Invariant solutions of the gas dynamics equations on 4-parametric three-dimensional subalgebras containing all translations in space and pressure translation

Abstract

The gas dynamics equations with pressure in the form of the sum of density and entropy functions are considered. The admissible group of transformations is expanded due to the pressure translation. The Lie algebra corresponding to the group is 12-dimensional. Invariant submodels of rank 1 generated by 3-dimensional 4-parameter subalgebras of all translations in space and pressure translation are constructed. Three families of exact solutions are found which describe the motion of particles with a linear velocity field with inhomogeneous deformation. The moment of time of the presence or absence of particles collapse for each family of solutions are found. In a particular case, the trajectories of particles motion are constructed. The volume of particles at the initial moment of time limited by the sphere is isolated. It has been proved that at any other time moments the volume turns into an ellipsoid and the particles volume value does not change with time.