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

在广义相对论中，引力不是一种力，而是四维宇宙在空间和时间中弯曲的结果。爱因斯坦方程将时空的曲率与其质量/能量含量联系起来。在局部坐标系下，它们可以写成耦合的非线性偏微分方程组。由于它们的复杂性，大多数研究都集中在精确解上，通过施加额外的对称性来获得，其中包括史瓦西（非旋转黑洞）和克尔（旋转黑洞）时空。一个自然的和高度非平凡的问题是，这些特殊的解决方案是否在小扰动下稳定。换句话说，假设在某个时刻，一个给定时空的曲率非常接近克尔时空的曲率，那么经过很长一段时间的演化，它最终会接近一个（可能不同的）克尔时空。该项目的主要目标是了解方程和方程组的解的行为，这些方程和方程组模拟爱因斯坦方程，但更简单。这些玩具问题的研究是必要的，以解决更困难的非线性稳定性问题上面提到的。最近有一系列关于史瓦西和克尔时空中线性波动方程衰变的研究，这是最简单的可能研究模型。即使这个问题是相当困难的，因为复杂的背景几何影响的色散特性在非平凡的方式。在紧凑的区域必须处理高频波包逗留沿着被困测地线很长一段时间，而在无穷大的非欧几里德字符的度量影响逐点的衰减率。被困集合的非常微妙（且不稳定）的性质特别需要来自调和分析、微分方程和微分几何的工具。然而，现在已经建立了测量衰变的可靠方法（例如局部能量估计和哈茨估计）。这些估计又可以用来解决非线性问题，如全局适定性（解的存在性和唯一性，以及初始数据的连续依赖性）。这样的非线性问题的一个例子是半线性波动方程的功率非线性小和大的初始数据。另一个例子，这是一个很好的模型爱因斯坦方程在调和坐标系中，是波动方程的一个度量，它本身取决于解决方案。这些技术也可用于解决麦克斯韦方程组解的最佳衰减问题，麦克斯韦方程组是描述史瓦西和克尔背景下电磁场演化的线性系统。最后，人们对来自弦理论的高维黑洞的研究很感兴趣，在低能量下，它可以用高维引力理论来描述。理解这些背景下波动方程的相关衰减性质也是非常有趣的。

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.