Critical regularity issues for the compressible Navier--Stokes system in bounded domains

Abstract

We are concerned with the barotropic compressible Navier-Stokes system in a bounded domain of ${\mathbb{R}}^d$ (with $d\ge 2$). In a critical regularity setting, we establish local well-posedness for large data with no vacuum and global well-posedness for small perturbations of a stable constant equilibrium state.Our results rely on new maximal regularity estimates - of independent interest - for the semigroup of the Lam\{é} operator, and of the linearized compressible Navier-Stokes equations.