A study on the generalized spectral theory and its applications to evolution equations

• Project/Area Number: 25800081
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Kyushu University
• Principal Investigator: Chiba Hayato 九州大学, マス・フォア・インダストリ研究所, 准教授 (70571793)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: スペクトル理論 / 蔵本モデル / 一般化スペクトル理論 / 一般化スペクトル

Outline of Final Research Achievements

In this project, the generalized spectral theory, which is a generalization of the standard spectral theory, is developed. By using this theory, we can estimate an asymptotic behavior of solutions of evolution equations whose linear part has a continuous spectrum.
This theory is applied to the Kuramoto model and the Schrodinger equation. In particular, for the Kuramoto model, the Kuramoto conjecture is proved.

Research Products

• [Journal Article] The first, second and fourth Painleve equations on weighted projective spaces (2016)
  • Author(s): H. Chiba
  • Journal Title: Journal of Differential Equations Volume: 260 Pages: 1263-1313
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Kovalevskaya exponents and the space of initial conditions of a quasi-homogeneous vector field (2015)
  • Author(s): H. Chiba
  • Journal Title: Journal of Differential Equations Volume: 259 Pages: 7681-7716
  • Acknowledgement Compliant
• [Journal Article] A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions (2015)
  • Author(s): H. Chiba
  • Journal Title: Advances in Mathematics Volume: 273 Pages: 324-379
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Continuous limit of the moments system for the globally coupled phase oscillators, (2013)
  • Author(s): H.Chiba
  • Journal Title: Discrete and Continuous Dynamical Systems Series A Volume: 33 Issue: 5 Pages: 1891-1903
  • DOI: 10.3934/dcds.2013.33.1891
  • Peer Reviewed
• [Journal Article] Reduction of weakly nonlinear parabolic partial differential equations (2013)
  • Author(s): H. Chiba
  • Journal Title: Journal of Mathematical Physics Volume: 54 Issue: 10 Pages: 101501-101501
  • DOI: 10.1063/1.4824014
  • Peer Reviewed
• [Presentation] A Spectral Theory of Linear Operators on a Gelfand Triplet and its Application to the Dynamics of Coupled Oscillators (2016)
  • Author(s): H. Chiba
  • Organizer: Patterns and Waves 2016
  • Place of Presentation: 北海道大学
  • Year and Date: 2016-08-01
  • Int'l Joint Research / Invited
• [Presentation] A Spectral Theory of Linear Operators on a Gelfand Triplet and its Application to the Dynamics of Coupled Oscillators (2016)
  • Author(s): H. Chiba
  • Organizer: Asian Mathematical Conference 2016
  • Place of Presentation: バリ島
  • Year and Date: 2016-07-25
  • Int'l Joint Research / Invited
• [Presentation] Generalized spectral theory and a proof of the Kuramoto conjecture (2015)
  • Author(s): H. Chiba
  • Organizer: Dynamics of Coupled Oscillators: 40 years of the Kuramoto Model
  • Place of Presentation: Dresden
  • Year and Date: 2015-07-30
  • Int'l Joint Research / Invited
• [Presentation] A Spectral Theory of Linear Operators on a Gelfand Triplet and its Application to the Dynamics of Coupled Oscillators (2014)
  • Author(s): H. Chiba
  • Organizer: SIAM conference on Nonlinear Waves and Coherent Structures
  • Place of Presentation: ケンブリッジ
  • Year and Date: 2014-08-11
  • Invited
• [Presentation] A Spectral Theory of Linear Operators on a Gelfand Triplet and its Application to Coupled Oscillators (2014)
  • Author(s): H. Chiba
  • Organizer: Dynamical Systems, Differential Equations and Applications
  • Place of Presentation: マドリッド
  • Year and Date: 2014-07-07
  • Invited
• [Presentation] A Spectral Theory of Linear Operators on a GelfandTriplet and its Application to Coupled Oscillators (2013)
  • Author(s): H. Chiba
  • Organizer: Joint US-Japan workshop for Young Researchers on Interactions among Localized Patterns in Dissipative Systems
  • Place of Presentation: ミネソタ大学
  • Invited