Problems in nonlinear hyperbolic equations

点击翻译按钮获取中文摘要

A synopsis of this research project is the question "Are black holes real?" Black holes loom large in the imagination as mysterious regions of space in which the force of gravity is so enormous that nothing, including light, can escape. However, black holes are mathematical constructs, first discovered as explicit solutions of the Einstein field equations that lie at the foundation of General Relativity. The best known family of such solutions, discovered by the applied mathematician R. Kerr, depends on two parameters. It was only later that physicists were able to relate some remarkable astrophysical objects, such as quasars, to these remarkable Kerr solutions. Yet, by definition, black holes cannot be detected by direct observation, and the claim that quasars are associated with massive black holes is based on indirect observations deemed consistent with the specific mathematical properties of the black hole solutions. This project investigates three fundamental questions concerning the mathematical theory of black holes, intimately tied to the issue of whether black holes can be real physical objects. They are the questions of rigidity, stability, and collapse. Stability, for example, concerns the question whether small perturbations of the Kerr solutions can grow arbitrarily large. If this were the case it would follow that the Kerr solutions are mathematical artifacts, with no physical reality. It has been conjectured that the Kerr solutions are stable, but despite some very important advances made recently by mathematicians using innovative partial differential equation methods, the problem remains wide open. The rigidity conjecture asserts that the Kerr family of solutions exhausts all possible stationary solutions of the Einstein field equations in vacuum, while the problem of collapse refers to the question of whether black holes can form in time, naturally, from configurations free of such objects. The project focuses on some of the most important nonlinear hyperbolic equations of mathematical physics. Its main part concerns three related problems in General Relativity at the heart of the theory of black holes: Uniqueness and stability of the Kerr solutions, and formation of black holes. It provides a specific strategy for making progress on the problem of non-linear stability for axially symmetric perturbations of Kerr spacetimes with small angular momentum. In addition, building on recent work on the resolution of the "Bounded Curvature Conjecture", the PI intends to continue the search for a scale invariant criterion for well-posedness, i.e., a scale invariant criterion that insures local existence and uniqueness of solutions. This is an important goal not just within General Relativity but for any of the basic hyperbolic equations, in fluids, elasticity, or relativity. The problems under study require new geometric and analytic ideas as well as the development of new techniques.