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

TL;DR

This paper proves the global existence and exponential decay of solutions to 2D compressible Navier-Stokes equations with large bulk viscosity and demonstrates their convergence to incompressible solutions as viscosity tends to infinity, even with vacuum initial data.

Contribution

It establishes the global existence and incompressible limit of solutions without restrictions on initial velocity divergence, using innovative time-layer techniques and flux estimates.

Findings

Solutions exist globally with large bulk viscosity.

Solutions converge to incompressible Navier-Stokes as viscosity increases.

Results hold even with vacuum initial density and non-divergence-free initial velocity.

Abstract

For periodic initial data with the density allowing vacuum, we establish the global existence and exponential decay of weak, strong and classical solutions to the two-dimensional(2D) compressible Navier-Stokes equations when the bulk viscosity coefficient is sufficiently large, without any extra restrictions on initial velocity divergence. Moreover, we demonstrate that when the bulk viscosity coefficient tends to infinity, these solutions converge to solutions to the inhomogeneous incompressible Navier-Stokes equations. For the incompressible limit of weak solutions, our results hold even without requiring the initial velocity field to be divergence-free. Our results are established by introducing time-layers to avoid imposing restrictions on the initial velocity divergence, along with estimates of $L^\infty$ norm of the effective viscous flux $G$ via a time-partitioning approach based on Gagliardo-Nirenberg inequality.