Pseudo-relativistic nonlinear Schroedinger equations

伪相对论非线性薛定谔方程

基本信息

批准号：
0702492
负责人：
Gigliola Staffilani
金额：
$ 11.87万
依托单位：
Massachusetts Institute of Technology
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2007
资助国家：
美国
起止时间：
2007-07-01 至 2010-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0702492&HistoricalAwards=false
关键词：
Pseudo relativistic nonlinear Schroedinger equations

Pseudo relativistic nonlinear Schroedinger equations

项目摘要

Pseudo-relativistic nonlinear Schroedinger EquationsAbstract of Proposed ResearchGigliola Staffilani and Enno LenzmannThis proposal sets forth work on a novel class of dispersive partial differential equations (PDEs) which are called pseudo-relativistic nonlinear Schroedinger equations. These equations, whose mathematical study is still in its infancy, have recently found a significant application as effective descriptions for the dynamical evolution of self-gravitating, relativistic matter. Based on this physical motivation, the PI's proposed research aims at understanding the qualitative behavior of solutions to these model equations. In particular, great emphasis is put on the pseudo-relativistic Hartree equation whose focusing nonlinearity is of critical strength in the sense that large initial data can lead to finite-time blow-up of the solution. Here it is of paramount interest to gain analytical insight into blow-up rates and dynamics, as well as to prove existence of non-radial blow-up solutions, thereby extending the PI's previous results on radial blow-up for the pseudo-relativistic Hartree equation (in collaboration with J. Froehlich). Apart from the issue of finite-time blow-up, the PI proposes a detailed study of global-in-time solutions and their asymptotic behavior as time tends to infinity. Dispersive partial differential equations provide meeting grounds for theoretical and empirical branches of the natural sciences. The mathematical study of pseudo-relativistic Schroedinger equations substantiates intuitive and numerical insights into physical models of theoretical astrophysics. One of the most outstanding problems that astrophysics is facing today is the so-called dark matter problem. Various theoretical models set forth the existence of boson stars as possible candidates for explaining this empirical puzzle. The pseudo-relativistic Hartree equation considered in this proposal serves as an effective description for the dynamical evolution of boson stars. Therefore, this nonlinear dispersive PDE provides analytical testing grounds for these theoretical objects

拟议的研究gigiola seftilani和enno lenzmannthis提案提出的提议提出的提议提出了一系列新型的分散偏微分方程（PDE），这些伪型非线性schroedinger方程提交了拟议的研究gigiola seftilani和enno lenzmannthis，这些方程式称为伪型非线性非线性schroeDinger Schroedinger Schroedinger Schroedinger Schroedinger Equations。这些方程式仍处于起步阶段的数学研究，最近发现了对自我赋予性，相对论问题的动态演变的有效描述的重要应用。基于这种身体动机，PI的拟议研究旨在了解解决这些模型方程的解决方案的定性行为。特别是，在大量初始数据可以导致解决方案有限的时间爆炸的意义上，其焦点非线性具有关键的强度，其焦点非线性具有关键的强度。在这里，获得对爆炸速率和动态的分析洞察以及证明存在非爆炸解决方案的最高兴趣，从而扩大了PI对伪偏见的Hartree方程（与J. Froehlich的协作）的径向爆炸的先前结果。除了有限的时间爆炸问题外，PI还提出了一项详细研究全球及时解决方案及其渐近行为的详细研究，因为时间往往是无穷大的。分散部分微分方程为自然科学的理论和经验分支提供了满足理由。伪尊敬的Schroedinger方程的数学研究证明了对理论天体物理学物理模型的直观和数值见解。今天，天体物理学面临的最杰出的问题之一就是所谓的暗物质问题。各种理论模型阐明了玻色颗星的存在是解释这种经验难题的可能候选者。该提案中考虑的伪忠实的Hartree方程是玻色子恒星动力学演化的有效描述。因此，这种非线性色散PDE为这些理论对象提供了分析测试场