Dynamics of Fluid and Nonlinear Waves

微分方程用于对物理、工程、生物学、金融等广泛背景下产生的问题进行建模。出于双重原因，人们付出了巨大的努力来严格理解这些数学模型。一方面，通过理论分析结果与实验数据的比较，建立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在一定程度上得到了现有实验观察的支持，对这些理想模型的理论研究就可以提供通过实验难以获得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属性。其中包括一些特殊的结构，如稳态、周期和准周期解、混沌轨道等，以及它们的稳定性等定性性质。一般来说，一方面，系统中只有稳定状态是物理上可观察到的，而理想但不稳定的状态由于对参数的极其敏感的依赖性而很难观察到。另一方面，不稳定状态也极其重要，部分原因是它们及其一些相关结构充当了系统中不同稳定状态集合的边界。在这个项目中，PI计划重点研究几个经典非线性偏微分方程系统中接近稳态的局部动力学，这些系统都属于非线性波的一般范畴。缺乏先验阻尼和复杂的非线性给数学分析带来了大部分挑战。 PI 计划严格研究不可压缩欧拉方程的局部动力学以及一类一般的拟线性哈密顿偏微分方程。特别是，对于模拟水等无粘性不可压缩流体的不可压缩欧拉方程，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，包括一些非常基础的问题仍然没有得到很好的理解。 PI 计划重点关注平衡附近的局部动态结构，包括稳定性/不稳定、局部不变流形、特殊解、分岔和奇异扰动。虽然这些方面是平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，它们的解图在无限维相空间中通常不具有足够的平滑性，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规律性分析之间的关系是非线性偏微分方程动力学的重要分析方面。理解和解决这些问题预计主要基于其特定的机械和几何结构，将在这些领域带来实质性的理论进步，并可能导致在底层系统中发现新的物理和数学现象。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标准进行评估，被认为值得支持。

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.