Solutions of Three-Dimensional Stationary Gas Dynamics Equations

TL;DR

The paper addresses the challenge of obtaining exact solutions to the three-dimensional stationary gas-dynamics equations for a polytropic gas by applying Lie symmetry methods. For the Chaplygin gas, it delivers a highly general solution family depending on three arbitrary functions, grounded in the infinite-dimensional symmetry of the potential-flow equation $\operatorname{div}\left(\frac{\nabla \phi}{|\nabla \phi|}\right)=0$ and the EOS $p = -\frac{a}{\rho} + b$. In the general adiabatic-index case, explicit solutions are constructed and parameterized by constants, with extensions to both potential and vortical flows, including Grad–Shafranov-type reductions and associated streamlines. The work expands the catalog of exact 3D solutions, highlights rich symmetry structures, and offers methods potentially applicable to broader nonlinear hydrodynamic models.

Abstract

This paper examines the three-dimensional stationary equations of a polytropic gas and employs symmetry methods to construct exact analytical solutions. In the Chaplygin gas case, the analysis yields a highly general solution family depending on three arbitrary functions, while the general adiabatic index formulation admits explicit solutions parameterized by several constants.