Critical Nonlinear Dispersive Equations

临界非线性色散方程

• 批准号: 1500424
• 负责人: Benjamin Dodson
• 金额: $ 16.97万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500424&HistoricalAwards=false
• 关键词: Critical; Nonlinear; Dispersive; Equations

This project will study dispersive partial differential equations. Such equations model many different phenomena, among them the propagation of various kinds of waves, such as water waves and laser light. These equations also model certain phenomena in particle physics. This project attempts to understand long-time behavior of such equations and related systems.There are a number of problems that will be studied in the course of this endeavor. The study will mainly revolve around the three well-known dispersive partial differential equations: wave, Schrodinger, and Korteweg de Vries. A great deal is unknown for the focusing Schrodinger and Korteweg de Vries (KdV) problems, particularly for the mass-critical problem. The PI intends to study the focusing, mass-critical Schrodinger problem for mass above the mass of the ground state, as well as the focusing gKdV problem for large mass below the mass of the ground state. The PI also plans to extend recent work with the I-method to the wave and Klein-Gordon equations. Finally, the PI will study the ultra hyperbolic Schrodinger equation.

该项目将研究分散部分微分方程。这样的方程模拟了许多不同的现象，其中包括各种波浪的传播，例如水波和激光。这些方程还模拟了粒子物理中的某些现象。该项目试图理解此类方程式和相关系统的长期行为。在这项工作过程中将研究许多问题。这项研究将主要围绕三个众所周知的分散部分微分方程：Wave，Schrodinger和Korteweg de Vries。专注于Schrodinger和Korteweg de Vries（KDV）问题，尤其是对于群众批量关键问题而言，这是未知的。 PI打算研究高于基质量质量的质量的聚焦，批判性的Schrodinger问题，以及对基态质量以下的大质量的聚焦GKDV问题。 PI还计划将最近的i方法扩展到Wave和Klein-Gordon方程式。最后，PI将研究超双曲线Schrodinger方程。

项目成果

Global well-posedness and scattering for nonlinear {S}chr\"{o}dinger equations with algebraic nonlinearity when {$d=2,3$} and {$u_0$} is radial

当 {$d=2,3$} 和 {$u_0$} 为径向时，具有代数非线性的非线性 {S}chr"{o}dinger 方程的全局适定性和散射

• 发表时间: 2019
• 期刊: Cambridge journal of mathematics
• 影响因子: 1.6
• 作者: Dodson, Benjamin

Global Well-Posedness for the Defocusing, Cubic, Nonlinear Wave Equation in Three Dimensions for Radial Initial Data in $\dot{H}^{s} \times \dot{H}^{s - 1}$, $s> \frac{1}{2}$

$dot{H}^{s} imes dot{H}^{s - 1}$, $s> 中径向初始数据的三维散焦三次非线性波动方程的全局适定性

• DOI: 10.1093/imrn/rnx323
• 发表时间: 2018
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Dodson, Benjamin