geometric structure of nonlinearity and singularity of solutions for wave equations

2013 Fiscal Year Final Research Report

• Project/Area Number: 22740088
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Nagoya University
• Principal Investigator: KOTARO Tsugawa 名古屋大学, 多元数理科学研究科, 准教授 (70402451)
• Project Period (FY): 2010-04-01 – 2014-03-31
• Keywords: 分散型方程式 / 非線形 / 偏微分方程式 / 慨周期関数 / 準周期関数 / 適切性 / シュレディンガー方程式 / ディラック方程式

Research Abstract

We studied the local and global well-posedness of the Cauchy problem for nonlinear dipersive equations and hyperbolic equations by the harmonic analysis. We improved the known results for a quadratic nonlinear Schrodinger equation and obtained the well-posedness result for low regularity data. The result is also applied to good Boussinesq equation. We showed the local well-posedness for nonlinear Dirac equation and Dirac-Klein-Gordon equation by using the property of null form. We also studied the Cauchy problem for the KdV equations with quasi periodic data.

Research Products

• [Journal Article] Local well-posedness of the KdV Equation with quasi-periodic initial data (2012)
  • Author(s): K. Tsugawa
  • Journal Title: SIAM J. Math. Anal Volume: 44-5 Pages: 3412-3428
  • Peer Reviewed
• [Journal Article] Well-posedness for nonlinear Dirac equations in one dimension (2010)
  • Author(s): S. Machihara, K. Nakanishi and K. Tsugawa
  • Journal Title: Kyoto J. Math Volume: 50, no.2 Pages: 403-451
  • Peer Reviewed
• [Journal Article] Local well-posedness for quadratic nonlinear Schrodinger equations and the "good" Boussinesq equation (2010)
  • Author(s): N. Kishimoto and K. Tsugawa
  • Journal Title: Differential Integral Equations Volume: 23, no.5-6 Pages: 463-493
  • Peer Reviewed
• [Presentation] Unconditional well-posedness of the fifth order modified KdV equation with perio dic boundary condition (2012)
  • Author(s): K. Tsugawa
  • Organizer: 研究集会「Workshop on Nonlinear Dispersive PDEs」
  • Place of Presentation: 東北大学大学, 仙台
  • Year and Date: 2012-08-29
• [Presentation] Global well posedness of a stochastic KdV equation (2012)
  • Author(s): K. Tsugawa
  • Organizer: 短期共同研究「非線形分散型方程式における最近の進展」
  • Place of Presentation: 京都大学数理解析研究所, 京都
  • Year and Date: 2012-05-23
• [Presentation] Well-posedness of the KdV equation with almost periodic initial data (2012)
  • Author(s): K. Tsugawa
  • Organizer: 京都大学 NLPDE セミナー
  • Place of Presentation: 京都大学, 京都
  • Year and Date: 2012-05-18
• [Presentation] Well-posedness of the KdV equation with almost periodic initial data (2012)
  • Author(s): K. Tsugawa
  • Organizer: 名古屋微分方程式セミナー
  • Place of Presentation: 名古屋大学, 名古屋
  • Year and Date: 2012-04-16
• [Presentation] Well-posedness of the KdV equation with almost periodic initial data (2012)
  • Author(s): K. Tsugawa
  • Organizer: 非線形偏微分方程式研究会
  • Place of Presentation: 福岡ガーデンバレス, 福岡
  • Year and Date: 2012-03-17
• [Presentation] Local well-posedness of the KdV equation with almost periodic initial data (2011)
  • Author(s): K. Tsugawa
  • Organizer: Analysis seminar
  • Place of Presentation: Princeton University, Princeton
  • Year and Date: 2011-11-14
• [Presentation] Local well-posedness of the KdV equation with almost periodic initial data (2011)
  • Author(s): K. Tsugawa
  • Organizer: Analysis and Applied Math Seminar
  • Place of Presentation: University of Toronto, Toronto
  • Year and Date: 2011-10-14
• [Presentation] Local well-posedness of the KdV equation with almost periodic initial data (2011)
  • Author(s): K. Tsugawa
  • Organizer: the 19th Workshop on Differential Equations and its Applications
  • Place of Presentation: National Cheng Kung University, Tainan, Taiwan
  • Year and Date: 2011-01-15