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

职业：偏微分方程稳定性、正则性和长期动力学的新机制

• 批准号: 1945179
• 负责人: Hao Jia
• 金额: $ 42.5万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1945179&HistoricalAwards=false
• 关键词: CAREER; New; Mechanisms; Stability; Regularity

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.

该项目专注于受流体力学和波传播启发的非线性偏微分方程组的数学分析。了解不可压缩流体的动力学，例如亚音速的水和空气，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。涡旋(涡)和剪切流等拟序结构是流体力学中的显著特征。为了减少科学和工程应用中的阻力、振荡和不稳定性，相干结构的形成、稳定性和演化是需要了解的关键流体现象。PI将发展新的、创新的数学方法来分析物理上重要的相干结构的动力学特性，这可以解决理论困难，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，波图与数学物理中的经典场论有着深刻的联系。拟议的项目为初级研究人员提供了一个理想的培训基础，让他们能够应用尖端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，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分析和傅立叶分析。在许多物理问题中，大型相干结构的分析需要对线性化流动进行精确的谱分析，而傅立叶分析已被证明是揭示微妙的非线性相互作用所不可或缺的。因此，该项目开发的技术可能在其他具有技术挑战性的摄动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，以努力了解涡度伸展和涡度输送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基础性的作用。对于波图方程，主要目的是将出射波的“能量通道”论点推广到这个具有技术挑战性的模型，以研究非微扰区域中孤子辐射的解耦。这些项目为研究生提供了广泛的问题，他们将学习使用频谱分析、傅立叶分析、动力系统和数值模拟的工具来研究物理上有重大意义的问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Nonlinear inviscid damping near monotonic shear flows

• DOI: 10.4310/acta.2023.v230.n2.a2
• 发表时间: 2020-01
• 期刊: Acta Mathematica
• 影响因子: 3.7
• 作者: A. Ionescu;H. Jia

Linear Vortex Symmetrization: The Spectral Density Function

线性涡旋对称化：谱密度函数

• DOI: 10.1007/s00205-022-01815-y
• 发表时间: 2022
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: Ionescu, Alexandru D.;Jia, Hao
• 通讯作者: Jia, Hao

On the Stability of Shear Flows in Bounded Channels, II: Non-monotonic Shear Flows

关于有界通道中剪切流的稳定性，II：非单调剪切流

• DOI: 10.1007/s10013-023-00661-z
• 发表时间: 2023
• 期刊: Vietnam Journal of Mathematics
• 影响因子: 0.8
• 作者: Ionescu, Alexandru D.;Iyer, Sameer;Jia, Hao
• 通讯作者: Jia, Hao

Uniform Linear Inviscid Damping and Enhanced Dissipation Near Monotonic Shear Flows in High Reynolds Number Regime (I): The Whole Space Case

高雷诺数状态下的均匀线性无粘阻尼和增强耗散近单调剪切流 (I)：整个空间案例

• DOI: 10.1007/s00021-023-00794-8
• 发表时间: 2023
• 期刊: Journal of Mathematical Fluid Mechanics
• 影响因子: 1.3
• 作者: Jia, Hao