Critical Dispersive Partial Differential Equations

临界色散偏微分方程

基本信息

批准号：
2153750
负责人：
Benjamin Dodson
金额：
$ 23.28万
依托单位：
Johns Hopkins University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2153750&HistoricalAwards=false
关键词：
Critical Dispersive Partial Differential Equations

项目摘要

The main objective of this project is to improve the understanding of dispersive partial differential equations. Dispersive partial differential equations include the wave, Schrodinger, and Korteweg de-Vries equation. These equations are ubiquitous in physics, modeling phenomena ranging from the behavior of subatomic particles to interstellar gravity waves. Of particular interest are questions of long time behavior of solutions. In other words, given certain initial data, does a solution to the equation exist? If a solution does exist, does it exist for all time? What is the behavior of the solution as time approaches either infinity or the maximum time for which the solution exists? Can we catalogue the various long time behaviors and obtain a complete description of the possible phenomena? The project provides research training opportunities for graduate students. In this project, the Principal Investigator and his collaborators study the long time behavior of dispersive partial differential equations with initial data in a critical norm. Many diverse dispersive partial differential equations have a scaling symmetry, and a solution to the equation gives an entire family of solutions. Often, the scaling symmetry completely describes the local behavior of the equation completely: the equation is well-posed for initial data in the critical space, but it is ill-posed for data in a less regular (subcritical) space. We wish to understand the long time behavior for such equations at the critical regularity, where it is known that local well-posedness occurs. Additionally, in many such equations, the set where ill-posedness occurs in a subcritical space is often a set of measure zero. Thus, we hope to describe the nature of the set of initial data for which ill-posedness occurs. The specific problems that are addressed in this project are the Schrodinger maps problem, the focusing, mass-critical nonlinear Schrodinger equation, the energy subcritical nonlinear wave and Schrodinger equations, and the one dimensional cubic nonlinear Schrodinger equation. For the defocusing, energy subcritical problems, we expect scattering to occur for initial data in the critical Sobolev space. Solitons are known to occur for the Schrodinger map problem and the mass-critical nonlinear Schrodinger equation. In both cases, scattering is known to occur for initial data below the soliton (for Schrodinger maps this is only in the equivariant case). We wish to understand the solution for initial data slightly above the soliton for these problems. Finally, for the one dimensional nonlinear Schrodinger equation, the aim is to understand the long time behavior for initial data that does not decay at infinity.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.

该项目的主要目的是提高对分散部分微分方程的理解。分散部分微分方程包括波浪，Schrodinger和Korteweg De-Vries方程。这些方程在物理学中无处不在，建模现象从亚原子颗粒的行为到星际重力波。特别有趣的是解决方案长期行为的问题。换句话说，鉴于某些初始数据，是否存在该方程的解决方案？如果解决方案确实存在，是否一直存在？当时间接近无穷大或解决方案的最大时间时，解决方案的行为是什么？我们可以分类各种长期行为并获得可能现象的完整描述吗？该项目为研究生提供了研究培训机会。在这个项目中，主要研究者及其合作者研究了具有关键规范的初始数据的分散部分微分方程的长时间行为。许多多样化的分散偏微分方程具有缩放对称性，对方程式的解决方案为整个解决方案提供了。通常，缩放对称性完全描述了方程的局部行为：方程在关键空间中的初始数据适当，但是在较不正常（亚临界）空间中的数据范围不足。我们希望在临界规律性下了解此类方程式的长时间行为，在那里知道当地的适合良好。此外，在许多这样的方程式中，在亚临界空间中发生不适的集合通常是一组量度零。因此，我们希望描述出现不良性的初始数据集的性质。该项目中解决的具体问题是Schrodinger地图问题，聚焦，质量至关重要的非线性Schrodinger方程，能量亚临界非线性波和Schrodinger方程以及一维立方非线性Schrodinger方程。对于散落的，能量亚临界问题，我们希望在关键Sobolev空间中进行初始数据发生散射。已知孤子是出于Schrodinger地图问题和质量至关重要的非线性Schrodinger方程而发生的。在这两种情况下，都知道在孤子下方的初始数据中发生散射（对于Schrodinger地图而言，这仅在均等的情况下）。我们希望理解这些问题的初始数据略高于Soliton的解决方案。最后，对于一维非线性Schrodinger方程式，目的是了解无限衰减的初始数据的长时间行为。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准通过评估来获得支持的。