2000 Fiscal Year Final Research Report Summary
Research of the Navier-Stokes equations by using the theory of Fourier analysis and semigroup theory
| Project/Area Number |
11640156
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Hitotsubashi University |
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Principal Investigator |
YAMAZAKI Masao Graduate School of Economics, Hitotsubashi University, Professor, 大学院・経済学研究科, 教授 (20174659)
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| Co-Investigator(Kenkyū-buntansha) |
YAMADA Hiromichi Graduate School of Economics, Hitotsubashi University, Professor, 大学院・経済学研究科, 教授 (50134888)
MACHIDA Hajime Graduate School of Commerce, Hitotsubashi University, Professor, 大学院・商学研究科, 教授 (40090534)
IWASAKI Shiro Graduate School of Economics, Hitotsubashi University, Professor, 大学院・経済学研究科, 教授 (00001842)
ISHIMURA Naoyuki Graduate School of Economics, Hitotsubashi University, Professor, 大学院・経済学研究科, 助教授 (80212934)
FUJITA Takahiko Graduate School of Commerce, Hitotsubashi University, Professor, 大学院・商学研究科, 教授 (50144316)
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| Project Period (FY) |
1999 – 2000
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| Keywords | Navier-Stokes equations / Lorentz spaces / Morrey spaces / Weak-* convergence / periodic solutions / almost periodic solutions / exterior domains / stability |
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| Research Abstract |
We studied the Navier-Stokes equation on the whole space or on exterior domains. We have already studied the unique existence and the stability under initieal perturbations of stationary solutions with external forces independent of time. In this research we considered the case where external force depends on the time-variable, and studied the unique existence and the stability of solutions. This research generalizes similar researches on time-periodic solutions and solutions almost periodic in time.
On the whole space we employ the Morrey spaces as the space of solutions, and succeeded in generalizing the results for usual L^p-spaces obtained by Professors Hideo Kozono, Mitsuhiro Nakao and Yasushi Taniuchi. On exterior domains we employ the weak-L^p spaces as the space of solutions, and we succeeded in relaxing the assumptions on the external forces very much, and firstly obtained conditions sufficient for the unique existence of solutions in 3-dimensional exterior domains.
The results for the Morrey spaces can be obtained in a manner similar to that employed in our previous study. On the other hand, in the proof of the results for weak-L^p spaces, it is essential to show that the integral of functions with values in a Banach space converges, where the integral is considered to diverge in general. In order to show this fact, we first consider the family of the Lorentz spaces which generalizes the weak-L^pA spaces, and we improved estimates of L^p-L^q type, which is often employed in previous studies, by using real interpolation, and the we used the duality property between the Lorentz spaces.
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