Linear Partial Differential Equations on Singular Spaces

奇异空间上的线性偏微分方程

• 批准号: 1265568
• 负责人: Jared Wunsch
• 金额: $ 19.5万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1265568&HistoricalAwards=false
• 关键词: Linear; Partial; Differential; Equations; Singular

This project focuses on the study of 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 or cone points, to large-scale structures "at infinity"--presents many open problems. The principal investigator will study the asymptotic behavior of waves propagating on certain curved spacetimes such as arise in the theory of general relativity. The goal is to understand the long-time behavior of the radiation pattern observed far away from a source. Recent work of the principal investigator has yielded results in spacetimes with a compactification similar to that of Minkowski space, and he will continue to pursue such results in a wider range of geometries. The project will also study the decay of waves near their source in different geometric settings. In particular, in the presence of corners or cone points, diffraction of waves is a potential obstruction to the rapid decay of waves; the principal investigator will continue his ongoing investigations in this area, especially into the question of resonances generated by cone points. Additionally, he will continue to study the behavior of eigenfunctions and quasimodes of the Laplace operator associated to completely integrable systems, in an effort to understand to what degree an eigenfunction can concentrate on a sub-torus of an Arnold-Liouville torus. These techniques should also yield new estimates for solutions to the linear Schrodinger equation in completely integrable settings.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. In particular, the principal investigator's work on decay of waves in curved spacetimes is closely related to problems of intense interest in the physics community, for instance as it gives a much simplified model of gravitational waves. His closely related work on 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.

这个项目的重点是奇异空间上的偏微分方程的研究，重点是光谱和散射理论。 波在平滑变化空间中的传播在许多方面都得到了很好的理解，但是与奇点的相互作用--范围可能从边界到角点或锥点，再到“无穷远”的大规模结构--提出了许多悬而未决的问题。首席研究员将研究波在某些弯曲时空中传播的渐近行为，例如广义相对论中出现的波。 目标是了解在远离源的地方观察到的辐射模式的长期行为。主要研究者最近的工作在时空中产生了与闵可夫斯基空间类似的紧化结果，他将继续在更广泛的几何形状中追求这些结果。该项目还将研究在不同几何设置下波在其源附近的衰减。 特别是，在角或锥点的存在下，波的衍射是波快速衰减的潜在障碍;首席研究员将继续在这一领域进行调查，特别是锥点产生的共振问题。此外，他将继续研究行为的本征函数和准模式的拉普拉斯运营商相关的完全可积系统，在努力了解到什么程度的本征函数可以集中在一个子环面的阿诺德刘维环面。这些技术也应该产生新的估计解决方案的线性薛定谔方程在完全可积settings. Geometrics影响的行为的解决方案波动方程在许多有趣的和微妙的方式。 根据牛顿的理论，我们知道光在许多情况下的行为就像是由微小的粒子组成的;另一方面，我们也知道光可以转弯（“转弯”），并且它倾向于分散。几何变化对波传播变化的影响（无论是光波、声波、水波、重力波，还是描述量子粒子的波函数）是该项目的中心焦点。特别是，首席研究员关于弯曲时空中波的衰变的工作与物理学界强烈感兴趣的问题密切相关，例如，它给出了一个简化的引力波模型。 他在线性薛定谔方程上的密切相关的工作不仅与非相对论量子粒子的物理学有关，而且与非线性薛定谔方程有关，该方程模拟了激光脉冲和超导性等不同的现象。