非線形特異摂動現象の数学解析と数値計算

• Project/Area Number: 07454032
• Research Category: Grant-in-Aid for General Scientific Research (B)
• Allocation Type: Single-year Grants
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Kyoto University
• Principal Investigator: 岡本 久 京都大学, 数理解析研究所, 教授 (40143359)
• Co-Investigator(Kenkyū-buntansha): 谷口 雅治 京都大学, 数理解析研究所, 助手 (30260623) ; 岩田 覚 京都大学, 数理解析研究所, 助手 (00263161) ; 竹井 義次 京都大学, 数理解析研究所, 助教授 (00212019) ; 室田 一雄 京都大学, 数理解析研究所, 教授 (50134466) ; 河合 隆裕 京都大学, 数理解析研究所, 教授 (20027379)
• Project Period (FY): 1995
• Project Status: Completed (Fiscal Year 1995)
• Budget Amount: ¥6,100,000 (Direct Cost: ¥6,100,000); Fiscal Year 1995: ¥6,100,000 (Direct Cost: ¥6,100,000)
• Keywords: 特異摂動 / 流体力学 / パンルベ方程式

Research Abstract

特異摂動理論のNavier-Stokes方程式への応用に関し、岡本久は同方程式の新しい厳密解を発見し,その流体力学的な性質を解明した.そのひとつは,TamadaとDorrepaalによる「斜めにぶつかる淀み点流」の3次元版であり,もうひとつはLerayによる相似解のスキームの解の発見である.
流体の数値シミュレーションに関しては,渦法による数値実験を岡本久が行った.せん断流と渦層の相互作用,あるいは2枚の渦層の相互作用による複雑な運動,特に特異点の発生とその構造に関して,これまで漠然と想像されてきたこととは違うメカニズムが存在することがわかった.
河合隆裕と竹井義次は代数解析の手法を用いて特異摂動理論を研究し,パンルベ方程式との興味る関係を導いた.
室田一雄と岩田覚は数値線形代数理論の研究を通じて本研究に貢献した.特異摂動問題を離散化すると条件数の大きな行列あるいは独特の構造を持つ行列が現れる.こういった行列に関する連立方程式の解法には離散数学の手法が役に立つ.室田は精度の改良や高速化に関する多くの方法を提唱し,その有用性をしめした.
谷口雅治は反応拡散方程式系において特異摂動状態にある定常解の安定性を論じた.これまで知られていた「解の存在証明法」を大幅に改良し,すっきりした理論体系を構築した.また2次元遷移層の安定性を特異摂動理論の枠組で厳密に取り扱うことが可能であることを示し,そのスケーリング指数を決定することに成功した.

