The relation between the quantitative properties of the solutions of partial differential equations and the geometrical structures of their characteristics

偏微分方程解的定量性质与其特征的几何结构之间的关系

• 批准号: 12640175
• 负责人: SUGIMOTO Mitsuru
• 金额: $ 2.18万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640175/
• 关键词: Oscillatory Integral; Fourier Integral Operator; Global boundedness; Schoedinger equation; Cauchy Problem; Lp-Analysis; Coronical transform; Smoothing effect; 双曲型方程式; 境界値問題; 除去可能特異点; 波動方程式; ポテンシャル

By the theory of microlocal analysis, the qualitative property of the solutions of partial differential equations, such as the position of their singularity, can be described completely in terms of their characteristics. The aim of this research project was to investigate to what extent the quantitative property is described as well, and to apply it to the problems of the theory of partial differential equations. Especially, I have obtained the following results:The boundedness of Fourier integral operators: A theory of the boundedness properties of Fourier integral operators has been constructed. Especially, we treated the operators which are used to express the canonical transformations, and established their boundedness on weighted L^2-spaces.A weak extension theorem for inhomogeneous equations: An Lp-extension theory of inhomogeneous partial differential equations on a punctured domain has been constructed. A new approach based on the microlocal analysis is used to obtain results, which cannot be covered by the method of Bochner, who first studied this problem for homogeneous equations.A smoothing effect of Schrodinger equations: A smoothing effect on the initial value problem of generalized Schroedinger equations has been investigated, and an unknown relation has been found between the characteristics of operators and the direction in which the solutions gain their extra regularity. This result will be applied to non-linear problem.

利用微局部分析理论，偏微分方程解的定性性质，如奇点的位置，可以完全用其特征来描述。这个研究项目的目的是调查在何种程度上的定量属性的描述，以及，并将其应用到偏微分方程理论的问题。特别地，我得到了以下结果：傅里叶积分算子的有界性：构造了傅里叶积分算子的有界性理论。特别地，我们处理了表示正则变换的算子，并建立了它们在加权L^2-空间上的有界性。非齐次方程的一个弱扩张定理：构造了穿孔域上非齐次偏微分方程的Lp-扩张理论。使用一种基于微局部分析的新方法来获得结果，这些结果无法被Bochner的方法所涵盖，Bochner首先针对齐次方程研究了这个问题。薛定谔方程的平滑效应：研究了广义Schroedinger方程初值问题的光滑效应，并发现了算子的特征与解获得额外正则性的方向之间的未知关系。这一结果也适用于非线性问题。

项目成果

杉本 充: "A Smoothing property of Schrodinger equations along the sphere."J.Anal.Math. Vol.89. 15-30 (2003)

Mitsuru Sugimoto：“薛定谔方程沿球面的平滑特性。”J.Anal.Math 第 89 卷（2003 年）。

松村 昭孝(F.Huang, X.Shiと共著): "Viscous shock wave and boundary layer solution to an inflow problem for compressible viscous gas"Commun.Math.Phys.. Vol.239. 261-285 (2003)

Akitaka Matsumura（与 F.Huang、X.Shi 合着）：“可压缩粘性气体流入问题的粘性冲击波和边界层解决方案”Commun.Math.Phys.. Vol.261-285 (2003)。

杉本充: "A weak extension theorem for inhomogeneous differential equations"Forum Math.. (発表予定).

Mitsuru Sugimoto：“非齐次微分方程的弱可拓定理”论坛数学..（待提交）。

西谷 達雄(A.Boveとの共著): "Necessary conditions for the well-posedness of the Cauchy problem for hyperbolic systems"Osaka J. Math.. Vol.39. 149-179 (2002)

Tatsuo Nishitani（与 A. Bove 合着）：“双曲系统柯西问题适定性的必要条件”Osaka J. Math.. Vol.39 (2002)。

松村 昭孝(N.Yamagataと共著): "Global weak solutions of the Navier-Stokes equations for multidimensional compressible flow subject to large external potential forces"Osaka J.Math. 38巻. 399-418 (2001)

Akitaka Matsumura（与 N.Yamagata 合着）：“受大外部势力影响的多维可压缩流的纳维-斯托克斯方程的全局弱解”Osaka J.Math Vol. 38. 399-418 (2001)