论文标题
“依赖的引导结构域”：波方程解的界限和时间衰减
"Bootstrap Domain of Dependence": Bounds and Time Decay of Solutions of the Wave Equation
论文作者
论文摘要
本文介绍了一个新颖的“引导域的依赖性”概念，根据该概念，在给定的任意持续时间的给定照明期之后，障碍物散布的波场在边界散射事件的历史上编码了与障碍物直径相等的障碍物的历史。产生的解决方案界限可根据短期历史记录提供对解决方案值的估计，并且它们在广泛的散射障碍物中建立了超质量快速衰减（即，衰减比任何负时间的衰减更快） - 包括某些类型的“捕获”易位，其周期性的诱捕轨道的周期性诱捕的轨道跨越了一系列正面的频率，这是一组正面的速度，并且以前的理论不可用。结果不依赖于对复杂平面中宽松的菲利普斯复合变量散射框架和相关的无谐振区域的考虑，仅利用真实值的频率，并遵循绿色函数和边界积分方程式在频率和时段中的使用，以及一定的$ q $ q $ - $ q $ - $ q $ -growth条件，在频率下构成了频率溶液的频率溶液均方体，即构型构型构型构型构型构型构型构型构型构型均值。
This article introduces a novel "bootstrap domain-of-dependence" concept, according to which, for all time following a given illumination period of arbitrary duration, the wave field scattered by an obstacle is encoded in the history of boundary scattering events for a time-length equal to the diameter of the obstacle, measured in time units. Resulting solution bounds provide estimates on the solution values in terms of a short-time history record, and they establish super-algebraically fast decay (i.e., decay faster than any negative power of time) for a wide range of scattering obstacle--including certain types of "trapping" obstacles whose periodic trapped orbits span a set of positive volumetric measure, and for which no previous fast-decay theory was available. The results, which do not rely on consideration of the Lax-Phillips complex-variables scattering framework and associated resonance-free regions in the complex plane, utilize only real-valued frequencies, and follow from use of Green functions and boundary integral equation representations in the frequency and time domains, together with a certain $q$-growth condition on the frequency-domain operator resolvent.
