Study of structure of solutions to Schrodinger equation from quantum-fluid point of view

• Project/Area Number: 24740108
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Global analysis
• Research Institution: Hiroshima University
• Principal Investigator: Masaki Satoshi 広島大学, 工学(系)研究科(研究院), 准教授 (20580492)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); 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: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 非線形シュレディンガー方程式 / 分散型方程式 / KdV 方程式 / 一般化KdV方程式 / 解の長時間挙動 / 最小化問題 / 最小爆発解 / 非線形偏微分方程式 / 一般化 KdV 方程式 / 散乱問題 / 最小非散乱解 / 非線型分散型方程式 / 非線型シュレディンガー方程式 / KdV方程式 / フーリエ制限問題 / シュレディンガー方程式 / 時間大域挙動 / 偏微分方程式 / 尺度不変空間 / 有限時間爆発 / 解の時間大域挙動

Outline of Final Research Achievements

The first intend of the research was to focus on quantum-fluid properties of Schrodinger equations and to clarify the correspondence between blowup of solution to a nonlinear Schrodinger equation and collapse of solution of a corresponding classical fluid equation. The intended method for this analysis is a contradiction argument that construct a virtual solution from failure of a target conclusion. In this decade, there is much progress on this kind of argument.
The research progresses unexpectedly and a special non-scattering solution which is minimal in a suitable sense is found for mass-subcritical equations. Behavior of the solution is completely unknown and different from any of those we knew before. I also consider generalized KdV equations and find that similar kind of threshold solution exists.

Research Products

• [Journal Article] On well-posedness of the generalized Korteweg-de Vries equation in scale critical ^Lr space (2016)
  • Author(s): Satoshi Masaki, Jun-ichi Segata
  • Journal Title: Analysis & PDE Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Existence and uniqueness of two dimensional Euler-Poisson system and WKB approximation to the nonlinear Schrodinger-Poisson system. (2015)
  • Author(s): Satoshi Masaki, Takayoshi Ogawa
  • Journal Title: Journal of Mathematical Physics Volume: 56 Issue: 12 Pages: 121502-121502
  • DOI: 10.1063/1.4936309
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation (2015)
  • Author(s): Satoshi Masaki
  • Journal Title: Communications on Pure and Applied Analysis Volume: 14 Issue: 4 Pages: 1481-1531
  • DOI: 10.3934/cpaa.2015.14.1481
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] An example of stable excited state on nonlinear Schrödinger equation with nonlocal nonlinearity (2013)
  • Author(s): Masaya Maeda, Satoshi Masaki
  • Journal Title: Differential Integral Equations Volume: 26
  • Peer Reviewed
• [Presentation] L2劣臨界一般化KdV方程式の長時間挙動 (2015)
  • Author(s): 眞崎聡 瀬片純一
  • Organizer: 日本数学会2015年度秋季総合分科会
  • Place of Presentation: 京都産業大学
  • Year and Date: 2015-09-13
• [Presentation] 一般化 Korteweg-de Vries 方程式の小さな初期値に対する大域解の存在 と散乱 (2015)
  • Author(s): 眞崎聡 瀬片純一
  • Organizer: 日本数学会2015年度秋季総合分科会
  • Place of Presentation: 京都産業大学
  • Year and Date: 2015-09-13
• [Presentation] Restriction estimate and its application to generalized KdV equation in ^Lp space (2015)
  • Author(s): Satoshi Masaki
  • Organizer: The 3rd East Asian Conference in Harmonic Analysis and Applications
  • Place of Presentation: Tokyo University
  • Year and Date: 2015-08-10
  • Int'l Joint Research
• [Presentation] 質量劣臨界非線型シュレディンガー方程式の解析 (2015)
  • Author(s): 眞崎聡
  • Organizer: 日本数学会2015年度年会
  • Place of Presentation: 明治大学
  • Year and Date: 2015-03-21 – 2015-03-24
  • Invited
• [Presentation] Global behavior of solutsions to mass-subcritical nonlinear Schrodinger equation (2014)
  • Author(s): Satoshi Masaki
  • Organizer: International workshop on PDEs and related topics in nonlinear problems
  • Place of Presentation: 広島大学
  • Invited
• [Presentation] Classical limit of some quantum Euler equations (2013)
  • Author(s): Satoshi Masaki
  • Organizer: The Second Pacific Rim Mathematical Association (PRIMA) Congress
  • Place of Presentation: 上海交通大学
  • Invited
• [Presentation] L2 劣臨界非線型Schrodinger方程式における最小爆発解について (2013)
  • Author(s): 眞崎聡
  • Organizer: 日本数学会２０１３年度秋季総合分科会
  • Place of Presentation: 愛媛大学
• [Presentation] L^2劣臨界非線形シュレディンガー方程式における最小非散乱解について (2012)
  • Author(s): 眞崎聡
  • Organizer: 日本数学会秋期総合分科会
  • Place of Presentation: 九州大学