Global well-posedness of strong solutions to the two-dimensional barotropic compressible Navier–Stokes equations with vacuum

Abstract

The aim of this paper is to establish the global well-posedness and large-time asymptotic behavior of the strong solution to the Cauchy problem of the two-dimensional compressible Navier–Stokes equations with vacuum. It is proved that if the shear viscosity \({\mu}\) is a positive constant and the bulk viscosity \({\lambda}\) is the power function of the density, that is, \({\lambda=\rho^{\beta}}\) with \({\beta \in [0,1],}\) then the Cauchy problem of the two-dimensional compressible Navier–Stokes equations admits a unique global strong solution provided that the initial data are of small total energy. This result can be regarded as the extension of the well-posedness theory of classical compressible Navier–Stokes equations [such as Huang et al. (Commun Pure Appl Math 65:549–585, 2012) and Li and Xin (http://arxiv.org/abs/1310.1673) respectively]. Furthermore, the large-time behavior of the strong solution to the Cauchy problem of the two-dimensional barotropic compressible Navier–Stokes equations had been also obtained.