Vortical and Self-similar Flows of 2D Compressible Euler Equations

TL;DR

This paper develops vortical and self-similar solutions for 2D compressible Euler equations, enhancing understanding of ocean vortices and providing benchmarks for numerical methods, with classifications of solution behaviors and conjectures for 3D cases.

Contribution

It introduces new vortical and self-similar solutions for 2D Euler equations, extending prior radial solutions and analyzing solution behaviors through a new Emden equation.

Findings

Classified conditions for blowup, periodicity, and global existence.

Provided new solutions that model ocean vortices.

Formulated a conjecture for 3D rotational solutions.

Abstract

This paper presents the vortical and self-similar solutions for 2D compressible Euler equations using the separation method. These solutions complement Makino's solutions in radial symmetry without rotation. The rotational solutions provide new information that furthers our understanding of ocean vortices and reference examples for numerical methods. In addition, the corresponding blowup, time-periodic or global existence conditions are classified through an analysis of the new Emden equation. A conjecture regarding rotational solutions in 3D is also made.