Generalized pitchfork bifurcation on a two-dimensional gaseous star with self-gravity and surface tension
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.
scientific article; zbMATH DE number 6375838
scientific article