Singularities and Long Time Dynamics in Nonlinear Dispersive Flows

这个项目的主要目标是研究一大类方程的解，这些方程可以最好地描述为非线性波。这些数学方程模拟了我们宇宙甚至日常生活中的一系列物理现象，包括海洋表面的水波和多粒子系统的量子力学相互作用，以及更大范围内的引力波及其在黑洞附近的演化。这些模型的共同特征在于波相互作用的强度，这对它们的短期和长期行为都有很大的影响。要研究的现象既包括波的短时击穿，也包括波的长时间持续和/或频散和衰减。这个项目的目标是解决非线性色散方程领域内的一系列具有挑战性的问题。要考虑的方程范围从半线性到完全非线性，要研究的性质类型从短程行为到长程行为，从散射到爆破问题。这项研究将偏微分方程式的思想和方法与几何、调和分析、复杂分析和微观局部分析联系起来。这一奖项反映了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。

The main goal of this project is to study the solutions to a wide class of equations that can best be described as nonlinear waves. These mathematical equations model a broad range of physical phenomena in our universe and even in our daily lives, including the water waves at the surface of our oceans and the quantum-mechanical interaction of multi-particle systems, as well as, on a broader scale, gravitational waves and their evolution in the proximity of black holes. The common feature in these models lies in the strength of the wave interactions, which drastically affects both their short-time and long-time behavior. Phenomena to be studied include both short-time breakdown of waves, as well as long-time persistence and/or dispersion and decay of waves. The objective of this project is to attack a broad array of challenging problems within the field of nonlinear dispersive equations. The equations to be considered range from semilinear to fully nonlinear, and the types of properties to be investigated span from short-range to long-range behavior, from scattering to blow-up questions. The research connects ideas and methods in partial differential equations to geometry, harmonic analysis, complex analysis, and microlocal analysis.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.