Studies on a new class of hyperbolic systems

一类新型双曲系统的研究

基本信息

批准号：
15340044
负责人：
NISHITANI Tatsuo
金额：
$ 6.46万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

项目摘要

We have obtained a definitive result about the classification of hyperbolic double characteristics.A hyperbolic double characteristic is called non effectively hyperbolic characteristic if the Hamilton map at the reference point admits only pure imaginary eigenvalues. A remaining fundamental question was whether the Cauchy problem around non effectively hyperbolic characteristic is C-infty well-posed?We classify hyperbolic double characteristics whether the behavior of null bicharacteristics around the reference double characteristic is stable with respect to the doubly characteristic manifold, that is whether there exists a null bicharacteristic with a limit point in the doubly characteristic manifold. We have obtained the following results:If the behavior of null bicharacteristics around the reference double characteristic then the principal symbol is elementary decomposable and the Cauchy problem is C-infty well-posed. On the other hand, if the behavior of null bicharacteristic is unstable then the principal symbol is not elementary decomposable and the Cauchy problem is not C-infty well-posed. We obtained more detailed results. In this unstable case the Cauchy problem is Gevrey 5 well-posed and this index 5 is optimal in the following sense; if there is a null bicharacteristic with a limit point in the doubly characteristic manifold then the Cauchy problem is not Gevrey s well-posed for any s>5.Based on the above results, we obtained the following result : assume that the codimension of the doubly characteristic manifold is 3 and the all eigenvalues of the Hamilton map remain to be pure imaginary then the Cauchy problem is Gevrey 5 well-posed.

我们已经获得了双曲双曲双特征分类的确切结果。如果参考点的汉密尔顿映射仅接受纯粹的假想特征值，则双曲双重特征被称为非有效双曲线特征。一个剩下的基本问题是，围绕非有效双曲线特征的库奇问题是否适合？我们对参考双重特征的无效双分细胞的行为对双重特征的行为进行分类，而对双重特征的表现是否稳定，那就是是否存在一个Null Bicharacteristic the null Bicharacteristic comptimitiant the limippoint the Dimplemist the Dormplay cormbletive doutivestiant。我们已经获得了以下结果：如果围绕参考双重特征的无效双分细胞的行为，那么主符号是基本的分解，而库奇问题的问题是c-infty posed良好。另一方面，如果零双分细胞的行为不稳定，则主要符号不是基本的分解，而Cauchy的问题也不是C-Infty afty poss。我们获得了更详细的结果。在这种不稳定的情况下，凯奇的问题是gevrey 5良好的问题，而该指数5在以下意义上是最佳的。如果在双重特征歧管中有一个无数的双分裂症，那么考奇的问题并不是gevrey在上述结果上的任何> 5.的范围，我们获得了以下结果，我们获得了以下结果：假设双重特征的歧管的编构态为3，而Hamilton Map的所有特征映射仍然是纯粹的想象中的所有问题。