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Study of Asymptotic Analysis of Systems of Quasilinear Partial Differential Equations

拟线性偏微分方程组渐近分析的研究

基本信息

批准号：
62460005
负责人：
YOSHIKAWA Atsushi
金额：
$ 3.58万
依托单位：
Kyushu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (B)
财政年份：
1987
资助国家：
日本
起止时间：
1987 至 1989
项目状态：
已结题

项目摘要

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.

本文的主要目的是通过描述拟线性双曲型偏微分方程系统的渐近解的性质，阐明拟线性与双曲性合流的影响。首先，我们发展了弱解的形式渐近理论。我们证明了，即使是高维的守恒律系统，当限制任何特定的相表面时，也能观察到一维守恒律的结构。另一方面，我们注意到扩展一维研究中基本的简单波的概念的可能性。其次，我们开始对小数据强解进行严格的渐近分析。形式解可以根据类似于线性情况的方法构造。然而，传统的估计只保证对假定的余数项和形式展开的顺序相同。为了克服这个困难，我们打算应用一个修正的硬隐函数定理，它在Sobolev尺度上是有效的。完整的细节还有待撰写，目前，已经编制了处理索博列夫尺度非线性所需的几个相关估计。证明了上述修正隐函数定理在实插值空间中是有意义的。