Stability Of Gaseous Stars In The Non-isentropic Case

The motion of the self-gravitational gaseous stars can be described by the EulerPoisson equations. It contains Euler equations for conservation of mass, momentum and energy, and Poisson equation through which the gravitational potential is determined by the density distribution of the gas itself:whereρ=ρ(x, t), v = v(x, t), S = S(x, t),P,g andφdenote the density, velocity, entropy, pressure, gravitational constant and gravitational potential, respectively. Here t≥0 denotes the time, space variables x∈R~3.The pressure satisfies the following equation of state:P = A_ρ~γe~S,where the constant A will be normalized to 1,γ> 1 is the adiabatic constant.Astrophysicists are interested in the stability question. The solutions and stability of Euler-Poisson equations are concerned with the entropy function and adiabatic constant. Recently, many results about the existence of steady solutions are obtained, and some stability results are obtained for isentropic flow. As we know, there are no results about the stability for non-isentropic flow.The main purpose of this paper is concerned with the nonlinear stability of gaseous stars and the existence of stationary solutions in the non-isentropic case. When 4/3 <γ< 2, S(x, t) is a smooth bounded function. First, we get the steady states are minimizers of the energy via concentration-compactness method; then using the variational approach we get the stability results of the non-isentropic flow.