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Research of applied analysis toward the global theory for nonlinear systems

Research Project

Project/Area Number 17340027
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionWaseda University

Principal Investigator

NISHIDA Takaaki  Waseda University, Faculty of Science and Engineering, Professor (70026110)

Co-Investigator(Kenkyū-buntansha) NAKAO Mitsuhiro  Kyushu University, Faculty of Mathematical Science, Professor (10136418)
KOKUBU Hiroshi  Kyoto University, Faculty of Science, Professor (50202057)
KAWANAGO Tadashi  Tokyo Institute of Technology, Faculty of Science and Engineering, Associate professor (20214661)
TANAKA kazunaga  Waseda University, Faculty of Science and Engineering, Professor (20188288)
Project Period (FY) 2005 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥10,900,000 (Direct Cost: ¥10,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2007: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2006: ¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2005: ¥4,000,000 (Direct Cost: ¥4,000,000)
KeywordsNonlinear Partial Differential Equation / Dynamical Systems / Global structure of solution space / Heat convection problems / Computer assisted proof / Free surface problems / 非線形波動
Research Abstract

Analysis for the heat convectin problems for the system of Oberbeck-Boussinesq equations in the horizontal strip domain. The existence of bifurcation curve of roll-type solutions is proved for the ten times Rayleigh number of critical Rayleigh number by a computer assisted proof . The second bifurcation point of stationary roll-type solutions is determined by a computer assisted proof. The hexagonal-type and rectangle-type solutions of 3-dimensional problems are also proved for the existence by a computer assisted proof at least for rather small Rayleigh numbers.
The cocoon bifurcation for the Michelson system is analyzed and proved for the existence bf infinitely many bifurcations of heteroclinic orbits to the saddle-node periodic orbit by a topological method and a computer assisted proof.
The driven-cavity problem of 2-dimensional Navier-Stokes equation is solved for rather large Reynolds numbers compared to the existing verified result by a Newton type computer assited proof.

Report

(4 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • Research Products

    (51 results)

All 2008 2007 2006 2005

All Journal Article (34 results) (of which Peer Reviewed: 17 results) Presentation (17 results)

  • [Journal Article] Standing waves of for nonlinear Schroedinger equations with A general nonlinearity : one and two dimensional cases2008

    • Author(s)
      K. Tanaka, et. al.
    • Journal Title

      Comm. Partial Differential Equation (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Standing waves for nonlinear Schroedinger equations with a general nonlinearity: one and two dimensional cases2008

    • Author(s)
      K.Tanaka, J.Byeon and L.Jeanjean
    • Journal Title

      Communication Partial Differential Equations 33

      Pages: 30-30

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On a linearized system arising in the study of Benard-Marangoni conv2007

    • Author(s)
      T.Nishida and Y.Teramoto
    • Journal Title

      Kokyuroku, Bessatsu, RIMS, Kyoto University 1

      Pages: 271-286

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Rigorous verification of the cocoon bifurcation in the Michelson system2007

    • Author(s)
      Hiroshi Kokubu, et. al.
    • Journal Title

      Nonlinearity 20

      Pages: 2147-2174

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Numerical verification of stationary solutions for Navier-Stokes problem2007

    • Author(s)
      M.T.Nakao, et. al.
    • Journal Title

      J.of Computational and Applied Mathematics 199

      Pages: 424-431

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Verified numerical computation of solutions for Navier-Stokes equation2007

    • Author(s)
      K.Nagatou, M.T.Nakao, et. al.
    • Journal Title

      Hokkaido Mathematical Journal 36

      Pages: 777-799

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Topological horseshoes of traveling waves for a fast-slow predator-prey2007

    • Author(s)
      H.Kokubu, K.Mischaikov, et. al.
    • Journal Title

      J.Dynamics and Differential Equations 19

      Pages: 623-654

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Improved convergence theorems of Newton's method for diff.eq2007

    • Author(s)
      T.Kawanago
    • Journal Title

      J.Comput.Appl.Math 199

      Pages: 365-371

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On a linearized system of Benard-Marangoni convection2007

    • Author(s)
      T. Nishida, Y. Teramoto
    • Journal Title

      Kokyuroku, Bessatsu, RIMS, Kyoto Univ vol.1

      Pages: 271-286

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Rigorous verfication of the cocoon bifurcation in the Michelson system2007

    • Author(s)
      H. kokubu, et. al.
    • Journal Title

      Nolinearity vol.2

      Pages: 2147-2174

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Numerical verification of stationary solutions for Navier Stokes problems2007

    • Author(s)
      M. T. Nakao, et. al.
    • Journal Title

      J. of Computational and Applied Mathematics vol.19

      Pages: 424-431

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Improved convergence theorems of Newton's method designed For the numerical verification for solutions of differential equations2007

    • Author(s)
      T. Kawanago
    • Journal Title

      J. of Computational and Applied Mathematics vol.19

      Pages: 365-371

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Rigorous verification of the cocoon bifurcation in the Michelson system2007

