Pseudo-relativistic nonlinear Schroedinger equations

拟相对论非线性薛定谔方程拟研究摘要Gigliola Staffilani和Enno Lenzman提出了一类新的色散偏微分方程(PDE)的研究工作，称为拟相对论非线性薛定谔方程。这些方程的数学研究仍处于初级阶段，最近已发现作为对自引力相对论物质动力学演化的有效描述的重要应用。基于这一物理动机，PI提出的研究旨在了解这些模型方程的解的定性行为。特别强调了伪相对论Hartree方程，它的聚焦非线性具有临界强度，因为大的初始数据会导致解的有限时间爆破。在这里，最重要的是获得对爆破率和动力学的分析洞察，以及证明非径向爆破解的存在性，从而推广PI先前关于伪相对论Hartree方程的径向爆破的结果(与J.Froehlich合作)。除了有限时间爆破问题外，PI还详细研究了全局时间解及其随着时间趋于无穷大的渐近行为。色散偏微分方程式为自然科学的理论和经验分支提供了相遇的场所。对伪相对论薛定谔方程的数学研究证实了对理论天体物理物理模型的直观和数值洞察力。今天天体物理学面临的最突出的问题之一就是所谓的暗物质问题。各种理论模型将玻色子恒星的存在作为解释这一经验谜题的可能候选者。本文所考虑的伪相对论Hartree方程是描述玻色子星动力学演化的有效方法。因此，这种非线性色散偏微分方程组为这些理论对象提供了分析测试依据

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