Symmetry analysis and new partially invariant solutions for the gas dynamics system with a special equation of state

TL;DR

This work extends symmetry analysis of the gas-dynamics system with the special state equation $P=f(\rho)+S$ by establishing the 12-dimensional Lie algebra $L_{12}$ and its automorphisms, and by classifying generating invariants for a broad set of four-dimensional subalgebras from an optimal list. It constructs a rank-1, defect-1 partially invariant submodel for one subalgebra and derives explicit solutions that lift to the original system, including isochoric and non-isochoric motion with detailed trajectory analyses. The paper also maps each four-dimensional subalgebra to its isomorphism class, providing a foundation for a hierarchical hierarchy of reduced systems and encouraging future exploration of invariant and partially invariant reductions in gas dynamics with nonstandard equations of state. Overall, the results yield new exact solutions, enriched invariant structures, and a roadmap for systematic submodel development in this class of PDEs.

Abstract

This paper is a contribution to the symmetry analysis of the gas dynamics system in the vein of the ''podmodeli'' (submodels) program outlined by Ovsyannikov (1994). We consider the case of the special state equation, prescribing pressure to be the sum of entropy and an arbitrary function of density. Such a system has a 12-dimensional symmetry Lie algebra. This work advances the study of its four-dimensional subalgebras, continuing the work started in Siraeva (2024). For a large subset of not previously considered, non-similar four-dimensional subalgebras from an optimal list in Siraeva (2014), we compute a complete set of generating invariants. For one of the subalgebras, we construct a partially symmetry-reduced system. We explicitly solve this reduced system (submodel). This leads to new families of explicit solutions of the original system. We analyze the trajectories of these solutions. Additionally, we match each of the subalgebras considered in this paper with its isomorphism class, planting a seed for future study of the hierarchy of the reduced systems.

Figures (2)

• Figure 1: The motion of the particle volume \ref{['eqn8']}, \ref{['eqn20']} with $\rho_0=1$, $k_0=1$, $m_0=1$. The volume is a sphere at $t=0$ and becomes ellipsoidal at $t=1.6$, $t=2$. The trajectories of particles \ref{['eqn20']} with initial coordinates $(x_0,y_0,z_0)=(0,0,0)$ (black dash curve); $(0,0,1)$ (red dash curve); $(0,0,-1)$ (green dash curve) are shown for $t=0$ to $t=3$.
• Figure 2: For fixed parameters $\rho_0=1$, $k_0=1$, $m_0=1$, we depict trajectories \ref{['eqn7']} of four particles starting from the same initial point $(-2,1,1)$ lying on the plane \ref{['eqn9']}, and emitted with four different initial velocities $u_0 = 0, 1, 2, 3$. At $t=3$, they are shown to lie on the same black dashed line parallel to the $x$-axis.