Pseudo-relativistic nonlinear Schroedinger equations

拟相对论非线性薛定谔方程研究建议摘要Gigliola Staffilani和Enno Lenzmanovich的建议阐述了一类新的色散偏微分方程（PDE）的工作，这类方程被称为拟相对论非线性薛定谔方程。这些方程，其数学研究仍处于起步阶段，最近发现了一个重要的应用作为自引力，相对论性物质的动力学演化的有效描述。基于这种物理动机，PI提出的研究旨在了解这些模型方程解的定性行为。特别是，非常重视的伪相对论性Hartree方程的聚焦非线性是临界强度的意义上，大的初始数据可以导致有限时间爆破的解决方案。在这里，它是最重要的利益，以获得分析洞察爆破率和动力学，以及证明存在的非径向爆破解决方案，从而扩大了PI的径向爆破的伪相对论哈特里方程（与J. Froehlich合作）的先前结果。除了有限时间爆破的问题，PI提出了一个详细的研究全球的时间解决方案和他们的渐近行为的时间趋于无穷大。色散偏微分方程为自然科学的理论和经验分支提供了交汇点。伪相对论薛定谔方程的数学研究证实了理论天体物理学物理模型的直观和数值见解。天体物理学目前面临的最突出的问题之一就是所谓的暗物质问题。各种各样的理论模型提出玻色子星的存在，作为解释这个经验之谜的可能候选者。在这个建议中考虑的伪相对论Hartree方程作为一个有效的描述玻色子星的动力学演化。因此，这种非线性色散偏微分方程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