高维双曲型方程的奇性与退化全局解的研究

项目名称： 高维双曲型方程的奇性与退化全局解的研究

项目编号： No.11471332

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 杨小舟

作者单位： 中国科学院武汉物理与数学研究所

项目金额： 68万元

中文摘要： 本项目对高维双曲守恒律方程的几个基础性的问题开展研究，包括：（1）通过对二维特征的包络面位置和形状的分析判断，来构造二维简化欧拉方程的全局解，并希望发现二维解的全局结构的新现象和典型现象；（2）构造二维等熵全欧拉方程的一类非自相似的二维稀疏波和一类二维全局理论解；（3）利用二维特征面的混合型包络面族的分类，来发现二维双曲型方程的解的新结构；（4）研究Chaplygin气体二维激波的绕射分析；（5）研究一类2×2的二维双曲型方程组的非自相似基本波的全局连接准则和连接方法。 本项目的特色，一是我们通过定义+包络、-包络和混合型包络，并通过对它们的分析、定位和求解，来发现二维全局解和二维基本波相互作用的典型结构及一般结构的分类,研究方法有新意和自己的东西；二是我们的研究将理论分析和数值计算结合在一起；三是我们要研究的问题是需要回答的一些基础性的问题，这些问题都具有很重要的意义和大的挑战性。

中文关键词： 二维全局解；二维基本波；相互作用；高维双曲型方程

英文摘要： Several foundermental problems of high dimensional hyperbolic conservation laws will be investigated in this project, these problems includ: (1) through the analysis of position and judgment of shape of the envelop surface for 2-D characteristics,we will construct the global solution two-dimensional simplified Euler equation, and hope to discover new phenomena and some typical phenomenon of 2-D global solution structure; (2) we will construct the 2-D non-selfsimilar rarefaction wave and a class of two dimensional global theoratical solutions of 2-D isentropic Euler equations; (3) we will discover the new global solution's structures by classifying mixed envelope family of the 2-D characteristics surface;(4) we will do the diffraction analysis of 2-D shock wave of Chaplygin gas; (5) we will study connection criteria and connection method of a class of non self similar global basic wave for 2 × 2 two dimensional hyperbolic equations . The characteristics of the project are, first one is that we defined + envelope, - envelope and mixed type envelope, and through the analysis of their positioning and shape to discover the basic classification of typical structure of two dimensional global solutions and the 2-D fundamental wave interaction and structure; the second one is that we will combine the theoretical analysis and numerical calculation together; the third one is that our research is the research of the problems of some foundermental problems which are very important with significance and challenge.

英文关键词： 2D Global solutions;2D Elementary Waves;Interaction;M-D Hyperbolic Equations