Study of Asymptotic Analysis of Systems of Quasilinear Partial Differential Equations

本研究的主要目的是通过描述拟线性双曲型偏微分方程组的渐近解的行为来阐明拟线性和双曲性之间的共存效应。首先，我们发展了弱解的形式渐近理论。我们证明了，即使对于高维守恒律系统，当约束任何特定的相面时，也能观察到一维守恒律的结构。另一方面，我们注意到了推广一维研究中基本的简单波概念的可能性。其次，我们用小数据开始了强解的严格渐近分析。形式解可以按照类似于线性情形的方法来构造。然而，传统的估计只保证假定的余项和形式展开式的阶数相同。为了克服这一困难，我们计划应用一个修正的硬隐函数定理，该定理在索波列夫尺度下是有效的。完整的细节还没有写出来，目前，已经汇编了处理索博列夫标度中的非线性所需的几个相关估计。认为上述修正的隐函数定理在实插值空间的框架下是有意义的。

The main purpose of the present study is to clarify effect of concurrence between quasilinearity and hyperboldicity through description of behaviors of asymptotic solutions of systems of quasilinear hyperbolic partial differential equations.In the first place, we developed formal asymptotic theory for weak solutions. We showed that, even for higher dimensional systems of conservation laws, the structure of 1-dimensional conservation law is observed when restricted any particular phase surface. On the other hand, we noticed possibility of extending the notion of simple waves which are basic in the 1-dimensional study.In the second place, we started rigorous asymptotic analysis of strong solutions with small data. Formal solutions can be constructed according to the method similar to the linear case. However, the conventional estimates only assure the same order for the presumed remainder term and the formal expansion. To overcome this difficulty, we are planning to apply a modified hard implicit function theorem which is valid in the Sobolev scale. The full details are yet to be written, and for the moment, several related estimates required for handling nonlinearity in the Sobolev scale are compiled. It is believed that the above mentioned modified implicit function theorem makes sense in the frame of real interpolation spaces.