Mathematical Sciences: Harmonic Analysis and Hyperbolic Partial Differential Equations

数学科学：调和分析和双曲偏微分方程

• 批准号: 9622875
• 负责人: Hart Smith
• 金额: $ 6.68万
• 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: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622875&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Harmonic; Analysis; Hyperbolic

ABSTRACT Smith 9622875 Smith will continue his research into the propagation of waves in nonhomogeneous mediums with ``rough'' sound speeds. The aim is to establish minimal differentiability assumptions on the metric determining the sound speed under which certain bounds on the solutions, known as the Strichartz estimates, hold. Under previous NSF funding, the investigator has shown that these bounds hold if the second order derivatives of the metric coefficients are bounded, and that this is the best possible condition of its sort that insures the validity of the Strichartz estimates. The proposed research includes replacing this condition by sharper, geometrically intrinsic ones. It is anticipated that the optimal condition is that the metric posess one bounded derivative, and in addition that the coefficients of the curvature tensor (certain combinations of second order derivatives) be bounded. The investigator is also continuing research into the development of function spaces adapted to oscillatory integral and hyperbolic problems. Previous NSF funding supported the development of a family of Hardy spaces adapted to fixed-time estimates for solutions of the wave equation. The proposed research includes developing similar spaces adapted to controlling time-averaged norms of solutions. The study of waves travelling in media with rough sound speeds (that is, where the speed may vary sharply from one point to another) is of both theoretical and practical significance. The theoretical interest comes from nonlinear equations, such as the Einstein equations for gravitation, in which the wave speed depends on the solution under consideration. Since rough solutions arise naturally, one needs to understand how waves travel in rough media to show that solutions exist, and to understand their properties. Physics predicts that energy should travel along geodesics, curves that represent the shortest path between points. Our research is sh owing that this is true even for rough sound speeds, under essentially the weakest assumptions that assure that geodesics exist and are well defined. The practical significance is that our research yields a mathematical construction for solving the wave equation, which is valid for rough sound speeds and stable under perturbations of the medium. It is clear that a computational algorithm based on such a construction will be inherently more stable than one requiring more restrictive conditions on the media. Our research would help establish that such algorithms are both stable and convergent.

摘要 Smith 9622875 史密斯将继续研究波在非均匀介质中以“粗糙”声速传播的情况。目的是对确定声速的度量建立最小可微性假设，在该假设下，解的某些界限（称为 Strichartz 估计）成立。在之前的 NSF 资助下，研究人员已经证明，如果度量系数的二阶导数有界，则这些界限成立，并且这是确保 Strichartz 估计有效性的最佳可能条件。 拟议的研究包括用更清晰的、几何固有的条件来代替这种条件。预计最佳条件是度量具有一个有界导数，并且此外曲率张量的系数（二阶导数的某些组合）是有界的。研究人员还在继续研究适应振荡积分和双曲问题的函数空间的开发。之前的 NSF 资助支持开发一系列适用于波动方程解的固定时间估计的 Hardy 空间。拟议的研究包括开发适合控制解决方案的时间平均规范的类似空间。 研究波在介质中以粗糙声速（即从一点到另一点速度可能急剧变化）传播的波具有理论和实际意义。理论兴趣来自非线性方程，例如爱因斯坦引力方程，其中波速取决于所考虑的解。由于粗糙解自然产生，因此需要了解波如何在粗糙介质中传播，以表明解存在，并了解它们的属性。物理学预测能量应该沿着测地线传播，测地线代表点之间的最短路径。我们的研究表明，即使对于粗略的声速来说，这也是正确的，本质上是在确保测地线存在并得到明确定义的最弱假设下。实际意义在于，我们的研究提出了求解波动方程的数学结构，该结构对于粗糙的声速有效并且在介质扰动下稳定。显然，基于这种结构的计算算法本质上比需要对介质有更多限制条件的算法更稳定。我们的研究将有助于确定此类算法既稳定又收敛。