Studies on a new class of hyperbolic systems

我们得到了关于双曲双特征的分类的一个确定的结果，如果在参考点的哈密顿映射只允许纯虚特征值存在，则称为非有效双曲特征。剩下的一个基本问题是围绕非有效双曲特征的柯西问题是否C适定？我们将双曲特征归类为围绕参考双特征的零双特征的行为相对于双特征流形是否稳定，即双特征流形中是否存在具有极限点的零双特征。我们得到了如下结果：如果零双特征的行为围绕参考双特征，则主符号是初等可分解的，并且柯西问题是C-适定的。另一方面，如果零双特征的行为是不稳定的，则主符号不是初等可分解的，柯西问题不是C-适定的。我们得到了更详细的结果。在这种不稳定的情况下，柯西问题是Gevrey5适定的，并且这个指标5在如下意义下是最优的；如果在双特征流形中存在一个具有极限点的零双特征标，那么对于任何S和Gt；5，柯西问题都不是Gevrey S适定的。基于上述结果，我们得到了如下结果：假设双特征流形的余维为3，且Hamilton映射的所有特征值仍然是纯虚数的，则柯西问题是Gevrey 5适定的。

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.