Mathematical Sciences: Harmonic Analysis and Hyperbolic Partial Differential Equations

摘要Smith 9622875 史密斯将继续研究波的传播 在具有“粗糙”声速的非均匀介质中。目的是 在度量上建立最小可微性假设， 声速下的某些界限的解决方案，被称为 哈茨估计，保持。在此前的NSF资助下，研究人员 已经表明，这些界限保持，如果二阶导数的 度量系数是有界的，并且这是确保Eschhartz估计的有效性的最佳可能条件。 拟议的研究包括用更锋利的， 几何上内在的。预计最佳条件 是度量拥有一个有界导数，此外， 曲率张量的系数（二阶的某些组合 衍生物）是有界的。调查人员还在继续研究 发展了适合振荡积分的函数空间， 双曲问题此前的NSF资金支持开发了一种 族的哈代空间适应于固定时间估计的解决方案的 波动方程拟议的研究包括开发类似的空间 适用于控制解决方案的时间平均范数。 对波在具有粗糙声速的介质中传播的研究（即， 其中速度可能从一点到另一点急剧变化）是两者的 理论和实践意义。理论上的兴趣来自于 非线性方程，如爱因斯坦引力方程， 其中波速取决于所考虑的解。以来 粗略的解决方案自然会出现，人们需要了解波是如何传播的。 在粗糙的介质中显示解决方案的存在，并了解其 特性.物理学预言能量应该沿着测地线传播， 代表点之间最短路径的曲线。我们的研究是 表明即使对于粗略的声速，在基本上 最弱的假设，确保测地线存在， 定义了其现实意义在于，我们的研究成果可以作为一种 解波动方程的数学构造，它对粗略的声速有效，在介质扰动下稳定。是 显然，基于这种构造的计算算法将是 本质上比需要更多限制性条件的更稳定 媒体.我们的研究将有助于建立这样的算法， 稳定和收敛。

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.