Global Existence And Asymptotic Behavior For A One-Dimensional Isentropic And Isothermal Model System Of Compressible Viscous Gas

In this paper we discuss the global existence and asymptotic behavior for a one-dimensional isentropic and isothermal model system of compressible viscous gas in a bounded region, with an external force and initial conditions as well as boundary conditions. The model we consider is as follows: v_t - u_x = 0, u_t + (av^-1)_x = μ,(u_x/v)_x + f(∫₀^xvdy,t) with initial conditions v(x,0) = v_o(x), u(x,0) = u_o(x) and boundary conditions u(0,t) = u(1,t) = 0.The main results are as follows: 1. The solution v{x,t) to our problem has uniform positive upper bound and lower bound. 2. The global existence and asymptotic behavior of solutions in H¹. 3. The global existence and asymptotic behavior of solutions in H².The results obtained in this paper are different with others: In case f = 0, the authors of references [3-8] have been obtained the existence and uniqueness of the uniformly boundary, global-in-time solution under various initial conditions and the equation of state which different with this paper. In reference [9], for arbitrary periodic external forces, S. Yanagi proves the existence of periodic solutions for the problem which same as this paper's. The authors of reference [10] prove that when the state function is: p(v) = av^-γ, (a > 0,is constant) and 1 < γ≤2, then for arbitrary large initial data and external forces, the problem has an uniformly bounded, global-in-time solutions. His proof is based on L²-energy estimates and a technique used in reference [9].The proof of this paper is based on the methods used by Yuming Qin in references [11-13,15-17], in which he proves the global existence, asymptotic behavior, exponential stability and universal attractor of solutions to a system of equations for a nonlinear one-dimensional viscous heat-conducting real gas.