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Comprehensive study of differential equations

Research Project

Project/Area Number 11304006
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionThe University of Tokyo

Principal Investigator

YAJIMA Kenji  The University of Tokyo , Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (80011758)

Co-Investigator(Kenkyū-buntansha) MAJIMA Hideyuki  Ochanomizu University, Faculty of Sciences, Professor, 理学部, 教授 (50111456)
IKAWA Mitsuru  Kyoto University, Graduate School of Sciences, Professor, 大学院・数理科学研究科, 教授 (80028191)
NAKAMURA Shu  The University of Tokyo , Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (50183520)
TSUTSUMI Yoshio  Tohoku University, Graduate School of Sciences, Professor, 大学院・理学研究科, 教授 (10180027)
GIGA Yoshikazu  Hokkaido University, Graduate School of Sciences, Professor, 大学院・理学研究科, 教授 (70144110)
Project Period (FY) 1999 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥27,400,000 (Direct Cost: ¥25,000,000、Indirect Cost: ¥2,400,000)
Fiscal Year 2001: ¥10,400,000 (Direct Cost: ¥8,000,000、Indirect Cost: ¥2,400,000)
Fiscal Year 2000: ¥8,000,000 (Direct Cost: ¥8,000,000)
Fiscal Year 1999: ¥9,000,000 (Direct Cost: ¥9,000,000)
KeywordsShrodinger equation / nonlinear PDE / linear PDE / nonlinear evolution equation / scattering theory / viscous solution / Navier-Stokes equation / Klein-Gordon equation / シュレーディンガー方程式 / シュレーディンガー作用素 / 超漸近解析 / 波動方程式 / 漸近安定性 / 偏微分方程式 / 初期値問題 / ランダムな磁場 / ショック解 / 超局所解析 / 代数解析 / スペクトル理論 / 非線型波動方程式 / ナビエ・ストークス方程式 / 超幾何関数
Research Abstract

We carried out an comprehensive study on linear and nonlinear partial and ordinary differential equations and obtained among others the following results:
1. Kenji Yajima and Shu Nakamura studied Schrodinger equations and obtained (1) the L^p-boundedness of wave operators of scattering and (2) the stability under subquadratic perturbations of the smooth and boundedness of fundamental solution; (3) clarified the relation between the local decay and the spectrum of Floquet operator for time periodic system; (4) constructed general theory of tunnenling in phase space and gave its applications; (5) defined the integrated density of states for random operators and proved Wegner estimates and the Lifschitz singularities.
2. Kiyoomi Kataoka studied the general theory of linear PDE and (1) gave an elementary definition of the holomorphic solutions complex of ε^R_x-modules and (2) reduced the branching of the singularities for Fuchsian elliptic boundary problems to that of the continuation of ODE.
3. Yoshikazu Giga studied nonlinear PDE and (1) gave a way to numerically compute the viscous solution of first order nonlinear PDE, (2) defined the proper visocous solutions and proved the existence and the uniquess of solutions and (3) proved the existence of solution of Navier-Stokcs equation with non-deaying initial conditions.
4. Yoshio Tsutsumi proved (1) the well-posedness of nonlinearly couples wave equations with different speeds, (2) the stability of constant solutions of nonlinear massive Klein-Gordon equation.
5. Hideyuki Majima produced new fundamental theorem on the asymptotic expansions from the point of view the super-asymptotic analysis.
6. Mitsuru Ikawa studied the scattering theory of wave equation by several convex bodies and obtained a precise asymptotic, formula for the poles of scattering matrix.

Report

(4 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • Research Products

    (42 results)

All Other

All Publications (42 results)

