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Research of Nonlinear Partial Differential Equations Using Variational Methods and Time-discretization Schemes (1999)

Research Project

Project/Area Number 09640177
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionScience University of Tokyo

Principal Investigator

TACHIKAWA Atsushi  Science University of Tokyo(S.U.T.), Faculty of Science and Technology, Associate Professor, 理工学部, 助教授 (50188257)

Co-Investigator(Kenkyū-buntansha) TAMIYA Takanori  Science University of Tokyo(S.U.T.), Fac.of Sci.and Tech.nology, Lecturer, 理工学部, 講師 (60183472)
TANAKA Ryuichi  Science University of Tokyo(S.U.T.), Fac.of Sci.and Tech.nology, Lecturer, 理工学部, 講師 (10112898)
KOBAYASHI takao  Science University of Tokyo(S.U.T.), Fac.of Sci.and Tech.nology, Associate Professor, 理工学部, 助教授 (90178319)
NAGASAWA Takeyuki  Tohoku Univ., Mathematical Institute, Associate Professor, 理学研究科, 助教授 (70202223)
KOTANI Kouichi  Science University of Tokyo(S.U.T.), Fac.of Sci.and Tech.nology, Assistant, 理工学部, 助手 (80183341)
細尾 敏男  東京理科大学, 理工学部, 助手 (30130339)
浜畑 芳紀  東京理科大学, 理工学部, 講師 (90260645)
吾郷 孝視  東京理科大学, 理工学部, 教授 (60112893)
Project Period (FY) 1997 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 1999: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 1998: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1997: ¥1,100,000 (Direct Cost: ¥1,100,000)
KeywordsVariational Method / Discretization / Harmonic Maps / 変分法 / 差分法 / 調和写像
Research Abstract

The aim of this research is to study the solutions of the nonlinear partial differential equations which are closely related to variational problems by using calculus of variations and time-discretization schemes. More precisely, for some variational functional F(u) we treat the partial differential equation ィイD7∂u(/)∂tィエD7-(the Euler-Language equation of F) = 0. To construct weak solutions for the above equation we proceed as follows. We consider the functional GィイD2nィエD2(u) = ∫ィイD7|u-uィイD2nィエD2-1|(/)2hィエD7dx+F(u), and define uィイD2nィエD2 as a minimizer of GィイD2nィエD2(u) successively. Combining {uィイD2nィエD2} by line segments we construct a approximate solution uィイD2hィエD2(x, t). Finally, under some conditions, we prove that uィイD2hィエD2(x, t) converge to a weak solution. On the other hand, id the limit of the sequence of maps {uィイD2hィエD2}converges to some map u(x, t), u(x, t) will be closely related to minimizing movement which is a new notion introduced by E.De Giorgi.
Using the above method … More , Tachikawa constructed weak solutions of the heat-type equations for harmonic maps (Eells-Sampson equation) from noncompact Riemannian manifolds into the n-dimensional spheres (ィイD7∂u(/)∂tィエD7-Δu-u|Du|ィイD12ィエD1 = 0). Moreover, he proved that the weak solutions are minimizing movements of the energy functionals.
Related to the above problem, Nagasawa and Tachikawa studied harmonic maps between noncompact complete Riemannian manifolds. Especially, they considered harmonic maps with a certain non-degeneracy condition and get the following nonexistence result. "Let N be a Handamard manifold whose sectional curvatures at a point p do not decay faster than distィイD1-2ィエD1"(p, pィイD20ィエD2) for some fixed point pィイD20ィエD2. Then there is no entire harmonic maps from RィイD1mィエD1 into N which satisfies a certain non-degeneracy condition."
Nagasawa constructed a weak solution of the Navier-Strokes equation on a Riemannian manifold using the above method. Moreover, he sharpened the energy estimates on the weak solutions constructed as above and got a new partial regularity estimates. He constructed weak solutions of the hyperbolic Ginzburg Landau equations too and studied them numerically. Less

Report

(4 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (28 results)

All Other

All Publications (28 results)

