Comprehensive study of differential equations

微分方程综合研究

• 批准号: 11304006
• 负责人: YAJIMA Kenji
• 金额: $ 17.54万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11304006/
• 关键词: Shrodinger equation; nonlinear PDE; linear PDE; nonlinear evolution equation; scattering theory; viscous solution; Navier-Stokes equation; Klein-Gordon equation; シュレーディンガー方程式; シュレーディンガー作用素; 超漸近解析; 波動方程式; 漸近安定性; 偏微分方程式; 初期値問題; ランダムな磁場; ショック

We carried out an comprehensive study on linear and nonlinear partial and ordinary differential equations and obtained among others the following results:1. Kenji Yajima and Shu Nakamura studied Schrodinger equations and obtained (1) the L^p-boundedness of wave operators of scattering and (2) the stability under subquadratic perturbations of the smooth and boundedness of fundamental solution; (3) clarified the relation between the local decay and the spectrum of Floquet operator for time periodic system; (4) constructed general theory of tunnenling in phase space and gave its applications; (5) defined the integrated density of states for random operators and proved Wegner estimates and the Lifschitz singularities.2. Kiyoomi Kataoka studied the general theory of linear PDE and (1) gave an elementary definition of the holomorphic solutions complex of ε^R_x-modules and (2) reduced the branching of the singularities for Fuchsian elliptic boundary problems to that of the continuation of ODE.3. Yoshikazu Giga studied nonlinear PDE and (1) gave a way to numerically compute the viscous solution of first order nonlinear PDE, (2) defined the proper visocous solutions and proved the existence and the uniquess of solutions and (3) proved the existence of solution of Navier-Stokcs equation with non-deaying initial conditions.4. Yoshio Tsutsumi proved (1) the well-posedness of nonlinearly couples wave equations with different speeds, (2) the stability of constant solutions of nonlinear massive Klein-Gordon equation.5. Hideyuki Majima produced new fundamental theorem on the asymptotic expansions from the point of view the super-asymptotic analysis.6. Mitsuru Ikawa studied the scattering theory of wave equation by several convex bodies and obtained a precise asymptotic, formula for the poles of scattering matrix.

我们对线性和非线性偏微分方程和常微分方程进行了全面的研究，得到了以下结果：Kenji Yajima和Shu Nakamura研究了薛定谔方程，得到了(1)散射波算符的L^p有界性和(2)基本解的光滑和有界性在次二次扰动下的稳定性；(3)阐明了时间周期系统的局部衰减与Floquet算子谱的关系；(4)建立了相空间隧道掘进的一般理论并给出了应用；(5)定义了随机算子的状态积分密度，证明了Wegner估计和Lifschitz奇点。Kiyoomi Kataoka研究了线性PDE的一般理论，(1)给出了ε^ r_x模的全纯解复的初等定义；(2)将Fuchsian椭圆边界问题的奇异分支化简为ode的延拓分支。Yoshikazu Giga对非线性偏微分方程进行了研究，(1)给出了一阶非线性偏微分方程粘性解的数值计算方法；(2)定义了适当的粘性解，并证明了解的存在性和唯一性；(3)证明了具有非衰减初始条件的Navier-Stokcs方程解的存在性。Yoshio Tsutsumi证明了(1)不同速度下非线性耦合波动方程的适定性，(2)非线性质量Klein-Gordon方程常解的稳定性。真岛秀之从超渐近分析的观点出发，提出了关于渐近展开式的新基本定理。Mitsuru Ikawa研究了几个凸体对波动方程的散射理论，得到了散射矩阵极点的精确渐近公式。

项目成果

Yoshikazu Giga: "A level set approach to semi-continuous viscosity solutions for Caucliy problems"Commun. Partial Differential Equations. 26. 813-839 (2001)

Yoshikazu Giga：“Caucliy 问题的半连续粘度解决方案的水平集方法”Commun。

Arne Jensen: "A remark on L^p-boundedness of wave operators for two dimensional Schrodinger operators"Commun. Mathenatical Physics. (to appear). (2002)

Arne Jensen：“关于二维薛定谔算子的波算子的 L^p 有界性的评论”Commun。

MAJIMA Hideyuki: "Quadratic relations for confluent hypergeometric functions"Tohoku Mathematical Journal. 52・4. 489-513 (2000)

MAJIMA Hideyuki：“合流超几何函数的二次关系”东北数学杂志 52・4（2000）。

Kenji Yajima: "L^P-boundedness of wave operators for two dimensional Schrodinger equations"Communications in Mathematical Phyoics. (発表予定).

Kenji Yajima：“二维薛定谔方程的波算子的 L^P 有界性”数学物理学通讯（待提交）。

儀我 美一: "非線型編微分方程式-解の漸立挙動と自亡相似解"共立出版. 300 (1990)

Yoshikazu Giga：“非线性版微分方程 - 解和自杀破坏性类似解的毕业行为”Kyoritsu Shuppan 300 (1990)。