Asymptotic behavior of solutions of initial-boundary value problems for 1D viscous and heat-conducting ionized gas

Abstract

Dynamics of the macroscopic motion of a compressible, viscous and heat-conducting, one-dimensional monatomic ionized gas can be described by a compressible Navier-Stokes type system. Compared with the corresponding compressible Navier-Stokes type systems modeling a viscous and heat conducting ideal polytropic gas [7,10,13–16,18,19,23,24,26,28,31,32], the real gases [17], nonlinear thermoviscoelasticity [5,6], and a viscous radiative and reactive gas [11,12,20,21,30], the dependence of the state of an ionized gas on the degree of ionization leads to the loss of concavity of the physical entropy in some small bounded domain. Such a property of the ionized gas makes it impossible to deduce the dissipative estimates on the first-order spatial derivative of the bulk velocity and the absolute temperature from the basic energy estimates obtained through the normalized entropy and, as a consequence, makes the construction of its global large solutions and the study of their large time behavior more difficult.By making full use of the intrinsic structure of equations, the existence of global large smooth solutions to two types of initial boundary value problems of one-dimensional, compressible Navier-Stokes system for a viscous and heat-conducting ionized gas is established in [22] for a class of density and/or temperature dependent transport coefficients, but since the estimates obtained there depend on the time variable, the problem on their large time behavior remains unsolved and the main purpose of this paper is to give the precise description of the large time behavior of the global solutions constructed in [22]. The key ingredient in our analysis is to deduce the uniform-in-time positive lower and upper bounds on the specific volume and the absolute temperature.