| Project/Area Number |
15340056
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Tohoku University (2004-2006)
Kyushu University (2003) |
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Principal Investigator |
OGAWA Takayoshi Tohoku University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (20224107)
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| Co-Investigator(Kenkyū-buntansha) |
ISHIGE Kazuhiro Tohoku University, Graduate School of Science, Associate Professor, 大学院理学研究科, 助教授 (90272020)
NAKAMURA Makoto Tohoku University, Graduate School of Science, Associate Professor, 大学院理学研究科, 助教授 (70312634)
KAWASHIMA Shuichi Kyushu University, Graduate School of Mathematics, Professor, 大学院数理学研究院, 教授 (70144631)
KOBAYASHI Takayuki Saga University, Department of Mathematics, Professor, 理工学部, 教授 (50272133)
KAGEI Yoshiyuki Kyushu University, Graduate School of Mathematics, Professor, 大学院数理学研究院, 教授 (80243913)
三沢 正史 熊本大学, 理学部, 助教授 (40242672)
小薗 英雄 東北大学, 大学院・理学研究科, 教授 (00195728)
加藤 圭一 東京理科大学, 理学部一部, 助教授 (50224499)
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| Project Period (FY) |
2003 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥12,900,000 (Direct Cost: ¥12,900,000)
Fiscal Year 2006: ¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2005: ¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2004: ¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2003: ¥3,400,000 (Direct Cost: ¥3,400,000)
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| Keywords | nonl-local parabolic equations / drift-diffusion equations / semiconductor simulation / Chemotaxis / nonlinear damped wave equations / Naveir-Stokes equations / Critical Sobolev inequality / mean curvature flow / 非局所放物型方程式 / 臨界密度 / 走化性粘菌モデル / 非線形波動方程式 / 圧縮性ナビエ-ストークス方程式 / 自己相似解 / 熱方程式 / 数値アルゴリズム / 粘性解 / 鉄磁性モデル / Schroedinger写像 / 対数型臨界Sobolev不等式 / ゲージ変換 / Navier-Stokes方程式 / 調和写像流 / Euler方程式 / 解の正則性条件 / Dirac方程式 / 複素Ginzburg-Landau方程式 / Besov空間 / 半導体素子方程式 / 複数Ginzburg-Landau方程式 |
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| Research Abstract |
The main researcher, Prof.Ogawa obtained the following results. He researched for the Sobolev type inequality of the critical type, especially for the real interpolation spaces such as Besov and Triebel-Lirzorkin spaces and generalized it for the abstract Besov and Lorentz space. Those inqualities involving the logarithmic interpolation order can be applied for the regularity and uniqueness criterion of the seimilinear partial differential equation. In a series of collaboration with the research colabolators, he shows that the reguarlity and uniquness criterion for the weak solution of the 3 dimensional Navier-Stokes equations and break down condition for the Euer equation. In a similar method, he also showed the regularity criterion for the smooth solution of the 2 dimensional harmonic heat flow into a sphere. In particular, for the weak solution of the harmonic heat flow, the similar regularity criterion is also holds. The result is obtained by establishing the "monotonicity formula"
… More
for the mean oscillation of the energy density of the solutions.
He also consider the asymptotic behavior of the solution for the semi-lineear parabolic equation of the non-local type. Those system appeared in a various Physical scaling such as semi-conductor simulation model, Chemotaxis model and the birth of star in Astronomy. The system is involving Poisson equation as the field generated by the dencity of the charge or mucous ameba and the non-local effect is essential for the analysis of the solution. He particulariy investigated to the critical situation, 2-dimensional case, and showed that there exists a time local solution in the critical Hardy space, time global solution upto the threshold initial density and finite time blow-up for the system of forcusing drift-diffusion case. Besides, the asymootitic behavior of the solution for small data is characterized by the heat kernel. Moreover if the field equation is purterbed in a certain nonlinear way, then there exist two solutions for the same initial data in a radially symmetric case.
He also studied for the asymptotic behavior of the solution for the semi-linear damped wave equation in whole and half spaces and exterior domains and show the small solution is going to be decomposed into the solutions of the linear heat equation, some combination of linear wave equation with nonlinear effect. This was shown for 1 and 3 dimensional cases before, however the mothod there could not be applicable for the 2dimensional case. Less
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