Properties of Solutions of Partial Differential Equations and Their Applications

偏微分方程解的性质及其应用

• 批准号: 63540134
• 负责人: OKANO Hatsuo
• 金额: $ 0.7万
• 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: 1988
• 资助国家: 日本
• 起止时间: 1988 至 1989
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-63540134/
• 关键词: hyperbolic equation; Fourier integral operator; propagation of singularity; Cauchy problem; cohomology; unitary representation; class field; lattice path combinatorics; 波面集合; コホモロジー; ユニタリー表現; 保型形式; ブール関数; E.R.積分; 発散級数

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) C exp[cxi^<1/k>], c > 0, 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 phenomena we define UWF^<{mu}>(u)(ultra wave front sets) for u that belongs to the space of ultradistributions S{k}' by (chi_0,xi_0) <not a member of> UWF^<{mu}>(u) <tautomer> *_<epsilon> > O*C ; |X^u(xi)| <less than or equal> exp[epsilon < xi >^<1/mu>], where X * S{k}*C^*_ and xi belongs to a conic neighborhood of xi_0. Then by using UWP^<{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 fundamental solutions of the Cauchy problem for degenerate hyperbolic operators (2) L_1 = THETA^2_ - t^2_ - at^kTHETA^x with 0 < k < j - 1 and (3) L_2 = THETA^2_ - x^<2j>THETA^2_ - aTHETA^x 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，基本解的符号呈指数增长，即不仅从上估计，而且从下估计： (1) C exp[cxi^<1 /k>], c > 0, (1) 中的常数 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，我们定义 UWF^<{mu}>。 (u)(超波前集）对于属于超分布 S{k}' 空间的 u (chi_0,xi_0) <不是成员> UWF^<{mu}>(u) <互变异构体> *_<epsilon> > O*C ; |X^u(xi)| <小于或等于> exp[epsilon < xi >^<1/mu>]，其中 X * S{k}*C^*_ 和 xi 属于圆锥邻域xi_0. 然后通过使用 UWP^<{mu}>(u) 我们可以描述非 C^* 适定柯西问题的解的非常高奇点的传播 我们还构造了退化柯西问题的基本解。双曲运算符 (2) L_1 = THETA^2_ - t^2_ - at^kTHETA^x 且 0 < k < j - 1 且 (3) L_2 = THETA^2_ - x^<2j>THETA^2_ - aTHETA^x 具有偶数整数 j，我们研究了其他相关主题。

项目成果

谷口和夫: "A fundamental solution for a degenerate hyperbolic operator of second order and Fourier integral operators of complex phase." 発表予定.

Kazuo Taniguchi：“二阶简并双曲算子和复相位傅立叶积分算子的基本解决方案。”

三谷佐孝: "On the compactness of extensions." Q and A in General Topology. 6. 103-106 (1988)

Sataka Mitani：“关于扩展的紧致性。” 6. 103-106 (1988)

石井伸郎: Acta Arith.54. (1990)

石井伸夫：算术学报.54 (1990)

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

一雄。

石井伸郎: "Ring class fields modulo 8 of Q(√)and the quartic character of units of Q(√)for m≡1 mod 8." Osaka J.Math.26. 625-646 (1989)

Nobuo Ishii：“域 Q(√<-m>) 的环类模 8 和 Q(√<m>) 单位的四次特征，其中 m≠1 mod 8。”Osaka J.Math.26。 (1989)