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。剩下的一个基本问题是，围绕非有效双曲线特征的库奇问题是否符合c- c-Infty？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了分类，相对于双重特征，这是稳定的是否存在一个无效的双分裂症，并具有双重特征歧管中的极限点。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，如果零双分裂症的行为不稳定，则主要符号不是基本的分解，而库奇的问题也不是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限度的双分裂症，那么凯奇问题对于上述结果的任何s> 5> cauchy问题并不是Gevrey a posp的良好，我们获得了以下结果：假设：假设：双重特征歧管为3，汉密尔顿图的所有特征值仍然是纯粹的想象，因此凯奇的问题是gevrey 5良好。