Symmetric flows for compressible heat-conducting fluids with temperature dependent viscosity coefficients

Abstract

We consider the Navier--Stokes equations for compressible heat-conducting ideal polytropic gases in a bounded annular domain when the viscosity and thermal conductivity coefficients are general smooth functions of temperature. A global-in-time, spherically or cylindrically symmetric, classical solution to the initial boundary value problem is shown to exist uniquely and converge exponentially to the constant state as the time tends to infinity under certain assumptions on the initial data and the adiabatic exponent $\gamma$. The initial data can be large if $\gamma$ is sufficiently close to 1. These results are of Nishida--Smoller type and extend the work [Liu et al., SIAM J. Math. Anal. 46 (2014), 2185--2228] restricted to the one-dimensional flows.