Problems in nonlinear hyperbolic equations

这个研究项目的概要是“黑洞是真实的吗？“黑洞在想象中显得很大，是一个神秘的空间区域，其中的引力是如此巨大，以至于包括光在内的任何东西都无法逃脱。然而，黑洞是数学概念，首先被发现是作为广义相对论基础的爱因斯坦场方程的显式解。应用数学家R. Kerr，取决于两个参数。只是到了后来，物理学家才能够将一些非凡的天体（如类星体）与这些非凡的克尔解联系起来。然而，根据定义，黑洞不能通过直接观测来检测，而类星体与大质量黑洞相关的说法是基于间接观测，被认为与黑洞解的特定数学性质一致。这个项目研究了有关黑洞数学理论的三个基本问题，这些问题与黑洞是否可以是真实的物理对象的问题密切相关。它们是刚性、稳定性和崩溃的问题。例如，稳定性涉及克尔解的小扰动是否可以任意增长的问题。如果是这样的话，那么克尔解就是数学伪像，没有物理真实性。克尔解是稳定的，但尽管最近数学家使用创新的偏微分方程方法取得了一些非常重要的进展，这个问题仍然是开放的。刚性猜想断言克尔族解穷尽了真空中爱因斯坦场方程的所有可能的定态解，而坍缩问题则涉及黑洞是否可以在时间上自然地从没有这样的物体的构型中形成的问题。该项目的重点是一些最重要的数学物理非线性双曲方程。它的主要部分涉及在黑洞理论的核心广义相对论的三个相关问题：克尔解的唯一性和稳定性，以及黑洞的形成。它提供了一个具体的战略取得进展的问题的非线性稳定性的轴对称扰动的小角动量克尔时空。此外，在最近关于解决“有界曲率猜想”的工作的基础上，PI打算继续寻找适定性的尺度不变标准，即，一个尺度不变的标准，确保当地的存在性和唯一性的解决方案。这不仅是广义相对论的一个重要目标，也是流体、弹性或相对论中任何基本双曲方程的一个重要目标。研究中的问题需要新的几何和分析思想以及新技术的发展。

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.