Research on geometric symmetry and singularity of solutions for nonlinear wave equations

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K17624
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Shinshu University
• Principal Investigator: Okamoto Mamoru 信州大学, 学術研究院工学系, 助教 (40735148)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 非線形波動方程式

Outline of Final Research Achievements

We completely determine the range of Sobolev regularity of initial data to be well-posed of the Cauchy problem for the Dirac-Klein-Gordon system. We also show that the modified scattering for the solution to the Cauchy problem for the short-pulse, fifth order KdV-type, and higher order KdV-type equations. We study the Cauchy problem for the nonlinear Schrodinger equation with nonalgebraic nonlinearity in five and six dimensions. We prove that almost sure global well-posedness holds for the Cauchy problem in a larger class than the energy space for the energy-critical nonlinear Schrodinger equation.

Free Research Field

関数方程式論

Academic Significance and Societal Importance of the Research Achievements

非適切性の証明では，非線形項を逐次近似して，滑らかな解の存在を示し，それを用いて，ノルムインフレーションを証明した．また，波束テスト法を用いた解の漸近挙動の解明は，高階KdV型方程式にも適用できることを示した．さらに，初期値を確率化することで，決定論的な反例が除外されることを示した．本研究で培ったこれらの手法は，より広範な方程式に援用することができるものと考えられ，今後の研究がさらに発展する基盤になると期待している．