RETRACTED ARTICLE: Solutions of the equilibrium equations with finite mass subject

Junguo Shi,Yadong Liao

doi:10.1186/s13660-015-0885-z

Abstract

AbstractIn this paper, we investigate the slow equilibrium equations with finite mass subject to a homogeneous Neumann type boundary condition. Based on an auxiliary function method and a differential inequality technique, the existence of equilibrium equations is obtained if the angular is bounded and the blow-up occurs in finite time.

Highlights

In -D space, the equilibrium equations for a self-gravitating fluid rotating about the x axis with prescribed velocity (r) can be written ∇P = ρ∇(– + r s (s) ds), ( . ) = πgρ.Here ρ, g, and denote the density, gravitational constant, and gravitational potential, respectively, P is the pressure of the fluid at a point x ∈ R, r = x + x
Miyamoto [ ] have found that there exists an equilibrium solution if the angular velocity is less than a certain constant and there is no equilibrium for large velocity
In more general conditions than in [ ], we prove that there exists an equilibrium solution under the following constraint set: AM := ρ ρ ≥, ρ is axisymmetric, ρ dx = M

Summary

Introduction

In -D space, the equilibrium equations for a self-gravitating fluid rotating about the x axis with prescribed velocity (r) can be written. In the study of this model, Auchmuty and Beals [ ] proved the existence of the equilibrium solution if the angular velocity satisfies certain decay conditions. In more general conditions than in [ ], we prove that there exists an equilibrium solution under the following constraint set: AM := ρ ρ ≥ , ρ is axisymmetric, ρ dx = M. A standard method to obtain steady states is to prescribe the minimizer of the stellar energy functional. The main problem is to show the steady state has finite mass and compact support. To approach this problem, we define the energy functional, g ρ(x)ρ(y). In Section , first we prove the existence of a minimizer of the energy functional F in AM.

Proof The interpolation inequality implies ρ

Lemma we easily know

For the induced potentials we have

Now we want to show that