Project/Area Number 
13640173

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  Nara Women's University 
Principal Investigator 
YANAGISAWA Taku Nara Women's University, Grasuate School of Humanities and Sciences, Associate Professor, 大学院・人間文化研究科, 助教授 (30192389)

Project Period (FY) 
2001 – 2003

Project Status 
Completed (Fiscal Year 2003)

Budget Amount *help 
¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2002: ¥700,000 (Direct Cost: ¥700,000)

Keywords  singular perturbation / vanishing viscosity limit / NavierStokes equations / Prandtl equation / vorticity / ホッジ分解 / 国際情報交換 / 中国 / 対称双曲性 / 圧縮性オイラー方程式 / 圧縮性ナビエ・ストークス方程式 / ブラントール方程式 / 非圧縮性ナビエ・ストークス方程式 
Research Abstract 
For the purpose of building the mathematical framework to investigate the boundary value problems for hyperbolic systems as the limit of singular perturbation, we have shown the following mathematical results through the consideration of concrete problems appearing in the fluidmechanics. 1)."Vanishing viscosity limit for the initial boundary value problems of the compressible NavierStokes equations in a domain with the boundary" We study the existence theorem for the initial boundary value of the Prandtl equation which appears as the first term of the boundary expansion of asymptotic solution to the compressible NavierStokes equations. By taking the FokkerPlanck type equation as the linearized equation, we can show the estimate with the improvement in the order of regularity. However, it is also observed that there should occur the phenomenon of the "loss of derivatives" for this linearized problem. Hence it is so far most likely to be difficult to show the existence theorem for the Prandtl equation in the Sobolev spaces. 2)."On the relation of the smoothness of the solutions of the 3D NavierStokes equations in a bounded domain with the vorticity" We have shown a new apriori estimate of the solutions to the 3D NavierStokes equations in a bounded domain which reveals that the maximum norm of the vorticity controls the smoothness of the solutions. Further we presented a generalized BiotSavart law on a bounded domain with the estimates of the Green's matrix of the Laplace operator, which was used in the proof of the new apriori estimate stated above.
