Hyperbolic operators with double characteristics, Hamilton map and Hamilton flow

• Project/Area Number: 23540199
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Osaka University
• Principal Investigator: NISHITANI Tatsuo 大阪大学, 理学(系)研究科(研究院), 教授 (80127117)
• Project Period (FY): 2011 – 2013
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000); Fiscal Year 2013: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000); Fiscal Year 2012: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2011: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: hyperbolic operator / Cauchy problem / Hamilton map / bicharacteristic / well-posedness / effectively hyperbolic / Hamilton flow / 非実効的双曲型作用素 / ハミルトン写像 / 零陪特性帯 / 初期値問題の適切性 / 基本分解 / 二次特性多様体 / スペクトル構造 / 実効的双曲型 / 初期値問題 / 適切性 / 適切生 / 実効果的双曲型

Research Abstract

Much progress has been achieved on the well-posedness of the Cauchy problem for linear hyperbolic operators with double characteristics. In particular in several transition cases from effectively hyperbolic to noneffectively hyperbolic, the relations between the spectral properties of the Hamilton map and the well-posedness conditions are clarified. I have published many such obtained results and also presented such results in several international meetings.

Research Products

• [Journal Article] On the Cauchy problem for hyperbolic operators with double characteristics, a transition case (2014)
  • Author(s): T.Nishitani
  • Journal Title: Fourier Analysis, Trends in Mathematics Pages: 311-334
  • Peer Reviewed
• [Journal Article] On the Cauchy problem for non effectively hyperbolic operators, a transition case (2013)
  • Author(s): T.Nishitani
  • Journal Title: Studies in Phase Space Analysis with Applications to PEDs Volume: vol 84 Pages: 259-290
  • Peer Reviewed
• [Journal Article] On the Cauchy problem for noneffectively hyperbolic operators, a transition case (2013)
  • Author(s): Tatsuo Nishitani
  • Journal Title: Studies in Phase Space Analysis with Applications to PDEs Volume: 84
  • Peer Reviewed
• [Journal Article] On the Cauchy problem for noneffectively hyperbolic operators, a transition case (2013)
  • Author(s): T.Nishitani
  • Journal Title: Studies in Phase Space Analysis with Applications to PDEs Volume: 84 Pages: 259-290
  • Peer Reviewed
• [Journal Article] Note on lower bounds of energy growth for solutions to wave equations (2012)
  • Author(s): S.Doi, T.Nishitani, H.Ueda
  • Journal Title: Osaka J. Math Volume: Vol 49 Pages: 1065-1085
  • NAID: 120005174161
  • Peer Reviewed
• [Journal Article] Note on lower bounds of energy growth for solutions to wave equations (2012)
  • Author(s): S.Doi, T.Nishitani, and H.Ueda
  • Journal Title: Osaka J. Math. Volume: 49 Pages: 1065-1085
  • NAID: 120005174161
  • Peer Reviewed
• [Journal Article] Some well-posed Cauchy problem for second order hyperbolic equations (2011)
  • Author(s): F.Colombini, T.Nishitani, N.Oruu, L.Pernazza
  • Journal Title: Osaka J. Math Volume: vol 48 Pages: 645-673
  • Peer Reviewed
• [Journal Article] On the Cauchy problem for noneffectively hyperbolic operators, the Gevrey 3 well-posedness (2011)
  • Author(s): E.Bernardi, T.Nishitani
  • Journal Title: J. Hyperbolic Differ. Equ Volume: vol 8 Pages: 615-650
  • Peer Reviewed
• [Journal Article] On the Cauchy problem for noneffectively hyperbolic operators, the Gevrey 4 well-posedness (2011)
  • Author(s): E.Bernardi, T.Nishitani
  • Journal Title: Kyoto J. Math Volume: Vol 51 Pages: 767-810
  • Peer Reviewed
• [Journal Article] A note on the zero free region of the Stokes multipliers for second order ordinary differential equations with cubic polynomial coefficients (2011)
