Global strong solutions to the vacuum free boundary problem for compressible Navier–Stokes equations with degenerate viscosity and gravity force

Abstract

For the vacuum free boundary problem of the one-dimensional compressible Navier–Stokes equations with the density dependent viscosity coefficient and gravity force, the global existence of the strong solution is proved, which is shown to converge to the steady solution with the same total mass when the initial datum is a small perturbation of the steady solution only at the basic level. The key is to establish the global-in-time regularity uniformly up to the vacuum boundary, which ensures the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of the steady solution with detailed convergence rates. Compared with previous studies, the novelty of this paper is the regularity of global solutions for the vacuum free boundary problems in viscous compressible fluids when the viscosity coefficient is not constant but degenerate.