GEOMETRY AND ANALYSIS FOR WAVE FIELDS

波场的几何和分析

• 批准号: 13304011
• 负责人: OZAWA Tohru
• 金额: $ 22.63万
• 依托单位: Hokkaido University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13304011/
• 关键词: nonlinear wave equations; nonlinear Dirac equations; nonlinear Klein-Gordon equations; nonlinear Schrodinger equations; scattering theory; 非線型分散型偏微分方程式; 非線型双曲型偏微分方程式; 非線型散乱理論; 非相対論的極限; 非線型散乱問題

In this research project, various space-time behavior of solutions to nonlinear dispersive equations, such as nonlinear Schrodinger equations (NLS) and KdV type equations, nonlinear hyperbolic equations, such as nonlinear wave and Klein-Gordon equations, and coupled systems of those equations, such as nonlinear field equations. The main results are the following :(1)Asymptotic completeness in the energy space H^1(R^3) for NLS with repulsive case has been proved.(2)A unified treatment for small data scattering for nonlinear field equations has been given in terms of critical and subcritical setting.(3)Existence and uniqueness of self-similar solutions for nonlinear wave equations have been proved in the framework of weak Lebesgue spaces.

本课题研究了非线性色散方程（如非线性薛定谔方程和KdV型方程）、非线性双曲型方程（如非线性波方程和Klein-Gordon方程）以及这些方程的耦合系统（如非线性场方程）解的各种时空行为。主要结果如下：(1)证明了具有排斥情形的NLS在能量空间H^1（R^3）上的渐近完备性。(2)给出了非线性场方程小数据散射在临界和亚临界条件下的统一处理方法。(3)在弱Lebesgue空间框架下，证明了非线性波动方程自相似解的存在唯一性。

项目成果

J.Colliander: "Almost conservation laws and global rough solutions to a Nonlinear Schrodinger equation"Mathematical Research Letters. 9. 1-24 (2002)

J.Colliander：“非线性薛定谔方程的几乎守恒定律和全局粗解”数学研究快报。

S. Machihara: "Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations"Mathematische Armalen. (印刷中).

S. Machihara：“非线性 Klein-Gordon 方程的能量空间中的非相对论极限”Mathematische Armalen（正在出版）。

J.Kato: "On solutions of the wave equation with homogeneous Cauchy data"Asymptotic Analysis. 37. 93-107 (2004)

J.Kato：“关于具有齐次柯西数据的波动方程的解”渐近分析。

N. Hayashi: "Global existence of small solutions to quadratic nonlinear Schrodinger equations in two space dimensions"SIAM Journal on Mathematical Analysis. 32. 1390-1403 (2001)

N. Hayashi：“二维空间二次非线性薛定谔方程小解的整体存在性”SIAM 数学分析杂志。

Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations

• DOI: 10.1007/s002080200008
• 发表时间: 2002-12
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Shuji Machihara;K. Nakanishi;T. Ozawa