Cauchy problem of nonlinear dispersive equations

• Project/Area Number: 25400158
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Nagoya University
• Principal Investigator: Tsugawa Kotaro 名古屋大学, 多元数理科学研究科, 准教授 (70402451)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000); Fiscal Year 2016: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000); Fiscal Year 2015: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2013: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: 分散型方程式 / 適切性 / 初期値問題 / 調和解析 / シュレディンガー方程式 / KdV方程式 / 非線形 / Schrodinger / KdV / Zakharov

Outline of Final Research Achievements

First, we proved the global well-posedness and asymptotic behavior of the Cauchy problem of Zakharov system in higher dimensions. We used the theory of Up, Vp space to prove it. The main difficulty comes from the difference of the properties of the linear equations. To overcome it, we used the intersection space of V2 space and Lebesgue space related to the end-point Strichartz estimate. Second, we categolized the fifth order KdV type equations. By using the normal form reduction, energy method and Bona-Smith approximation, we proved that the local well-posedness and parabolic smoothing effect.

Research Products

• [Journal Article] Local well-posedness and parabolic smoothing effect of fifth order dispersive equations on the torus (2017)
  • Author(s): K. Tsugawa
  • Journal Title: RIMS Kokyuroku Bessatsu, Volume: 印刷中
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Scattering and well-posedness for the Zakharov system at a critical space in four and more spatial dimensions, (2017)
  • Author(s): I. Kato and K. Tsugawa,
  • Journal Title: Differential and Integral equations Volume: 印刷中
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] Local well-posedness of derivative nonlinear Schr¨odinger equations on the torus (2017)
  • Author(s): K. Tsugawa
  • Organizer: 研究集会「Critical Exponents and Nonlinear Evolution Equation」,
  • Place of Presentation: 東京理科
  • Int'l Joint Research / Invited
• [Presentation] Local well-posedness and parabolic smoothing efect of semilinear fifth order dispersive equations on the torus (2016)
  • Author(s): Kotaro TSUGAWA
  • Organizer: Harmonic Analysis, Geometric Analysis and PDE Workshop
  • Place of Presentation: 埼玉大学
  • Year and Date: 2016-03-03
• [Presentation] Parabolic smoothing effect and local well-posedness of nonlinear dispersive equations on the torus (2016)
  • Author(s): Kotaro TSUGAWA
  • Organizer: 波動セミナー(ワークショップ)
  • Place of Presentation: 北海道大学
  • Year and Date: 2016-02-08
• [Presentation] Parabolic smoothing effect and local well-posedness of fifth order dispersive equations on the torus (2016)
  • Author(s): Kotaro TSUGAWA
  • Organizer: Parabolic smoothing effect and local well-posedness of fifth order dispersive equations on the torus
  • Place of Presentation: 愛媛大学
  • Year and Date: 2016-02-06
• [Presentation] Parabolic smoothing effect and local well-posedness of semilinear fifth order dispersive equations on the torus (2016)
  • Author(s): K. Tsugawa
  • Organizer: 2016 Taiwan-Japan Workshop on Dispersion, Navier Stokes, Kinetic, and Inverse Problems
  • Place of Presentation: 国立成功大学(台湾)
  • Int'l Joint Research / Invited
• [Presentation] Local well-posedness and parabolic smoothing effect of semilinear dispersive equations (2016)
  • Author(s): K. Tsugawa
  • Organizer: 第３回神 楽坂非線形波動研究会
  • Place of Presentation: 東京理科大
  • Invited
• [Presentation] 5階の分散型方程式の局所適切性と放物型平滑化効果 (2015)
  • Author(s): Kotaro TSUGAWA
  • Organizer: 京都大学数理解析研究所談話会
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2015-11-11
• [Presentation] Refined energy estimate and local well-posedness of fifth order dispersive equations on the torus (2015)
  • Author(s): Kotaro TSUGAWA
  • Organizer: 熊本大学応用解析セミナー
  • Place of Presentation: 熊本大学
  • Year and Date: 2015-08-01
• [Presentation] Refined energy estimate and local well-posedness of fifth order dispersive equations on the torus (2015)
  • Author(s): Kotaro TSUGAWA
  • Organizer: RIMS研究集会「調和解析と非線形偏微分方程式
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2015-07-06
• [Presentation] Refined energy estimate and local well-posedness of fifth order dispersive equations on the torus (2015)
  • Author(s): 津川 光太郎
  • Organizer: 研究集会「Nonlinear Evolution Equations and Related Topics」
  • Place of Presentation: 京都大学
  • Year and Date: 2015-03-10
  • Invited
• [Presentation] Refined energy estimate and local well-posedness of fifth order dispersive equations on the torus (2014)
  • Author(s): 津川 光太郎
  • Organizer: 九州関数方程式セミナー
  • Place of Presentation: 九州大学
  • Year and Date: 2014-12-12
  • Invited
• [Presentation] Refined energy estimate and local well-posedness of fifth order dispersive equations on the torus (2014)
  • Author(s): 津川 光太郎
  • Organizer: 研究集会「The 12th Linear and Nonlinear Waves」
  • Place of Presentation: ピアザ淡海滋賀県立県民交流センター
  • Year and Date: 2014-11-12
  • Invited
• [Presentation] Refined energy estimate and local well-posedness of fifth order dispersive equations on the torus (2014)
  • Author(s): 津川 光太郎
  • Organizer: 第1回神楽坂非線形波動研究会
  • Place of Presentation: 東京理科大
  • Year and Date: 2014-10-23
  • Invited
• [Presentation] 5階の非線形分散型方程式の局所適切性 (2014)
  • Author(s): 津川 光太郎
  • Organizer: 日本数学会秋季総合分科会特別講演
  • Place of Presentation: 広島大学
  • Year and Date: 2014-09-27
  • Invited
• [Presentation] A cancellation property and the well-posedness of fifth order KdV type equations on the torus (2014)
  • Author(s): K. Tsugawa
  • Organizer: 研究集会「Critical Exponents and Nonlinear Evolution Equation」
  • Place of Presentation: 東京理科大学神楽坂キャンパ ス
• [Presentation] Well-posedness of the KdV equation with almost periodic initial data (2013)
  • Author(s): K. Tsugawa
  • Organizer: 第9 回非線型の諸問題
  • Place of Presentation: 高知大学
• [Presentation] On the normal form reduction (2013)
  • Author(s): K. Tsugawa
  • Organizer: 2013 Participating School in Analysis of PDE\Global theory of nonlinear dispersive equations"
  • Place of Presentation: Jeju National University Park
• [Funded Workshop] The 9th Nagoya Workshop on Differential Equations (2017)
  • Place of Presentation: 名古屋大学
  • Year and Date: 2017-03-21
• [Funded Workshop] Nonlinear Wave and Dispersive Equations, Kyoto 2016 (2016)
  • Place of Presentation: 京都大学
  • Year and Date: 2016-09-06
• [Funded Workshop] 第８回 名古屋微分方程式研究集会 (2016)
  • Place of Presentation: 名古屋大学
  • Year and Date: 2016-02-23