DECAY AND WELL-POSEDNESS OF SOLUTIONS TO HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

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In the theory of General Relativity gravity is seen not as a force, but as the result of the bending of a four dimensional universe in both space and time. Einstein's Equations connect the curvature of spacetime to its mass/energy content. In local coordinates they can be written as the system of coupled nonlinear partial differential equations. Due to their complexity, most studies have focused on exact solutions, obtained by imposing additional symmetries, which include Schwarzschild (non-rotating black holes) and Kerr (rotating black holes) spacetimes. A natural and highly nontrivial problem is whether these particular solutions are stable under small perturbations. In other words, assuming that at a moment in time the curvature of a given spacetime is very close to the curvature of a Kerr spacetime, it is expected that after evolving for a long time it will eventually approach a (potentially different) Kerr spacetime. The main goal of this project is understanding the behavior of solutions to equations and systems of equations that model Einstein's Equations, but are simpler. The study of these toy problems is necessary in order to tackle the much harder nonlinear stability problem mentioned above. There has recently been a flurry of activity with regard to understanding the decay for the linear wave equations on Schwarzschild and Kerr spacetimes, which is the simplest possible model to study. Even this problem is quite difficult, since the complicated background geometry affects the dispersion properties in nontrivial ways. In compact regions one must deal with high frequency wave packets that linger along trapped geodesics for a long time, while at infinity the non-Euclidean character of the metric affects the pointwise rates of decay. The very delicate (and unstable) nature of the trapped set in particular requires tools coming from harmonic analysis, differential equations, and differential geometry. Nevertheless, robust ways of measuring decay (e.g. local energy estimates and Strichartz estimates) have now been established. These estimates can in turn be used to tackle nonlinear problems like global well-posedness (existence and uniqueness of solutions, and continuous dependence of initial data). One example of such nonlinear problems is the semilinear wave equation with power nonlinearities for both small and large initial data. Another example, which is a good model of Einstein's Equations in harmonic coordinates, is the wave equation for a metric that itself depends on the solution. These techniques can also be used to settle the problem of optimal decay of solutions to the Maxwell Equations, the linear system which describes the evolution of an electromagnetic field on Schwarzschild and Kerr backgrounds. Finally, there is interest in the study of higher dimensional black holes coming from string theory, which at low energies can be described by higher-dimensional theories of gravity. Understanding the relevant decay properties for the wave equation on these backgrounds would also be quite interesting.