Products of distributions and the problem of galactic rotation

Abstract

In the present paper we study the nonlinear system ut + [ϕ(u)]x + v = 0, vt + ψ(u)vx = 0 as a model for the one-dimensional dynamics of dark matter. We prove that under certain conditions this system, such as the Gurevich-Zybin system, can also explain why the observed rotation speed (relative to the galactic center) of stars near galactic halos do not coincide with what it is expected in classical mechanics. The solutions are obtained in fully explicit formulas, in a convenient space of distributions, without using any result within the classical framework. For such purpose we use the α-solution concept which is defined within a product of distributions. Such a concept generalizes the classical solution concept and for evolution equations may also be seen as an extension of the weak solution concept to the nonlinear setting.