaborator's greatly simplified existence proof for Einstein's equations and its generalizations and refinements will have a large impact. Moreover the detailed asymptotic behavior they prove in harmonic coordinates will be useful also for the physics community. The methods the PI has developed for the free boundary problem of fluids work also for the compressible case and also with vorticity since it uses interior equations and not just equations on the boundary. The methods PI and collaborators are developing to deal with nonlinear equations with variable coefficients will hopefully also be useful to show stability of perturbations of large solutions.