Mathematical modelling of galaxies in MOND (Modified Newtonian Dynamics)

该研究项目的应用背景和动机是所谓的旋涡星系中的“质量缺失问题”。 在此类星系的外部区域，观察到的恒星绕银河系中心的大致圆形轨道上的速度太高，与根据观察到的物质计算出的势不相容。此外，这个旋转速度并不像开普勒近似所预期的那样，作为离中心距离的函数而下降，而是在测量的范围内保持恒定。今天大多数天体物理学家所赞成的解释是这样一种假设，即每个星系都被一个球形的暗物质晕所包围。米尔格罗姆提出了另一种解释，即牛顿动力学在非常低的加速度下被修改。这种MOND范式可以通过适当修改引力势的泊松方程来实现为物理理论。修正的场方程是非线性的，但在引力场不太弱的情况下，它的预言与牛顿的预言是一致的，本项目的目标是将MONDian场方程与Vlasov或无碰撞Boltzmann方程耦合得到非线性的偏微分方程组。当把通常的泊松方程作为场方程时，所得到的系统是Vlasov-Poisson系统，它是描述星系动力学的标准模型。它已被广泛研究的数学文献中，它的MONDian对应到目前为止已（几乎）完全忽略。拟议的项目的目的是了解这个系统的稳态解的稳定性，以及一般的初始数据是否导致全球的时间解决方案或引力坍缩或其他类型的分解的解决方案的问题。特别是，其目的是发现可能的定性差异之间的经典Vlasov泊松系统和MONDian对应这些问题。

The applications background and motivation for the proposed research project is the so-called "missing mass problem" in spiral galaxies. In the outer regions of such galaxies the observed velocities of stars on their roughly circular orbits about the galactic center are far too high to be compatible with the potential computed from the observed matter. Moreover, this rotational velocity does not fall off as a function of the distance from the center, as would be expected by a Keplerian approximation, but remains constant as far out as it is measured. The explanation which is favored by the majority of astrophysicists today is the hypothesis that each galaxy is surrounded by a spherical halo of dark matter.Some 30 years ago, M.~Milgrom proposed an alternative explanation, where Newtonian dynamics is modified at very low accelerations. This MOND paradigm can be implemented as a physical theory by suitably modifying the Poisson equation for the gravitational potential. The modified field equation is non-linear, but its predictions agree with the Newtonian ones, when the gravitational field is not too weak.The objective of the proposed project is the non-linear system of partial differential equations obtained by coupling this MONDian field equation to the Vlasov or collisionless Boltzmann equation. When the usual Poisson equation is taken as field equation, the resulting system is the Vlasov-Poisson system, which is the standard model for describing the dynamics of galaxies. It has been extensively studied in the mathematics literature, where its MONDian counterpart has so far been (almost) completely ignored.The aim of the proposed project is to understand the stability properties of steady state solutions of this system and the question whether general initial data lead to global-in-time solutions or to gravitational collapse or some other type of break down of solutions. In particular, the aim is to uncover possible qualitative differences between the classical Vlasov-Poisson system and its MONDian counterpart with respect to these issues.