On universality of homogeneous Euler equation

TL;DR

This paper explores the universal properties of the multidimensional homogeneous Euler equation, showing how its solutions relate to various hydrodynamic equations across different dimensions.

Contribution

It demonstrates that subclasses of solutions to the Euler equation can generate solutions to diverse hydrodynamic systems, revealing a unifying framework.

Findings

Solutions of the Euler equation relate to shallow water equations

Connections established with integrable systems in Riemann invariants

Multidimensional equations describe isoenthalpic and polytropic motions

Abstract

Master character of the multidimensional homogeneous Euler equation is discussed. It is shown that under restrictions to the lower dimensions certain subclasses of its solutions provide us with the solutions of various hydrodynamic type equations. Integrable one dimensional systems in terms of Riemann invariants and its extensions, multidimensional equations describing isoenthalpic and polytropic motions and shallow water type equations are among them.