Linear Partial Differential Equations on Singular Spaces

奇异空间上的线性偏微分方程

• 批准号: 1001463
• 负责人: Jared Wunsch
• 金额: $ 19.81万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001463&HistoricalAwards=false
• 关键词: Linear; Partial; Differential; Equations; Singular

This project will study partial differential equations on singular spaces, with an emphasis on spectral and scattering theory. The propagation of waves on smoothly varying spaces is well understood in many respects, but the interaction with singularities--which might range from boundaries, to corners, to structures "at infinity"--presents many open problems. One of the project's components is related to the understanding of wave propagation on Kerr spacetimes (i.e., near rotating black holes). The principal investigator will study the distribution of quasi-normal modes, the ways that the black hole may "ring" with damped oscillations. The initial goal of this project is to obtain a rigorous description of the exponential decay rate of high frequency modes. The project also includes work on problems of local energy decay on Riemannian manifolds, for which the geometry of the infinite ends turns out to have a profound effect on the low frequency phenomena that may dominate energy decay. Another aspect of wave propagation of interest to the principal investigator is the infinite-speed propagation occurring in solutions to the Schrodinger equation. In settings in which geometric rays are trapped in a bounded region, little is known about the regularity of solutions. The principal investigator is intent on studying the effects of these trapped rays, as well as the effects of geometric singularities such as cone points on the propagation.Geometry influences the behavior of solutions to wave equations in many interesting and subtle ways. Following Newton, we know that light behaves in many regimes as if made of tiny particles. On the other hand, we also know that light can turn corners ("diffract") and that it tends to disperse. The effect of changes in geometry to changes in propagation of waves (be they light or sound or water or gravity waves, or the wave-functions describing quantum particles) is the central focus of this project's research. In particular, the principal investigator's work on quasi-normal modes for Kerr spacetimes is closely related to problems of intense interest in the physics community, as these modes are part of the signature of gravitational waves. The principal investigator's study of the linear Schrodinger equation is related to applications not just to the physics of nonrelativistic quantum particles, but also to the nonlinear Schrodinger equation, which models such disparate phenomena as laser pulses and superconductivity.

该项目将研究奇异空间的部分微分方程，重点是光谱和散射理论。在许多方面都可以很好地理解了在平稳变化的空间上的波的传播，但是与奇点的相互作用 - 从边界到角落到“无穷大”的结构，这是许多开放的问题。该项目的组件之一与对Kerr空间的波传播的理解有关（即近旋转黑洞）。主要研究者将研究准正常模式的分布，即黑洞可以用阻尼振荡“响起”的方式。该项目的最初目标是获得高频模式指数衰减率的严格描述。该项目还包括有关局部能量衰减问题的研究，对无限末端的几何形状证明对低频现象产生了深远的影响，这可能主导能量衰减。主要研究者感兴趣的波传播的另一个方面是在Schrodinger方程的溶液中发生的无限速传播。在将几何射线捕获在有界区域的设置中，对解决方案的规律性知之甚少。主要研究者的目的是研究这些被困的光线的影响，以及几何奇异性（例如锥体点）对传播的影响。几何影响以许多有趣且微妙的方式影响解决方案对浪潮方程的行为。在牛顿之后，我们知道光在许多方面都表现得好像是由微小粒子制成的。另一方面，我们还知道光可以转弯（“衍射”），并且倾向于散布。几何形状变化对波的传播变化的影响（无论是光，声音，水或重力波，还是描述量子颗粒的波浪函数）的影响是该项目研究的中心重点。特别是，主要研究者在KERR空间的准正常模式上的工作与物理界的强烈兴趣问题密切相关，因为这些模式是引力波的标志的一部分。首席研究者对线性schrodinger方程的研究不仅与非递归量子颗粒的物理学有关，还与非线性schrodinger方程有关，该方程模拟了激光脉冲和超导性等不同现象。