Mathematical modelling of galaxies in MOND (Modified Newtonian Dynamics)

点击翻译按钮获取中文摘要

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.