On Euler-Poisson Equations

This dissertation investigates Euler-Poisson equations:where t ≥ 0 denotes time, x ∈ Ωt is a spacial variable and ft C R~n (n ≥ 3) is a bounded smooth domain, ρ = ρ(t, x) is the density, v = v(t, x) ∈ R~n, S = S(t, x) and Φ = Φ(t, x) stand for the velocity, entropy function and the self-gravitational potential respectively, g denotes the gravitational constant and ω_n is the volume of the unit ball in R~n. P is the pressure which satisfies the following equation of state: P = P(ρ, S) = e~S ρ~γ with γ > 1 being an adiabatic constant. This system of Euler-Poisson equations comes from astrophysics. When n = 3, it is a model to describe the hydrodynamic evolution of the internal structure of the self-gravitational gaseous stars. 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.We are interested in stationary solutions of the system, i.e. solutions independent of time t. In detail, the first and the third equations are automatically satisfied under some conditions, for example, the star rotates around some axis [34] or the velocity v is zero [17]. Thus we only need to consider the second and the fourth equations.First, we simply consider the case that domain Ω = B_r(0), which is a ball in R~n centered at the origin with radius R > 0. After transformation:with K = n(n - 2)ω_ng , the second and the fourth equations can be turned into a semi-linear elliptic one, div -div(v ? Vv) = σ. Here, the main difficulties that the strong maximum principle fails to hold are needed to be overcomed, since σ > 0. Under some conditions, we obtain the existence, depending on 7, and the properties of spherically symmetric solutions of the above equation with the boundary data = 0, by a shooting method. Furthermore, this paper also discuss non-existence of spherically symmetric solutions under come conditions.Then, we consider a more general velocity field v on a more general bounded smooth domain fi C I". Transformation u = -^jespy~1 turns the second and the fourth equations into the following semi-linear equation with a parameter a > 0,S S S S X Sdi\(e~TVu)--(AS)e~TU+Ke~^e~z'UT:zi—af(x)e~:y — 0, where JQ \f(x)\2dx — 1. When f(x) and AS change sign, the strong maximum principle can not be directly used here, so it is necessary to overcome the difficulties caused by them. For the case when f(x) and AS change sign, under various conditions, the existence, multiplicity and uniqueness of positive solutions of problem div(e" Vm) — -(AS)e~iu +SSI S'^e'^v^1 — of{x)e~i =0,iGO;u\qq = 0 are studied.