Asymptotic Compactness Of Global Trajectories Generated By The Navier-Stokes-Possion Equations Of A Compressible Fluid

In this paper, we consider the Navier-Stokes-Possion equations on I(?)R andΩ(?)R₃ a bounded domain with C^2,θ boundedαΩ(0 <θ< 1). Then we prove that positive trajectories by solutions of the Navier-Stokes-Possion equations are precompact.The scientists regard the Navier-Stokes equations of a viscous compressible fluid without the Possion term as important. The Navier- Stokes equations may reflect the motion of the fluids. But they do not consider the infection of the self-gravition to fluids. The Navier-Stokes-Possion equations mainly describe the motion of gasous stars with viscosity and self-gravitation. Therefore, the Navier-Stokes-Poisson equations are the more rigorous and better models that reflect the motion of gasous stars than the Navier-Stokes equations and the Euler-Poisson equations.In[11] [12], the authors show the existence of a global attractor for compressible isothermal fluid in one space dimension. In [2], on I= R,Ω(?)R₃ a bounded domain with Lipschitz boundary, Feireisl and companions show that positive trajectories generated by the solutions are precompacted in suitably chose topologies under the sole assumption that the driving force is a bounded and measurable function on R⁺.In this paper, that the Navier-Stokes equations gain the Possion term makes the estimates change, and increases the difficulty. In view of means used by Feireisl[2], the main idea that is based on careful analysis of the " defect measure " can overcome difficulties.This paper is divided into three chapters.Chapter 1: Introduction. It includes researched history, backgrounds and the result of the paper. Chapter 2: The proof of Theorem 1.1--The existence of bounded aborbing sets. Itis divided into three. It includes Preliminaries ? Decay estimates and Conclusion-the proof of Theorem 1.1 .Chapter 3: The proof of Theorem 1.2. It is divided into five.§3.1: Basic estimates and convergence of the sequence of time shifts. In this section, we show several inequalities independent of n forρ_n, u_n,Φ_n.§3.2: More about the density. We introduce functions M_k(z) and T_k(z). Approximating M_k by a sequence of smooth functions, we obtain more about the density.§3.3: The effective viscous flux and its properties. We introduce the quantity p(ρ)-(λ+ 2μ)divu called usually the effective viscous flux and some properties.§3.4: Compactness of the density. First, we will introduce a functionΨand its property. Then, we show that D(t) enjoys some properties. Therefore, we we indirectly show the compactness of the density.§3.5: Compactness of the momenta. Using the convergence ofρ_n,tn,u_n,tn, we get that the momenta is compact.