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Symmetric systems and strongly hyperbolic systems

对称系统和强双曲系统

基本信息

批准号：
07454027
负责人：
NISHITANI Tatsuo
金额：
$ 1.98万
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：
1995
资助国家：
日本
起止时间：
1995 至 1996
项目状态：
已结题

项目摘要

Our research project has been organized as follows :(i) Clarify the structure of strongly hyperbolic systems which can not be symmetrizable.(ii) Study the stability of symmetrizable systems under hyperbolic perturbations.As for (i) we got the following results. Let L be a m*m system of patial differential operators of first order. Denoting by h the determinant of the principal symbol of L the general picture of our necessary condition for strong hyperbolicity of L could be stated as : if L is strongly hyperbolic then the Cauchy problem for h+k is correctly posed for every m-1-th minor k of L.Moreover if the reference characteristic z is involutive and the system is strongly hyperbolic then KerL (z) * ImL (z) = {0}. Thus the Taylor expansion of L along KerL starts with a linear term L_Z called the localization of L.Let z, w be characteristics of the original system and of the localization respectively. If (z, w) is involutive then KerL_z (w) * ImL_z (w) = {0}.As for (ii) we formulated non degenerate characteristic for first order system. We say that z is non degenerate if KerL (z) * ImL (z) = {0}, the dimension of L_Z is maximal and L_Z (w) is diagonalizable for every w. Then the main result is that every hyperbolic system is symmetrizable near non degenerate characteristic. From this we can derive stability of non degenerate characteristics. Namely we can not remove non degenerate characteristics by hyperbolic perturbations.We proceed this study and got the following result. Let L be a m*m sysmmetric first order hyperbolic system. Then if the dimension of L is greater than m (m+1) /2-m+2 then genericaly, every hyperbolic perturbation is trivial that is every hyperbolic system near L can be symmetrized.

我们的研究项目组织如下：（i）阐明不可对称的强双曲系统的结构。（ii）研究可对称系统在双曲扰动下的稳定性。对于（i）我们得到了以下结果。设 L 是一阶偏微分算子的 m*m 系统。用 h 表示 L 的主符号的行列式，L 的强双曲性的必要条件的一般情况可以表述为：如果 L 是强双曲性，则对于 L 的每个 m-1 次 k，正确提出 h+k 的柯西问题。此外，如果参考特征 z 是对合的并且系统是强双曲性的，则 KerL (z) * ImL (z) = {0}。因此，L 沿 KerL 的泰勒展开以线性项 L_Z 开始，称为 L 的局部化。设 z、w 分别为原始系统和局部化的特征。如果 (z, w) 是对合的，则 KerL_z (w) * ImL_z (w) = {0}。对于 (ii)，我们为一阶系统制定了非简并特征。如果 KerL (z) * ImL (z) = {0}，L_Z 的维数最大且 L_Z (w) 对于每个 w 可对角化，则我们称 z 是非退化的。则主要结果是每个双曲系统在非简并特性上都是可对称的。由此我们可以得出非简并特性的稳定性。即我们不能通过双曲扰动去除非简并特征。我们进行了这项研究并得到了以下结果。设 L 为 m*m 对称一阶双曲系统。那么如果 L 的维数大于 m (m+1) /2-m+2 ，那么一般来说，每个双曲扰动都是微不足道的，即 L 附近的每个双曲系统都可以对称。