Study on global property of holomorphic vector fields

全纯向量场的全局性质研究

• 批准号: 06640181
• 负责人: ITO Toshikazu
• 金额: $ 1.15万
• 依托单位: Ryukoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1994
• 资助国家: 日本
• 起止时间: 1994 至 1995
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-06640181/
• 关键词: singular points of holomorphic vector fields; strongly hyperbolic system; reaction diffusion system; asymptotic stability; dissipative dynamical system; inertial manifold; Lorenz attractors; piecewise linear vector fields; 正則ベクトル場; ザイフェルト予想; 偏微分

T.Ito counted the number of compact leaves of one dimensional non-singular foliation on the 2n-1 dimensional sphere S^<2n-1> associated with holomorphic vector fields, using Poincare-Dulac polynomialization Theorem at isolated singular point.W.Matsumoto has studied the characterization of the strongly hyperbolic systems of partial differential equations. He showed that in case where the coefficients depend only on the time variable, the strongly hyperbolic systems can be transformed to a Fuchsian with diagonal principal part at each point of the fiber space. This condition is the necessary and sufficient for the strong hyperbolicity when the dimension of x-space is one. On the other hand, in higher dimension case, he proposed a conjecture that the uniformity on the dual variables of transforming matrix might be necessary, and he showed that Petrovski's non-strongly hyperbolic system is unstable and easily change to a strongly hyperbolic system by a small perturbation.S.Yotsutani has st … More udied the asymptotic behavior and the rate of the convergence of solutions to a practical mathematical model of an chemical interfacial reaction, which is a parabolic system with nonlinear bondary conditions. Moreover, he has also studied some class of semilinear elliptic equations, and obtained a strong classification theorem of structure of positive radial solutions. The theorem claims that structure of solution is closely related with zeros of a function determined by coefficients and power of a nonlinear term.H.Oka showed that chaotic attractor of Lorenz-type is generated locally by an unfolding of some degenerate singularity. Also, she studied some piecewise linear vector fields and found a bifurcation including infinitely many doubling of codimension 2 homoclinic orbits which is called orbit-flip type.Y.Morita obtained with H.Ninomiya and E.Yanagida a theorem on the existence of an inertial manifold for a reaction-diffusion equation with nonlinear boundary condition. Moreover they gave the reduced finite-dimensional equation on the manifold together with applications to specific equations. Here the inertial manifold means a finite-dimensional invariant manifold which attracts every orbit in a phase space. Y.Morita studied in a joint work with K.Mischaikow a global structure of the dissipative dynamical system for the Ginzburg-Landau equation in a bounded interval and they showed a Morese structure of the dynamics. Y.Morita with S.Jimbo obtained an instability theorem of non-constant equilibrium solutions to the Ginzburg-Landau equation in any convex domain while they constructed stable non-constant solutions in an annulus domain. Less

T.Ito利用孤立奇点处的Poincare-Dulac多项式化定理计算了2n-1维球面S^上与全纯向量场相关的一维非奇异叶理的紧叶数<2n-1>，W.松本研究了强双曲型偏微分方程组的特征.他表明，在案件的系数只取决于时间变量，强双曲系统可以转化为一个Fuchsian与对角主部分在每一点的纤维空间。当x-空间的维数为1时，这个条件是强双曲性的充要条件。另一方面，在高维情形下，他提出了变换矩阵的对偶变量的一致性可能是必要的猜想，并证明了Petrovski的非强双曲方程组是不稳定的，很容易通过一个小的扰动变为强双曲方程组。 ...更多信息 研究了一类具有非线性边界条件的抛物型化学界面反应数学模型解的渐近性态和收敛速度。此外，他还研究了一类半线性椭圆型方程，得到了正径向解结构的一个强分类定理。该定理认为解的结构与由非线性项的系数和幂确定的函数的零点密切相关。H.Oka证明了Lorenz型混沌吸引子是由退化奇点的开折局部产生的。Y.Morita与H.二宫和E.Yanagida一起得到了一类具有非线性边界条件的反应扩散方程的惯性流形存在性定理。此外，他们还给出了流形上的降维有限维方程及其在具体方程中的应用。这里的惯性流形是指吸引相空间中所有轨道的有限维不变流形。Y.森田研究了在联合工作K.Mischaikow的整体结构的耗散动力系统的金斯堡-朗道方程在有界区间，他们显示了Morese结构的动态。Y.Morita和S.Jimbo在构造环域上的稳定的非常数解的同时，得到了Ginzburg-Landau方程在任意凸域上的非常数平衡解的不稳定性定理.少

项目成果

E.Yanagida and S.Yotsutani: "Existence of positive radial solutions to Δu+k(|x|)|u|^<P-1>u=0 in IR^n" J.Differential Equations. 115. 477-502 (1995)

E.Yanagida 和 S.Yotsutani：“IR^n 中 Δu+k(|x|)|u|^<P-1>u=0 的正径向解的存在”J.微分方程 115。477-502 (1995)

Waichiro Matsumoto: "On the strong hyperbolicity of systems with coefficients depending only on the time variable" International Conference Differential Equations. (to appear.).

Waichiro Matsumoto：“关于系数仅取决于时间变量的系统的强双曲性”国际会议微分方程。

Shoji Yotsutani: "Existence of nodal fast-decay solutions to DELTAu+K(|X|)|u|^<p-1>u=0 in R^n" Nonlinear Analysis TMA. 22. 1005-1015 (1994)

Shoji Yotsutani：“R^n 中 DELTAu K(|X|)|u|^<p-1>u=0 的节点快速衰减解的存在”非线性分析 TMA。

T.Ito: "A Poincare-Bendixson type theorem for holomorphic vector fields" 京都大学数理解析研究所講究録. 878. 1-9 (1994)

T.Ito：“全纯向量场的庞加莱-本迪克森型定理”京都大学数学科学研究所 Kokyuroku 878. 1-9 (1994)。

W.Matsumoto: "The strong hyperbolicity of first order systems with coefficients depending only on the time variable" International Conference Differntial Equations. (to appear).

W.Matsumoto：“系数仅取决于时间变量的一阶系统的强双曲性”国际会议微分方程。