Symmetric systems and strongly hyperbolic systems

对称系统和强双曲系统

• 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
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07454027/
• 关键词: symmetrizable system; strong hyperbolicity; involutive; non degenerate; hyperbolic perturbation; localization; 対称化; 非退化特異点

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.

我们的研究项目如下：（i）阐明了不能对称化的强双曲方程组的结构。(ii)研究可对称化系统在双曲扰动下的稳定性，对于（i）我们得到了如下结果。设L是m × m的一阶空间微分算子系统.用h表示L的主符号的行列式，我们的L的强双曲性的必要条件的一般描述可以表述为：如果L是强双曲的，那么对于L的每一个m-1次子式k，h+k的柯西问题是正确的。此外，如果参考特征z是对合的，并且系统是强双曲的，那么KerL（z）* ImL（z）= {0}。因此，L沿着KerL的泰勒展开从称为L的局部化的线性项L_Z开始。若（z，w）是对合的，则KerL_z（w）* ImL_z（w）= {0}.对于（ii），我们给出了一阶系统的非退化特征.如果KerL（z）* ImL（z）= {0}，L_Z的维数是最大的，且L_Z（w）对任意w都可对角化，则称z是非退化的.主要结果是每一个双曲型方程组在非退化特征附近都是可对称化的。由此我们可以导出非退化特征的稳定性。即不能用双曲型摄动来消除非退化特征。设L是一个m × m对称一阶双曲型方程组.则如果L的维数大于m（m+1）/2-m+2，则一般地，每个双曲型扰动都是平凡的，即在L附近的每个双曲型方程组都可以对称化.

项目成果

T. Nishitani: "On localization of a class of strongly hyperbolic systems" Osaka J. Math.32・1. 41-69 (1995)

T. Nishitani：“关于一类强双曲系统的定位”Osaka J. Math.32・1（1995）。

T.Nishitani: "On localization of a class of strongly hyperbolic systems" Osaka J.Math.32・1. 41-69 (1995)

T.Nishitani：“关于一类强双曲系统的定位”Osaka J.Math.32・1（1995）。

Tatsuo Nishitani: "On localizations of a class of strongly hyperbolic systems" Osaka Journal of Mathematics. 32. 41-69 (1955)

Tatsuo Nishitani：“关于一类强双曲系统的本地化”大阪数学杂志。

T.Nishitani: "Symmetrization of hyperbolic systems with non degenerate characteristics" J.Func.Analysis. 132・2. 251-272 (1995)

T.Nishitani：“具有非简并特征的双曲系统的对称性”J.Func.Analysis 132・2（1995）。

T. Nishitani: "Stubility of symmetric systems urder hyperbolic perturbations" Hokkaido Math. J.26・1. (1997)

T. Nishitani：“双曲扰动的对称系统的稳定性”北海道数学 J.26・1（1997）。