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

[401819]该奖支持的工作涉及对严格双曲型偏微分方程解的性质的几个问题的研究。这些方程是双曲方程理论的核心——经典波动方程的衍生。将推导线性情况下的精细估计，以分析物理上更现实的非线性问题的解的性质。要考虑的问题包括：估计解的大小的适当措施的时间衰减和分析与波传播发生的非均匀介质相对应的非常系数问题。这种估计也将被考虑在有边界的域上求解波动方程，例如在障碍物的外部，以找到在数据上最弱的限制下可能的最佳时间衰减。最后，将讨论由数据的相应性质决定的非线性波动方程解的平滑性。满足非线性方程的波的一般平滑传播将在初始平滑度较低的情况下重新考虑。将考虑在非光滑表面出现的情况下形成非线性奇点的例子，包括内部域和存在边界的情况。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创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。＊＊＊

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. ***