Communications in Mathematical Sciences

Volume 18 (2020)

Number 7

On the free boundary problem of 1D compressible Navier–Stokes equations with heat conductivity dependent of temperature

Pages: 2039 – 2057

DOI: https://dx.doi.org/10.4310/CMS.2020.v18.n7.a9

Authors

Zilai Li (School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, China)

Yulin Ye (School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, China)

Abstract

The free boundary problem of one-dimensional heat conducting compressible Navier–Stokes equations with large initial data is investigated. We obtain the global existence of strong solution under stress-free boundary condition along the free surface, where the heat conductivity depends on temperature $(\kappa = \overline{\kappa} \theta^b , b \in (0, \infty))$ and the viscosity coefficient depends on density $(\mu = \overline{\mu} (1 + \rho^a) , a \in [ 0, \infty))$. Moreover, the large-time behavior of the free boundary for the full compressible Navier–Stokes equations is also considered when the viscosity is constant and it is first shown that the interfaces which separate the gas from vacuum will expand outwards at an algebraic rate in time for all $\gamma \gt 1$.

Keywords

compressible Navier–Stokes equations, temperature-dependent heat conductivity, free boundary, global strong solution, large-time behavior

2010 Mathematics Subject Classification

35Q30, 35R35, 76N10

Full Text (PDF format)

Li is supported by the NSFC (No. 11701443, No.11601128, No.11671319, No.11931013), Fund of HPU (No.B2016-57). Ye is partially supported by NSFC (No.11701145, No.11971147) and Project funded by China Postdoctoral Science Foundation (No. 2020M672196).

Received 4 September 2018

Accepted 21 May 2020

Published 11 December 2020