Global Classical Solutions to the Compressible Navier-Stokes Equations with Navier-type slip Boundary Condition in 2D Bounded Domains

TL;DR

This paper proves the global existence of classical solutions to the 2D compressible Navier-Stokes equations with Navier slip boundary conditions, even with vacuum states initially, and analyzes the long-term behavior of density oscillations.

Contribution

It is the first to establish global classical solutions with vacuum and Navier slip boundary conditions in 2D bounded domains, using new boundary estimates.

Findings

Global existence of classical solutions with small initial energy

Density oscillations grow exponentially if vacuum initially present

First result for Navier slip boundary conditions with vacuum in 2D domains

Abstract

We study the barotropic compressible Navier-Stokes equations with Navier-type boundary condition in a two-dimensional simply connected bounded domain with $C^{\infty}$ boundary $\partial\Omega.$ By some new estimates on the boundary related to the Navier-type slip boundary condition, the classical solution to the initial-boundary-value problem of this system exists globally in time provided the initial energy is suitably small even if the density has large oscillations and contains vacuum states. Futhermore, we also prove that the oscillation of the density will grow unboundedly in the long run with an exponential rate provided vacuum (even a point) appears initially. As we known, this is the first result concerning the global existence of classical solutions to the compressible Navier-Stokes equations with Navier-type slip boundary condition and the density containing vacuum initially for general 2D bounded smooth domains.