Mathematical Sciences: Size and Regularity Estimates for Solutions to Hyperbolic Equations

数学科学：双曲方程解的大小和规律性估计

• 批准号: 9401819
• 负责人: R. Michael Beals
• 金额: $ 12万
• 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: 1994
• 资助国家: 美国
• 起止时间: 1994-05-15 至 1997-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401819&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Size; Regularity; Estimates

9401819 Beals The work supported by this award involves the study of several questions on properties of solutions to strictly hyperbolic partial differential equations. The equations are an outgrowth of the classical wave equation which lies at the heart of the theory of hyperbolic equations. Refined estimates in the linear case will be derived, to analyze the properties of solutions to the physically more realistic nonlinear problems. Among the questions to be considered are: estimates of time decay for appropriate measures of the size of a solution and analysis of nonconstant coefficient problems corresponding to nonuniform media in which wave propagation takes place. Such estimates will also be considered for solutions to wave equations on domains with boundary, such as in the exterior of an obstacle, to find the best possible time decay subject to the weakest restrictions on the data. Finally, smoothness properties of solutions to nonlinear wave equations as determined by the corresponding properties of the data will be treated. The general propagation of smoothness in a wave satisfying a nonlinear equation will be reconsidered in a setting of less initial smoothness. Examples will be considered where the formation of nonlinear singularities develop in cases in which nonsmooth surfaces appear, both for interior domains and in the presence of a boundary. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***

小行星9401819 该奖项支持的工作涉及严格双曲型偏微分方程解的性质的几个问题的研究。 这些方程是经典波动方程的产物，而经典波动方程是双曲方程理论的核心。 在线性情况下的精细估计将被导出，以分析物理上更现实的非线性问题的解决方案的属性。 其中要考虑的问题是：估计的时间衰减的适当措施的大小的解决方案和分析的非常数系数的问题，对应于非均匀介质中的波传播发生。 这样的估计也将被认为是解决方案的波动方程的边界域，如在外部的障碍，找到最好的可能的时间衰减受到最弱的限制的数据。 最后，光滑性的非线性波动方程的解决方案所确定的相应属性的数据将被处理。 在初始光滑度较小的情况下，重新考虑满足非线性方程的波的一般光滑性传播。 将考虑的例子中，非线性奇点的形成发展的情况下，出现非光滑表面，无论是内部域和存在的边界。 偏微分方程是物理世界数学建模的基础。 数学分析的作用与其说是建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。 ***