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.

在该研究项目中，非线性色散方程解决方案的各种时空行为，例如非线性Schrodinger方程（NLS）和KDV类型方程，非线性双曲线方程，例如非线性波和Klein-Gordon方程，以及这些方程式的系统，例如非线性场方程。主要结果是：（1）已证明具有排斥性NL的能量空间中的渐变性完整性已证明。（2）对非线性和亚临界设置的非线性现场方程的小数据散射进行了统一的处理。（3）对非线性波动的自动溶液的存在和独特的范围已被证明是在不平衡的范围内已被证明是everage evere cagie nefter的范围。

项目成果

J.Kato: "Weighted Strichartz estimates and existence of self-similar solutions for semilinear wave equations"Indiana University Mathematical Journal. 52. 1615-1630 (2003)

J.Kato：“半线性波动方程的加权 Strichartz 估计和自相似解的存在性”印第安纳大学数学杂志。

M.Nakamura: "Small date scattering for nonlinear Schrodinger, wave and klein-Gordon equations"Annali della Scuola Normale Superiore di Pisa, V. 1・2. 435-460 (2002)

M. Nakamura：“非线性薛定谔、波和克莱因-戈登方程的小数据散射”Annali della Scuola Superiore di Pisa，V. 1・2 (2002)。

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.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。

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