Stability of solitons and long-term dynamics of fluids

孤子的稳定性和流体的长期动力学

• 批准号: 2007008
• 负责人: Alexandru Ionescu
• 金额: $ 32.49万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2023-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2007008&HistoricalAwards=false
• 关键词: Stability; solitons; long; term; dynamics

The main goal of the project is to study solutions of several important partial differential equations. These solutions describe physical phenomena, such as the dynamics of fluids, the motion of water waves, and gravitation, which can be observed in daily life. We aim to analyze these solutions rigorously and to recover quantitative and qualitative information about their behavior as mathematical theorems. The research project involves graduate students and postdoctoral associates, and connects ideas and methods in partial differential equations, numerical analysis, geometry, and harmonic analysis.The PI will study the long term dynamics of solutions of several important evolution equations, such as the Euler equations in 2 and 3 dimensions, water wave models, and the Einstein field equations of General Relativity. Six problems of interest are explicitly identified as part of the project. These problems concern the global stability of shear flows and vortices for the 2D Euler and the generalized SQG equations, the long term dynamics of water waves in a large box in 2D and 3D, and the global stability of solutions of the Einstein equations with matter.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.

该项目的主要目标是研究几个重要的偏微分方程的解。这些解描述了日常生活中可以观察到的物理现象，例如流体动力学、水波运动和引力。我们的目标是严格分析这些解决方案，并将有关其行为的定量和定性信息恢复为数学定理。该研究项目涉及研究生和博士后，并将偏微分方程、数值分析、几何和调和分析的思想和方法联系起来。PI将研究几个重要演化方程解的长期动力学，例如欧拉方程二维和三维、水波模型和广义相对论的爱因斯坦场方程。该项目明确确定了六个感兴趣的问题。这些问题涉及 2D 欧拉和广义 SQG 方程的剪切流和涡流的全局稳定性、2D 和 3D 大盒子中水波的长期动力学以及爱因斯坦物质方程解的全局稳定性。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。

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

Polynomial sequences in discrete nilpotent groups of step 2

步骤 2 的离散幂零群中的多项式序列

• DOI: 10.1515/ans-2023-0085
• 发表时间: 2023
• 期刊: Advanced Nonlinear Studies
• 影响因子: 1.8
• 作者: Ionescu, Alexandru D.;Magyar, Ákos;Mirek, Mariusz;Szarek, Tomasz Z.
• 通讯作者: Szarek, Tomasz Z.

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

Axi‐symmetrization near Point Vortex Solutions for the 2D Euler Equation

• DOI: 10.1002/cpa.21974
• 发表时间: 2019-04
• 期刊: Communications on Pure and Applied Mathematics
• 影响因子: 3
• 作者: A. Ionescu;H. Jia

Polynomial averages and pointwise ergodic theorems on nilpotent groups

幂零群上的多项式平均值和逐点遍历定理

• DOI: 10.1007/s00222-022-01159-0
• 发表时间: 2023
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: Ionescu, Alexandru D.;Magyar, Ákos;Mirek, Mariusz;Szarek, Tomasz Z.
• 通讯作者: Szarek, Tomasz Z.