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.

我们的研究项目的组织如下：（i）澄清强烈双曲系统的结构，这些系统无法对称。（ii）研究双曲线扰动下可对称系统的稳定性。（i）我们得到以下结果。令L为一阶的Patial差异操作员的M*M系统。用h表示L的主要符号的决定因素，我们的强烈双曲性l的一般状况可以说为：如果L强夸张，那么H+K的Cauchy问题对于每个M-1-1-1-then Minor k的L. More primover pormortivation z的每个M-1-1-次要k都正确地构成。因此，L沿KERL的泰勒膨胀以线性项为l_z开始，称为L.LET Z的定位，分别是原始系统和定位的特征。如果（z，w）是涉及的，则kerl_z（w） * iml_z（w）= {0}。我们说，如果kerl（z） * iml（z）= {0}，z是不退化的。然后主要的结果是，每个双曲系统在非退化特征附近都可以对称。由此，我们可以得出非退化特征的稳定性。即，我们无法通过双曲线扰动来消除非退化特征。我们继续进行这项研究并得到以下结果。令L为M*M Sysmmetric一级双曲线系统。那么，如果L的尺寸大于M（M+1） /2-M+2，则一般，每个双曲线扰动都是微不足道的，可以对称L附近L附近的每个双曲线系统。