Rotational and Self-similar Solutions for the Compressible Euler Equations in R^3

TL;DR

This paper derives rotational and self-similar solutions for the 3D compressible Euler equations, introducing a new Emden dynamical system, and analyzes their blowup and global existence, offering valuable examples for vortex modeling.

Contribution

It presents novel rotational and self-similar solutions for the 3D Euler equations, extending previous irrotational and 2D solutions, and introduces a new Emden dynamical system for analysis.

Findings

Identification of blowup phenomena in solutions.

Conditions for global existence of solutions.

Concrete examples of vortices in fluid dynamics.

Abstract

In this paper, we present rotational and self-similar solutions for the compressible Euler equations in R^3 using the separation method. These solutions partly complement Yuen's irrotational and elliptic solutions in R^3 [Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 4524-4528] as well as rotational and radial solutions in R^2 [Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 2172-2180]. A newly deduced Emden dynamical system is obtained. Some blowup phenomena and global existences of the responding solutions can be determined. The 3D rotational solutions provide concrete reference examples for vortices in computational fluid dynamics.