Project/Area Number  09440047 
Research Category 
GrantinAid for Scientific Research (B).

Section  一般 
Research Field 
解析学

Research Institution  Waseda University 
Principal Investigator 
SHIBATA Yoshihiro Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (50114088)

CoInvestigator(Kenkyūbuntansha) 
檀 和日子 筑波大学, 数学系, 助手 (40251029)
KOBAYASHI Takayuki Kyushu Institute of Techinics, Department Engineering, Assistant Professor, 工学部, 講師 (50272133)
KAJI Hajime Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (70194727)
MOROYA Yoshiaki Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (90063718)
KOJIMA Kiyofumi Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (30063689)
KUBO Akisata Fujita Health University, Associated Professor, 助教授 (60170023)
杉山 由恵 早稲田大学, 理工学部, 助手 (60308210)
清水 扇丈 静岡大学, 工学部, 助教授 (50273165)

Project Fiscal Year 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥13,700,000 (Direct Cost : ¥13,700,000)
Fiscal Year 1999 : ¥4,900,000 (Direct Cost : ¥4,900,000)
Fiscal Year 1998 : ¥4,300,000 (Direct Cost : ¥4,300,000)
Fiscal Year 1997 : ¥4,500,000 (Direct Cost : ¥4,500,000)

Keywords  3 dim. Exterior Domain / NavierStokes Equation / Physically Reasonable Solution / Stability / 2 dim. Exterior Domain / LィイD2pィエD2LィイD2qィエD2 estimate / Stokes semigroup / Oseen semigroup / 3次元外部領域 / NavierStokes方程式 / physically reasonable solution / 安定性 / 2次元外部領域 / L_pL_q評価 / Stokes半群 / Oseen半群 / 非圧縮性粘性流体 / 圧縮性粘性流体 / L_<p,∞>空間 / diffusion wave / 2相問題 / GinzburgLandau方程式 / 非定常問題の安定性 / 定常問題 / 楕円型評価 / Besov空間 / LittlewoodPayley分解 / 外部問題 / L_pL_q decay評価 / 漸近挙動 / 実解析 
Research Abstract 
We studied the initialboundary value problem for NavierStokes equation in 3 dimensional exterior domain which describes the motion of incompressible viscous fluids. In 1930, J. Leray proved the existence of its weak solutions without uniqueness. But, we can not get any qualitative property of the motion of fluid from Leray's solutions. In 1950, R. Finn started to study the qualitative property of soltuions to the stationary NavierStokes equation, which was termed physically reasonable solutions (prs) by him. He proved the unique existence theorem of prs for small external force and the velocity uィイD2∞ィエD2 at infinity. After Finn, Heywood proved the stability of prs in the LィイD22ィエD2 framework, which now become one of the most important base of numerical investigation of the fluid motion. But, prs does not belong to LィイD22ィエD2 space, and therefore we have to study the stability of prs in the class to which prs belongs. This problem remained more than 30 years. In our present study, w
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e solved this problem. We used the following argument. We took the Oseen approcimation in the 3 dim. Exterior domain, and then we investigated the local energy estimate near the boundary of optimal order of the Oseen equation by showing the fractional differentiablity of Oseen resolvent near the origin. Combinig this and LィイD2pィエD2LィイD2qィエD2 estimate in the whole space by cutoff technique, we proved the optimal LィイD2pィエD2LィイD2qィエD2 estimate of the Oseen semigroup in 3 dim. Exterior domain. The most important point is that all the constant appearing in the estimate is independent of uィイD2∞ィエD2. By using this estimate, we could solve the stability problem of Finn's prs. (2) In 2 dimensinal case, we know the unique existence of Leray's weak solutions. But, since the Stokes fundamental solution has logarihmic singularity, we know less property of solution to NS in the exterior domain or even the whole space compared with 3 dim. Case. We could obtain the asymptotic expansion of Stokes resolvent near the origin and we found that the logarithmic singularity is canceled out by the reflection phenomenon near the bounday, and therefore we obtained the optimal LィイD2pィエD2LィイD2qィエD2 estimate (1< q≦ p≦ ∞) of Stokes semigroup in a 2 dim. Exterior domain by using the similar argument to the 3 dim. Case. Applying this estimate, we could show the best convergent rate of weak solutions to NS when time goes to infinity. (3) We considered the motion of the compressible viscous fluid too. Extending the method developed in study (1), we obtained the optimal LィイD2pィエD2LィイD2qィエD2 estimate of solutions to linearized eqautions in the 3 dim. Exterior domain, which is applied to obtain the optimal convergent rate of solutions to the original nonlinear probolem at time infinity. Less
