Generalized pitchfork bifurcation on a two-dimensional gaseous star with self-gravity and surface tension
type: Item
instance of: scholarly article
title: Generalized pitchfork bifurcation on a two-dimensional gaseous star with self-gravity and surface tension
zbMATH Open document ID: 1311.35019
author: Shengfu Deng
published in: Discrete and Continuous Dynamical Systems
publication date: 2014-12-02T00:00:00Z
review text: The author considers two-dimensional inviscid, irrotational incompressible gaseous star occupying a spatial domain \(\Omega_{t}\) with a constant density \(\rho\) subjected to the self-gravity and surface tension forces with the aim to look for solutions, here the domain \(\Omega_{t}\) and the velocity field \(\bar{\upsilon}(t,x)\), \(x \in \Omega_{t}\) of the purely rotation about a center with a constant speed. The velocity field \(\bar{\upsilon}(t,x)\) in the incompressible Euler-Poisson system inside the moving field domain \(\Omega_{t}\) satisfies the system \[ \rho [ \overline{\upsilon}_{t}+(\bar{\upsilon},\nabla)\overline{\upsilon}]+\nabla P+\rho \nabla \Phi=0, \quad\mathrm{div}\phantom{.}\bar{\upsilon}=0, \] where \(P\) is the pressure and \(\Phi\) is the self-generated gravitational potential defined by the Poisson equation \(\triangle \Phi=2\pi g \rho \chi_{\Omega_{t}}\) on \(\mathbb{R}^{2}\), \(\chi_{\Omega_{t}}\) is the characteristic function of \(\Omega_{t}\). The indicated Euler-Poisson has a system rotational symmetry, and the rotating waves are the relative equilibria of the energy with respect to the angular momentum. The author studies the traveling wave solutions with a small amplitude and a small velocity at the usage of bifurcation theory methods.
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 76B15
Mathematics Subject Classification ID: 85A05
Mathematics Subject Classification ID: 35C07
zbMATH DE Number: 6375838
zbMATH Keywords: Dirichlet-Neumann operator
zbMATH Keywords: periodic orbits
zbMATH Keywords: Lyapunov-Schmidt method
zbMATH Keywords: Euler-Poisson system
zbMATH Keywords: rotating waves
zbMATH Keywords: relative equilibria
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
full work available at URL: dcds.2014.34.3419
OpenAlex ID: W2334884132
cites work: A theory of solitary water-waves in the presence of surface tension
cites work: On periodic water-waves and their convergence to solitary waves in the long-wave limit
cites work: Exact solitary water waves with capillary ripples at infinity
cites work: The sub-harmonic bifurcation of Stokes waves
cites work: A multiplicity result for solitary gravity-capillary waves in deep water via critical-point theory
cites work: A plethora of solitary gravity-capillary water waves with nearly critical Bond and Froude numbers
cites work: Exact steady periodic water waves with vorticity
cites work: Traveling Two and Three Dimensional Capillary Gravity Water Waves
cites work: Traveling gravity water waves in two and three dimensions.
cites work: Exact Theory of Three-Dimensional Water Waves at the Critical Speed
cites work: Two-dimensional travelling waves over a moving bottom
cites work: Three-dimensional gravity-capillary waves on water -- small surface tension case
cites work: An existence theory for three-dimensional periodic travelling gravity-capillary water waves with bounded transverse profiles
cites work: Three‐dimensional travelling gravity‐capillary water waves
cites work: A bifurcation theory for three-dimensional oblique travelling gravity-capillary water waves
cites work: A dimension–breaking phenomenon in the theory of steady gravity–capillary water waves
cites work: A spatial dynamics approach to three-dimensional gravity-capillary steady water waves
cites work: Fully localised solitary-wave solutions of the three-dimensional gravity -- capillary water-wave problem
cites work: Q4544271
cites work: Q3486279
cites work: Water waves for small surface tension: an approach via normal form
cites work: Perturbed homoclinic solutions in reversible 1:1 resonance vector fields
cites work: Asymmetrical three-dimensional travelling gravity waves
cites work: On the existence theory for irrotational water waves
cites work: On the theory of steady-state waves of finite amplitude
cites work: Orbits homoclinic to exponentially small periodic orbits for a class of reversible systems. Application to water waves
cites work: Three-dimensional, nonlinear wave interaction in water of constant depth
cites work: Geometry and a priori estimates for free boundary problems of the Euler's equation
cites work: Existence of a generalized solitary wave solution for water with positive Bond number less than 1/3
cites work: Existence of large amplitude periodic waves in two-fluid flows of infinite depth
DOI: 10.3934/DCDS.2014.34.3419
description: scientific article; zbMATH DE number 6375838