2006 Fiscal Year Final Research Report Summary
Study of the stability of solutions to some nonlinear evolution equations based on recent development of real analysis
| Project/Area Number |
15340204
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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 |
Basic analysis
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| Research Institution | Waseda University |
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Principal Investigator |
SHIBATA Yoshihiro Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 教授 (50114088)
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| Co-Investigator(Kenkyū-buntansha) |
TANAKA Kazunaga Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 教授 (20188288)
YAMAZAKI Masao Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 教授 (20174659)
KOBAYASHI Takayuki Saga Univ., Faculty of Science and Engineering, Professor, 理工学部, 教授 (50272133)
SHIMIZU Senjo Shizuoka Univ., Faculty of Engineering, Assistant Professor, 工学部, 助教授 (50273165)
HISHIDA Toshiaki Nigata Univ., Faculty of Engineering, Assistant Professor, 工学部, 助教授 (60257243)
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| Project Period (FY) |
2003 – 2006
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| Keywords | Stokes equation / Neumann problem / Maximal regularity / Navier-Stokes equation / free boundary problem / flow past rotating body / perturbed half-space / Oseen equation |
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| Research Abstract |
We study the spectral analysis of Stokes equations based on the recent development of the real analysis, Fourier analysis and functional analysis and its application to the Naveir-Stokes equations in several different situations arising from the mathematical physics.
1) We studied an inncompressible viscous flow past a rigid body, which is mathematically described by Oseen equations. We studied the decay properties of the Oseen semigroup in the exterior domain and showed global in time stability Navier-Stokes flow past a rigid body
2) We studied an inncompressible viscous flow in a perturbed half-space which describes for example flow past high buildings. We studied an optimal decay properties of solutions to the Stokes equations in a perturbed half-space and proved a global in time unique existence of solutions to the Navier-Stokes equations in a perturbed half-space with small initial data.
3) We proved the maximal regularity of solutions to the Stokes equation with the Neumann boundry
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condition in a bonded domain. We use some recent development of the operator-valued Fourier analysis by Weis and Denk-Hieber-Pruss. Our method is very simple compared with previous results and seems to be applicable to linear evoulution equations of parabolic type. Moreover, we proved local in time unique existence of strong solutions with arbitrary initial data and global in time unique existence of strong solutions with some small initial data of the free boundary problem of Navier Stokes equations which describes the transient motion of an isolated volume of viscous incompressible fluid
4) We studied an inncompressible viscous flow past a rotating rigid body. This problem was already studied by Galdi and Galdi and Silvestre in the L_2 framework. Our main contribution is to show the decay estimate of the continuous semigroup associated with linearized problem. The main difficulty comes from first order differential system with polynomially growing coefficients which is not subordinated by the Laplacian. We developed new technique to investigate the high frequency part of the spectrum. Less
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