Properties of Pseudo-differential Operators and Their Applications to Hyperbolic Equations

伪微分算子的性质及其在双曲方程中的应用

• 批准号: 01540151
• 负责人: TANIGUCHI Kazuo
• 金额: $ 1.02万
• 依托单位: University of Osaka Prefecture
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1989
• 资助国家: 日本
• 起止时间: 1989 至 1990
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-01540151/
• 关键词: hyperbolic equation; ultra wave front set; Gevrey class; propagation of singularity; Cauchy problem; Fourier integral operator; complex phase function; pourus media equation; フ-リエ積分作用素; ジェヴレイ関数; 波面集合; 擬微分作用素

The fundamental solution of the Cauchy problem for a hyperbolic operator is given in the form of Fourier integral operator. As shown below, when the problem is not C^* well-posed, the symbol of the fundamental solution has exponential growth, that is, it is estimated not only from above but also from below by (1) <numerical formula> The constant kappa in (1) corresponds to the constant in the necessary and sufficient condition for the well-posedness in Gevrey classes. In order to study this phenomenon we define UWF^<(mu)>(u) (ultra wave front sets) for u that belongs to the space of ultradistributions S{kappa}' by <numerical formula> where X * S{kappa}*C^*_ and xi belongs to a conic neighborhood of xi_<omicron>. Then by using UWF^<(mu)>(u) we can state the propagation of very high singularities for the solution of not C^* well-posed Cauchy problem. We also construct the ndamental solutions of the Cauchy problem for degenerate hyperbolic operators (2) <numerical formula> (3) <numerical formula> with an even integer j and we investigated other related topics.

双曲算子柯西问题的基本解以傅立叶积分算子的形式给出。如下图，当问题不是C^*适定时，基本解的符号具有指数增长，即不仅从上估计，而且从下估计，通过 (1) <数值公式> 常数(1) 中的 kappa 对应于 Gevrey 类适定性的充分必要条件中的常数。为了研究这种现象，我们通过 <数值公式> 定义属于超分布 S{kappa}' 空间的 u 的 UWF^<(mu)>(u)（超波前集），其中 X * S{kappa} *C^*_ 和 xi 属于 xi_<omicron> 的圆锥邻域。然后，通过使用 UWF^<(mu)>(u)，我们可以描述非 C^* 适定柯西问题解的非常高奇点的传播。我们还构造了具有偶数 j 的简并双曲算子 (2) <数值公式> (3) <数值公式> 的柯西问题的基本解，并研究了其他相关主题。

项目成果

新開 謙三: "Stokes multipliers and a weekly hyperbolic operators"

Kenzo Shinkai：“斯托克斯乘数和每周双曲算子”

新開謙三: "Stokes multipliers and a weakly hyperbolic operator."

Kenzo Shinkai：“斯托克斯乘数和弱双曲算子。”

Kenzo. SHINKAI and Kazuo. TANIGUCHI: "On ultra wave front sets and Fourier integral operators of infinite order." Osaka J. Math.27. 709-720 (1990)

贤三.

Kazuo. TANIGUCHI,: "A fundamental solution for a degenerate hyperbolic operator of second order and Fourier integral operators of complex phase,"

一雄。

新開謙三: "Fundamental solution for a degenerate hyperbolic operator in Gevrey classes"

Kenzo Shinkai：“Gevrey 类中简并双曲算子的基本解决方案”