Integrable and Non-Integrable Dispersive Partial Differential Equations

可积和不可积色散偏微分方程

基本信息

批准号：
1763074
负责人：
Monica Visan
金额：
$ 27万
依托单位：
University of California-Los Angeles
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1763074&HistoricalAwards=false
关键词：
Integrable Non Dispersive Partial Differential

项目摘要

One of the models the PI is proposing to investigate is the Korteweg-de Vries (KdV) equation. This equation was derived more that a hundred years ago to explain the behavior of long waves in channels of shallow water. In the 1960s, researchers at Princeton's Plasma Physics Laboratory demonstrated that this equation exhibits a wealth of novel features, which have sparked the interest of mathematicians and physicists alike. However, despite all the attention it has received over the years, existence of solutions under minimal assumptions has been proved only recently by the PI and her collaborators. One ingredient in their work is the recent discovery of new conservation laws. This project outlines several additional problems that can now be attacked using this discovery. Another major impetus behind this project is to prove that complicated transient dynamics resolve into simple dynamics in the distant future. The physical significance of this phenomenon relies on its stability under perturbations. While in the past, the PI has investigated deterministic perturbations to the equations, the current project takes this theme in a new direction by considering stability in the presence of (random) noise. The project focuses on several problems that lie at the intersection of nonlinear dispersive partial differential equations, completely integrable systems, and stochastic partial differential equations. The PI's discovery of new microscopic conservation laws for KdV has opened the door to treating three seemingly unrelated problems of long-standing interest regarding KdV on the line: optimal regularity well-posedness, symplectic non-squeezing, and invariance of white noise. In addition, the PI is proposing a coherent plan for establishing invariance of the Gibbs measure for the Landau-Lifshitz model and invariance of white noise for the focusing cubic Nonlinear Schr\"odinger Equation (NLS). This program involves establishing the analogous statements for the physical atomic models associated with these problems (which the PI has successfully completed) and then taking the continuum limit for the corresponding rough data. This should reveal the physical renormalizations for the Landau-Lifshitz and the cubic NLS models that would ensure well-posedness for such data. As the Gibbs measure for the Landau-Lifshitz model corresponds to Brownian motion on the sphere, this problem is also interesting from a purely probabilistic point of view as yielding a Hamiltonian measure-preserving flow on such paths.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.

PI 提议研究的模型之一是 Korteweg-de Vries (KdV) 方程。这个方程是在一百多年前导出的，用于解释浅水河道中长波的行为。 20 世纪 60 年代，普林斯顿等离子体物理实验室的研究人员证明，这个方程表现出丰富的新颖特征，引起了数学家和物理学家的兴趣。然而，尽管多年来它受到了广泛的关注，但 PI 和她的合作者直到最近才证明了最小假设下解决方案的存在。他们工作的一个组成部分是最近发现的新守恒定律。该项目概述了现在可以使用此发现来解决的几个其他问题。该项目背后的另一个主要推动力是证明复杂的瞬态动力学在遥远的未来可以分解为简单的动力学。这种现象的物理意义取决于其在扰动下的稳定性。虽然过去，PI 研究了方程的确定性扰动，但当前项目通过考虑（随机）噪声存在下的稳定性，将这一主题推向了新的方向。该项目重点研究非线性色散偏微分方程、完全可积系统和随机偏微分方程交叉点的几个问题。 PI 对 KdV 的新微观守恒定律的发现，为解决 KdV 线上长期关注的三个看似无关的问题打开了大门：最优正则性适定性、辛非压缩性和白噪声不变性。此外，PI 还提出了一个连贯的计划，用于建立 Landau-Lifshitz 模型的吉布斯测度不变性和聚焦三次非线性 Schr\"odinger 方程 (NLS) 的白噪声不变性。该计划涉及建立以下类似语句：与这些问题相关的物理原子模型（PI 已成功完成），然后对相应的粗略数据进行连续统限制，这应该揭示了物理重整化。 Landau-Lifshitz 和三次 NLS 模型将确保此类数据的适定性，由于 Landau-Lifshitz 模型的吉布斯测度对应于球体上的布朗运动，因此从纯概率的角度来看，这个问题也很有趣。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。