Vortical and self-similar flows of 2D compressible Euler equations

Abstract

a b s t r a c tThis 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.

Rotational and self-similar solutions for the compressible Euler equations in R3

“…(See Lemma 5 in [18].). However, it is trivial to see that it is not possible to have the 3D timeperiodic solutions (3)-(5) as functionb(t) < 0.…”

“…These novel solutions (3)-(5) partly complement Yuen's irrotational and elliptic solutions in 3D [17] as well as rotational and radial solutions in 2D [18]. A newly deduced Emden dynamical system…”

“…Remark 3 For the compressible Euler equations (1) in R 3 , the rotational solutions (3)- (5) correspond to Yuen's irrotational and elliptic solutions in R 3 [17] as well as rotational and radial solutions in R 2 [18].…”

“…For rotational flows, Zhang and Zheng constructed explicitly rotational solutions for the Euler equations with γ = 2 and N = 2 in 1997 [19]. In 2014, Yuen obtained a class of rotational solutions for the compressible Euler equations (1) for γ > 1 in 2D in [18]:…”

“…Based on the works in [17] and [18], we obtain novel rotational and self-similar solutions for the 3D compressible Euler equations (1).…”