Multiscale Wave Dynamics in Nonlinear Balance Laws

摘要DMS-0505975，H Liu，爱荷华州州立大学题目：非线性平衡律中的多尺度波动动力学本项目将研究由几个非线性的、与时间相关的偏微分方程如欧拉-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，需要新的数学和数值方法。 具体的主题包括尺度转换，图案的形成以及基本物理尺度的解决方案的时间规律性。数学分析将包括比例限制的合理性，波型的稳定性，以及计算这些问题的解决方案的方案的数值分析。这一研究在等离子体鞘层的形成、高频波的传播以及各种物理过程中的色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，包括但不限于物理学、材料科学和移动的电信技术。我们的目标是开发新的数学工具和计算方法，这将显着推进最先进的国家。该理论将被使用，并通过确定的实际应用驱动。 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.