This project will concentrate on studying the well-posedness of certain nonlinear systems of hyperbolic partial differential equations (PDEs). One class of problems is the free boundary problem for the motion of the surface of a fluid in a vacuum. Another project is to study global solutions of equations related to Einstein's equations of general relativity. The basic questions for these equations include: (i) Do we have local existence and uniqueness of solutions in a certain class? (ii) Do we have blow-up of solutions? (e.g. black holes in general relativity) (iii) What is the long time behavior of solutions?
The answers to these questions may provide further understanding of gravitational waves in the universe. Understanding the properties of, and controlling the interface between, two fluids may have industrial applications.
