Symmetric systems and strongly hyperbolic systems

我们的研究项目安排如下：(I)澄清不能对称化的强双曲系统的结构；(Ii)研究双曲扰动下可对称化系统的稳定性。设L是一阶m*m次微分算子系。用h表示L主符号的行列式，我们得到的L强双曲性必要条件的一般图式可以表述为：如果L是强双曲的，则对L的每一个m-1次项k，都能正确地提出h+k的柯西问题。此外，如果参考特征z是对合的，且系统是强双曲的，则Kerl(Z)*iml(Z)={0}。因此，L沿着克尔展开的泰勒展开式从一个线性项L_Z开始，称为L的局部化。设z，w分别是原系统和局部化系统的特征。如果(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的维度极大，L Z(W)对角化，则称z为非退化的，则主要结果是每个双曲组都是可对称化的近非退化特征.由此我们可以得到非退化特征的稳定性。也就是说，我们不能用双曲摄动来消除非退化特征。我们进行了这项研究，得到了如下结果。设L是m×m对称一阶双曲组。然后，如果L的维数大于m(m+1)/2-m+2，则一般而言，每个双曲摄动都是平凡的，即L附近的每个双曲组都可以对称化。

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.