An Investigation of symmetries and structure of solutions for nonlinear dispersive equation

• Project/Area Number: 21684003
• Research Category: Grant-in-Aid for Young Scientists (A)
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Hokkaido University
• Principal Investigator: TAKAOKA Hideo 北海道大学, 大学院・理学研究院, 教授 (10322794)
• Project Period (FY): 2009 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥19,500,000 (Direct Cost: ¥15,000,000、Indirect Cost: ¥4,500,000); Fiscal Year 2012: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2011: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000); Fiscal Year 2010: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2009: ¥5,200,000 (Direct Cost: ¥4,000,000、Indirect Cost: ¥1,200,000)
• Keywords: 非線形分散型 / 適切性 / 初期値問題 / 非線形波動 / 大域存在 / 非線形分散型方程式 / 非線形分散型方程

Research Abstract

In this study, I have developed the local and global well-posedness for the initial value problem related to a class of nonlinear dispersive equations, such as the KdV equations and the nonlinear Schrodinger equations. I investigated the propagation of singularities of solutions using Fourier analysis, and obtained the local in time unique existence of solutions to the modified KdV equation.
Moreover, I proved the global energy estimate for the nonlinear Schrodinger equations, and proved global in time unique existence of solutions and invariance of the associated Gibbs measure to the derivative nonlinear Schrodinger equation.

