Study on asymptotic behavior and singularities of solutions for nonlinear dispersive equations by using geometric symmetry

• Project/Area Number: 20K14342
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Osaka University
• Principal Investigator: Mamoru Okamoto 大阪大学, 大学院理学研究科, 准教授 (40735148)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Project Status: Completed (Fiscal Year 2023)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2022: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2021: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 適切性 / 漸近挙動 / 初期値問題 / 非線形分散型方程式

Outline of Research at the Start

本研究では、幾何学的対称性を用いて、非線形波動・分散型方程式の解の挙動と特異性の解析を行う。非線形波動・分散型方程式は、物理的な背景を持ち、様々な保存量や対称性を有している。これらの保存量や対称性を用いて、非線形波動・分散型方程式の初期値問題の適切性や解の漸近挙動の解析を行うことで、解の特異性が発生する構造を捉え、非線形性を制御する手法を発展させることを目指す。

Outline of Final Research Achievements

We study on the singularity and asymptotic behavior of solutions for the nonlinear dispersive equations. We obtain an almost sharp well-posedness result of the Cauchy problem for a system of nonlinear Schrodinger equations. We also study the asymptotic behavior of solutions for a fourth order nonlinear Schrodinger equation with critical nonlinearity in the sense of scattering. Moreover, we prove the ill-posedness of the Cauchy problem for the nonlinear wave equation.

Academic Significance and Societal Importance of the Research Achievements

非線形項に微分を含む非線形シュレディンガー方程式において、逐次近似法を用いる限りほとんど最良な適切性を得ることができた。そこで培った手法により、非線形相互作用の制御手法の方針が得られた。また、漸近挙動を解明し、散乱の意味で臨界な状況における解の振る舞いを明らかにした。さらに、非適切性の証明により、解の特異性がいかに発生するかを詳細に調べることができた。

