Analysis of fundamental properties of elastic equations

弹性方程基本性质分析

基本信息

批准号：
15540152
负责人：
SOGA Hideo
金额：
$ 1.86万
依托单位：
IBARAKI UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540152/
关键词：
elastic equations wave equations scattering theory inverse problems partial differential equations hyperbolic equations energy decay Rayleigh wave 一意接続性

项目摘要

In this research project we aimed initially to know properties of solutions of the elastic equations and to clarify roots of those properties. And, using the clarification, we intended to study individual topics concerned with elastic waves.We have seen that the elastic wave equations are near the scalar-valued wave equation although they are one of kinds of hyperbolic systems, and that this is because of positivity and symmetry of the elastic operators. As one of main results concerning this, we have proved that the elastic operators can be expressed of product form of first order operators in the same way as the scalar-valued elliptic operators. This expression cannot be expected for systems of the general form.It is known that there exists the Rayleigh wave in the elastic equations, which does not occur in the scalar-valued equations. We have shown synthetically how this existence affects scattering theories and have constructed a general scattering theory of the Lax-Phillips type accounting the Rayleigh wave.Furthermore, we have studied individual topics in this theory, and have obtained a representation of the scattering kernel. This representation is a fundamental and useful formula to solve inverse scattering problems.A special result on the energy decay has been got also for the scalar-valued equation. Namely it is proved that the total energy does not decay necessarily if the dissipative term is added of non isotropy. This gives an interesting suggestion on behavior of the elastic waves.

在该研究项目中，我们最初旨在了解弹性方程解决方案的属性，并阐明这些属性的根源。而且，使用澄清，我们打算研究与弹性波有关的个体主题。我们已经看到，弹性波方程是接近标量值的波动方程，尽管它们是多种双曲线系统之一，这是由于弹性和弹性运算符的阳性和对称性。作为有关此的主要结果之一，我们证明了弹性运营商可以以与标量值椭圆运算符相同的方式以一阶操作员的产品形式表达。对于一般形式的系统，无法期望这种表达。众所周知，弹性方程中存在瑞利波，而标量值不存在。我们已经表明，这种存在如何影响散射理论，并构建了一种宽度菲利普斯类型的一般散射理论，该理论占瑞利波的占主导地位。Furthermore，我们已经研究了该理论中的个体主题，并获得了散射核心的表示。这种表示是解决反向散射问题的基本和有用的公式。对于标量值的方程，能量衰减的特殊结果也得到了特殊的结果。也就是说，如果添加非各向同性的耗散项，总能量不一定是衰减的。这给出了关于弹性波的行为的有趣建议。