Global Classical Solutions to 3D Compressible Navier-Stokes System with Vacuum in Bounded Domains under Non-Slip Boundary Conditions

Abstract

This paper studies the global well-posedness of classical solutions to the isentropic compressible Navier-Stokes equations in 3D domains D under non-slip boundary conditions. D will separate into the inner and boundary parts along a free surface: In the inner part, the density is allowed to vanish and the gradient of it grows with an exponential rate when vacuum appears initially; while in the boundary part, no vacuum forms and the higher order derivatives of the density remain uniformly bounded. We utilize the Lagrangian coordinates introduced by Christodoulou-Lindblad to study such dichotomy.