Study of Asymptotic Analysis of Systems of Quasilinear Partial Differential Equations

本文的主要目的是通过对拟线性双曲型偏微分方程组渐近解的性质的描述，阐明拟线性与双曲性共存的效应。我们证明了，即使对于高维守恒律系统，当限制任何特定的相表面时，也可以观察到一维守恒律的结构。另一方面，我们注意到推广一维研究中基本的简单波概念的可能性。其次，我们开始了小数据下强解的严格渐近分析。形式解可以根据类似于线性情况的方法构造。然而，传统的估计只能保证假定余项和形式展开式的阶数相同。为了克服这个困难，我们计划应用修改后的硬隐函数定理，这是有效的索伯列夫规模。完整的细节还没有写出来，目前，在索博列夫尺度中处理非线性所需的几个相关估计被编译。我们认为，在真实的插值空间的框架下，上述修正的隐函数定理是有意义的。

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.