Study On Some Topics For The Euler-Poisson Equations

In this paper, we consider the Euler-Poisson equationswhere t > 0. x ∈ Ω. Ω is a collected open subset of RN. .V > 3. p = p(t.x) is the density of gaseous stars, v = u(t.x) ∈ RN is the velocity. $ = $(t.x) is the energy potential of the self-gravitational force, g is the gravitational constant. N is the measure of the unit ball in RN. and P is the pressure satisfying the following equation:where K and are positive constants. > 1 with the entropy function S = S(t.x) in R- RN .The system (1.1) is compressible Euler equations: the gravitational potential is determined by the density distribution of the gas itself through the Poisson equation f 1.2). In fact, the existence, uniqueness and stability of solutions for the equations strongly depend on the exponent . Euler-Poisson equations have been studied by many scholars all over the world, such as [2], [4], [6], [9], [11] and [12]. The authors in "6] have studied the stationary solutions of the equations when N = 3: Moreover, it is interesting that the solutions can blow up in finite time, which has been obtained in [14] when N = 3. In our paper, we studied the stationary solutions and blowup solutions of the equations in the general N-dimensional space, which depend on .In our paper, the existence and non-existence of stationary- solutions for the Euler-Poisson equations in C2(Ω) have been first obtained by Mountain PassTheorem (see [1] or [3]) and iteration method etc. And then, the existence of blowup solutions in C2(BR(0)) for non-isentropic flow and for isentropic flow has been obtained by constructive method. Lastly, we got the existence of non-blowup solutions in C2(RN) for isentropic flow by Fourier transformation and Hardy-Littlewood-Paley inequality (see [16]).