Introduction Of Research Advance For The Euler-Poisson Equations

The time evolution of the gaseous stars can be modelled by the system of the Euler-Poissonequation which consists of the Euler equations for the conservation of mass,momentumand energy,and the Poisson equation for the gravitational potentialwhere t≥0,x∈R₃,ρ=ρ(t,x) is the density,v = V(t,x)∈R³ in the velocity.S is the entropy,φ=φ(t, x)is the potential function of the self-gravitational force,gis the gravitationalconstant and P(ρ, S )is the pressure.This system has been studied extensively since the nineteenth century mainly becauseof its relation to astrophysics.For example,for the stability of nonmoving stationary solutionswith radial symmetry for the barotropic gas,there are well known Chandrasekhar and Eddington principles giving the stability when the adiabatic constant greater than (?)and instabilityotherwise,cf.[9].The nonlinear justification of this stability criteria was proved in[16] for theadiabatic constant greater than (?).Recently, there are some works on the existence of stationarysolutions with or without rotation around an axis in [16, 38].As a continuation in this direction, in this paper, we will consider the existence of multiple solutions and the exactmultiplicity of solutions in a more general setting where the velocity field may not be just arotation around an axis. The uniqueness and multiplicity of solutions for different velocity fields give rich solution phenomena to this classical system.In the following discussion, we will concentrate on the barotropic gas,i.e.,where the constant factor k will be normalized to 1 in the sequel, andγ> 1 is the adiabatic constant. For the Euler-Poisson system,it is interesting that the stability and existence of stationary solutions crucially depend on the adiabatic constant. In general, the heavier gas corresponds to smaller y. For example, the adiabatic constantγis (?) for monatomic gas and(?)for diatomic gas. For the significance of the adiabatic constant on the existence, stability, uniqueness and boundary behavior of solutions, please refer to[10,16, 39,32] and references therem.For a stationary solution with a given velocity field v(x), the momentum equation can be written asTaking the divergence on both sides givesCombining this with the Poisson equationwhereTo satisfy the conservation laws of mass and energy,the velocity field cannot be arbitrary.In fact, if v(x) is a rotation around the x₃ axis with a prescribed time independent angular velocityΩ=Ω(η) as a function ofη(x) = (?). then the functions (ρ, v, S,φ) given bysatisfy both the conservation laws of mass and energy, that is,In this case,the solution to the elliptic equation gives a solution to the Euler-Poisson system which was studied in[38].Moreover,the function f(x) in this case takes the formIn this paper, we will consider the case when the function f(x) is not identically zero.In other physical situations, the gaseous star may not rotate just around an axis, so that the function (?)(x) defined can be a general function of space variables. The main purpose of this paper is to study the effect of f(x) coming from the velocity field on the multiplicity of the solutions. In the following discussion, we assume that the conservations of mass and energy are satisfied by a given velocity field. Thus,we can concentrate on the elliptic equation. In fact, the problem on the system coupled with the conservation of mass and energy is interesting which is almost open.