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

## TL;DR

This paper proves the global existence, uniqueness, and exponential stability of symmetric solutions to the compressible Navier--Stokes equations with temperature-dependent viscosity in bounded domains, extending previous one-dimensional results.

## Contribution

It extends Nishida--Smoller type results to multi-dimensional symmetric flows with temperature-dependent coefficients, allowing large initial data near certain parameters.

## Key findings

- Existence and uniqueness of global symmetric solutions
- Exponential convergence to equilibrium state
- Extension from 1D to multi-dimensional symmetric flows

## 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.