Lebesgue Space Estimates for Solutions to Hyperbolic Equations

9706840 比尔斯 该调查涉及到严格双曲型偏微分方程（“波”）的解决方案的某些性质的研究。 为了分析非线性问题解的性质，将导出线性情形下新的齐次估计和光滑估计。 将获得对解的大小的适当测量的时间衰减的估计。 经典的波动方程的空间域是外部的凸障碍的结果将扩展到处理更一般的自然条件的障碍的边界。 关键估计值对应于找到最佳可能的时间衰减，该时间衰减受数据上最弱的大小和平滑度限制。 更一般的障碍，其中某些类型的衰减估计是已知的，但没有明确表示的近似解决方案，也将被视为。 将分析类似的全空间问题（无障碍物）。 在这些情况下的主要问题涉及高阶方程或系统，以及波在其中传播的介质的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来源之一，包括关于扰动势所需的最小正则性的问题。 在非均匀介质中的传播（例如，非恒定的波速），其中涉及显式时间估计的边界相对稀疏，也将被考虑。 (The经典边界涉及那些根据经典能量来测量尺寸的边界，经典能量通常不会随时间衰减。 在模型问题和一般情况下，将考虑在最佳可能条件下对其他自然空间测量的数据大小的最佳时间估计。 波已经被经典地研究了，就像能量一样，它通常是守恒的，或者以一种容易理解的方式表现出来。 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才能在这种背景下得到充分的理解--那些大尺度行为仅仅是小尺度行为的简单放大的模型问题（例如自由空间中的无扰波）。 在真正的物理问题中，波遇到障碍物并被反射或吸收;它们在不均匀的介质中传播;它们与自身以及传播介质相互作用。 将分析这种波的大小行为，特别是它们的振幅随时间的衰减，以便理解从很远（或在波产生很久之后）测量这种大小的量可以推导出关于波源的信息。

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).