  • [Publications] Yajima, K: "L^p-boundedness of wave operators for two Dimensional Schrodinger operators"Commun. Math. Physics. 208. 125-152 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Martinez, A.: "On the fundamental solution of semi-classical schrodinger equations at resenaut times"Commun. Math. Physics. 216. 357-373 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Nakamura, S.: "Lifschitz tail for Schrodinger Operators Lvitti randam Magnetic field"Commun. Math. Physics. 214. 565-572 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kataoka, K.: "Formal symbol type solutions of Fuchsion microdifferential equations"J. Math. Sci. University of Tokyo. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Fukutani, S.: "Special polynomials cesscciated with the rational solutions and the Hirota bilinear relations…"Nagoya Math. Journal. 159. 179-200 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Giga, M.-H.: "Generalized motion by nonlocalcurvature in the plane"Arch. Rat. Mech. Anal. 159. 259-333 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Jensen, Arne: "Ripples in Mathematics-The discrete wavelet transform"Springer-Verlag. 246 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] YAJIMA, Kenji: "L^p-boundedness of wave operators for two dimensional Schrodinger operaters"Commun. Math. Phys.. 208. 125-152 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Martinez, Andre and YAJIMA, Kenji: "On the fundamental solutions of semi-classical Schrodinger equations at resonant times"Commun. Math. Phys.. 216. 357-373 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] NAKAMURA, Shu: "Lifschitz tail for Schrodinger operators with random magnetic field"Commun. Math. Phys.. 214. 565-572 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] KATAOKA, Kiyoomi: "Formal symbol type solutions of Fuchsian micro-differential equations"J. Math. Sci. Univ. Tokyo. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Fukutani, S.: "Special polynomials associated with the rational solutions and the Hirota bilinear relations of the 2nd and the 4th Painleve equations"Nagoya Math. J.. 159. 179-200 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] GIGA, M.-H.: "Generalizeed motion by nonlocal curvature in the plane"Arch. Rat. Mech. Anol.. 159. 259-333 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] JENSEN, A.: "Ripples in Mathematics - The discrete wavelet transform -"Springer-Verlag (Heidelberg-New York-Tokyo). 246 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Andre Martinez: "On the fundamental solution of semi-classical schrodinger equations at rescnant times"Commun. Math. physics. 216. 357-373 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Kenji Yajima: "On the behavior at infinity of the fundamental solution of time dependent Schrodinger equations"Reviews in Mathematical Physics. 13. 891-920 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Kenji Yajima: "Smoothing property for Schrodinger equations with potentials superquadrare at infinity"Commun. Mathenatical Physics. 221. 573-590 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Arne Jensen: "A remark on L^p-boundedness of wave operators for two dimensional Schrodinger operators"Commun. Mathenatical Physics. (to appear). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Shu Nakamura: "A remark on the Direchlet-Newman clecoupling and the integrates dencity of Stated"Journal of Functional Analysis. 179. 136-152 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Jean-Michel Combes: "The L^p-theary of special shift-functions the wegner estimate, and the integrated density of states"Commun. Math. physics. 218. 113-130 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Andre Martinez: "Phase space tunneling and multistate scatteving"Journal of Functional Analsis. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] Yoshikazu Giga: "A level set approach to semi-continuous viscosity solutions for Caucliy problems"Commun. Partial Differential Equations. 26. 813-839 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Yoshikazu Giga: "On the motion of singular interfucial surface"Japanese S Ind. Appl. Math.. 18. 231-248 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Miho Giga: "Genevclized motion by non-local curvature in the plane"Arch. Rational Mecl. Anol. 159. 295-333 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Miho Giga: "Very Singular diffusion equations"Advanced Studies in Pure Math.. 31. 93-125 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Yoshikazu Giga: "Remarks on the uniqueness of bounded. solutions of the Navier States equations"Non-linear Analysis. 47. 4151-4156 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] J. Escher: "On a limiting motion and self-intersections of curves moved by the intermediate surface diffusion flow"Non-linear Analysis. 47. 3717-3728 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Yoshikazu Giga: "Global existence of two dimensional Navies-stokes flow with nondecaying initial velocity"J. Math. Fluid Mech.. 3. 302-315 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 堤 誉志雄: "非線型波動方程式の解の大城有在と爆発"「数学」(日本数学会). 53. 139-156 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] GRAFFI Sandro: "Absolute Continuity of the Floquet Spectrum for a Nonlinearly Forced Harmonic Oscillator"Communications in Mathematical Physics. 215. 245-250 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] MARTINEZ Andre: "On the fundamental Solution of Semi-classical Schrodinger equations at resonant times"Communications in Mathematical Physics. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] ARTBAZAR,Galtlayer: "The L'-continuity of wave operators for one dimensional Schrodinger Operators"Journal of Mathematical Science,Univ.of Tokyo. 7・2. 221-240 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] NAKAMURA Shu: "Lifshitz tail for Schrodinger Operators with random magnetic field"Communications in Mathematical Physics. 214. 565-572 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] OZAWA Tohru: "On the coupled system of nonlinear wave equations with different propagation speeds"Banach Center Publications. 52. 181-188 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] MAJIMA Hideyuki: "Quadratic relations for confluent hypergeometric functions"Tohoku Mathematical Journal. 52・4. 489-513 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Kenji Yajima: "L^P-boundedness of wave operators for two dimensional Schrodinger equations"Communications in Mathematical Phyoics. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] Kenji Yajima: "On the behavior at infinity of the fundamental solution for time-dependent Schridinger equations"Reviews in Mathematical Phyoics. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] Shu Nakamura: "Lifshitz tail for 2D discrete Schrodinger operator with random magnetic field"Ann. Henri Poincare. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] Yishikazu Ciga: "On the Cauchy problem for the Navier-Stokes equations with nondecaying initial data"Quaderni di Mathmatica. 4. 27-67 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] A. de Bouard: "White noise driven Korteweg-de Vries equation"Journal of Functional Analysis. 169. 532-558 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Hideyuki Majima: "Quadratic relations for confluent hypergeometric functions"Tohoku Mathematical Journal. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] 儀我 美一: "非線型編微分方程式-解の漸立挙動と自亡相似解"共立出版. 300 (1990)

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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