Project/Area Number  07454027 
Research Category 
GrantinAid for Scientific Research (B)

Section  一般 
Research Field 
解析学

Research Institution  Osaka University 
Principal Investigator 
NISHITANI Tatsuo Osaka University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80127117)

CoInvestigator(Kenkyūbuntansha) 
杉本 充 大阪大学, 理学部, 講師 (60196756)
榎 一郎 大阪大学, 理学部, 助教授 (20146806)
NAMBA Makoto Osaka University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (60004462)
長瀬 道弘 大阪大学, 理学部, 教授 (70034733)
USUI Sampei Osaka University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90117002)
TAKAHASHI Satoshi Osaka University, Graduate School of Science, Lecturer, 大学院・理学研究科, 講師 (70226835)
TAKEGOSHI Kensho Osaka University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (20188171)
SAKUMA Makoto Osaka University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (30178602)

Project Fiscal Year 
1995 – 1996

Project Status 
Completed(Fiscal Year 1996)

Budget Amount *help 
¥3,100,000 (Direct Cost : ¥3,100,000)
Fiscal Year 1996 : ¥1,500,000 (Direct Cost : ¥1,500,000)
Fiscal Year 1995 : ¥1,600,000 (Direct Cost : ¥1,600,000)

Keywords  symmetrizable system / strong hyperbolicity / involutive / non degenerate / hyperbolic perturbation / localization / 対称化可能系 / 強双曲系 / 包合的 / 非退化特性点 / 双曲摂動 / 局所化 / 対称化 / 非退化特異点 
Research Abstract 
Our research project has been organized as follows : (i) Clarify the structure of strongly hyperbolic systems which can not be symmetrizable. (ii) Study the stability of symmetrizable systems under hyperbolic perturbations. As for (i) we got the following results. Let L be a m*m system of patial differential operators of first order. Denoting by h the determinant of the principal symbol of L the general picture of our necessary condition for strong hyperbolicity of L could be stated as : if L is strongly hyperbolic then the Cauchy problem for h+k is correctly posed for every m1th minor k of L.Moreover if the reference characteristic z is involutive and the system is strongly hyperbolic then KerL (z) * ImL (z) = {0}. Thus the Taylor expansion of L along KerL starts with a linear term L_Z called the localization of L.Let z, w be characteristics of the original system and of the localization respectively. If (z, w) is involutive then KerL_z (w) * ImL_z (w) = {0}. As for (ii) we formulated non degenerate characteristic for first order system. We say that z is non degenerate if KerL (z) * ImL (z) = {0}, the dimension of L_Z is maximal and L_Z (w) is diagonalizable for every w. Then the main result is that every hyperbolic system is symmetrizable near non degenerate characteristic. From this we can derive stability of non degenerate characteristics. Namely we can not remove non degenerate characteristics by hyperbolic perturbations. We proceed this study and got the following result. Let L be a m*m sysmmetric first order hyperbolic system. Then if the dimension of L is greater than m (m+1) /2m+2 then genericaly, every hyperbolic perturbation is trivial that is every hyperbolic system near L can be symmetrized.
