CAREER: Dynamics of Nonlinear Dispersive Partial Differential Equations

职业：非线性色散偏微分方程的动力学

• 批准号: 1945615
• 负责人: Sung-Jin Oh
• 金额: $ 42.5万
• 依托单位: University of California-Berkeley
• 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: 2020-05-15 至 2025-04-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1945615&HistoricalAwards=false
• 关键词: CAREER; Dynamics; Nonlinear; Dispersive; Partial

From the dynamics of subatomic particles to electromagnetism, fluids, plasmas and gravity among astronomical bodies, nature is governed by nonlinear dispersive partial differential equations at all scales. The objective of this NSF CAREER project is to enhance the rigorous understanding of the long-term behavior and singularities of solutions to nonlinear dispersive partial differential equations of physical origin by attacking strategically chosen open problems for a wide array of such equations. The theoretical advances from this project will lead to a clearer and more effective understanding of highly nonlinear phenomena in physics, such as solitons, gravitational singularities in general relativity and magnetic reconnection in plasma physics. The project will train undergraduate, graduate, and post-doctoral researchers by developing accessible expositions of modern research topics, organizing summer workshops and mentoring research projects.Specifically, for three classes of nonlinear dispersive equations at different levels of complexity, the following goals will be pursued: (i) for the energy-critical wave maps and (hyperbolic) Yang–Mills equations, which are examples of relativistic field theories, to prove a new forward-in-time scattering criterion for “initially outgoing” solutions, with a view towards resolving the soliton resolution conjecture in the one-soliton regime, and along a sequence of times in general; (ii) for linear and nonlinear wave equations on a black hole background, to develop a Fourier-based approach that rigorously justifies the sharp late-time asymptotics along the event horizon, which will be a starting point for investigation of the linear and nonlinear instability of the Kerr Cauchy horizon, and ultimately the strong cosmic censorship conjecture in the vicinity of the Kerr spacetimes; (iii) for the Hall-magnetohydrodynamics equations in plasma physics, to establish a general local wellposedness theory, which is interesting in view of the recent work of the PI that proved illposedness of the Cauchy problem in the vicinity of the trivial solution. Insights from various disciplines of mathematics, such as differential geometry, calculus of variations, harmonic analysis, spectral theory and microlocal analysis, will naturally enter in attaining the above goals.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.

从亚原子粒子的动力学到天体间的电磁、流体、等离子体和引力，自然界在所有尺度上都受到非线性色散偏微分方程的控制。这个NSF CAREER项目的目标是通过攻击战略选择的开放问题，为广泛的此类方程，以提高对物理起源的非线性色散偏微分方程的长期行为和奇异性的严格理解。该项目的理论进展将导致对物理学中高度非线性现象的更清晰和更有效的理解，例如孤子，广义相对论中的引力奇点和等离子体物理学中的磁场重联。该项目将通过开发现代研究主题的无障碍博览会，组织夏季研讨会和指导研究项目来培养本科生，研究生和博士后研究人员。具体来说，对于三类不同复杂程度的非线性色散方程，将追求以下目标：（i）对于能量临界波图，（双曲）杨-米尔斯方程，这是相对论场论的例子，证明“初始向外”解的一个新的时间向前散射准则，以期解决一般情况下单孤子状态和沿着时间序列的孤子分辨猜想;（ii）对于黑洞背景下的线性和非线性波动方程，开发一种基于傅立叶的方法，严格证明沿着事件视界的尖锐的后期渐近性，这将是研究克尔柯西视界的线性和非线性不稳定性的起点，并最终在克尔时空附近的强宇宙审查猜想;（iii）霍尔磁流体力学方程在等离子体物理，建立一个一般的局部适定性理论，这是有趣的，鉴于最近的工作PI证明不适定性的柯西问题在附近的平凡的解决方案。来自不同数学学科的见解，如微分几何、变分法、调和分析、谱理论和微局部分析，将自然地参与实现上述目标。该奖项反映了NSF的法定使命，并被认为值得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来支持。

项目成果

The threshold conjecture for the energy critical hyperbolic Yang–Mills equation

能量临界双曲Yang–Mills方程的阈值猜想

• DOI: 10.4007/annals.2021.194.2.1
• 发表时间: 2021
• 期刊: Annals of Mathematics
• 影响因子: 4.9
• 作者: Oh, Sung-Jin;Tataru, Daniel
• 通讯作者: Tataru, Daniel

On the Cauchy Problem for the Hall and Electron Magnetohydrodynamic Equations Without Resistivity I: Illposedness Near Degenerate Stationary Solutions

• DOI: 10.1007/s40818-022-00134-5
• 发表时间: 2019-02
• 期刊: Annals of PDE
• 影响因子: 2.8
• 作者: In-Jee Jeong;Sung-Jin Oh

The Yang-Mills heat flow and the caloric gauge

• DOI: 10.24033/ast.1179
• 发表时间: 2017-09
• 期刊: Astérisque
• 作者: Sung-Jin Oh;D. Tataru

A Scattering Theory Approach to Cauchy Horizon Instability and Applications to Mass Inflation

柯西视界不稳定性的散射理论方法及其在大规模通货膨胀中的应用

• DOI: 10.1007/s00023-022-01216-7
• 发表时间: 2023
• 期刊: Annales Henri Poincaré
• 作者: Luk, Jonathan;Oh, Sung-Jin;Shlapentokh-Rothman, Yakov
• 通讯作者: Shlapentokh-Rothman, Yakov

Global Nonlinear Stability of Large Dispersive Solutions to the Einstein Equations

• DOI: 10.1007/s00023-021-01148-8
• 发表时间: 2021-08
• 期刊: Annales Henri Poincaré
• 作者: J. Luk;Sung-Jin Oh