Problems in nonlinear hyperbolic equations

这个研究项目的一个概要是“黑洞是真的吗？”黑洞在想象中隐约可见，是太空中的神秘区域，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，最初是作为爱因斯坦场方程的显式解被发现的，而爱因斯坦场方程是广义相对论的基础。由应用数学家R.Kerr发现的最著名的这类解依赖于两个参数。直到后来，物理学家才能够将一些非凡的天体物理物体，如类星体，与这些非凡的克尔解决方案联系起来。然而，根据定义，黑洞不能通过直接观测来探测，类星体与大质量黑洞有关的说法是基于被认为与黑洞解的特定数学属性相一致的间接观测。这个项目研究了关于黑洞数学理论的三个基本问题，这三个问题与黑洞是否可以是真实的物理对象密切相关。它们是僵化、稳定和崩溃的问题。例如，稳定性涉及科尔解的小扰动是否可以变得任意大的问题。如果是这样的话，科尔解就是数学上的产物，没有物理真实性。人们猜想Kerr解是稳定的，但尽管最近数学家们利用创新的偏微分方程方法取得了一些非常重要的进展，这个问题仍然是悬而未决的。刚性猜想断言，克尔解族在真空中耗尽了爱因斯坦场方程组的所有可能的定态解，而坍塌问题指的是黑洞是否可以在没有这样的物体的情况下及时自然地形成。该项目专注于数学物理中一些最重要的非线性双曲型方程。它的主要部分涉及作为黑洞理论核心的广义相对论中的三个相关问题：克尔解的唯一性和稳定性，以及黑洞的形成。为研究具有小角动量的Kerr时空的轴对称扰动的非线性稳定性问题提供了一种具体的策略。此外，在最近解决“有界曲率猜想”的基础上，PI打算继续寻找良定性的标度不变准则，即确保解的局部存在和唯一性的标度不变准则。这不仅是广义相对论中的一个重要目标，而且对于流体、弹性或相对论中的任何基本双曲型方程都是如此。正在研究的问题需要新的几何和解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.