Theory of hyperloobic systems

我们已经取得了很多成果。我们在这里提到的一些主要结果。1.对于两个独立变量的2 × 2系统，得到了Cauchy问题适定性的一个充分必要条件.该条件使用牛顿多面体表示。在本研究中，我们发现了一个特例，它是远离初始平面的严格双曲型，因为柯西问题对任何低阶项都不适定.引入了伪对称双曲组的概念，推广了可对称化双曲组的概念。本文证明了具有一个空间变量的拟对称双曲型方程组的柯西问题是适定的。对于多个空间变量的伪对称系统，这个问题仍然是开放的.我们成功地获得了一个必要条件的低阶项的柯西问题是适定的一般，双曲型系统使用的行列式的非交换域的本地化生活：。全符号局部化的非交换行列式的前半部分与主符号经典行列式的主半部分重合.当降维足够高时，冻结系统在每个空间点的对称性蕴涵着原系统的对称性。特别地，如果每个冻结系统是弦双曲的，那么当降维很高时，原系统也是强双曲的。

We have obtained a lot of results. We refer here some of the main results.1. For 2 x 2 systems with two independent variables, we obtained a necessary and sufficient condition in order that the Cauchy problem is well posed. The condition is expressed using the Newton polyhedron. In this study we found a peculiar example which is strictly hyperbolic apart from the initial plane for that the Cauchy problem is not well posed for any lower order term.2. We introduced a new notion "pseudo-symmetric hyperbolic systems" which extends the symmetrizable hyperbolic systems. We proved that the Cauchy problem for pseudo-symmetric hyperbolic systems with one space variable is well posed. The question is still open for pseudo-symmetric systems with several space variables.3. We succeeded in obtaining a necessary condition on lower order terms for the Cauchy problem is well posed for general, hyperbolic systems using the determinant on a non commutative field where the localization lives : the. leading part of the non commutative determinant of the localization of the total symbol coincides with the principal part of the classical determinant of the principal symbol.4. The symmetrizability of the frozen system at every space point implies the symmetrizability of the original systm if the reduced dimension is enough high. In particular if the every frozen system is stringly hyperbolic then the original system is also strongly hyperbolic if the reduced dimension is high.