An analysis of the 2-D isentropic Euler Equations for a generalized polytropic gas law

TL;DR

This paper analyzes 2-D isentropic Euler equations for generalized polytropic gases, focusing on rotational subsonic flows, deriving a Bernoulli equation, establishing an ellipticity principle, and exploring quasi-potential flows.

Contribution

It introduces novel applications of an ellipticity principle and Bernoulli equation for generalized gases, including the first analysis of potential flow in pseudo-subsonic regimes for gamma<1.

Findings

Derived a Bernoulli type equation for self-similar rotational flows.

Established an ellipticity principle for generalized polytropic gases.

Analyzed quasi-potential flows and their behavior.

Abstract

In this paper we developed an analysis of the compressible, isentropic Euler equations in two spatial dimensions for a generalized polytropic gas law. The main focus is rotational flows in the subsonic regimes, described through the framework of the Euler equations expressed in self-similar variables and pseudo-velocities. A Bernoulli type equation is derived, serving as a cornerstone for establising a self-similar system tailored to rotational flows. We also developed an Ellipticity Principle for generalized polytropic gases, which is applied twice in this paper. To the best of the authors' knowledge, both applications appear for the first time. In particular, the analysis of the potential flow problem in the pseudo-subsonic regime is nontrivial for generalized polytropic gases when gamma< 1. In this setting, refined techniques, such as the Moser iteration method combined with suitable a priori estimates, are required. In the final section, the study extends to an analysis of a perturbed model, introducing the concept of quasi-potential flows, offering insights into their behavior and implications.