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On Gevrey property and asymptotic analysis for divergent solutions of analytic partial differential equations

解析偏微分方程发散解的Gevrey性质及渐近分析

基本信息

批准号：
07454024
负责人：
MIYAKE Masatake
金额：
$ 3.01万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1995
资助国家：
日本
起止时间：
1995 至 1996
项目状态：
已结题

项目摘要

The purpose of this project is to study Gevrey property and asymptotic analysis for divergent power series solutions of analytic partial differential equations.The first result is that we established the Toeplitz operator method in analytic partial differential equations which enables us to give a precise notion of Fredholmness in the Goursat problem in various Gevrey spaces which has not been studied in explicite way in early studies. We proved further that an index formula for ordinary differnetial operator on Gevrey space is nothing but the geometrical index fornmula for a Toeplitz sumbol associated with the Gevrey filtration n the ring of ordinary differential operators.The second result is that we gave a necessary and sufficient condition for the Borel summability for divergent power series solution of the Cauchy problem of the heat equation, and we proved that the Borel sum is just expressed by an integral with the heat kernel. The condition for the C data we obtained is the well known condition for the uniqueness of solutions of the Cauchy problem. In proving this we made clear that the problem of Borel summability in partial differential equations is not local property of solutions, whereas the notion of the Borel summability is only local one. We also made clear that this problem provides a new kind of problems in partial differential equations.

本项目的目的是研究解析偏微分方程发散幂级数解的Gevrey性质和渐近分析，第一个结果是建立了解析偏微分方程的Toeplitz算子方法，使我们能够在各种Gevrey空间中给出Goursat问题的Fredholmness的精确概念，这在早期的研究中还没有明确的研究。进一步证明了Gevrey空间上常微分算子的指数公式就是常微分算子环上与Gevrey滤子相联系的Toeplitz求和的几何指数公式.第二个结果是给出了热方程Cauchy问题的发散幂级数解的Borel求和的充要条件，证明了Borel和正好可以用热核积分表示。我们得到的C数据的条件是著名的柯西问题解的唯一性条件。在证明这一点时，我们明确了偏微分方程的Borel可和性问题不是解的局部性质，而Borel可和性的概念只是局部性质。我们也清楚地表明，这个问题提供了一类新的偏微分方程问题。