Global regularity and incompressible limit of 2D compressible Navier-Stokes equations with large bulk viscosity

TL;DR

This paper establishes the global regularity of large vacuum solutions to 2D compressible Navier-Stokes equations with large bulk viscosity and derives an explicit convergence rate to the incompressible limit as viscosity increases.

Contribution

It corrects a flaw in previous global estimates and proves the incompressible limit with vacuum, providing explicit convergence rates and handling vacuum states.

Findings

Global regularity of solutions with vacuum for large bulk viscosity

Correction of a key estimate in prior work

Explicit convergence rate to incompressible limit as viscosity tends to infinity

Abstract

In this paper, we study the global regularity of large solutions with vacuum to the two-dimensional compressible Navier-Stokes equations on $\mathbb{T}^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2}$, when the volume (bulk) viscosity coefficient $\nu$ is sufficiently large. It firstly fixes a flaw in \cite[Proposition 3.3]{Danchin2023}, which concerns the $\nu$-independent global $t$-weighted estimates of the solutions. Amending the proof requires non-trivially mathematical analysis. As a by-product, the incompressible limit with an explicit rate of convergence is shown, when the volume viscosity tends to infinity. In contrast to \cite[Theorem 1.3]{Danchin2019} and \cite[Corollary 1.1]{DM2017} where vacuum was excluded, the convergence rate of the incompressible limit is obtained for the global solutions with vacuum, based on some $t$-growth and singular $t$-weighted estimates.