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Research on Systems of Partial Differential Equations Appling the Abstract Algebra

应用抽象代数研究偏微分方程组

基本信息

批准号：
13640196
负责人：
MATSUMOTO Waichiro
金额：
$ 1.09万
依托单位：
Ryukoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640196/
关键词：
formal symbol normal form of systems Cauchy problem Caufhy-Kowalevskaya theorem strongly hyperbolic systems system of Fuchs type Kac problem minimal curvature energy curve 偏微分方程式系の標準形フックス型偏微分方程式系 j曲率方程式

项目摘要

We have three themes. The first is to establish the necessary and sufficient condition for the Cauchy-Kowalevskaya theorem for systems of partial differential equations including the generalization to the, Nagumo type. On the original Cauchy-Kowalevskaya theorem, we obtained clearer proof. On the theorem of Nagumo type, we obtained a proof on the necessity and also a proof on the sufficiency in a special case. The second is the characterization of the strong hyperbolicity on systems. On the example which is pointwisely diagonalizable but not strongly hyperbolic by Petrovsky, we have already known that it changes to a strongly hyperbolic system by any hyperbolic perturbation. We showed that this phenomenon occurs generally for the systems with time-dependent coefficients. To show this, we apply the solvability of the Cauchy problem for the systems of Fuchs type. We also succeeded the generalization of the structure of the solvability. The third is the solvability of the Cauchy problem for p-parabolic systems to the future. Unfortunately, we obtained an idea to solve this problem, but finally we cannot achieve it as the complete form. In these researches, the comparison between the calculation on the non-commutative ring of the meromorphic formal symbols and that on the holomorphic pseude-differential operators has played an essential role.At first, the Kac problem is not the theme of this project. We obtained a good knowledge on the non-commutative groups through the research on the determinatnt theory on non-commutative ring and it brings a viewpoint on the framework of the existence of the counterexamples on the Kac problem by Sunada. As a result, we obtained concrete example of the domains which change from convex to nonconvex smoothly and for which the Kac problem is affirmative by proving mathematically Watanabe's conjecture by the numerical try. This is the first offer of a nonconvex domain for which the Kac problem is affirmative.

我们有三个主题。首先，建立了偏微分方程组的Cauchy-Kowalevskaya定理成立的充分必要条件，包括对Nagumo型的推广。在原来的Cauchy-Kowalevskaya定理上，我们得到了更清晰的证明。在Nagumo型定理上，我们得到了在一个特殊情况下的必要性证明和充分性证明。二是系统强双曲性的表征。对于Petrovsky的点对角化但不是强双曲的例子，我们已经知道它在任何双曲扰动下都会变成强双曲系统。我们证明了这种现象通常发生在具有时间相关系数的系统中。为了证明这一点，我们将柯西问题的可解性应用于Fuchs型系统。我们还成功地推广了可解性的结构。第三是p-抛物型方程组的柯西问题的可解性。不幸的是，我们得到了一个解决这个问题的想法，但最终我们无法实现它作为一个完整的形式。在这些研究中，亚纯形式符号非交换环上的计算与全纯伪微分算子上的计算的比较起到了重要的作用。首先，Kac问题并不是这个项目的主题。通过对非交换环上的决定理论的研究，我们对非交换群有了较好的认识，并提出了Sunada关于Kac问题反例存在性的框架观点。通过对Watanabe猜想的数学证明，得到了从凸到非凸平滑变化且Kac问题是肯定的区域的具体实例。这是第一次提出一个Kac问题是肯定的非凸域。