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Study of Diophantine Phenomena Appearing in Asymptotic Analysis of Nonlinear Partial Differential Equations

非线性偏微分方程渐近分析中的丢番图现象研究

基本信息

批准号：
11640183
负责人：
YOSHINO Masafumi
金额：
$ 1.54万
依托单位：
CHUO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2002
项目状态：
已结题

项目摘要

We have obtained two main results 1) and 2).1) We reveal the role of Diophantine conditions when solving a system of nonlinear partial differential equations or difference equations in the normal form theory. Namely we havea) The simultaneous normal form of a commuting system of maps under Bruno condition is presented. This is also valid for a system of circle maps in a Gevrey class.b) The solvability of nonlinear Fuchsian equations of general variables in a class of finitely smooth functions is proved. This yields a so-called Grobman-Hartmann type theorem.c) The global normal form of a pseudodifferential operator on tori which is a perturbation of a system of constant vector fields is presented. Necessary and sufficient conditions are given in terms of a Diophantine condition.d) To make clear the relations between a simultaneous Diophantine condition of a commuting system of maps or vector fields and the Diophantine conditions for every generator of the system.2) We show that a Riemann-Hilbert problem is closely related to the solvability of equations without Diophantine conditions. Namely we havea) Solvability of nonlinear Fuchsian equations of general variables under a so-called Riemann-Hilbert condition is proved. This is applied to a mixed Monge- Ampere equation.b) Solvability of the same equations as in a) in a class of singular functions under a Poincare condition is presented. This can be regarded as a Frobenius theorem for a partial differential equation. The results are partly reported in ICM conference in Beijing in 2002.

我们得到了两个主要结果：(1)和(2)。(1)揭示了在范式理论中求解非线性偏微分方程或差分方程系统时丢芬图条件的作用。a)给出了布鲁诺条件下映射交换系统的同时正规形式。这对于Gevrey类中的圆形映射系统也是有效的。b)证明了一类有限光滑函数中一般变量非线性Fuchsian方程的可解性。c)给出了环面上伪微分算子的全局范式，该算子是常向量场系统的摄动。用丢番图条件给出了充分必要条件。d)明确映射或向量场交换系统的同时丢番图条件与系统各发生器的丢番图条件之间的关系。2)我们证明了Riemann-Hilbert问题与无丢芬图条件的方程的可解性密切相关。即a)证明了一般变量非线性Fuchsian方程在Riemann-Hilbert条件下的可解性。这适用于混合蒙日-安培方程。b)在Poincare条件下，给出了与a)相同的方程在一类奇异函数中的可解性。这可以看作是偏微分方程的Frobenius定理。2002年在北京召开的ICM会议上部分报告了研究结果。