Multiscale Wave Dynamics in Nonlinear Balance Laws

摘要 DMS-0505975，H Liu，爱荷华州立大学标题：非线性平衡定律中的多尺度波浪动力学 该项目将研究由几个非线性、与时间相关的偏微分方程（例如欧拉-泊松系统和欧拉方程）控制的问题中的多尺度波浪动力学。这些方程模拟了涉及多个长度尺度的各种物理过程，需要全新的数学和数值方法。 具体主题包括基本物理尺度上的尺度转变、模式形成以及解的时间规律性。数学分析将包括尺度限制的合理性、波浪模式的稳定性以及这些问题的计算解决方案的数值分析。这项研究有许多应用，包括各种物理过程中的等离子体鞘层形成、高频波传播和色散波动力学。拟议的工作预计不仅会对这些领域产生重大影响，而且还会对处理多尺度波浪动力学的其他领域产生重大影响，包括但不限于地球物理学、材料科学和移动电信技术。目标是开发新的数学工具和计算方法，以显着推进现有技术水平。该理论将由已确定的实际应用来使用和驱动。 预计这项研究将对非线性偏微分方程的一般理论具有根本性的重要性。

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.