1996 Fiscal Year Final Research Report Summary
Symmetric systems and strongly hyperbolic systems
| Project/Area Number |
07454027
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (B)
|
| Allocation Type | Single-year Grants |
|---|
| 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)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
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)
NAMBA Makoto Osaka University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (60004462)
USUI Sampei Osaka University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90117002)
|
|---|
| Project Period (FY) |
1995 – 1996
|
| Keywords | symmetrizable system / strong hyperbolicity / involutive / non degenerate / hyperbolic perturbation / localization |
|---|
| 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 m-1-th 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) /2-m+2 then genericaly, every hyperbolic perturbation is trivial that is every hyperbolic system near L can be symmetrized.
|