    • Author(s)
      H.Kokubu, D.Wilczak, and P.Zgliczynski
    • Journal Title

      Nonlinearity 20

      Pages: 2147-2174

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Improved convergence theorems of Newton's method designed for the numerical verification of differential equations2007

    • Author(s)
      Tadashi Kawanago
    • Journal Title

      Journal of Computational and Applied Mathematics 199

      Pages: 365-371

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Numerical verification ofstationary solutions for Navier-Stokes problems2007

    • Author(s)
      K.Nagatou, K.Hashimoto and M.T.Nakao
    • Journal Title

      Journal of Computational and Applied Mathematics 199

      Pages: 424-431

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On a linearized system arising in the study of Benard-Marangoni convection2007

    • Author(s)
      Takaaki Nishida, Yoshiaki Teramoto
    • Journal Title

      Proc. Intern. Conf. on Navier-Stokes Equations and Applications, Kokyuroku New Series, RIMS, Kyoto Univ. 1

      Pages: 1-16

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Improved convergence theorems of Newton's method designed for the numerical verification for solutions of differential equations2007

    • Author(s)
      Tadashi Kawanago
    • Journal Title

      J. Comput. Appl. Math. 199

      Pages: 365-371

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Some computer assisted proofs on 3-D heat convection problems2006

    • Author(s)
      M.T.Nakao, T.Nishida, et. al.
    • Journal Title

      Proc.of Nonlinear Theory and its Appl.NOLTA NOLTA'06

      Pages: 427-430

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] A numerical verification of bifurcation points for heat convection2006

    • Author(s)
      M.T.Nakao, T.Nishida, et. al.
    • Journal Title

      Proc.Conf.on Scientific Computing, Athen 2nd Proc.

      Pages: 8-8

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Cocoon bifurcation in 3-D reversible vector fields2006

    • Author(s)
      F.Dumortier, H.Kokubu, et. al.
    • Journal Title

      Nonlinrarity 19

      Pages: 305-328

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] The Conley index for fast-slow system II:Multi-D slow variable2006

    • Author(s)
      H.Kokubu, H.Oka, et. al.
    • Journal Title

      J.Differential Equations 225

      Pages: 242-307

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] A unified theorem describing Hopf and pitchfork bifurcation2006

    • Author(s)
      T.Kawanago
    • Journal Title

      ICNAAM 2006, Wiley-VCH Verlag ICNAAM

      Pages: 174-177

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Mutiple stable patterns for some reaction-diffusion equation2006

    • Author(s)
      K.Tanaka, T.Ide, et. al.
    • Journal Title

      Discrete Contin.Dyn.Sys. 14

      Pages: 93-116

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] A numerical verification of bifurcation points for nonlinear heat convection problems2006

    • Author(s)
      M.T.Nakao, Y.Watanabe, N.Yamamoto, T.Nishida
    • Journal Title

      Proc. 2nd Intern. Conf. on From Scientific computing to Computational Engineering, Athen, July 5-8, 2006 2

      Pages: 1-8

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Cocoon bifurcation in three dimensional reversiblevector fields2006

    • Author(s)
      F.Dumortier, S.Ibanez, Hiroshi Kokubu
    • Journal Title

      Nonlinearity 19

      Pages: 305-328

    • Related Report
      2006 Annual Research Report
  • [Journal Article] High frequency chaotic solutions of differential equations2006

    • Author(s)
      P.Felmer, S.Martinez, Kazunaga Tanaka
    • Journal Title

      Ergodic Theory and Dynamical Systems 26,2

      Pages: 379-407

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Cocoon Bifurcations in Three Dimensional Reversible Vector Fields2006

    • Author(s)
      Hiroshi Kokubu, Freddy Dumortier, Santiago Ibanez
    • Journal Title

      Nonlinearity 19

      Pages: 305-328

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Improved Convergence Theorems of Newton's Method Designed For The Numerical Verification For Solutions of Differential Equations2006

    • Author(s)
      Tadashi Kawanago
    • Journal Title

      Journal Computational and Applied Mathematics (to appear)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Hopf bifurcation in viscous incompressible flow down an inclined plane2005

    • Author(s)
      T.Nishida, Y.Teramoto, et. al.
    • Journal Title

      J.Mathematical Fluid Mechanics 7

      Pages: 29-71

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] A computational approach to constructive a priori and a posteriori est2005

    • Author(s)
      M.T.Nakao, K.Hashimoto, et
    • Journal Title

      Proc.of the 4th JSIAM-SIMAI Seminar Gakuto J.

