Dynamical System Approach in Partial Differential Equations

偏微分方程中的动力系统方法

• 批准号: 1645673
• 负责人: Wenxian Shen
• 金额: $ 8.84万
• 依托单位: Auburn University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2021-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1645673&HistoricalAwards=false
• 关键词: Dynamical; System; Approach; Partial; Differential

The main goal of this project is to study dynamical behavior of special solutions for some hyperbolic type partial differential equations (PDEs), which include nonlinear wave equation, nonlinear Schrodinger equation and compressible Euler equation. These equations are widely used to model various problems arising from physics, engineering, biology, finance, etc. The PI aims to construct certain special solutions for those aforementioned equations and their perturbations to understand the modeled phenomena. The perturbations considered in this project have either different scales or randomness, which requires new mathematical ideas and tools to handle. The successfully constructed solutions will not only provide new theoretical results but also stimulate efficient numerical algorithms to compute solutions in multi-scale and stochastic dynamical systems. More specifically, the PI intends to study the following problems rigorously. The first category of problems is to construct special dynamical structures for three kinds of hyperbolic PDEs, which include a system of nonlinear wave equations, the cubic focusing Schrodinger equation and quasilinear Schrodinger equation in high dimensions. All these problems have multiple spatio-temporal scales, which requires new mathematical ideas and analytical tools to handle. The next problem is to study the persistence of Sine-Gordon breather under multiplicative noise perturbation. The main aim is to understand the effect of stochastic forcing. A new method stemming from invariant manifold, random dynamical system and ergodic theory needs to be developed. The last problem in this project is to study the asymptotic behavior of solutions near some steady states for isentropic and non-isentropic compressible flows by using ideas of invariant manifold. The PI will extend existing ideas and develop techniques both on geometric and analytic aspects to provide not only a solution but also a new perspective of this problem.

本课题的主要目的是研究一类双曲型偏微分方程（包括非线性波动方程、非线性薛定谔方程和可压缩欧拉方程）的特解的动力学行为。这些方程被广泛用于模拟物理、工程、生物、金融等领域的各种问题。PI旨在为上述方程及其扰动构造某些特殊解，以理解所模拟的现象。在这个项目中考虑的扰动要么有不同的尺度，要么有随机性，这需要新的数学思想和工具来处理。成功构建的解不仅提供了新的理论结果，而且为求解多尺度随机动力系统提供了有效的数值算法。更具体地说，PI打算严格研究以下问题。第一类问题是构造三种双曲偏微分方程的特殊动力结构，包括非线性波动方程系统、三次聚焦薛定谔方程和高维拟线性薛定谔方程。所有这些问题都具有多重时空尺度，需要新的数学思想和分析工具来处理。下一个问题是研究正弦戈登呼吸器在乘性噪声扰动下的持久性。主要目的是了解随机强迫的影响。从不变流形、随机动力系统和遍历理论出发，提出了一种新的求解方法。本课题的最后一个问题是利用不变流形的思想研究等熵和非等熵可压缩流在某些稳态附近解的渐近行为。PI将扩展现有的思想和发展几何和分析方面的技术，不仅提供解决方案，而且提供这个问题的新视角。