Asymptotic behavior of spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations with large initial data

Abstract

In this paper, we study the asymptotic behavior of global spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations for the viscous heat conducting ideal polytropic gas flow with large initial data in $ H^1 $, when the heat conductivity coefficient depends on the temperature, practically, $ \kappa(\theta) = \tilde{\kappa}_1+\tilde{\kappa}_2\theta^q $ where constants $ \tilde{\kappa}_1>0 $, $ \tilde{\kappa}_2>0 $ and $ q>0 $ (as to the case of $ \tilde{\kappa}_1 = 0 $, please refer to the Appendix). In addition, the exponential decay rate of solutions toward to the constant state as time tends to infinity for the initial boundary value problem in bounded domain is obtained. The mass density and temperature are proved to be pointwise bounded from below and above, independent of time although strong nonlinearity in heat diffusion. The analysis is based on some delicate uniform energy estimates independent of time.