Research Products

• [Journal Article] Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation (2010)
  • Author(s): J. Colliander, M. Keel, G. Staffilani, H .Takaoka,T.Tao
  • Journal Title: Invent. Math Volume: 181 Pages: 39-113
  • Peer Reviewed
• [Journal Article] Local well-posedness in low regularity of the mKdV equation with periodic boundary condition (2010)
  • Author(s): K.Nakanishi,H.Takaoka,Y.Tsutsumi
  • Journal Title: Discrete Contin. Dyn. Syst. Volume: 28 Pages: 1635-1654
  • Peer Reviewed
• [Journal Article] Local well-posedness in low regularity of the mKdV equation with periodic boundary condition (2010)
  • Author(s): K.Nakanishi, H.Takaoka, Y.Tsutsumi
  • Journal Title: Discrete Contin.Dyn.Syst. Volume: 28 Pages: 1635-1654
  • Peer Reviewed
• [Journal Article] Transfer of energy to high frequencies in the cubic defocusing nonlinear Schr\"{o}dinger equation (2010)
  • Author(s): J.Colliander, M.Keel, G.Staffilani, H.Takaoka, T.Tao
  • Journal Title: Invent.Math. Volume: 181 Pages: 39-113
  • Peer Reviewed
• [Journal Article] Stability in H1/2 of the sum of K solitons for the Benjamin-Ono equation (2009)
  • Author(s): S. Gustafson, H.Takaoka and T.-P. Tsai
  • Journal Title: J. Math. Phys. Volume: 50 Pages: 14-14
  • Peer Reviewed
• [Journal Article] Unique local existence of solution in low regularity space of the Cauchy problem for the mKdV equation with periodic boundary condition (2009)
  • Author(s): K. Nakanishi, H.Takaoka and Y. Tsutsumi
  • Journal Title: Seminaire: Equations aux Derivees Partielles Volume: 2007-2008 Pages: 7-7
  • Peer Reviewed
• [Journal Article] Stability in H^<1/2> of the sum of K solitons for the Benjamin-Ono equation (2009)
  • Author(s): S.Gustafson, H.Takaoka, T.-P.Tsai
  • Journal Title: J.Math.Phys. 50 Pages: 14-14
  • Peer Reviewed
• [Journal Article] Unique local existence of solution in low regularity space of the Cauchy problem for the mKdV equation with periodic boundary condition (2009)
  • Author(s): K.Nakanishi, H.Takaoka, Y.Tsutsumi
  • Journal Title: Seminaire : Equations aux Derivees Partielles, 2007-2008 17 Pages: 7-7
  • Peer Reviewed
• [Presentation] Almost sure global well-posedness for the periodic derivative NLS (2013)
  • Author(s): 高岡秀夫
  • Organizer: 第5回名古屋微分方程式研究集会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2013-03-12
• [Presentation] Almost sure global well-posedness for the periodic derivative NLS (2013)
  • Author(s): 高岡秀夫
  • Organizer: 名古屋微分方程式研究集会
  • Place of Presentation: 名古屋大学（名古屋市）
  • Invited
• [Presentation] Almost sure globalwell-posedness for the periodic derivative NLS equation (2012)
  • Author(s): 高岡秀夫
  • Organizer: Nonlinear Dispersive Equations and Fluid Mechanics
  • Place of Presentation: 東北大学
  • Year and Date: 2012-12-12
• [Presentation] Low regularitywell-posedness and applications to almost sure global well-posedness of the periodic derivative NLS equation (2012)
  • Author(s): 高岡秀夫
  • Organizer: 弘前非線形方程式研究
  • Place of Presentation: 弘前大学
  • Year and Date: 2012-11-03
• [Presentation] Low regularitywell-posedness theory and applications to almost sure global well-posedness of theperiodic derivative NLS equation: Part 2 (2012)
  • Author(s): 高岡秀夫
  • Organizer: Topics on Nonlinear Partial Differential Equations
  • Place of Presentation: 浦項工科大学校(韓国)
  • Year and Date: 2012-09-27
• [Presentation] Low regularitywell-posedness theory and applications to almost sure global well-posedness of theperiodic derivative NLS equation: Part 1 (2012)
  • Author(s): 高岡秀夫
  • Organizer: Topics on Nonlinear Partial Differential Equations
  • Place of Presentation: 浦項工科大学校(韓国)
  • Year and Date: 2012-09-26
• [Presentation] On the derivative nonlinear Schrodinger equation with the periodic boundary (2012)
  • Author(s): 高岡秀夫
  • Organizer: Nonlinear Wave and Dispersive Equation
  • Place of Presentation: 京都大学
  • Year and Date: 2012-03-07
• [Presentation] On the derivative nonlinear Schrodinger equation with the periodic boundary condition (2012)
  • Author(s): 高岡秀夫
  • Organizer: Nonlinear Wave and Dispersive Equation
  • Place of Presentation: 京都大学(京都市)(招待講演)
  • Year and Date: 2012-03-07
• [Presentation] Low regularity well-posedness theory and applications to almost sure global well-posedness of the periodic derivative NLS equation: Part 1, Part 2 (2012)
  • Author(s): 高岡秀夫
  • Organizer: Topics on Nonlinear Partial Differential Equations
  • Place of Presentation: 浦項工科大学校（韓国）
  • Invited
• [Presentation] Low regularity well-posedness and applications to almost sure global well-posedness of the periodic derivative NLS equation (2012)
  • Author(s): 高岡秀夫
  • Organizer: 弘前非線形方程式研究会
  • Place of Presentation: 弘前大学（弘前市）
  • Invited
• [Presentation] Almost sure global well-posedness for the periodic derivative NLS equation (2012)
  • Author(s): 高岡秀夫
  • Organizer: Nonlinear Dispersive Equations and Fluid Mechanics
  • Place of Presentation: 東北大学（仙台市）
  • Invited
• [Presentation] Low regularity solutions to the derivative nonlinear Schrodinger equation (2011)
  • Author(s): 高岡秀夫
  • Organizer: 熊本大学応用解析セミナー
  • Place of Presentation: 熊本大学
  • Year and Date: 2011-12-10
• [Presentation] Low regularity solutions to the derivative nonlinear Schrodinger equation (2011)
  • Author(s): 高岡秀夫
  • Organizer: 熊本大学応用解析セミナー
  • Place of Presentation: 熊本大学(熊本市)(招待講演)
  • Year and Date: 2011-12-10
• [Presentation] Weak solutions for the derivative nonlinear Schrodinger equations (2011)
  • Author(s): 高岡秀夫
  • Organizer: The 7th HU and SNU symposium on Mathematics
  • Place of Presentation: ソウル大学(韓国)
  • Year and Date: 2011-11-16
• [Presentation] Weak solutions for the derivative nonlinear Schrodinger equations (2011)
  • Author(s): 高岡秀夫
  • Organizer: The 7th HU and SNU symposium on Mathematics
  • Place of Presentation: ソウル大学(韓国)(招待講演)
  • Year and Date: 2011-11-16
• [Presentation] A priori estimates and weak solutions for the derivative nonlinear Schrodinger equations on torus (2011)
  • Author(s): 高岡秀夫
  • Organizer: Workshop on differential equations and its applications
  • Place of Presentation: National Cheng Kung University(台湾)
  • Year and Date: 2011-01-15
• [Presentation] A priori estimates and weak solutions for the derivative nonlinear Schr￥"{o}dinger equations on torus (2011)
  • Author(s): H.Takaoka
  • Organizer: Workshop on differential equations and its applications
  • Place of Presentation: National Cheng Kung University, Taiwan(招待講演)
  • Year and Date: 2011-01-15
• [Presentation] Well-posedness of the derivative nonlinear Schrodinger equation (2010)
  • Author(s): 高岡秀夫
  • Organizer: NLPDE seminar
  • Place of Presentation: 京都大学
  • Year and Date: 2010-11-26
• [Presentation] Well-posedness of the derivative nonlinear Sch￥"{o}dinger equation (2010)
  • Author(s): H.Takaoka
  • Organizer: NLPDE seminar
  • Place of Presentation: 京都大学(招待講演)
  • Year and Date: 2010-11-26
• [Presentation] 非線形シュレディンガー方程式の大域解と解のソボレフノルム評価 (2010)
  • Author(s): 高岡秀夫
  • Organizer: 偏微分方程式セミナー
  • Place of Presentation: 北海道大学
  • Year and Date: 2010-04-19
• [Presentation] 非線形シュレディンガー方程式の大域解と解のソボレフノルム評価 (2010)
  • Author(s): H.Takaoka
  • Organizer: 偏微分方程式セミナー
  • Place of Presentation: 北海道大学(招待講演)
  • Year and Date: 2010-04-19
• [Presentation] 非線形シュレディンガー方程式に対するエネルギー転換と解のエネルギー評価 (2010)
  • Author(s): 高岡秀夫
  • Organizer: 解析セミナー
  • Place of Presentation: 神戸大学
  • Year and Date: 2010-02-15
• [Presentation] Growth of Sobolev norms of solutions for the cubic NLS on T2 (2009)
  • Author(s): 高岡秀夫
  • Organizer: 第9回名古屋国際数学コンファレンス「Harmonic Analysis and Partial Differential Equations」
  • Place of Presentation: 名古屋大学
  • Year and Date: 2009-09-29
• [Presentation] Growth of Sobolev norms of solutions for the cubic NLS on T^2 (2009)
  • Author(s): 高岡秀夫
  • Organizer: 第9回名古屋国際数学コンファレンス「Harmonic Analysis and Partial Differential Equations」
  • Place of Presentation: 名古屋大学
  • Year and Date: 2009-09-29