Multiscale Wave Dynamics in Nonlinear Balance Laws

DMS-0505975，H Liu，Iowa State University题目：非线性平衡律中的多尺度波动动力学这个项目将研究由几个非线性、依赖时间的偏微分方程组(如欧拉-泊松系统和欧拉方程)控制的问题中的多尺度波动动力学。这些方程模拟了涉及多个长度尺度的各种物理过程，需要显著新的数学和数值方法。具体主题包括标度转换、模式形成以及基本物理标度下解的时间规律性。数学分析将包括标度极限的证明，波形的稳定性，以及计算这些问题的解的方案的数值分析。这一研究在各种物理过程中的等离子体鞘形成、高频波传播和频散波动力学等方面有着广泛的应用。预计拟议的工作不仅将对这些领域产生重大影响，而且还将对涉及多尺度波动动力学的其他领域产生重大影响，包括但不限于地球物理、材料科学和移动通信技术。我们的目标是开发新的数学工具和计算方法，以显著提高最先进的水平。这一理论将由已确定的实际应用来使用和推动。这一研究对非线性偏微分方程组的一般理论也具有重要意义。

Abstract DMS-0505975, H Liu, Iowa State UniversityTitle: Multi-scale Wave Dynamics in Nonlinear Balance Laws This project will study multi-scale wave dynamics in problems governed by several nonlinear, time-dependent partial differential equations such as the Euler-Poisson system and Euler equations. These equations model a variety of physical processes involving multiple length scales, requiring significantly new mathematical and numerical approaches. Specific topics include the scale transition, pattern formation as well as the time regularity of solutions in terms of fundamental physical scales. Mathematical analysis will include justification of scaling limits, stability of wave patterns, and numerical analysis of schemes for computing solutions of these problems. This research has many applications including plasma sheath formation, high frequency wave propagation and dispersive wave dynamics in a variety of physical processes. The proposed work is expected to have a significant impact not only on these areas, but also to other fields dealing with multi-scale wave dynamics including, but not restricted to geophysics, materials science and mobile telecommunication technologies. The goal is to develop new mathematical tools and computational methods that will significantly advance the state of the art. The theory will be used and driven by identified practical applications. It is also expected that this study will be of fundamental importance to the general theory of nonlinear partial differential equations.