      Pages: 5-5

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] A positive solution for a nonlinear Schroedinger equation on R^N2005

    • Author(s)
      K.Tanaka and L.Jeanjean
    • Journal Title

      Indiana Univ.Math.J. 54

      Pages: 443-464

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Hopf Bifurcation in Viscous Incompressible Flow Down an Inclined Plane2005

    • Author(s)
      Takaaki Nishida, Yoshiaki Teramoto, Hideaki Yoshihara
    • Journal Title

      Journal of Mathematical Fluid Mechanics 7

      Pages: 29-71

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A Numerical Method to Verify the Invertibility of Linear Elliptic Operators with Applications to Nonlinear Problems2005

    • Author(s)
      M.T.Nakao, K.Hashimoto, Y.Watanabe
    • Journal Title

      Computing 75

      Pages: 1-14

    • NAID

      120000981466

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Multi-clustered High Energy Solutions for a Phase Transition Problem2005

    • Author(s)
      K.Tanaka, P.Felmer, S.Martinez
    • Journal Title

      Proceeding Royal Society Edinburgh 135A

      Pages: 731-765

    • Related Report
      2005 Annual Research Report
  • [Presentation] Numerical verification of bifurcating solutions with multipeaks for 3-dimensional Rayleigh-Benard convection2007

    • Author(s)
      M.T.Nakao, T.Nishida, et. al.
    • Organizer
      NOLTA 2007
    • Place of Presentation
      Vancouver, Canada
    • Year and Date
      2007-09-16
    • Related Report
      2007 Annual Research Report
  • [Presentation] Numerical verification of bifurcating solutions for 3-D Rayleigh-Benard2007

    • Author(s)
      M.T.Nakao
    • Organizer
      NOLTA2007, Intern. Sympo.Nonlinear Theory
    • Place of Presentation
      Vancouver,Canada
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Secondary bifurcations of heat convection systems2007

    • Author(s)
      Takaaki Nishida
    • Organizer
      Intern. Workshop on Numerical Verification
    • Place of Presentation
      Waseda University
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Bifurcation problems of heat convection systems2007

    • Author(s)
      Takaaki Nishida
    • Organizer
      Symposium on Evolution Equations
    • Place of Presentation
      Tokai University
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] A positive solutions of a scalar field equation2007

    • Author(s)
      Kazunaga Tanaka
    • Organizer
      Chile-Japan PDE Symposium, Univ. of Chile
    • Place of Presentation
      Santiago,Chile
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Numerical verfication of bifurcating Solutions for 3-dimensional Rayleigh-Benard convection2007

    • Author(s)
      M. T. Nakao
    • Organizer
      NOLTA2007, Intern. Sympo. On Nonliner Teory and its Appl.
    • Place of Presentation
      Vancouver, Canada
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Constructive error estimates in finite element method of nonlin.PDE2006

    • Author(s)
      M.T.Nakao
    • Organizer
      Finite Element Method TEC 2006
    • Place of Presentation
      El Paso,Texas,USA
    • Year and Date
      2006-12-01
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] A unified theorem describing Hopf and pitchfork bifurcation2006

    • Author(s)
      Tadashi Kawanago
    • Organizer
      ICNAAM 2006
    • Place of Presentation
      Crete,Greece
    • Year and Date
      2006-09-16
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Numerical verification of solutions for heat convection problems2006

    • Author(s)
      M.T.Nakao
    • Organizer
      Czech-Japanese Seminar in Applied Math.
    • Place of Presentation
      Prague,Czech
    • Year and Date
      2006-09-01
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Bifurcation problems of Benard-Marangoni heat convection2006

    • Author(s)
      Takaaki Nishida
    • Organizer
      Intern. Conf.on Navier-Stokes Equations
    • Place of Presentation
      RIMS,Kyoto University
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Prescribed energy problem for a singular Hamiltonian system2006

    • Author(s)
      Kazunaga Tanaka
    • Organizer
      Current Trends in Nonlinear Analysis
    • Place of Presentation
      Otranto,Italia
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Numerical verification for Hopf and pitchfork bifurcation points2005

    • Author(s)
      Tadashi Kawanago
    • Organizer
      Workshop on Numerical Analysis of Flow Problem
    • Place of Presentation
      長崎ワシントンホテル
    • Year and Date
      2005-11-22
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On numerical verification of solutions for 3-D heat convection problems2005

    • Author(s)
      M.T.Nakao
    • Organizer
      Dagstuhl Seminar on Numerical Algorithms
    • Place of Presentation
      Dagstuhl,Germany
    • Year and Date
      2005-09-26
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 流体方程式系の解析・計算機援用解析2005

    • Author(s)
      西田 孝明
    • Organizer
      日本数学会2005年度関数方程式分科会特別講演
    • Place of Presentation
      岡山大学
    • Year and Date
      2005-09-20
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Cocoon bifurcation in reversible 3-D vector fields2005

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      Dynamical System Seminar, Georgia Inst. Tech.
    • Place of Presentation
      Atlanta,USA
    • Year and Date
      2005-09-01
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Computer assisted analysis of heat convection problems2005

    • Author(s)
      Takaaki Nishida
    • Organizer
      Intern. Conf.on Appl.Math., Tamkang University
    • Place of Presentation
      Taipei,Taiwan
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Topological singular perturbation theory based on Conley index2005

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      Prof. P.Brunovsky's 70th birthday conference
    • Place of Presentation
      CIRM,Luminy,France
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary

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Published: 2005-04-01   Modified: 2016-04-21  

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