Hyperbolic systems with double characteristics
Hyperbolic systems with double characteristics
复制标题
具有双重特性的双曲系统
DOI：
10.1002/cpa.3160460207
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发表时间：
1993
影响因子：
3
通讯作者：
L. Hörmander
中科院分区：
文献类型：
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作者：
L. Hörmander
约翰(见[4])有了一个令人惊讶的发现，尽管标量双曲型微分方程总是严格双曲型的极限情况(参见[4])。Nuij，[6])，有这样的双曲型系统，其没有小的扰动是严格双曲的。他的结果涉及具有三个空间变量和三个未知数的系统，这些系统与弹性力学方程密切相关。John证明了在二阶实系数系统的空间中，具有实系数的邻近双曲型系统形成余维为4的流形，并且没有这样的系统是严格双曲的。在本文中，我们将给出John在[4]中的结果的另一种证明和推广。特别是，我们将证明，当复系数被允许时，情况是完全不同的。John所考虑的对称化系统就是严格双曲型系统的极限(定理5.7)。然而，当允许有四个空间变量和复系数时，他观察到的三个空间变量的现象再次出现。在我们的公式中，[4]的结果依赖于圆上存在一个非平凡的实线丛，即莫比乌斯带。复数情形下的类似结果依赖于二维球面S2上存在非平凡复数线丛这一事实。第二节致力于利用这些事实的稳定性定理。约翰(见[4])的思想是第三节、第四节和第五节中研究各类非对称双曲组的基本思想。每一节都从讨论一类对称双曲组开始，然后考察其非对称扰动。第三节主要研究具有三个空间变量和3×3矩阵系数的一阶实系数线性双曲组。在对称的情况下，它们一般具有四个或两个非退化双重特征。具有四个双重特征的那些在某种意义上是等价的，它们只是通过变量的改变和可逆矩阵的左右相乘而不同。即使是非对称双曲摄动也不再是一般的(定理3.5)。具有两个非退化双特征的系统依赖于一个参数，其非对称双曲摄动依赖于两个参数。它们被明确地列出(定理3.6)。第四节是一阶复系数线性双曲组的类比研究，主要具有四个空间变量和3×3个矩阵系数。在厄米特对称的情况下，我们
F. John (see [4]) made the surprising discovery that although scalar hyperbolic differential equations are always limiting cases of strictly hyperbolic ones (cf. Nuij,[6]), there are hyperbolic systems for which no small perturbation is strictly hyperbolic. His result concerned systems with three space variables and three unknowns, closely related to the equations of elasticity. John proved that nearby hyperbolic systems with real coefficients form a manifold of codimension 4 in the space of second-order systems with real coefficients, and that no such systems are strictly hyperbolic. In this paper we shall give alternative proofs and extensions of the results of John in [4]. In particular, we shall show that the situation is quite different when complex coefficients are allowed. The symmetrizable systems considered by John are then limits of strictly hyperbolic ones (Theorem 5.7). However, the phenomenon he observed for three space variables reappears when there are four space variables and complex coefficients are allowed. The results of [4] depend in our formulation on the fact that there is a non-trivial real line bundle on the circle, the Mobius strip. The analogous results in the complex case depend on the fact that there are non-trivial complex line bundles on the two-dimensional sphere S2. Section 2 is devoted to stability theorems exploiting these facts.The ideas of John (see [4]) as developed in Section 2 are basic in the study of various classes of non-symmetric hyperbolic systems in Sections 3, 4, and 5. Each of these sections begins with a discussion of a class of symmetric hyperbolic systems whose non-symmetric perturbations are then examined. Section 3 is mainly devoted to first-order linear hyperbolic systems with real coefficients, primarily with three space variables and 3 x 3 matrix coefficients. In the symmetric case they have generically either four or two non-degenerate double characteristics. Those with four double characteristics are equivalent in the sense that they only differ by a change of variables and left and right multiplication by invertible matrices. Even non-symmetric hyperbolic perturbations are no more general (Theorem 3.5). The systems with two nondegenerate double characteristics depend on one parameter, and their nonsymmetric hyperbolic perturbations depend on two parameters. They are listed explicitly (Theorem 3.6). Section 4 is an analogous study ofjrst-order linear hyperbolic systems with complex coefficients, primarily with four space variables and 3 x 3 matrix coefficients. In the hermitian symmetric case we