Global Classical Solutions to the 3D Density-Dependent Viscosity Compressible Navier-Stokes Equations with Navier-Slip Boundary Condition in a Simply Connected Bounded Domain

TL;DR

This paper proves the global existence of classical solutions to the 3D density-dependent compressible Navier-Stokes equations with slip boundary conditions, allowing for vacuum states and large initial density oscillations in bounded domains.

Contribution

It is the first to establish global classical solutions for these equations with vacuum and density-dependent viscosity in general 3D bounded smooth domains.

Findings

Global classical solutions exist under small initial energy.

Initial density can have large oscillations and include vacuum.

New techniques for boundary integral estimates are developed.

Abstract

This paper concerns the global existence for classical solutions problem to the 3D Density-Dependent Viscosity barotropic compressible Navier-Stokes in $\Omega$ with slip boundary condition, where $\Omega$ is a simply connected bounded $C^{\infty}$ domain in $\mathbb{R}^3$ and its boundary only has a finite number of 2-dimensional connected components. By a series of a priori estimates, we show that the classical solution to the system exists globally in time under the assumption that the initial energy is suitably small. The initial density of such a classical solution is allowed to have large oscillations and contain vacuum states. We also adopt some new techniques and methods to obtain necessary a priori estimates, especially the boundary integral terms estimates. This is the first result concerning the global existence of classical solutions to the compressible Navier-Stokes equations with density containing vacuum initially and viscosity coefficients depending on density for general 3D bounded smooth domains.