Harmonic Analysis and Hyperbolic Partial Differential Equations

调和分析和双曲偏微分方程

• 批准号: 9970407
• 负责人: Hart Smith
• 金额: $ 6.5万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970407&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Hyperbolic; Partial; Differential

DMS-9970407AbstractThe investigator is continuing his research into the behavior ofwave propagation in nonhomogeneous media with "rough" wave speeds.The goal is to obtain estimates of Strichartz and nullform typeon solutions to variable coefficient wave equations underminimal differentiability assumptions on the coefficients.Under current NSF funding, the investigator has shown thatthe Strichartz estimates hold if the curvature tensor of the wavespeed metric is bounded and measurable. The proposed researchincludes extending these results to obtain estimates for bilinearnull-forms of solutions, as well as estimates for eigenfunctionsof the Laplacian on spaces with bounded curvature. The investigatoris also involved in joint work studying the Strichartz and bilinearnullform estimates for waves which reflect from convex obstacles.Previous joint research has obtained such estimates in bounded regions of space-time. The proposed research involves combining these local methods with known energy decay estimates to obtain suchestimates valid globally in space and time. These results are thenbeing applied to establish long time existence for certain nonlinearwave equations on the region exterior to a convex obstacle.The development of techniques to handle partial differential equationswith low regularity coefficients is of both theoretical and practicalinterest. The theoretical interest comes from the study ofnonlinear equations in math and physics, such as Einstein's equationsfor the gravitational field, or the equations of fluid dynamics. A central question for such equations is the "long time" existenceof solutions; that is, showing that solutions do not blow upat some future time. The main technique for doing this is to showthat one can control the size of the nonlinear part of the equationin terms of quantities that are known to remain controlled, suchas the energy of the solution. For Einstein's equations, thisrequires understanding the wave equation with rough wave speeds.The practical importance of such studies is that a theoreticalunderstanding of the propagation of waves in rough media shouldlead to more efficient algorithms for the numerical computationof solutions. Our work involves splitting the solution intoelementary pieces, each of which behaves in a particularly simplemanner. This is analogous to the decomposition of functions intowavelets, much of the modern theory of which was developed tounderstood nonlinear equations of stable-state phenomena. Waveletshave in turn led to simple and stable algorithms for solving such equations, as well as to powerful algorithms in the fields of imageand signal processing.

DMS-9970407AbstractThe investigator is continuing his research into the behavior ofwave propagation in nonhomogeneous media with "rough" wave speeds.The goal is to obtain estimates of Strichartz and nullform typeon solutions to variable coefficient wave equations underminimal differentiability assumptions on the coefficients.Under current NSF funding, the investigator has shown thatthe Strichartz estimates hold if the curvature波动量指标的张量是有界和可测量的。提出的研究包括扩展这些结果，以获得溶液的双线性null形式的估计，以及在具有有界曲率的空间上的laplacian的本征征的估计。研究人员还参与了研究strichartz和Biinearnullform的联合工作，这些波浪反映了从凸障碍的波浪进行估计。提出的联合研究已在时空的有界区域获得了此类估计。拟议的研究涉及将这些局部方法与已知能量衰减估计值相结合，以使其在空间和时间上在全球范围内获得有效的估计值。然后将这些结果应用于在凸障碍区域外部的某些非线波方程的长时间存在。处理偏微分方程的技术在低规律性系数方面处理部分微分方程的技术既具有理论上的兴趣和实用性。理论上的兴趣来自数学和物理学中的非线性方程的研究，例如重力场的爱因斯坦方程或流体动力学方程。这种方程式的一个核心问题是解决方案的“长期”存在。也就是说，这表明解决方案不会爆发未来的时间。这样做的主要技术是示出一个人可以控制已知能保持控制的数量术语的非线性部分的大小，因此是解决方案的能量。对于爱因斯坦的方程式，这种重点是以粗糙的波速来理解波动方程。这种研究的实际重要性是，在粗糙介质中，波浪传播的理论理解应该对解决方案的数值计算进行更有效的算法。我们的工作涉及将溶液分为元素，每种溶液都表现在一个特别的简单者中。这类似于函数的函数插图的分解，其现代理论的大部分是稳定现象的tounderst tounderstnereare equations。弹药又导致了用于求解此类方程的简单稳定算法，以及在ImageAnd信号处理领域的强大算法。