  • Author(s): T.Nishitani
  • Journal Title: Funkcialaj Ekvacioj Volume: vol 54 Pages: 473-483
  • NAID: 130001271654
  • Peer Reviewed
• [Journal Article] Some well-posed Cauchy problem for second order hyperbolic equations (2011)
  • Author(s): F.Colombini, T.Nishitani, N.Orru, L.Pernazza
  • Journal Title: Osaka J.Math. Volume: 48 Pages: 645-673
  • Peer Reviewed
• [Journal Article] On the Cauchy problem for noneffectively hyperbolic operators: Gevrey 4 well-posedness (2011)
  • Author(s): E.Bernardi, T.Nishitani
  • Journal Title: Kyoto J. Math. Volume: 51 Pages: 767-810
  • Peer Reviewed
• [Presentation] Local and microlocal Cauchy problem for noneffectively hyperbolic operators (2013)
  • Author(s): T.Nishitani
  • Organizer: 9-th International ISSAC Congress
  • Place of Presentation: クラコフ, ポーランド
• [Presentation] A remark on the local and microlocal Cauchy problem for noneffectively hyperbolic operators (2013)
  • Author(s): T.Nishitani
  • Organizer: Linear and Nonlinear Hyperbolic Equations
  • Place of Presentation: ピサ, イタリア
• [Presentation] 二次特性点を持つ偏微分方程式の初期値問題 (2013)
  • Author(s): 西谷 達雄
  • Organizer: 2013年日本数学会企画特別講演
  • Place of Presentation: 京都大学
• [Presentation] A renark on the local and microlocal Cauchy problem for noneffectively hyperbolic operators (2013)
  • Author(s): Tatsuo Nishitani
  • Organizer: Linear and Nonlinear Hyperbolic Equations
  • Place of Presentation: ピサ（イタリア）
  • Invited
• [Presentation] Local and microlocal Cauchy problem for noneffectively hyperbolic operators (2013)
  • Author(s): Tatsuo Nishitani
  • Organizer: 9-th International ISSAC Congress
  • Place of Presentation: クラコフ（ポーランド）
  • Invited
• [Presentation] On the Cauchy Problem for noneffectively hyperbolic operators, a transition case (2012)
  • Author(s): T.Nishitani
  • Organizer: Fourier Analysis and Pseudo-Differential Operators
  • Place of Presentation: ヘルシンキ, フィンランド
• [Presentation] On the Cauchy problem for noneffectively hyperbolic operators--a transition case (2012)
  • Author(s): T.Nishitani
  • Organizer: Fourier Analysis and Pseudo-Differential Operators
  • Place of Presentation: Helsinki
  • Invited
• [Presentation] The Gevrey well-posedness of the Cauchy problem for noneffectively hyperbolic operators (2011)
  • Author(s): T.Nishitani
  • Organizer: 2011 NCTS Taiwan-Japan Workshop on PDEs and Geometric Analysis
  • Place of Presentation: 台北, 台湾
  • Year and Date: 2011-12-19
• [Presentation] On the Cauchy problem for noneffectively hyperbolic operators, a transition case (2011)
  • Author(s): T.Nishitani
  • Organizer: Perspectives in Phase Space Analysis on PDEs
  • Place of Presentation: Bertinoro, イタリア
  • Year and Date: 2011-09-29
• [Presentation] On the Cauchy problem for noneffectively hyperbolic operators, a transition case (2011)
  • Author(s): T.Nishitani
  • Organizer: Perspectives in Phase Space Analysis on PDE's(招待講演)
  • Place of Presentation: Bertinoro (イタリア)
• [Presentation] The Gevrey well-posedness of the Cauchy problem for noneffectively hyperbolic operators (2011)
  • Author(s): T.Nishitani
  • Organizer: 2011 NCTS Taiwan-Japan Workshop on PDEs and Geometric Analysis(招待講演)
  • Place of Presentation: 台北 (台湾)
• [Book] Hyperbolic Systems with Analytic Coefficients (2014)
  • Author(s): T.Nishitani
  • Publisher: Springer
• [Book] Cauchy problem for Noneffectively Hyperbolic Operators (2013)
  • Author(s): T.Nishitani
  • Publisher: Mathematical Society of Japan
• [Book] Cauchy Problem for Noneffectively Hyperbolic Operators (2013)
  • Author(s): Tatsuo Nishitani
  • Total Pages: 170
  • Publisher: Mathematical Society of Japan