Gevrey strong hyperbolicity and the structure of Hamilton map and flow

• Project/Area Number: 26400167
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Osaka University
• Principal Investigator: Nishitani Tatsuo 大阪大学, その他部局等, 名誉教授 (80127117)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2015: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2014: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: Gevrey 強双曲性指数 / 初期値問題 / Gevrey クラス / 適切性 / 伝播錐 / 横断的 / 余接空間 / Gevrey強双曲型 / 局所化 / 横断的強双曲系 / Gevrey適切性 / 包合的特性多様体 / シンプレクティック特性多様体 / 局所化系 / 強双曲系 / Gevrey強双曲性指数 / 法束上で狭義双曲型 / 包合的 / Gevrey強双曲性 / Bronshteinの定理

Outline of Final Research Achievements

Several fundamental results on the strong Gevrey hyperbolicity index have been obtained. In particular, for homogeneous hyperbolic differential operators of order m of which characteristic set is a smooth manifold, the Cauchy problem is Gevrey m/(m-2) well posed for any lower order term if the localization is strictly hyperbolic polynomial on the conormal space and the propagation cone is transverse to the characteristic manifold.

Research Products

• [Journal Article] Note on strongly hyperbolic systems with involute characteristics (2018)
  • Author(s): G.Metivier and T.Nishitani
  • Journal Title: Kyoto J.Math. Volume: 印刷中
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] On the Cauchy problem for differential operators with double characteristics, A transition from non-effective to effective characteristics (2018)
  • Author(s): T.Nishitani
  • Journal Title: Publ. RIMS Koyto Uiniv. Volume: 54 Pages: 317-349
  • Peer Reviewed / Open Access
• [Journal Article] On the Gevrey strong hyperbolicity (2017)
  • Author(s): T.Nishitani
  • Journal Title: Osaka J. Math. Volume: 54 Pages: 383-408
  • NAID: 120006312693
  • Peer Reviewed / Open Access
• [Journal Article] Note on strongly hyperbolic systems with involute characteristics (2017)
  • Author(s): G.Metivier and T.Nishitani
  • Journal Title: Kyoto J. Math. Volume: 印刷中
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] A simple proof of the existence of tangent bicharacteristics for noneffectively hyperbolic operators (2015)
  • Author(s): T.Nishitani
  • Journal Title: Kyoto Journal of Mathematics Volume: 55 Pages: 281-297
  • Peer Reviewed / Open Access
• [Journal Article] Counterexamples to $ C^{\infty}$ well posedness for some hyperbolic operators with triple characteristics (2015)
  • Author(s): E.Bernardi and T.Nishitani
  • Journal Title: Proc. Japan Acad. Ser A. Volume: 91 Pages: 19-24
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Parametric resonance in wave map (2014)
  • Author(s): T.Nishitani and K.Yagdjian
  • Journal Title: Funkcialaj Ekvacioj Volume: 57 Pages: 351-374
  • NAID: 130004927739
  • Peer Reviewed / Open Access
• [Journal Article] Local and micro local Cauchy problem for non-effectively hyperbolic operators (2014)
  • Author(s): T.Nishitani
  • Journal Title: J.Hyperbolic Differ. Equ. Volume: 11 Pages: 185-213
  • Peer Reviewed
• [Journal Article] 二次特性的双曲型偏微分法廷式の初期値問題の適切性 (2014)
  • Author(s): 西谷達雄
  • Journal Title: 数学 Volume: 66 Pages: 366-391
  • Peer Reviewed
• [Presentation] On the Cauchy problem for $D_t^2-b(t)D_xa(x)D_x$ (2017)
  • Author(s): T.Nishitani
  • Organizer: Simposio di Analisi Matematica in occasione dei 70 anni di Ferruccio Colombini
  • Int'l Joint Research / Invited
• [Presentation] On the Gevrey strong hyperbolicity (2016)
  • Author(s): T.Nishitani
  • Organizer: Seminar on PDE, University of Pisa
  • Place of Presentation: Department of Math. University of Pisa (イタリア)
  • Invited
• [Presentation] On the Gevrey strong hyperbolicity (2015)
  • Author(s): 西谷達雄
  • Organizer: 第２２回超局所解析と古典解析
  • Place of Presentation: 富山県民会館
  • Year and Date: 2015-12-05
  • Invited
• [Presentation] On the Gevrey strong hyperbolicity (2015)
  • Author(s): Tatsuo Nishitani
  • Organizer: Seminaire sur equations hyperboliques et physique mathematique
  • Place of Presentation: IHP 研究所 パリ
  • Year and Date: 2015-06-02
  • Invited
• [Presentation] Weyl-Hormander calculus in the Gevrey classes (2014)
  • Author(s): T.Nishitani
  • Organizer: Global Properties of PDE's on Manifolds
  • Place of Presentation: Cagliari (Italy)
  • Year and Date: 2014-09-17 – 2014-09-19
  • Invited
• [Book] Cauchy problem for differential operators with double characteristics (2017)
  • Author(s): T.Nishitani
  • Total Pages: 211
  • Publisher: Springer
  • ISBN: 9783319676111
• [Book] 線形双曲型偏微分方程式 (2015)
  • Author(s): 西谷達雄
  • Total Pages: 286
  • Publisher: 朝倉書店
• [Book] Hyperbolic systems with analytic coefficients (2014)
  • Author(s): T.Nishitani
  • Total Pages: 237
  • Publisher: Springer