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

TL;DR

This paper proves local and global well-posedness results for the compressible Navier-Stokes system in bounded domains using new maximal regularity estimates, advancing understanding of the system's behavior in critical regularity settings.

Contribution

It introduces novel maximal regularity estimates for the Lamé operator and linearized equations, enabling well-posedness results for large data without vacuum and small perturbations.

Findings

Established local well-posedness for large data with no vacuum.

Proved global well-posedness for small perturbations of equilibrium.

Developed new maximal regularity estimates for key operators.

Abstract

We are concerned with the barotropic compressible Navier-Stokes system in a bounded domain of $\mathbb{R}^d$ (with $d\geq2$). 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\{\'e} operator, and of the linearized compressible Navier-Stokes equations.