Research Products

• [Int'l Joint Research] The University of Edinburgh(英国)
• [Int'l Joint Research] Ecole Normale Superieure de Lyon(フランス)
• [Int'l Joint Research] Univ. of Edinburgh(英国)
• [Int'l Joint Research] Univ. Bielefeld(ドイツ)
• [Journal Article] Sharp well-posedness for the Cauchy problem of the two dimensional quadratic nonlinear Schr?dinger equation with angular regularity (2024)
  • Author(s): Hirayama Hiroyuki、Kinoshita Shinya、Okamoto Mamoru
  • Journal Title: Journal of Differential Equations Volume: 395 Pages: 181-222
  • DOI: 10.1016/j.jde.2024.02.037
  • Peer Reviewed / Open Access
• [Journal Article] Uniqueness and non-uniqueness of the Gaussian free field evolution under the two-dimensional Wick ordered cubic wave equation (2024)
  • Author(s): Tadahiro Oh, Mamoru Okamoto, Nikolay Tzvetkov
  • Journal Title: Ann. Inst. Henri Poincare Probab. Stat. Volume: -
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] A remark on the well-posedness for a system of quadratic derivative nonlinear Schr?dinger equations (2022)
  • Author(s): Hirayama Hiroyuki、Kinoshita Shinya、Okamoto Mamoru
  • Journal Title: Communications on Pure and Applied Analysis Volume: 21 Issue: 10 Pages: 3309-3334
  • DOI: 10.3934/cpaa.2022101
  • Peer Reviewed / Open Access
• [Journal Article] Well-posedness for a system of quadratic derivative nonlinear Schr"odinger equations in almost critical spaces (2021)
  • Author(s): Hirayama Hiroyuki、Kinoshita Shinya、Okamoto Mamoru
  • Journal Title: Journal of Mathematical Analysis and Applications Volume: 499 Issue: 2 Pages: 125028-125028
  • DOI: 10.1016/j.jmaa.2021.125028
  • Peer Reviewed / Open Access
• [Journal Article] Long-time behavior of solutions to a fourth-order nonlinear Schroedinger equation with critical nonlinearity (2021)
  • Author(s): Mamoru Okamoto, Kota Uriya
  • Journal Title: Journal of Evolution Equations Volume: 21 Issue: 4 Pages: 4897-4929
  • DOI: 10.1007/s00028-021-00737-8
  • Peer Reviewed / Open Access
• [Journal Article] Comparing the stochastic nonlinear wave and heat equations: a case study (2021)
  • Author(s): Oh Tadahiro、Okamoto Mamoru
  • Journal Title: Electronic Journal of Probability Volume: 26 Issue: none
  • DOI: 10.1214/20-ejp575
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] A remark on norm inflation for nonlinear wave equations (2020)
  • Author(s): Forlano Justin、Okamoto Mamoru
  • Journal Title: Dynamics of Partial Differential Equations Volume: 17 Issue: 4 Pages: 361-381
  • DOI: 10.4310/dpde.2020.v17.n4.a3
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Well-Posedness for a system of quadratic derivative nonlinear schr¨odinger equations with radial initial data (2020)
  • Author(s): Hirayama Hiroyuki、Kinoshita Shinya、Okamoto Mamoru
  • Journal Title: Annales Henri Poincare Volume: 21 Issue: 8 Pages: 2611-2636
  • DOI: 10.1007/s00023-020-00931-3
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] A remark on triviality for the two-dimensional stochastic nonlinear wave equation (2020)
  • Author(s): Oh Tadahiro、Okamoto Mamoru、Robert Tristan
  • Journal Title: Stochastic Processes and their Applications Volume: 130 Issue: 9 Pages: 5838-5864
  • DOI: 10.1016/j.spa.2020.05.010
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Presentation] Uniqueness of the Gaussian free field evolution under the Wick ordered nonlinear Klein-Gordon equation (2024)
  • Author(s): Mamoru Okamoto
  • Organizer: New trends in nonlinear dispersive equations
  • Int'l Joint Research / Invited
• [Presentation] On the solutions to the two-dimensional Wick ordered nonlinear Klein-Gordon equation (2024)
  • Author(s): Mamoru Okamoto
  • Organizer: The 41th Kyushu Symposium on Partial Differential Equations
  • Int'l Joint Research / Invited
• [Presentation] Gauss型の初期値をもつ非線形Klein-Gordon方程式のほとんど確実な解について (2023)
  • Author(s): 岡本葵, Tadahiro Oh, Nikolay Tzvetkov
  • Organizer: 日本数学会秋季総合分科会
• [Presentation] 2次の非線形項をもつ非線形Klein-Gordon方程式のほとんど確実な大域的適切性 (2023)
  • Author(s): 岡本葵
  • Organizer: 広島数理解析セミナー
  • Invited
• [Presentation] 2次の非線形項をもつ2 次元非線形Schrodinger方程式の角度正則性を課したSobolev空間での適切性 (2023)
  • Author(s): 平山浩之、木下真也、岡本葵
  • Organizer: 日本数学会年会
• [Presentation] Stochastic quantization of the Phi^3_3-model (2022)
  • Author(s): Mamoru Okamoto
  • Organizer: Nonlinear and Random Waves
  • Int'l Joint Research / Invited
• [Presentation] On the Phi^3_3 measure (2022)
  • Author(s): Mamoru Okamoto
  • Organizer: the 42nd conference on Stochastic Processes and their Applications
  • Int'l Joint Research / Invited
• [Presentation] Phi^3_3-測度について (2022)
  • Author(s): 岡本葵
  • Organizer: 信州若里数理解析研究会
  • Invited
• [Presentation] 2次の非線形項を持つ非線形Schrodinger方程式の初期値問題の適切性について (2022)
  • Author(s): 岡本葵
  • Organizer: Dispersive and wave equations
  • Invited
• [Presentation] Almost sure global well-posedness for the quadratic nonlinear Klein-Gordon equation (2022)
  • Author(s): 岡本葵
  • Organizer: 第13回 名古屋微分方程式研究集会
  • Invited
• [Presentation] 非線形 Klein-Gordon 方程式のほとんど確実な大域的適切性 (2022)
  • Author(s): 岡本葵
  • Organizer: 神戸大学解析セミナー
  • Invited
• [Presentation] Stochastic quantization of the $\Phi^3_3$-model (2021)
  • Author(s): Mamoru Okamoto
  • Organizer: Rigorous statistical mechanics and related topics
  • Int'l Joint Research / Invited
• [Presentation] 空間3次元における非線形Klein-Gordon方程式のほとんど確実な大域的適切性 (2021)
  • Author(s): 岡本葵
  • Organizer: 日本数学会秋季総合分科会
  • Invited
• [Presentation] 微分型非線形Schrodinger方程式系のほとんど最良なSobolev空間における適切性 (2021)
  • Author(s): 岡本葵
  • Organizer: 第57回南大阪応用数学セミナー
  • Invited
• [Presentation] 吸引的なHartree型$\Phi^4_3$量子場モデルの不変測度とその流れ (2021)
  • Author(s): 岡本葵
  • Organizer: Saga Workshop on Partial Differential Equations
  • Invited
• [Presentation] Almost sure global well-posedness for the focusing nonlinear wave equation with a Hartree-type nonlinearity (2021)
  • Author(s): Mamoru Okamoto
  • Organizer: Oberseminar Analysis
  • Int'l Joint Research / Invited
• [Presentation] 吸引的なHartree型非線形波動方程式のほとんど確実な大域的適切性 (2021)
  • Author(s): 岡本葵，Tadahiro Oh，Leonardo Tolomeo
  • Organizer: 日本数学会年会
• [Presentation] 微分型非線形シュレディンガー方程式系の適切性に対する最良ソボレフ指数について (2021)
  • Author(s): 平山浩之，木下真也，岡本葵
  • Organizer: 日本数学会年会
• [Presentation] 球対称な初期値を持つ微分型非線形Schr\"odinger方程式系の適切性 (2020)
  • Author(s): 岡本葵
  • Organizer: 第16回 非線型の諸問題
  • Invited
• [Presentation] 吸引的なHartree型$\Phi^4_3$測度と非線形波動方程式のほとんど確実な大域的適切性 (2020)
  • Author(s): 岡本葵
  • Organizer: 微分方程式セミナー
  • Invited
• [Presentation] 非線形波動方程式の初期値問題の非適切性について (2020)
  • Author(s): Justin Forlano, 岡本葵
  • Organizer: 日本数学会秋季総合分科会
• [Funded Workshop] Stochastics and Nonlinear Partial Differential Equations (2022)