  • [Publications] Atsushi TACHIKAWA: "Weak Solution to Evolution Problems of Harmonic Maps from Non compact manifolds"Rendiconti di Matematica(掲載予定). (未定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takao KOBAYASHI: "Singular Solutions and Propagation of Holomorpinic solutions to Nonlinear Differential Equations"Publ. RIMS. Kyoto Univ.. 31. 43-63 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takao KOBAYASHI: "Global Hypoellipticity of Suhelliptic Opevntors on Closed Manifolds"Hokkaido Math. J.. 28. 613-633 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kouichi KOTANI: "Simultaneous Point and Interval Predictions in the Weibull Distribution"Statistica anno LCII. 221-235 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takeyuki NAGASAWA: "Navier-Stokes flow on Riemannian Manifolds"Nonlinear Anal.. 30. 825-832 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takeyuki NAGASAWA: "Initial-Final Value Problems for Ordinary Differential Equations and Appiications to Equivariant Harmonn Mops"J. Math. Soc. JAPAN. 50. 545-555 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takeyuki NAGASAWA: "Construction of Weak Solutions of Navier-Stokes equations on Riemannian Manitoidny Minimizing Varictional Functionals"Adv. in Muth. Sci. Appl. 9. 51-71 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takeyuki NAGASAWA: "Numerical Analysis for Hyperbolic Ginzbury-Landau System"Nonlinear Anal.(掲載予定). (未定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Atsushi TACHIKAWA: "Weak Solution to Evolution Problems of Harmonic Maps from Noncompact Manifolds."Rendiconti di Matematica. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takao KOBAYASHI: "Singular solutions and propagation of holomorphic solutions to nonlinear differential equations."Publ.RIMS, Kyoto Univ.. 31. 43-63 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takao KOBAYASHI and Hideki OMORI: "Global hypoellipiticity of subelliptic operators on closed manifolds."Hokkaido Mathematical Journal. 28. 613-633 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kouichi KOTANI, T.ISHIKAWA and Takanori TAMIYA: "Simultaneous point and interval predictions in the Weibull distribution."Statistica anno LVII. 221-235 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takeyuki NAGASAWA: "Navier-Stokes flow on Riemannian manifolds."Nonlinear Anal.. 30(2). 825-832 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takeyuki NAGASAWA and Keisuke Ueno: "Initial-final value problems for ordinary differential equations and applications to equivariant harmonic maps."J.Math.Soc.Japan. 50(3). 545-555 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takeyuki NAGASAWA: "Construction of weak solutions of the Navier-Stokes equations on Riemannian manifold by minimizing variational functionals."Adv.in Math.Sci.Appl.. 9(1). 51-71 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takeyuki NAGASAWA, Kazuaki NAKANE and Seiro OMATA: "Numerical analysis for hyperbolic Ginzburg landau system."Nonlinear Anal.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Atsushi TACHIKAWA: "Weak Solution to Evolution Problems of Harmonic maps from Noncompact Manifolds"Rendiconti di Matematica. (掲載予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] Takao KOBAYASHI: "Global hypoellipticity of subelliptic operators on closed manifolds"Hokkaido Mathematical Journal. 28. 613-633 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Takeyuki NAGASAWA: "Construction of Weak Solutions of Navier-Stokes equations on Riemannian manifold by minimizing Variational Functionals"Adv.in Math.Sci.Appl.. 9. 51-71 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 長澤壮之・上野慶介: "Initial-final value problems for ordinary differential equations and applications to equivaliant harmonic maps" Journal of Mathematical Society of Japan. 50(3). 545-555 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 長澤壮之: "Construction of Weak Solutions of Navier-Stokes equations on Riemannian manifold by minimizing variational functionals" Advances in Mathematical Science and Applications. 9(未定). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 小林隆夫: "Singular solutions and Propagation of Holomorphic Solutions to Nonlinear Differential Equations" Publ.RIMS,Kyoto Univ. 31. 43-63 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 立川篤: "Weak Solutions for the Evolution Problems of Harmonic Maps on Nondecreasing Domains" Multhematical Science and Applications. 10. 475-480 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 立川篤: "Nonexistence Problems on Harmonic Maps between oncompact omplete Riemannian Manifolds" Proceedings of the Sixth Tokyo Conference on Nonlinear PDE. 53-61 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 長澤壯之: "Navier-Stokes Flow on Riemannian Manifolds" Nonlinear Aralysis,Theory,Methods & Applications. 30. 825-832 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 吾郷孝視: "Fermat Questions for Composite Moduli" Journal of Number Theory. 66. 29-50 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 浜畑芳紀: "Modular 3-folds Obtained from Quaternion uniary Groups of degree 2" Journal of Muthematical Sciences. 4. 385-401 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 細尾敏男: "Automorphism Groups of Cubic Surfaces" Journal of Algebra. 192. 651-677 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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