One-Dimensional Compressible Heat-Conducting Gas with Temperature-Dependent Viscosity

TL;DR

This paper proves the global existence and uniqueness of solutions for a one-dimensional compressible heat-conducting gas model with temperature-dependent viscosity and heat conductivity, accommodating large initial data under certain conditions.

Contribution

It is the first to establish global solutions for the system with viscosity depending on both density and temperature, allowing large initial data when a parameter is small.

Findings

Global existence and uniqueness of solutions are proven.

Solutions exist for large initial data under specific conditions.

The model handles general adiabatic exponents.

Abstract

We consider the one-dimensional compressible Navier--Stokes system for a viscous and heat-conducting ideal polytropic gas when the viscosity $\mu$ and the heat conductivity $\kappa$ depend on the specific volume $v$ and the temperature $\theta$ and are both proportional to $h(v)\theta^{\alpha}$ for certain non-degenerate smooth function $h$. We prove the existence and uniqueness of a global-in-time non-vacuum solution to its Cauchy problem under certain assumptions on the parameter $\alpha$ and initial data, which imply that the initial data can be large if $|\alpha|$ is sufficiently small. Our result appears to be the first global existence result for general adiabatic exponent and large initial data when the viscosity coefficient depends on both the density and the temperature.