Global classical solutions of free boundary problem of compressible Navier–Stokes equations with degenerate viscosity

Abstract

This paper concerns with the one dimensional compressible isentropic Navier–Stokes equations with a free boundary separating fluid and vacuum when the viscosity coefficient depends on the density. Precisely, the pressure P and the viscosity coefficient μ are assumed to be proportional to ${\rho }^{\gamma }$ and ${\rho }^{\theta }$ respectively, where ρ is the density, and γ and θ are constants. We establish the unique solvability in the framework of global classical solutions for this problem when $\gamma \ge \theta >1$. Since the previous results on this topic are limited to the case when $\theta \in (0,1]$, the result in this paper fills in the gap for $\theta >1$. Note that the key estimate is to show that the density has a positive lower bound and the new ingredient of the proof relies on the study of the quasilinear parabolic equation for the viscosity coefficient by reducing the nonlocal terms in order to apply the comparison principle.