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-9970407摘要研究人员正在继续研究波在具有“粗糙”波速的非均匀介质中传播的行为。目标是在系数的最小可微性假设下获得对变系数波动方程解的 Strichartz 和零型类型的估计。在当前 NSF 下在资金支持下，研究人员证明，如果波速度量的曲率张量有界且可测量，则斯特里查茨的估计成立。拟议的研究包括扩展这些结果以获得双线性零形式解的估计，以及有界曲率空间上拉普拉斯本征函数的估计。研究人员还参与了研究凸障碍物反射波的 Strichartz 和双线性零形式估计的联合工作。之前的联合研究已经在时空有界区域中获得了此类估计。拟议的研究涉及将这些局部方法与已知的能量衰减估计相结合，以获得在空间和时间上全局有效的估计。然后将这些结果应用于凸障碍物外部区域上某些非线性波动方程的长期存在性。开发处理低正则系数偏微分方程的技术具有理论和实际意义。理论兴趣来自对数学和物理非线性方程的研究，例如爱因斯坦的引力场方程或流体动力学方程。此类方程的一个核心问题是解的“长期”存在性。也就是说，表明解决方案不会在未来某个时间崩溃。这样做的主要技术是证明人们可以根据已知保持受控的量（例如解的能量）来控制方程非线性部分的大小。对于爱因斯坦方程，这需要理解具有粗糙波速的波动方程。此类研究的实际重要性在于，从理论上理解波在粗糙介质中的传播应该导致更有效的算法来数值计算解。我们的工作涉及将解决方案分成基本部分，每个部分都以特别简单的方式运行。这类似于将函数分解为小波，其许多现代理论是为了理解稳态现象的非线性方程而发展的。小波反过来又带来了解决此类方程的简单而稳定的算法，以及图像和信号处理领域的强大算法。