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

在广义相对论中，引力不被视为一种力，而是四维宇宙在空间和时间上弯曲的结果。爱因斯坦方程将时空的曲率与其质量/能量含量联系起来。在局部坐标下，它们可以写成耦合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解在小扰动下是否稳定。换句话说，假设在某一时刻，给定时空的曲率非常接近克尔时空的曲率，那么在经过很长一段时间的演化后，它最终将接近（可能不同的）克尔时空。这个项目的主要目标是理解方程和方程组的解的行为，这些方程和方程组是爱因斯坦方程的模型，但更简单。为了解决上面提到的更难的非线性稳定性问题，对这些问题的研究是必要的。最近出现了一系列关于理解史瓦西和克尔时空中线性波动方程衰减的活动，这是可能研究的最简单的模型。即使这个问题也是相当困难的，因为复杂的背景几何以不同的方式影响色散特性。在紧致区域，人们必须处理沿被困测地线长时间徘徊的高频波包，而在无穷远处，度规的非欧几里得特性影响逐点衰减率。捕获集的非常微妙（和不稳定）的性质特别需要来自谐波分析，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Strichartz估计）。这些估计可以反过来用于解决非线性问题，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）。这类非线性问题的一个例子是对于小数据和大数据具有幂非线性的半线性波动方程。另一个例子，是爱因斯坦方程在调和坐标系中的一个很好的模型，是度规的波动方程，它本身依赖于解。这些技术也可以用来解决麦克斯韦方程组解的最优衰减问题，麦克斯韦方程组是描述电磁场在史瓦西和克尔背景下演化的线性系统。最后，人们对来自弦理论的高维黑洞的研究感兴趣，弦理论在低能量下可以用高维引力理论来描述。理解在这些背景下波动方程的相关衰减特性也将是相当有趣的。

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.