Solvability of nonlinear dispersive equations with complicated resonance structure

2022 Fiscal Year Final Research Report

• Project/Area Number: 17K14220
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: University of Miyazaki
• Principal Investigator: Hirayama Hiroyuki 宮崎大学, 教育学部, 准教授 (90748328)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Keywords: 非線形分散型方程式 / 初期値問題 / 適切性 / 共鳴

Outline of Final Research Achievements

For the Cauchy problem of system of nonlinear Schrodinger equations, we almost completely characterized the Sobolev indexes which allows the well-posedness by the conditions focused on resonance structure. We also considered the Cauchy problem of nonlinear fourth order Schrodinger equations which have the polynomial nonlinearity with third order or lower spatial derivatives. We obtained the results for the well-posedness of this problem, which contain the improvement of the previous works. In particular, for the case that the equation has scale invariance, we obtained the well-posedness in the scaling critical Sobolev spaces. Furthermore, we clarified that the structure of Zakharov-Kuznetsov-Burgers equation, which is one of dispersive-dissipative model, is better than the structure of Zakharov-Kuznetsov equation, which has dispersion but not dissipation. To obtain this result, we prove the well-posedness of the Cauchy problem of Zakharov-Kuznetsov-Burgers equation.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

本研究で扱った方程式のほとんどは空間2次元以上のモデルであり, 共鳴構造が複雑であるという特徴を持つ.そのような特徴は物理現象などを背景としたモデルにも現れるため, その解析は数学だけでなく現象の立場においても重要である. 実際, 本研究で扱った方程式も物理現象を背景としているものが多い. また, 本研究の主題にもなっている分散性は波の伝播を記述するモデルに多く見られる性質であり, 分散性と非線形性による影響は共鳴構造に依存する. そのため, 共鳴構造を精密に調べることは, 非線形分散型方程式の性質を明らかにするために重要な役割を果たす.