In this talk, I will present new results on the symmetry reduction of gas dynamic systems of PDEs following the general framework presented by  Lev Ovsyannikov in his article  “The “podmodeli” program. Gas dynamics”

https://www.sciencedirect.com/science/article/pii/0021892894901376

The gas dynamics systems of equations, with an arbitrary state equation, has an 11-dimensional Lie algebra of symmetries which generates a group of space translations, time translation, rotations, Galilean translations, and uniform dilations. In this investigation, we consider a special state equation, where the pressure is the sum of two functions – the first function depends on density, and the second function depends on entropy. Such a system of equations has an additional symmetry, pressure translation, and thus admits a 12-dimensional Lie algebra of symmetries. I will present a classification of subalgebras of this 12-dimensional Lie algebra up to similarity and show subgraphs of their embeddings. I will demonstrate how using the process of symmetry reduction one can transform a gas dynamic system of this type into a system with one independent variable (an ODE system), which can be solved exactly leading to new particular solutions of the original system. A description of the motion of particles for these particular solutions will be also presented.