Abstract
In this paper, we are concerned with the compressible Euler-Poisson system coupled to a magnetic field in the three-dimensional space. Based on a variational method and the exact expression of the Green’s function for an elliptic equation in spherical coordinates, we prove the existence of stationary star solutions.
Highlights
The Euler-Poisson system of compressible fluids coupled to a magnetic field is given by
Coupling to the magnetic field, Federbush, Luo, and Smoller [ ] first proved the existence of axisymmetric stationary solutions of system ( . ). They utilized a variational method and proved the existence of a stationary solution expressed by density, which is a minimizer of the associated energy functional
Motivated by the paper [ ], we show the existence of stationary star solutions ( . ) of system ( . ) in spherical coordinates by using the variational methods
Summary
The Euler-Poisson system of compressible fluids coupled to a magnetic field is given by. Were concerned with three-dimensional spherically symmetric solutions of the compressible Navier-Stokes-Poisson equations with free boundary condition They proved the nonlinear asymptotic stability of the Lane-Emden solutions for spherically symmetric motions of viscous gaseous stars if the adiabatic constant α lies in the stability range They utilized a variational method and proved the existence of a stationary solution expressed by density, which is a minimizer of the associated energy functional. = πGρ H = , where ρ is the density, H = (H , H , H ) is the magnetic field, p is the pressure function, is the Newtonian potential, G is the gravitational constant, and μ is the permeability of vacuum. It follows from the definition of cross product ( . ) and ( . ) that (∇ × H) × H
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