Invariant submodels of the model of thermal motion of gas in a rarefied space

A model describing the thermal motion of a gas in a rarefied space is investigated. This model can be used in the study of the motion of gas in outer space, and the processes occurring inside the tornado, and the state of the medium behind the shock front of the wave after a very intense explosion. For a given initial pressure distribution, a special choice of mass Lagrange variables leads to a reduced system of differential equations describing this motion, in which the number of independent variables is one less than the original system. This means that there is a stratification of a highly rarefied gas with respect to pressure. Namely, in a strongly rarefied space for each given initial pressure distribution, at each instant of time all gas particles are localized on a two-dimensional surface moving in this space. At each point of this surface, the acceleration vector is collinear with its normal vector. The resulting system admits an infinite Lie transformation group. All significantly various submodels that are invariant with respect to the subgroups of its eight-parameter subgroup generated by the transfer, extension, rotation, and hyperbolic rotation operators (the Lorentz operator) are found. For invariant submodels of rank 1, the basic mechanical characteristics of the gas flow described by them are obtained. Conditions for the existence of these submodels are given. For invariant submodels of rank 2, integral equations describing these submodels are obtained. For some submodels, the problem of describing the gas flow from the initial location of its particles and the distribution of their velocities has been investigated.