The approach of an analytic semigroup for free boundary problems of viscous compressible fluids
| Project/Area Number |
17540156
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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 | Shizuoka University |
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Principal Investigator |
SHIMIZU Senjo Shizuoka University, Faculty of Engineering, Associate professor, 工学部, 助教授 (50273165)
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| Co-Investigator(Kenkyū-buntansha) |
SHIBATA Yoshihiro Waseda University, School of Science and Engineering, Professor, 理工学術院, 教授 (50114088)
KIKUCHI Koji Shizuoka University, Faculty of Engineering, Professor, 工学部, 教授 (50195202)
HOSHIGA Akira Shizuoka University, Faculty of Engineering, Associate professor, 工学部, 助教授 (60261400)
ADACHI Shinji Shizuoka University, Faculty of Engineering, Associate professor, 工学部, 助教授 (40339685)
NAKAJIMA Toru Shizuoka University, Faculty of Engineering, Associate professor, 工学部, 助教授 (50362182)
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| Project Period (FY) |
2005 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2006: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2005: ¥1,700,000 (Direct Cost: ¥1,700,000)
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| Keywords | Free boundary problems / Analytic semigroups / Lp-maximal regularity / Stokes equations / Navier-Stokes equations / R-boundedness / Fourier multiplier theorem / 非圧縮性粘性流体 / 最大正則性 / ストークス方程式 / 準線形方程式 / 2相流体 / ノイマン境界条件 |
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| Research Abstract |
In this research, we consider a free boundary problem for the Navier-Stokes equation which describes the motion of an isolated finite volume of viscous incompressible fluid without taking surface tension into account By using the Lagrange coordinates, free boundary problem is written by the quasi-linear equations on the fixed boundary. Our purpose is to prove a local in time unique existence theorem for any initial data and a global in time unique existence theorem for some small initial data.
To treat quasi-linear equations, first we prove the Lp-Lq maximal regularity of solutions to the linearized problem, which is described by the Neumann problem for the Stokes equation. We consider this problem by analytic semigroup approach. Our main issues is to use R-boundedness and operator valued Fourier multiplier theorem which are recently developed by Weis ('01, Math.Ann.) and Denk-Hieber-Pruss ('03, Mem.AMS).
Based on the Lp-Lq maximal regularity result of the linearized problem, by using the contraction mapping principle, we proved a local in time unique existence theorem for any initial data and external force and a global in time unique existence theorem for some small initial data which are orthogonal to the rigid space in the case where external force vanishes.
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Report
(3 results)
Research Products
(32 results)
