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

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.