On One-dimensional Compressible Navier-Stokes Equations with Degenerate Viscosity and Constant State at Far fields

TL;DR

This paper proves the global existence and describes the long-term behavior of solutions to one-dimensional compressible Navier-Stokes equations with density-dependent viscosity, extending previous results to broader parameter ranges.

Contribution

It establishes global weak solutions for all positive viscosity exponents and pressure exponents, enlarging the known parameter ranges and improving prior results.

Findings

Global existence of weak solutions for all lpha>0 and mma>1

Asymptotic behavior of solutions with constant far-field density

Extension of previous results to broader parameter ranges

Abstract

In this paper, we are concerned with the Cauchy problem for one-dimensional compressible isentropic Navier-Stokes equations with density-dependent viscosity $\mu(\rho)=\rho^\alpha (\alpha>0)$ and pressure $P(\rho)=\rho^{\gamma}\ (\gamma>1)$. We will establish the global existence and asymptotic behavior of weak solutions for any $\alpha>0$ and $\gamma>1$ under the assumption that the density function keeps a constant state at far fields. This enlarges the ranges of $\alpha$ and $\gamma$ and improves the previous results presented by Jiu and Xin. As a result, in the case that $0<\alpha<\frac12$, we obtain the large time behavior of the strong solution obtained by Mellet and Vasseur when the solution has a lower bound (no vacuum).