Theory of hyperloobic systems

我们已经取得了很多成果。我们在这里引用一些主要的结果。对于具有两个自变量的2 × 2系统，我们得到了柯西问题适定的一个充分必要条件。该条件用牛顿多面体表示。在这个研究中，我们发现了一个特殊的例子，除了初始平面外，它是严格双曲的，因为柯西问题对于任何低阶项都不能很好地提出。引入了伪对称双曲系统的概念，扩展了可对称双曲系统的概念。证明了具有一个空间变量的伪对称双曲系统的Cauchy问题是适定的。这个问题对于具有多个空间变量的伪对称系统仍然是开放的。我们利用非交换域上的行列式，成功地得到了柯西问题在一般双曲型系统中被很好地定域的一个低阶项的必要条件：总符号的局部非交换行列式的前导部分与主符号的经典行列式的主部重合。如果降维足够高，冻结系统在每个空间点的对称性就意味着原始系统的对称性。特别是，如果每个冻结系统都是弦双曲的，那么如果降维很高，原始系统也是强双曲的。

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.