Project/Area Number  09640164 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
解析学

Research Institution  Hitotsubashi University 
Principal Investigator 
YAMAZAKI Masao Graduate School of Economics, Associate Professor, 大学院・経済学研究科, 助教授 (20174659)

CoInvestigator(Kenkyūbuntansha) 
ISHIMURA Naoyuki Hitotsubashi University, Graduate School of Economics, Associate Professor, 大学院・経済学研究科, 助教授 (80212934)
FUJITA Takehiko Hitotsubashi University, Faculty of Commerce, Professor, 商学部, 教授 (50144316)
YAMADA Hiromichi Hitotsubashi University, Graduate School of Economics, Professor, 大学院・経済学研究科, 教授 (50134888)
IWASAKI Shiro Hitotsubashi University, Graduate School of Economics, Professor, 大学院・経済学研究科, 教授 (00001842)
永島 孝 一橋大学, 大学院・経済学研究科, 教授 (00017526)
MACHIDA Hajime Hitotsubashi University, Faculty of Commerce, Professor, 商学部, 教授 (40090534)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥1,900,000 (Direct Cost : ¥1,900,000)
Fiscal Year 1998 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1997 : ¥1,100,000 (Direct Cost : ¥1,100,000)

Keywords  NavierStokes equations / stationary solutions / stability / weak L^n spaces / exterior domains / NavierStokes方程式 / 定常解 / 安定性 / 弱L^n空間 / 外部領域 / Lorentz空間 
Research Abstract 
We studied the existence, the uniqueness and the stability of stationary solutions of the NavierStokes equations in exterior domains, which is regarded as a model describing the motion of fluids, by functional analytic methods. For ndimensional spaces, the function space L^n is mainly employed in previous works on this direction. We showed during the period from April, 1997 to March, 1998 that, when the space dimension is 3, solutions in the function space L^3 exist only in very limited cases, and hence we must consider a somewhat larger space L^<3, *> as the function space in which solutions exist. Moreover, for the NavierStokes equation in the whole space, we gave a condition on the external forces sufficient for the unique existence of small stationary solutions belonging to the Morrey spaces, which is strictly larger than the space L^<n, *>. We furthermore showed that the stationay solutions above are stable under small initial perturbation in function spaces which contain distributions other than Radon measures. We studied the exterior problem of the NavierStokes equations in the space for n<greater than or equal>3 during the period from April, 1998 to March, 1999. We then showed the unique existence of small stationary solutions in the function space under the condition that the external forces are given as the first order derivatives of potentials small in the function space L^<n/2, *>. We also showed that the stationary solutions above are stable under small initial perturbations in the function space L^<n, *>. This result is applicable to external forces more general than those which can be treated by previous results obtained by potential theoretical methods, and is applicable to the 3dimensional case which could not be treated by functional analytic methods before.
