Analysis of fundamental properties of elastic equations
Project/Area Number 
15540152

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  IBARAKI UNIVERSITY 
Principal Investigator 
SOGA Hideo IBARAKI Univ., the College of Education, Professor, 教育学部, 教授 (40125795)

CoInvestigator(Kenkyūbuntansha) 
KAIZU Satoshi IBARAKI Univ., the College of Education, Professor, 教育学部, 教授 (80017409)
NOZAKI Hideaki IBARAKI Univ., the College of Education, Professor, 教育学部, 教授 (60208337)
NAKAMURA Gen HOKKAIDO Univ., College of Science, Professor, 理学研究科, 教授 (50118535)
ITO Hiroya Univ.of ElectroComm., Dept Math., Assoc.Professor, 電気通信学部, 助教授 (30211056)
UMEZU Kenitiro Maehashi Tech.Univ., College of Tech., Assoc.Professor, 工学部, 助教授 (00295453)
代田 健二 茨城大学, 理学部, 助手 (90302322)

Project Period (FY) 
2003 – 2004

Project Status 
Completed (Fiscal Year 2004)

Budget Amount *help 
¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 2004: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2003: ¥1,400,000 (Direct Cost: ¥1,400,000)

Keywords  elastic equations / wave equations / scattering theory / inverse problems / partial differential equations / hyperbolic equations / energy decay / Rayleigh wave / 一意接続性 
Research Abstract 
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 scalarvalued 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 scalarvalued 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 scalarvalued equations. We have shown synthetically how this existence affects scattering theories and have constructed a general scattering theory of the LaxPhillips 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 scalarvalued 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.

Report
(3 results)
Research Products
(3 results)