CAREER: New Mechanisms for Stability, Regularity and Long Time Dynamics of Partial Differential Equations

该项目侧重于非线性偏微分方程的数学分析，受到流体动力学和波传播的启发。了解不可压缩流体（如亚音速下的水和空气）的动力学对于从飞机、船只和发动机的设计到海洋和大气的研究等各种应用都很重要。相干结构，如涡旋（漩涡）和剪切流，是流体动力学的突出特征。在科学和工程应用中，为了减少阻力、振荡和不稳定性，相干结构的形成、稳定和演化是需要理解的关键流体现象。PI将开发新的，创新的数学方法来分析物理上重要的相干结构的动态特性，这可以解决理论难题，并为实际应用提供强大的数学工具。PI还将在波图的背景下研究辐射和粒子的相互作用，这与数学物理中的经典场论有着深刻的联系。建议的项目为初级研究人员提供了一个理想的训练基地，以应用前沿的数学分析来研究流体动力学和波传播中的复杂物理现象。研究生将积极参与这些研究项目。PI和合作者的目标是开发新的方法，可以在流体动力学的非线性渐近稳定性问题的背景下有效地结合精确的谱和傅立叶分析。在许多物理问题中，分析大型相干结构需要对线性化流动进行精确的谱分析，而傅里叶分析在揭示微妙的非线性相互作用方面已被证明是不可或缺的。因此，项目中开发的技术可能在其他技术上具有挑战性的微扰问题中有更广泛的应用。PI还将研究更简单的流体方程模型，以努力理解涡度拉伸和涡度输运效应之间的相互作用和平衡，这在三维欧拉方程的正则性理论中起着重要作用。对于波映射方程，主要目标是将传出波的“能量通道”论点扩展到这个技术上具有挑战性的模型，以研究非摄动状态下辐射与孤子的解耦。这些项目为研究生提供了广泛的问题，他们将学习使用光谱分析、傅立叶分析、动力系统和数值模拟等工具来研究物理上的重大问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project focuses on mathematical analysis of nonlinear partial differential equations that are inspired by fluid dynamics and wave propagation. Understanding the dynamics of incompressible fluids, such as water and air at subsonic speed, is important for a variety of applications, ranging from the design of airplanes, boats and motors, to the study of oceans and the atmosphere. Coherent structures, such as vortices (eddies) and shear flows, are prominent features in fluid dynamics. The formation, stability, and evolution of coherent structures are critical fluid phenomena to understand in order to reduce drag, oscillation, and instability in scientific and engineering applications. The PI will develop new, innovative mathematical methods to analyze the dynamic properties of physically important coherent structures, which can resolve theoretical difficulties as well as provide powerful mathematical tools for practical applications. The PI will also study the interaction of radiation and particles in the context of wave maps, which have a deep connection to the classical field theories from mathematical physics. The proposed projects provide an ideal training ground for junior researchers in applying cutting edge mathematical analysis to study sophisticated physical phenomena in fluid dynamics and wave propagation. Graduate students will be actively involved in these research projects. The PI and collaborators aim to develop new methods that can effectively combine precise spectral and Fourier analysis in the context of nonlinear asymptotic stability problems of fluid dynamics. In many physical problems, the analysis of large coherent structures requires precise spectral analysis for the linearized flow, while Fourier analysis has proved indispensable in uncovering delicate nonlinear interactions. Thus, the techniques developed in the project may have a wider range of applications in other technically challenging perturbative problems. The PI will also study simpler models of fluid equations in an effort to understand the interaction and balance between vorticity stretching and vorticity transportation effects, which play a fundamental role in the regularity theory of three-dimensional Euler equations. For the wave maps equation, the main goal is to extend the "channel of energy" argument for outgoing waves to this technically challenging model to study the decoupling of radiation from solitons in a non-perturbative regime. These projects provide a wide range of problems for graduate students, who will learn to use tools from spectral analysis, Fourier analysis, dynamical systems, and numerical simulation, in the study of physically significant problems.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.