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Dynamics of Fluid and Nonlinear Waves

流体动力学和非线性波

基本信息

批准号：
1900083
负责人：
Chongchun Zeng
金额：
$ 30.22万
依托单位：
Georgia Tech Research Corporation
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-15 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900083&HistoricalAwards=false
关键词：
Dynamics Fluid Nonlinear Waves

项目摘要

Differential equations are used to model problems arising from a broad background including physics, engineering, biology, finance, etc. Great efforts are made to understand these mathematical models rigorously for a twofold reasons. On the one hand, the validity and the relevance of these ideal models are established through the comparison between the results from theoretical analysis and the experimental data. On the other hand, once the meaningfulness of a mathematical model is supported by available experimental observations to certain degree, the theoretical studies on these ideal models can provide properties and predictions of the original problems that are hard to be obtained through experiments. For systems involving the temporal evolution, of particular interests are those structural and asymptotic properties. These include some special structures, such as steady states, periodic and quasi-periodic solutions, chaotic orbits etc, as well as their qualitative properties like stability etc. In general, on the one hand, only stable states are physically observable in a system, while the ideal, but unstable, states are hardly observed due to their extremely sensitive dependence on the parameters. On the other hand, unstable states are also extremely important, partly due to the fact that they and some of their associated structures serve as the boundaries separating different collections of stable states in a system. In this project, the PI plans to focus on the local dynamics near steady states in several classical nonlinear partial differential equation systems, which all belong to the general category of nonlinear waves. The lack of a priori damping and the complicated nonlinearity pose most of the challenges in their mathematical analysis. The PI plans to rigorously investigate the local dynamics of the incompressible Euler equation as well as a general class of quasi-linear Hamiltonian partial differential equations. In particular, for the incompressible Euler equation, which models inviscid incompressible fluids such as water, the proposed problems include fluids in rigid containers and fluids with free surfaces like ocean waves. Even though there have been extensive studies on these systems and great progresses have been made in recent year, due to their very complex nature, many issues including some very fundamental ones are still not well understood after years of efforts. The PI plans to focus on their local dynamic structures near equilibria, including stability/instability, local invariant manifolds, special solutions, bifurcations, and singular perturbations. While these aspects are standard notions in the theory of smooth dynamical systems, due to the highly nonlinear nature of these partial differential equations, their solution maps often do not have sufficient smoothness in the infinite dimensional phase spaces for the classical theory to apply directly. In contrast to ordinary differential equations, the relationship between the qualitative structures and the regularity analysis of these nonlinear partial differential equations is an essential analytical aspect of nonlinear partial differential equation dynamics. Understanding and solving these problems, expected to be largely based on their specific mechanical and geometric structures, would result in substantial theoretical advances in these areas and possibly lead to the discovery of new physical and mathematical phenomena in the underlying systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

微分方程被用来模拟问题所产生的广泛背景，包括物理，工程，生物学，金融等作出巨大努力，以了解这些数学模型严格的双重原因。一方面，通过理论分析结果与实验数据的对比，验证了这些理想模型的有效性和相关性。另一方面，一旦数学模型的意义得到了实验观测的一定程度的支持，对这些理想模型的理论研究就可以提供实验难以得到的原始问题的性质和预测。对于涉及时间演化的系统，特别感兴趣的是那些结构和渐近性质。这些包括一些特殊的结构，如稳定状态，周期和拟周期解，混沌轨道等，以及它们的定性性质，如稳定性等。一般来说，一方面，只有稳定的状态是物理上可观察的系统，而理想的，但不稳定的状态，很难观察到由于其极其敏感的参数依赖。另一方面，不稳定态也是非常重要的，部分原因是它们和它们的一些相关结构作为系统中不同稳定态集合的边界。在这个项目中，PI计划专注于几个经典的非线性偏微分方程系统中的稳态附近的局部动力学，它们都属于非线性波的一般范畴。先验阻尼的缺乏和复杂的非线性对其数学分析提出了最大的挑战。PI计划严格研究不可压缩欧拉方程的局部动力学以及一般类的准线性哈密顿偏微分方程。特别是，对于不可压缩的欧拉方程，它模拟无粘不可压缩流体，如水，所提出的问题包括刚性容器中的流体和具有自由表面的流体，如海浪。尽管近年来人们对这些系统进行了广泛的研究并取得了很大的进展，但由于其非常复杂的性质，经过多年的努力，许多问题包括一些非常基本的问题仍然没有得到很好的理解。PI计划专注于平衡点附近的局部动态结构，包括稳定性/不稳定性，局部不变流形，特殊解，分叉和奇异摄动。虽然这些方面是光滑动力系统理论中的标准概念，但由于这些偏微分方程的高度非线性性质，它们的解映射在无限维相空间中通常不具有足够的光滑性，无法直接应用经典理论。与常微分方程相比，这些非线性偏微分方程的定性结构和正则性分析之间的关系是非线性偏微分方程动力学的一个重要分析方面。理解和解决这些问题，预计将在很大程度上基于其特定的机械和几何结构，该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查进行评估，被认为值得支持的搜索.