Dynamics of inviscid fluids and nonlinear waves

偏微分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。严格理解这些数学模型的努力有两个目的。一方面，通过理论分析结果与实验观测结果的比较，建立了这些理想模型的物理相关性和有效性。另一方面，一旦数学模型的意义在一定程度上得到现有实验数据的支持，对这些理想模型的理论研究就可以提供原始物理问题难以通过实验获得的性质。对于涉及时间演化的物理系统，特别感兴趣的是那些结构和渐近性质。这包括一些特殊的结构，如稳态、周期和准周期轨道、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，在一个系统中，只有稳定的状态在物理上是可观察到的，而理想的，但不稳定的状态，由于它们对参数的极其敏感的依赖，很难被观察到。另一方面，不稳定状态也很重要，部分原因是它们和它们的一些相关结构作为分隔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在稳态附近的局部动力学，这些系统都属于非线性波的一般类别。先验阻尼的缺乏和复杂的非线性给其数学分析带来了很大的挑战。更具体地说，PI建议严格研究以下偏微分方程系统的局部动力学。第一个是不可压缩欧拉方程，它模拟了非粘性和不可压缩的流体，如水。提出的问题包括刚性容器中的流体和具有自由表面（如海浪）的流体。提案中的第二个偏微分方程是弗拉索夫-泊松系统，该系统模拟无碰撞等离子体，它由具有速度和电荷的粒子组成。第三种是准线性波动方程。尽管近年来人们对这些系统进行了广泛的研究并取得了许多重要进展，但由于它们的非线性特性，许多问题，包括一些基本问题，经过多年的努力仍然没有得到很好的理解。PI计划关注它们在平衡点附近的局部动力结构，包括稳定/不稳定、局部不变流形、特殊解和分岔。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，它们的解映射通常在相空间中没有足够的光滑性，无法直接应用经典理论。相对于常微分方程，这些非线性偏微分方程的定性结构与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。解决和理解这些问题，预计将主要基于它们特定的机械和几何结构，将导致这些领域的实质性理论进步，并可能导致在相应系统中发现新的物理和数学现象。

Partial differential equations are widely used to model various problems arising from physics, engineering, biology, finance, etc. The aims of the efforts to understand these mathematical models rigorously are twofold. On the one hand, the physical relevance and the validity of these ideal models are established through the comparison between the results from theoretical analysis and the experimental observations. On the other hand, once the meaningfulness of a mathematical model is supported by available experimental data to certain extent, the theoretical studies on these ideal models can provide properties of the original physical problems that are hard to be obtained through experiments. For physical systems involving temporal evolution, of particular interests are those structural and asymptotic properties. These include some special structures, such as steady states, periodic and quasi-periodic orbits, chaotic orbits etc, and 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 very 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. More specifically, the PI proposes to study rigorously the local dynamics of the following partial differential equation systems. The first one is the incompressible Euler equation, which models non-viscous and incompressible fluids like water. The proposed problems include fluids in rigid containers and fluids with free surfaces like ocean waves. The second partial differential equation in the proposal is the Vlasov-Poisson system that models the collisionless plasma, which consists of particles with both velocity and electrical charge. The third one is the quasi-linear waves equation. Even though there have been extensive research on these systems and many important progresses have been made in recent year, due to their very nonlinear nature, many issues including some fundamental ones are still not well understood after years of efforts. The PI plans to focus on their local dynamic structures near equillibria, including stability/instability, local invariant manifolds, special solutions and bifurcations. 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 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. Solving and understanding 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 corresponding systems.