Spacetime estimates for dispersive equations and applications to nonlinear prooblems

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K03370
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Shimane University
• Principal Investigator: Wada Takeshi 島根大学, 学術研究院理工学系, 教授 (70294139)
• Co-Investigator(Kenkyū-buntansha): 中村 誠 大阪大学, 大学院情報科学研究科, 教授 (70312634) ; 北 直泰 熊本大学, 大学院先端科学研究部(工), 教授 (70336056)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: 非線形偏微分方程式 / 分散型方程式 / 時空間評価 / Schrodinger 方程式 / 適切性 / 解の時間大域的挙動

Outline of Final Research Achievements

Waves whose phase speed varies with wavelength are said to be dispersive. This study is mainly devoted to the nonlinear Schrodinger equation, which is a typical example of nonlinear dispersive equations, and the Maxwell-Schrodinger system, which is a coupled system of dispersive and hyperbolic equations. We studied the solvability and solution behavior of these equations by functional analytic methods. We proved modifications of Strichartz and Kato type smoothing estimates, which are fundamental tools in the analysis of nonlinear dispersive equations, and improved the results on the solvability of the initial value problem of nonlinear Schrodinger equations. Furthermore, we clarified time global behavior of solutions of the Maxwell-Schrodinger and Benjamin-Ono equations.

Free Research Field

非線形偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

本研究における非線形 Schrodinger 方程式の初期値問題の適切性に関する結果は初期データが小さい場合には最良と考えられるものであり，これによって同方程式の数学的構造が明確になったといえる。