Mathematical Sciences: Nonlinear Waves, Nonlinear Materials and Chaotic Mixing

9500568格罗夫这个项目寻求开发一种研究混合层、非线性材料和流体混沌的基本方法。双曲型守恒律中的非线性波通常对问题的形式和数值求解算法表现出敏感的依赖性。一个完整的双曲波理论应该包括以下几个部分：(1)描述波对其所在流动的影响的跳跃条件，(2)层的宽度或增长率，(3)内部结构的完整理论，以及(4)对波的所有可能的敏感依赖的系统展开。分析中使用的数学工具和理论多种多样，包括：偏微分方程(双曲守恒定律)、随机场、微扰理论(普通的和重整化的)、重整化群方法、行波分析、分叉理论和常微分方程组的几何理论。一个典型的混合层问题是由流体界面的不稳定性引起的加速度驱动层。这种混合层出现在许多基础科学和技术领域，包括超新星、惯性约束聚变和内燃机化油器中的喷射射流。详细的数学建模和分析将寻求修改和改进描述这种流动的定义数学方程，并更好地理解这些方程的结构。这一分析将被用来发展高分辨率数值方法来求解这些方程。我们研究的非线性材料波纹图描述了重要的金属成形过程，如冲切(剪切带)和注塑成型塑料部件(粘弹性材料)的流动不稳定性。分析非线性材料中的波使用类似的综合方法：建模、理论、计算和应用。与合作者的互动将允许将这项技术转移到适当的应用物理和工程社区。

9500568 Grove This project seeks to develop a fundamental approach to the study of mixing layers, nonlinear materials, and fluid chaos. Nonlinear waves in hyperbolic conservation laws commonly display sensitive dependence to the problem formulation and to numerical solution algorithms. A completed theory of a hyperbolic wave should contain the following ingredients: (1) jump conditions describing the influence of the wave on the flow in which it is embedded, (2) the width or growth rate of the layer, (3) a complete theory of the internal structure, and (4) a systematic unfolding of all possible sensitive dependencies for the wave. The mathematical tools and theories used in the analysis are varied, and include: partial differential equations (hyperbolic conservation laws), random fields, perturbation theory (ordinary and renormalized), renormalization group methods, traveling wave analysis, bifurcation theory, and the geometric theory of ordinary differential equations. A representative mixing layer problem is acceleration driven layers, arising in instabilities of a fluid interface. Such mixing layers arise in many areas of basic science and technology, including supernovae, inertial confinement fusion, and injection jets in carburetors for internal combustion engines. Detailed mathematical modeling and analysis will seek to modify and refine the defining mathematical equations describing such flows and develop an improved understanding of the structure of these equations. This analysis will in turn be used to develop high resolution numerical methods for the solution of these equations. The nonlinear material wave patterns we study describe important metal forming processes such as punching and cutting (shear bands) and flow instabilities in forming plastic components by injection molding (viscoelastic materials). The analysis of waves in nonlinear materials uses a similar integrated approach: modeling, theor y, computations, and applications. Interactions with collaborators will allow transfer of this technology to appropriate applied physics and engineering communities.