1998 Fiscal Year Final Research Report Summary
Study on symmetric positive systems
| Project/Area Number |
09440059
|
|---|
| 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, Grad.Sch.of Sci., Professor, 大学院・理学研究科, 教授 (80127117)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
MANDAI Takeshi Gifu University, Fac.of Eng., Associate Professor, 工学部, 助教授 (10181843)
ICHINOSE Wataru Shinsyu University, Fac.of Sci., Professor, 理学部, 教授 (80144690)
KAJITANI Kunihiko Tsukuba University, Fac.of Math., Professor, 数学系, 教授 (00026262)
SUGIMOTO Mitsuru Osaka University, Grad.Sch.of Sci., Associate Professo, 大学院・理学研究科, 助教授 (60196756)
MATSUMURA Akitaka Osaka University, Grad.Sch.of Sci., Professor, 大学院・理学研究科, 教授 (60115938)
|
|---|
| Project Period (FY) |
1997 – 1998
|
| Keywords | symmetric system / boundary matrix / weight function / a priori estimate / blow up |
|---|
| Research Abstract |
(i) We have studied symmetric positive systems on a bounded open set assuming the following conditions on the boundary matrix. That is, outside a embedded submanifold of codimension 1 in the boundary, the rank of the boundary matrix is constant. In this case, we found a suitable weight function which is positive outside the above mentioned submanifold so that an a priori estimate for solutions to the boundary value problem is obtained. Using this a priori estimate, we have proved the existence of solution to the boundary value problem which is regular with respect to the normal direction. This is very important to applications to non-linear perturbations. With the aid of this a priori estimate and the existence of smooth solutions, we succeeded to get the behavior of weak solutions near the reference embedded submanifold, which is very sharp as several examples show. These results, as far as concerning two dimentional domains, are fairly satisfactory. If we apply this result to so called Triconi's equation, we get another proof of the uniqueness of solution. As for 3 dimentional domains, there remains one fundamental case which we cound not treat.
(ii) In the case that the boundary matrix is zero on the submanifold where the rank of the boundary matrix changes, we clarified the structure of the boundary value problem. On the blown up manifold along the submanifold, we can get an a priori estimate with a simple weight function. Using this a priori estimate in the blown up space, we proved the existence of smooth solution even with respect to the normal direction. Applying the same method that we employed in (i), we are able to examine the behavior of weak solutions near the submanifold. Applying this result we obtained a priori estimate for the linealized MHD equation under some boundary condition which has not been treated before.
|
