Two by two strongly hyperbolic systems and Gevrey classes
Two by two strongly hyperbolic systems and Gevrey classes
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二乘二强双曲系统和 Gevrey 类
DOI：
10.1007/bf02826087
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发表时间：
1999
期刊：
影响因子：
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通讯作者：
T. Nishitani
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文献类型：
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作者：
F. Colombini;T. Nishitani
本文讨论一阶系统L的柯西问题。若L（x，D）+ B（x）的Cauchy问题对每个B（x）都是C~适定的，则称L是强双曲的.对于一阶常系数系统，已知L是强双曲的当且仅当L是一致可对角化的（[7]，[11]）。这里我们说L是一致可对角化的，如果在每一个~处，符号L（~）可被S（~）对角化，且具有与~无关的一致界IIS（~）-1II，IIS（~）ll。我们的研究在这说明是出于这一事实，我们的主要兴趣在这说明是调查，在何种程度上，一致对角化意味着强双曲性的情况下，变系数。在[6]中，他证明了：如果L（x，~）是一致可对角化的，则柯西问题对~（2）中的任何B（x）都是适定的。另一方面，在[5]中，他们引入了伪对称系统的概念，并证明了若系统的系数仅解析地依赖于t和真实的，则当系统是伪对称的时，Cauchy问题在某个Sobolev空间中是适定的。我们还记得，t9 cRn中的C~函数f（x）属于Gevrey类~（s）（~ 9），si> 1，如果对任意紧集Kc~ 9有C~> 0使得
In this note the Cauchy problem for a first order system L will be discussed. If the Cauchy problem for L (x, D)+ B (x) is C~ well posed for every B (x) then L is called to be strongly hyperbolic. In case of first order systems with constant coefficients it is known that L is strongly hyperbolic if and only if L is uniformly diagonalizable ([7],[11]). Here we say that L is uniformly diagonalizable if at every~, the symbol L (~) is diagonalizable by S (~) with uniform bounds IIS (~)-1 II, IIS (~) ll independent of~. Our study in this note is motivated by this fact and our main interest in this note is to investigate, in what extent, uniform diagonalizability implies strong hyperbolicity in case of variable coefficients. In [6], he proved that if L (x,~) is uniformly diagonalizable then the Cauchy problem is well posed for any B (x) in~(2). On the other hand, in [5] they introduced a notion,, pseudosymmetric system-and proved that if the coefficients depend only on t and real analytically, then the Cauchy problem is well posed in some Sobolev space provided the system is pseudosymmetric. We recall that a C~ functionf (x) in t9 cR n belongs to the Gevrey class~(s)(~ 9), s I> 1 if for any compact set Kc~ 9 there is C~> 0 such that
T.Nishitani：“具有两个自变量的二乘二系统的双曲性” Comm.P.D.Es.23(5
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