Research on Systems of Partial Differential Equations Appling the Abstract Algebra

我们有三个主题。第一个是建立了偏微分方程组Cauchy-Kowalevskaya定理的充分必要条件，包括对Nagumo型的推广。在原有的Cauchy-Kowalevskaya定理的基础上，得到了更清晰的证明。在Nagumo型定理的基础上，给出了一个必要性证明，并在一种特殊情况下给出了充分性证明。第二部分是系统强双曲性的刻画。对于Petrovsky提出的点态可对角化但非强双曲的例子，我们已经知道，它通过任何双曲扰动都变成强双曲系统。我们发现，这种现象普遍发生的系统与时间相关的系数。为了证明这一点，我们应用的Cauchy问题的Fuchs型系统的可解性。我们还成功地推广了可解性的结构。第三个问题是p-抛物方程组柯西问题的未来可解性。不幸的是，我们得到了一个想法来解决这个问题，但最终我们不能实现它作为完整的形式。在这些研究中，亚纯形式符号的非交换环上的计算与全纯伪微分算子上的计算的比较起了至关重要的作用。通过对非交换环上决定性理论的研究，我们对非交换群有了很好的认识，并对Sunada关于Kac问题的反例的存在性框架提出了看法。通过数值试验，对Watanabe猜想进行了数学证明，得到了从凸到非凸光滑变化的区域的具体例子，并证明了Kac问题是肯定的。这是第一次提供一个非凸域的卡茨问题是肯定的。

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.