Singularity theoretical research on Partial Differential Equations and Differential Geometry

偏微分方程与微分几何的奇异性理论研究

• 批准号: 10304003
• 负责人: IZUMIYA Shyuichi
• 金额: $ 12.12万
• 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: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10304003/
• 关键词: The Eikonal equations; Gravitation lens; Minkowski space; ruled surface; four vertices theorem; hyperbolic Gauss map; Lagrangian immersion; Weak solution; ヘルムホルツ方程式; 漸近解; ハミルトン・ヤコビ方程式; 衝撃波; 特異点; ダルブー可展開面; コンピュータグラフィックス; 特性曲線; ローレンツ微分幾何学

In this research project, we established the fundamental results on the propagation of singualnties (or shock waves) for weak solutions of partial differential equations and the construction on new invariants in Differential Geometry as an application of Singularity theory. Those results could not be studied by using the main frame of 20th century's Mathematics. Those results contain the classification of singularities for solutions of the Eikonal equation which appears in the theory of Ocean acoustics, construction of the generalized notion of weak solutions which is a generalization of both of the entropy and the viscosity solutions, the unified treatment on four vertices theorems of curves, the method to construct many mean curvature constant surfaces, construction of the symplectic framework for multiple-plane garvitation lensing and the study of sinuglarities of hyperbolic Gauss maps and lightcone Gauss maps. In the final year, we have given a classification of singular plane curves by symplectic diffeomorphisms and discovered that the difference from the classification by ordinary diffeomorphisms is a symplectic invariant. We have also found relations between special space curves and ruled surfaces. Moreover, we have studied line congruences and have given a characterization of normal line congruences by the notion of Lagrangian congruences.

在这个研究项目中，我们建立了偏微分方程弱解的奇点（或冲击波）传播以及作为奇点理论的应用的微分几何中新不变量的构造的基本结果。这些结果无法用20世纪数学的主要框架来研究。这些结果包括海洋声学理论中出现的Eikonal方程解的奇点分类、熵解和粘性解的广义弱解概念的构造、四个顶点的统一处理曲线定理、多平均曲率常数曲面的构造方法、多平面引力透镜辛框架的构造以及双曲高斯图和光锥的奇异性研究高斯图。去年，我们用辛微分同胚给出了奇异平面曲线的分类，发现它与普通微分同胚分类的区别是辛不变量。我们还发现了特殊空间曲线和直纹曲面之间的关系。此外，我们研究了直线同余，并通过拉格朗日同余的概念给出了法线同余的表征。

项目成果

M.Tsuji: "Construction of singularities for nonlnear wave equations" Proceedings of the Eighth International Colloquium on Diff-Equ.423-430 (1998)

M.Tsuji：“非线性波动方程奇点的构造”第八届国际 Diff-Equ.423-430 座谈会论文集（1998 年）

K.Ono: "On Arnold's conjecture for symplectic fixed points" Banach Center Publications. 45. 13-24 (1998)

K.Ono：“论阿诺德对辛不动点的猜想”巴拿赫中心出版物。

M.-H.Giga: "Generalized motion by nonlocal curvature in the plane"Arch. Rational Mech. Anal.. 159. 295-333 (2001)

M.-H.Giga：“平面内非局部曲率的广义运动”Arch。

T. Ozawa: "Linearizations of ordinary differential equations by area preserving maps"Nagoya Math. J.. 156. 109-122 (1999)

T. Ozawa：“通过面积保留映射对常微分方程进行线性化”名古屋数学。

S. IZUMIYA: "A time-like surface in Minkowski 3-space which contains light-like lines"Journal of Geometry. 64. 95-101 (1999)

S. IZUMIYA：“Minkowski 3 空间中的类时间表面，其中包含类光线”《几何杂志》。