Lebesgue Space Estimates for Solutions to Hyperbolic Equations

双曲方程解的勒贝格空间估计

• 批准号: 9706840
• 负责人: R. Michael Beals
• 金额: $ 6.55万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706840&HistoricalAwards=false
• 关键词: Lebesgue; Space; Estimates; Solutions; Hyperbolic

9706840 Beals The investigation involves the study of certain properties of solutions to strictly hyperbolic partial differential equations ("waves"). New qize and smoothness estimates in the linear case will be derived, in order to analyze the properties of solutions to nonlinear problems. Estimates of time decay for appropriate measures of the size of a solution will be obtained. Results for the classical wave equation on a spatial domain which is exterior to a convex obstacle will be extended to handle more general natural conditions at the boundary of the obstacle. The key estimates correspond to finding the best possible time decay subject to the weakest size and smoothness restrictions on the data. More general obstacles, for which certain types of decay estimates are known but for which explicit representations of approximations to solutions are absent, will also be treated. Analogous full-space problems (no obstacles) will be analyzed. The primary questions in these cases involve higher order equations or systems, and the effects of interactions with changes in the properties of the medium in which the waves are propagating. Nonsmooth perturbations in the medium are one source of problems, including questions about the minimal regularity needed for the perturbing potential. Propagation in inhomogeneous media (for example, nonconstant wave speeds) where bounds involving explicit time estimates are relatively sparse, will also be considered. (The classical bounds involve those which measure size in terms of the classical energy, which will not typically exhibit decay in time.) The optimal time estimates under the best possible conditions on the size of the data measured in terms of other natural spaces will be considered, in model problems and in general. Waves have been studied classically in terms of a size quantity like energy, which is typically conserved or behaves in a fashion which is easy to understand. In order to go beyond the simpl est mathematical approximations to understand the small- and large-scale properties of true physical waves, other means of measuring the wave size must be used. Only the simplest model problems have been fully understood in this context in the past - those for which the large-scale behavior is merely a simple magnification of the small-scale behavior (such as unperturbed waves in free space). In true physical problems, waves encounter obstacles and are reflected or absorbed; they travel in media which are not uniform; they interact with themselves and with the media in which they propagate. The size behavior of such waves, and in particular the decay of their amplitudes in time, will be analyzed, in order to understand what information can be deduced about the wave source from measurements of such size quantities made far away (or long after the wave is generated).

9706840 Beals本研究涉及对严格双曲型偏微分方程解的某些性质的研究。为了分析非线性问题解的性质，我们将在线性情形下导出新的Qize和光滑性估计。对于溶液大小的适当度量，将获得时间衰减的估计。对于凸形障碍物外的空间域上的经典波动方程，其结果将被推广到处理更一般的障碍物边界上的自然条件。关键估计对应于在数据的最弱大小和光滑度限制下找到可能的最佳时间衰减。也将讨论更一般的障碍，对于这些障碍，某些类型的衰减估计是已知的，但对于这些障碍，其解的近似没有显式表示。将分析类似的全空间问题(无障碍)。这些情况下的主要问题涉及高阶方程或系统，以及与波在其中传播的介质性质变化的相互作用的影响。介质中的非光滑扰动是问题的一个来源，包括关于微扰势所需的最小正则性的问题。在包含显式时间估计的边界相对稀疏的非均匀介质(例如，非恒定波速)中的传播也将被考虑。(经典界限涉及那些用经典能量来衡量大小的界限，经典能量通常不会表现出时间衰变。)在模型问题和一般情况下，将考虑以其他自然空间衡量的数据大小的最佳可能条件下的最佳时间估计。波的大小，如能量，通常是守恒的，或者以一种容易理解的方式来研究波。为了超越最简单的数学近似来理解真实物理波的小尺度和大尺度特性，必须使用其他测量波大小的方法。在过去，只有最简单的模型问题在这种背景下才被完全理解--对于那些大尺度行为仅仅是小尺度行为的简单放大的那些问题(例如自由空间中的无扰波)。在真正的物理问题中，波遇到障碍并被反射或吸收；它们在不均匀的介质中传播；它们与自己以及它们传播的介质相互作用。将分析这种波的大小行为，特别是它们的幅度在时间上的衰减，以便理解从远距离(或在波产生后很长一段时间)对这种大小的测量可以推断出关于波源的哪些信息。