Research Products

• [Publications] 岡本 久: "Axisymmetric stagnation flow converging obliquely on a cylinder" J. Phys. Soc. Japan. 64. 2714-2717 (1995)
• [Publications] 岡本 久: "Numerical computation of vortex sheet roll-up in the background shear flow" to appear in Fluid Dynam. Res.17. (1996)
• [Publications] 岡本 久: "The resonance of modes in the problem of capillary gravity waves" to appear in Physica D.
• [Publications] 岡本 久: "Two Dimensional, Periodic, Capillary-Gravity Waves With Negative Surface Tension" Proc. IUTAM Conf. Structure and Dynamics of Non-linear Waves in Fluids, World Scientific. 363-369 (1995)
• [Publications] Takahiro Kawai, T. Aoki and Y. Takei: "Algebraic analysis of singular perturbations" to appear in Sugaku Exposition.
• [Publications] Takahiro kawai and Henry P. Stapp: "On infrared singularities" RIMS preprint 1055. (1995)
• [Publications] J. van der Woude and Kazuo Murota: "Disturbance decoupling with pole placement for structured systems: a graph-theoretic approach" SIAM Journal on Matrix Analysis and Applications. 16. 922-942 (1995)
• [Publications] Kazuo Murota: "Finding optimal minors of valuated bimatroids" Applied Mathematics Letters. 8. 37-42 (1995)
• [Publications] Yoshitsugu Takei: "On a WKB-theoretic approach to the Painleve transcendents" Conf. Proc. "XIth International Congress of Math. Physics (Unesco-Sorbonne-Paris), Intl. Press. 533-542 (1995)
• [Publications] Takahiro Kawai and Yoshitsugu Takei: "WKB analysis of Painleve transcendents with a large parameter. I" preprint (RIMS-1007), Adv. in Math.
• [Publications] Takashi Aoki, Takahiro Kawai and Y. Takei: "WKB analysis of Painleve transcendents with a large parameter. II. -Multiple-scale analysis of Painleve transcendents" preprint (RIMS-1038), "Structure of Solutions of Differential Equations (Katata/Kyoto 1995).
• [Publications] C. A. Berenstein, Y. Takei: "Exponential representation of a holomorphic solution of a system of differential equations associated with ....." preprint (RIMS-1039), "Structure of Solutions of Differential Equations, World Scientific.
• [Publications] Takahiro Kawai, C. A. Berenstein, D. C. Struppa: "Interpolating varieties and the Fabry-Ehrenpreis-Kawai gap theorem" RIMS preprint 1056. (1999)
• [Publications] Kazuo Murota: "Computing the degree of determinants via combinatorial relaxation" SIAM Journal on Computing. 24. 765-796 (1995)
• [Publications] Kazuo Murota: "Combinatorial relaxation algorithm for the maximum degree of subdeterminats: Computing Smith-McMillan form at infinity...." Applicable Algebra in Engineering, Communication and Computing. 6. 251-273 (1995)
• [Publications] Kazuo Murota: "Comparison of conventional and 'invariant' schemes of fundamental solutions method for annular domains" Japan Journal of Industrial and Applied Mathematics. 12. 61-85 (1995)
• [Publications] Kazuo Murota: "An identity for matching and skew-symmetric determinant" Linear Algebra and Its Applications. 218. 1-27 (1995)
• [Publications] Kazuo Murota: "An identity for bipartite matching and symmetric determinant" Linear Algebra and Its Applications. 221. 261-274 (1995)
• [Publications] Satoru Iwata and Kazuo Murota: "A Theorem on the Principal STructure for Independent Machings" Descrete Appl. Math.61. 229-244 (1995)
• [Publications] Satoru Iwata and Kazuo Murota: "A Minimax Theorem and a Dulmage-Mendelsohn Type Decomposition for a Class of Generic Partitioned Matrices" SIAM J. Matrix Anal. Appl.16. 719-734 (1995)
• [Publications] Satoru Iwata: "Princeipal Structure of Submodular Systems and Hitchcock-type Independent Flows" Combinatorica. 15. 515-532 (1995)
• [Publications] M. Shigeno and Satoru Iwata: "A dual Approximation Approach to Weighted Matroid Intersection" Oper. Res. Lett. 18. 153-156 (1995)
• [Publications] Masaharu Taniguchi and Y. Nishiura: "Stability and characteristic wavelength of planar interfaces in the large diffusion limit of the inhibitor" Proc. Roy. Soc. Edinburgh. (to appear).
• [Publications] Masaharu Taniguchi: "A remark on singular perturbation methods via the Lyapunov-Schmidt reduction" Publ. Res. Inst. Math. Sci.(to appear).
• [Publications] Masaharu Taniguchi: "An improvement in construction of internal transition layers" Proceedings of the China-Japan Symposium on Reaction-Diffusion Equations....., Fudan Univ.(1994)
• [Publications] 藤井 宏、岡本 久: "非線型力学、岩波講座「応用数学」" 岩波書店, (1995)
• [Publications] 室田 一雄: "離散構造とアルゴリズムIV" 近代科学社, (1995)