Nonlinearly stability of solutions on the outer Pressure Problem of Compressible Navier-Stokes System with Temperature-Dependent Heat Conductivity

TL;DR

This paper establishes the nonlinear stability and convergence of solutions to the one-dimensional compressible Navier-Stokes equations with temperature-dependent heat conductivity under outer pressure boundary conditions, extending previous results.

Contribution

It proves boundedness, existence, and stability of solutions with less restrictive initial data assumptions, improving upon prior work by Nagasawa.

Findings

Bounded specific volume and temperature over time

Existence of local and global strong solutions

Convergence to stationary state and stability

Abstract

In this paper, the one-dimensional compressible Navier-Stokes system with outer pressure boundary conditions is investigated. Under some suitable assumptions, we prove that the specific volume and the temperature are bounded from below and above independently of time, and then give the local and global existence of strong solutions. Furthermore, we also obtain the convergence of the global strong solution to a stationary state and the nonlinearly stability of its convergence. It is worth noticing that all the assumptions imposed on the initial data are the same as Takeyuki Nagasawa [Japan.J.Appl.Math.(1988)]. Therefore, our work can be regarded as an improvement of the results of Takeyuki.