Detailed Description of the Evolution Mechanism for Singularities in the System of Pressureless Gas Dynamics

Abstract

The system of pressureless gas dynamics is a hydrodynamically justified generalization of the system consisting of the Burgers vector equation in the limit of vanishing viscosity and the mass conservation law. The latter system of equations was intensively used, in particular, in astrophysics to describe the large-scale structure of the Universe. The solutions of the vector Burgers equation involve interesting dynamics of singularities, which can describe concentration processes. However, this dynamics does not satisfy the law of momentum conservation, which prevents us from treating it as dynamics of material objects. In this paper, momentum-conserving dynamics of singularities is investigated on the basis of the pressureless gas dynamics system. Such dynamics turns out to be more diverse and complex, but it is also possible to formulate a variational approach, for which the basic principles and relations are obtained in the work.