On one-dimensional compressible Navier-Stokes equations with degenerate viscosity and constant state at far fields

Abstract

In this paper, we are concerned with the Cauchy problem for one-dimensional compressible isentropic Navier-Stokes equations with density-dependent viscosity μ(ρ) = ρα(α > 0) and pressure P(ρ) = ργ (γ > 1). We will establish the global existence and asymptotic behavior of weak solutions for any α > 0 and γ > 1 under the assumption that the density function keeps a constant state at far fields. In particular, in the case that \documentclass[12pt]{minimal}\begin{document}$0<\alpha <\frac{1}{2}$\end{document}0<α<12, we obtain the large time behavior of the strong solution obtained by Mellet and Vasseur when the solution has a lower bound (no vacuum).