CAREER: Dynamics of Nonlinear Dispersive Partial Differential Equations

从亚原子粒子的动力学到天体之间的电磁、流体、等离子体和重力，自然界在所有尺度上都受到非线性色散偏微分方程组的支配。这个NSF职业项目的目标是通过对一系列此类方程的战略性选择的开放问题的攻击，来加强对物理起源的非线性色散偏微分方程解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中高度非线性现象的更清晰和更有效的理解，如孤子、广义相对论中的引力奇点和等离子体物理中的磁重联。该项目将培养本科生、研究生和博士后研究人员，通过对现代研究主题的可访问性阐述、组织暑期研讨会和指导研究项目。具体地说，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)对于相对论场论的例子--能量临界波图和(双曲)Yang-Mills方程，证明一种新的“初始传出”解的时间正向散射准则，以期在单孤子体制下和一般沿时间序列解决孤子分辨猜想；(Ii)对于黑洞背景上的线性和非线性波动方程，发展一种基于傅立叶的方法，严格证明沿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，并最终研究克尔时空附近的强宇宙审查猜想；(Iii)对于等离子体物理中的霍尔-磁流体动力学方程，建立一般的局部适定性理论，这是有趣的，因为PI最近的工作证明了柯西问题在平凡解附近的不适定性。来自不同数学学科的见解，如微分几何、变分法、调和分析、频谱理论和微局部分析，将自然地进入到实现上述目标的过程中。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。

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.