Global well-posedness of the Cauchy problem of two-dimensional compressible Navier–Stokes equations in weighted spaces

Abstract

In this paper, we study the global well-posedness of classical solution to 2D Cauchy problem of the compressible Navier–Stokes equations with large initial data and vacuum. It is proved that if the shear viscosity μ is a positive constant and the bulk viscosity λ is the power function of the density, that is, λ(ρ)=ρβ with β>3, then the 2D Cauchy problem of the compressible Navier–Stokes equations on the whole space R2 admits a unique global classical solution (ρ,u) which may contain vacuums in an open set of R2. Note that the initial data can be arbitrarily large to contain vacuum states. Various weighted estimates of the density and velocity are obtained in this paper and these self-contained estimates reflect the fact that the weighted density and weighted velocity propagate along with the flow.