2012 Fiscal Year Final Research Report
Theory of global well-posedness on the nonlinear partial differential equations
| Project/Area Number |
20224013
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| Research Category |
Grant-in-Aid for Scientific Research (S)
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| Allocation Type | Single-year Grants |
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| Research Field |
Basic analysis
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| Research Institution | Waseda University (2011-2012)
Tohoku University (2009-2010) |
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Principal Investigator |
KOZONO Hideo 早稲田大学, 理工学術院, 教授 (00195728)
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| Co-Investigator(Kenkyū-buntansha) |
YANAGIDA Eiji 東京工業大学, 大学院・理工学研究科, 教授 (80174548)
ISHIGE Kazuhiro 東北大学, 大学院・理工学研究科, 教授 (90272020)
NAKAMURA Makoto 山形大学, 理学部, 教授 (70312634)
KUBO Hideo 北海道大学, 大学院・理学研究科, 教授 (50283346)
KANEDA Yukio 愛知工業大学, 工学部, 教授 (10107691)
ISHIHARA Takashi 名古屋大学, 大学院・工学研究科, 准教授 (10262495)
YOSHIMATSU Katsunori 名古屋大学, 大学院・工学研究科, 助教 (70377802)
KAGEI Yoshiyuki 九州大学, 大学院・数理学研究院, 教授 (80243913)
EI Shinichro 九州大学, マス・フォア・インダストリ, 教授
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| Project Period (FY) |
2008 – 2012
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| Keywords | 非線形解析学 / 偏微分方程式論 / 調和解析学 / 関数解析学 |
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| Research Abstract |
We investigate the local existence of strong solutions and their blow-up within a finite time in arbitrary dimensional domains. The life-span of local solutions is characterized in terms of the L^1 and L^p-norms of the given initial data. Simultaneously, it is clarified that the total mass and the second momentum of the initial data together with the coefficient of the system of equations have a great influence on the blow-up phenomena. As an application, we prove that the blow-up solution either exhibits a definite blow-up rate determined by p, or oscillates in L^1 with the larger amplitude than the absolute constant. Furthermore, in multi-connected domains, it is still an open question whether there does exist a solution of the stationary Navier-Stoeks equations with the inhomogeneous boundary data whose total flux is zero. The relation between the nonlinear structure of the equations and the topological invariance of the domain plays an important role for the solvability of this problem. We prove that if the harmonic part of solenoidal extensions of the given boundary data associated with the second Betti number of the domain is orthogonal to non-trivial solutions of the Euler equations, then there exists a solution for any viscosity constant. The relation between Leary's inequality and the topological type of the domain is also clarified.
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