The gas dynamics equations with the state equation of a general form have the following symmetries: space translations, time translation, rotations, Galilean translations, uniform dilatation. In this investigation the state equation is a pressure equal to the sum of two functions - the first function depends on density, and the second function depends on entropy [1]. Such system of equations has additional symmetry – pressure translation. The system admits a 12-dimensional Lie algebra. An optimal system of dissimilar subalgebras of the 12-dimensional Lie algebra was constructed in [2].

Invariant submodels of rank 1 are calculated for 3-dimensional subalgebras [2]. The submodels are the systems of ordinary differential equations. Exact solutions were found for some submodels [3]. The motion of particles and volumes according to the exact solution is considered.

The author was supported by the Russian Foundation for Basic Research (project no. 18-29-10071) and partially from the Federal Budget by the State Target (project no. 0246-2019-0052).

References

[1] Ovsyannikov L.V. The “podmodeli” program. Gas dynamics // Journal of Applied Mathematics and Mechanics, 1994, vol. 58, no. 4, pp. 601-627. doi:10.1016/0021-8928(94)90137-6

[2] Siraeva D.T. Optimal system of non-similar subalgebras of sum of two ideals // Ufa Mathematical Journal. 2014. Vol. 6, No 1. Pp.90-103. Doi:10.13108/2014-6-1-90

[3] Siraeva D.T. Two invariant submodels of rank 1 of the hydrodynamic type equations and exact solutions // Journal of Physics: Conference Series, 1666 012049 (2020). Doi: 10.1088/1742-6596/1666/1/012049