Global Existence and Incompressible Limit for Compressible Navier-Stokes Equations in Bounded Domains with Large Bulk Viscosity Coefficient and Large Initial Data

TL;DR

This paper proves the global existence and decay of solutions for compressible Navier-Stokes equations with large bulk viscosity in bounded domains, and shows convergence to incompressible flow as viscosity increases.

Contribution

It establishes the global existence and decay of solutions with large bulk viscosity and initial data of arbitrary size, and demonstrates the incompressible limit in bounded domains.

Findings

Global existence and exponential decay of solutions

Convergence to incompressible Navier-Stokes equations as viscosity tends to infinity

Use of logarithmic interpolation inequality and compensated compactness

Abstract

We investigate the barotropic compressible Navier-Stokes equations with the Navier-slip boundary conditions in a general two-dimensional bounded simply connected domain. For initial density that is allowed to vanish, we establish the global existence and exponential decay of weak, strong, and classical solutions when the bulk viscosity coefficient is suitably large, without any restrictions on the size of the initial data. Furthermore, we prove that when the bulk viscosity coefficient tends to infinity, the solutions of the compressible Navier-Stokes equations converge to those of the inhomogeneous incompressible Navier-Stokes equations. The key idea is to utilize the logarithmic interpolation inequality on general bounded domains and apply the compensated compactness lemma.